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# Exploring Object-Aware Attention Guided Frame Association for RGB-D SLAM
Ali Caglayan\*1, Nevrez Imamoglu\*1, Oguzhan Guclu\*2, Ali Osman Serhatoglu\*3 Ahmet Burak Can3, Ryosuke Nakamura1 1 National Institute of Advanced Industrial Science and Technology, Tokyo, Japan 2 Sahibinden, Istanbul, Turkiye, 3 Hacettepe University, Ankara, Turkiye {ali.caglayan, nevrez.imamoglu, r.nakamura}@aist.go.jp guclu.oguzhan@outlook.com, {aoserhatoglu, abc}@cs.hacettepe.edu.tr
## Abstract
Attention models have recently emerged as a powerful approach, demonstrating significant progress in various fields. Visualization techniques, such as class activation mapping, provide visual insights into the reasoning of convolutional neural networks (CNNs). Using network gradients, it is possible to identify regions where the network pays attention during image recognition tasks. Furthermore, these gradients can be combined with CNN features to localize more generalizable, task- specific attentive (salient) regions within scenes. However, explicit use of this gradient- based attention information integrated directly into CNN representations for semantic object understanding remains limited. Such integration is particularly beneficial for visual tasks like simultaneous localization and mapping (SLAM), where CNN representations enriched with spatially attentive object locations can enhance performance. In this work, we propose utilizing task- specific network attention for RGB- D indoor SLAM. Specifically, we integrate layer- wise attention information derived from network gradients with CNN feature representations to improve frame association performance. Experimental results indicate improved performance compared to baseline methods, particularly for large environments.
## 1 Introduction
Attention mechanisms have recently gained significant popularity in deep learning, enhancing performance in various computer vision tasks, including object detection [1] and tracking [2], image generation [3], keypoint selection [4], person re- identification [5], as well as odometry [6] and segmentation [7] in point cloud data. Deep learning methods have also become essential components in machine vision applications for autonomous systems, particularly SLAM, a crucial capability for robots and self- driving vehicles [8]. However, as emphasized by [9], there is still considerable room for improvement in deep learning- based SLAM, especially
<center>Figure 1. Overview of the RGB-D SLAM framework utilizing attention-guided deep features for enhanced frame association. </center>
in tasks involving geometric reasoning or frame association. For instance, CNN features from a pre- trained model were successfully utilized in [9] to address loop closure detection within an RGB- D SLAM framework, achieving improved performance over state- of- the- art methods on the TUM RGB- D benchmark [10].
Visualization techniques such as class activation mapping (CAM) enable the understanding of CNN decisions by highlighting image regions where the network is most attentive [11]. Gradient- based methods further enhance these visual explanations by leveraging network gradients to identify the most influential visual regions contributing to network predictions [12]. Typically, these regions correspond to high- level semantic features crucial for network decisions, making gradient- based attention methods valuable for tasks such as weakly- supervised detection and segmentation [13]. Inspired by this, recent studies utilize attention information to reduce the need for large- scale training data labeled at pixel- level, thus improving performance across various weakly- supervised visual tasks [13].
In [14], class activation mapping (CAM) modules [11] are explicitly integrated into CNNs as attention branches to directly learn and modulate network attention. Although these methods provide effective attention maps that enhance network recognition per
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formance, they introduce additional trainable parameters into the network. In contrast, gradient- based approaches such as Grad- CAM [12] can also obtain network attention maps without adding extra parameters. For example, inspired by [12] and [13], the method presented in [15], identifies attention regions for generalized object localization in a weakly- supervised manner. By integrating gradient information with CNN features, this approach effectively highlights attention- relevant regions for different objects, enabling better performance on various visual tasks.
Although supervised attention mechanisms have been effectively applied to various vision tasks [16, 5, 4], explicit utilization of gradient- based attention information, beyond visualization, to enrich CNN representations with object semantics remains relatively limited, especially in complex tasks such as SLAM. In fact, gradient- based attention obtained from network layers (without additional training or fine- tuning) could potentially guide CNN features toward more effective representation of object semantics. This approach can suppress irrelevant regions and emphasize distinctive objects, enhancing scene understanding. Such integration is particularly valuable for visual tasks like RGB- D SLAM, as demonstrated by [9], where CNN representations of spatially attentive object regions significantly improved frame association performance.
In this work, we propose to explicitly leverage task- specific network attention to enhance RGB- D indoor SLAM performance (see Figure 1). Specifically, we integrate CNN semantic layer representations with gradient- based, layer- wise attention maps generated by an ImageNet- pretrained network [17] as in [15]. These attention- guided representations emphasize distinctive object- aware regions with suppressed background, enabling more robust frame associations for improved loop closure detection compared to the RGB- D SLAM approach proposed in [9]. Although our attention- based approach currently focuses on frame association using color images, it can potentially be extended to other tasks, such as motion estimation or efficient keyframe/keypoint selection. Experimental results demonstrate promising initial improvements in mapping performance through this attention- enhanced representation approach.
## 2 Proposed Method
### 2.1 SLAM Framework
The SLAM system in [9] is a graph- based framework that utilizes feature- based odometry estimation and a deep feature indexing mechanism for loop closure detection. The system builds a pose graph by inserting nodes for each incoming frame and estimates odometry and loop closures through feature- based matching and deep feature indexing, respectively.
For odometry estimation, the transformation between consecutive frames is computed by detecting and matching keypoints, then applying RANSAC to estimate robust transformations. Loop closure detection, on the other hand, employs a deep feature- based mechanism integrated with task- specific network attention (see Section 2.2). Unlike [9], we propose an enhanced approach where CNN layer representations are modulated by gradient- based attention maps, effectively highlighting objects of interest and suppressing background noise. Specifically, deep features extracted from semantic layers are modulated using network gradients to encode object- aware attention information. These attention- guided features are subsequently passed through random recursive neural networks (RNNs) to produce compact, semantic- rich representations for indexing (see Figure 2).
Deep features extracted from keyframes are indexed into a priority search k- means tree [18]. During the loop closure search, the indexed deep features are queried, and candidate matches are identified based on feature similarity. An adaptive thresholding step is then applied to eliminate outliers. Finally, each candidate frame goes through a motion estimation procedure (the same as in the odometry estimation step) relative to the current frame, and loop closures are determined based on the quality of the resulting transformations.
The loop closure search process is crucial for map accuracy, as incorrect loop closure detection can lead to graph optimization failure, resulting in an inaccurately constructed map. Our proposed integration of gradient- based attention into CNN features provides a more robust frame representation, resulting in improved scene understanding and more accurate loop closures (e.g., up to 10 to 20 cm in large environments of the TUM RGB- D benchmark [10]).
### 2.2 Attention Guided CNN Features
The proposed attention- guided deep feature extraction module (Figure 2) provides semantically rich representations tailored for improved RGB- D loop closure detection. Specifically, we leverage a task- specific salient object detection approach that combines forward and backward features from an ImageNet- pretrained VGG network, as introduced in [15]. In our approach, deep representations from selected CNN layers (i.e., block 5, see Figure 2) are modulated using gradient- based, layer- wise attention maps. These gradients highlight object- aware regions, effectively suppressing irrelevant background information. This process enables the extraction of more discriminative CNN features for improved scene representation [19].
Unlike methods such as Grad- CAM [12] or distinct class saliency [13], which initialize gradients by setting a specific class to 1 and others to 0; our approach follows [15] and directly utilizes the actual class prediction scores from the softmax output of the network. These
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<center>Figure 2. Detailed view of the proposed attention-guided, object-aware feature extraction process. </center>
prediction scores are used as initial gradients for backpropagation to compute object saliency values, capturing the attentive regions for all objects at the desired network layer \(\mathbf{L}_l\) , independent of specific class labels. The gradients of the predicted class scores at a selected layer are formulated as in Eq. 1:
\[\mathbf{G}_l = \frac{\partial\mathbf{S}}{\partial\mathbf{L}_l} \quad (1)\]
where \(\mathbf{G}_l\) represents the gradient of object scores \(\mathbf{S}\) with respect to the feature activations at \(\mathbf{L}_l\) [13]. During backpropagation, we employ partially guided backpropagation between separated blocks at max- pooling layers for computational efficiency. Specifically, negative gradients are suppressed only at these transitions, unlike the method in [13], which sets all negative gradients to 0 across all layers. Once the gradient \(\mathbf{G}_l\) is obtained, we compute the attention- guided feature representation \(\mathbf{F}_l\) as follows:
\[\mathbf{F}_l = \delta (\mathbf{L}_l,\mathbf{G}_l) \quad (2)\]
where \(\delta\) represents the fusion function that combines the feed- forward CNN layer features \(\mathbf{L}_l\) with gradient- derived attention maps \(\mathbf{G}_l\) , highlighting the most salient object regions. For a given layer \(l\) , we explore multiple fusion strategies to integrate object attention features \((\mathbf{G}_l)\) with forward activations \((\mathbf{L}_l)\) , effectively suppressing background clutter. These strategies include (i) directly applying the normalized gradient tensor (Eq.3, Eq.4) and (ii) generating a global object saliency map by summing the gradient tensor across channels (Eq.5, Eq.6). We denote these attention strategies as direct attention modulation (DAM), exponential attention modulation (EAM), global attention fusion (GAF), and exponential global attention (EGA), corresponding to the following formulations in Eq. 3, 4, 5, and 6, respectively.
\[\begin{array}{rl} & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N(\mathbf{G})}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N(\mathbf{G})}}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}} \end{array} \quad (6)\]
Here, \(\odot\) denotes the Hadamard product, and \(N(.)\) represents the normalization function, which scales \(\mathbf{G}\) to the range [0,1] to serve as an attention mask for \(\mathbf{L}\) . Unlike [15], where gradients are normalized for general feature enhancement, we normalize gradients specifically to suppress activations related to background clutter, ensuring a stronger focus on salient objects. This approach produces attention- guided features where activations corresponding to object regions remain dominant, improving representation quality for scene understanding.
### 2.3 Random RNN for Feature Encoding
After obtaining object attention- guided CNN features from block 5 (L5 following [19]), the next step is to encode these representations into a more compact space. Directly using these high- dimensional features for frame- to- frame comparison can degrade SLAM performance due to the curse of dimensionality. To address this, we employ RNNs [20] to pool the features into a lower- dimensional, compact, and separable representation, as in [9]. Unlike [9], we first apply average pooling before reshaping the CNN activations. To adapt high- dimensional VGG L5 features, we merge every two activation maps by averaging pixels, reducing the feature size to \(7 \times 7 \times 256\) . We then reshape the activations to \(14 \times 14 \times 64\) for RNN processing. RNNs recursively merge adjacent vectors into parent
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Table 1. Accuracy comparison of attentionguided models against the baseline [9], measured in RMS-ATE (m), on the fr1 (small) and fr2 (large) sequences.
<table><tr><td></td><td>baseline [9]</td><td>GAF</td><td>EAM</td><td>EGA</td><td>DAM</td></tr><tr><td>360</td><td>0.056</td><td>0.054</td><td>0.056</td><td>0.051</td><td>0.053</td></tr><tr><td>desk</td><td>0.020</td><td>0.020</td><td>0.019</td><td>0.020</td><td>0.020</td></tr><tr><td>desk2</td><td>0.030</td><td>0.030</td><td>0.028</td><td>0.031</td><td>0.028</td></tr><tr><td>floor</td><td>0.029</td><td>0.029</td><td>0.030</td><td>0.029</td><td>0.029</td></tr><tr><td>plant</td><td>0.035</td><td>0.036</td><td>0.035</td><td>0.036</td><td>0.038</td></tr><tr><td>room</td><td>0.047</td><td>0.049</td><td>0.050</td><td>0.049</td><td>0.049</td></tr><tr><td>teddy</td><td>0.038</td><td>0.040</td><td>0.039</td><td>0.038</td><td>0.039</td></tr><tr><td>average</td><td>0.0364</td><td>0.0369</td><td>0.0367</td><td>0.0363</td><td>0.0366</td></tr><tr><td>large_no_loop</td><td>0.355</td><td>0.242</td><td>0.179</td><td>0.139</td><td>0.137</td></tr><tr><td>large_with_loop</td><td>0.357</td><td>0.342</td><td>0.348</td><td>0.353</td><td>0.357</td></tr><tr><td>pioneer_360</td><td>0.150</td><td>0.137</td><td>0.152</td><td>0.160</td><td>0.150</td></tr><tr><td>pioneer_slam</td><td>0.428</td><td>0.398</td><td>0.417</td><td>0.395</td><td>0.355</td></tr><tr><td>pioneer_slam2</td><td>0.160</td><td>0.163</td><td>0.166</td><td>0.164</td><td>0.158</td></tr><tr><td>pioneer_slam3</td><td>0.282</td><td>0.265</td><td>0.267</td><td>0.264</td><td>0.271</td></tr><tr><td>average</td><td>0.289</td><td>0.258</td><td>0.255</td><td>0.246</td><td>0.238</td></tr></table>
vectors using tied weights and a tanh activation function [20]. We employ the one- level structured RNN from [19], where each RNN outputs a \(k\) - dimensional feature vector ( \(k = 64\) ). Following [9], we use 16 RNNs, producing a final 1024- dimensional feature vector ( \(64 \times 16 = 1024\) ).
## 3 Experiments
We evaluated the performance of the proposed approach on the popular TUM RGB- D dataset [10], using the fr1 and fr2 sequences to assess performance in both medium- and large- scale indoor environments. The fr2 sequences, recorded in a large industrial halls with more challenging conditions, provide a more rigorous evaluation than the fr1 sequences.
Table 1 presents the RMS- ATE (root mean square of absolute trajectory error in meters) for different attention fusion strategies compared to the baseline [9]. On the fr1 sequences, object- attentive features do not show a significant improvement over the baseline. This is likely because the small- scale sequences contain fewer distinctive objects, limiting the advantage of semantic attention. When the scene is centered around a single object, low- level features may provide more reliable frame associations than high- level object- aware attention. Moreover, if the sequence of sample data is around one particular object, it is neither easy nor feasible for the network to distinguish foreground object and background clutter using the proposed object attentive gradients. Consequently, attention- guided features offer no clear benefit in these cases. However, both the baseline and attention- based models achieve high accuracy, with errors close to the ground truth, indicating that attention integration does not negatively impact performance in small- scale settings.
In contrast, the fr2 sequences show a clear per formance gain with object- attentive features, supporting the idea that attention- based SLAM can enhance large- scale mapping by prioritizing object regions over background clutter. As seen in Table 1, all attention- based models significantly reduce RMS- ATE compared to the baseline. The observed drift errors range between 10 cm and 35 cm, which is acceptable for these highly challenging large- scale sequences. These improvements demonstrate that attention- guided feature representations can generalize well to complex, real- world environments, making them promising for large- scale autonomous navigation tasks. Our ablative study on different attention fusion strategies confirms that the direct attention modulation (DAM) method consistently outperforms other approaches, yielding the best accuracy across most sequences. Figure 3 visualizes sample estimated trajectories using DAM- based object attention on fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3. The proposed model effectively minimizes RMS- ATE errors, producing trajectory maps closely aligned with ground truth results. The results show that leveraging object attention in SLAM can reduce cumulative drift and improve long- term trajectory consistency, particularly in environments with rich semantic content.
<center>Figure 3. Comparison of estimated trajectories using the DAM attention model against ground truth for the fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3 sequences. </center>
## 4 Conclusion
We proposed a gradient- based object- attentive approach for loop closure detection in RGB- D SLAM, integrating attention- guided features by modulating CNN representations with object- attentive gradients. To our knowledge, this is the first attempt to incorporate attention mechanisms in a SLAM system this way. Experimental results demonstrate the effectiveness of our approach, particularly in large- scale environments. The strong performance on the fr2 sequences suggests that attention- guided features could also be beneficial for outdoor mapping applications. Future work includes using eye- fixation trained networks, exploring attention- based keypoint detection and keyframe selection, and extending the method to a multi- modal RGB- D setting for enhanced performance.
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## References
[1] W. Wang, Q. Lai, H. Fu, J. Shen, H. Ling, and R. Yang, "Salient object detection in the deep learning era: An in- depth survey," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 6, pp. 3239- 3259, 2022. [2] Z. Zhou, W. Pei, X. Li, H. Wang, F. Zheng, and Z. He, "Saliency- associated object tracking," in Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 9866- 9875. [3] Y. Zhang, N. Wu, C. Z. Lin, G. Wetzstein, and Q. Sun, "Gazefusion: Saliency- guided image generation," ACM Transactions on Applied Perception, vol. 21, no. 4, pp. 1- 19, 2024. [4] G. Tinchev, A. Penate- Sanchez, and M. Fallon, "Skd: Keypoint detection for point clouds using saliency estimation," IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3785- 3792, 2021. [5] X. Ren, D. Zhang, X. Bao, and Y. Zhang, "S2- net: Semantic and salient attention network for person re- identification," IEEE Transactions on Multimedia, pp. 1- 1, 2022. [6] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Attention- guided lidar segmentation and odometry using image- to- point cloud saliency transfer," Multimedia Systems, vol. 30, no. 4, p. 188, 2024. [7] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Salidar: Saliency knowledge transfer learning for 3d point cloud understanding," in BMVC, 2022, p. 584. [8] R. Mur- Artal and J. D. Tardos, "Orb- slam2: an open- source slam system for monocular, stereo and rgb- d cameras," IEEE Transactions on Robotics, vol. 33, no. 5, pp. 1255- 1262, 2017. [9] O. Guclu, A. Caglayan, and A. B. Can, "Rgb- d indoor mapping using deep features," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2019. [10] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers, "A benchmark for the evaluation of rgb- d slam systems," in Proc. of the International Conference on Intelligent Robot Systems (IROS), October 2012.
[11] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, "Learning deep features for discriminative localization," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016. [12] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, "Grad- cam: Visual explanations from deep networks via gradient- based localization," in Proceedings of the IEEE International Conference on Computer Vision (ICCV), Oct 2017. [13] W. Shimoda and K. Yanai, "Distinct class- specific saliency maps for weakly supervised semantic segmentation," in Proceedings of the European Conference on Computer Vision (ECCV) Workshops, October 2016. [14] H. Fukui, T. Hirakawa, T. Yamashita, and H. Fujiyoshi, "Attention branch network: Learning of attention mechanism for visual explanation," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019. [15] N. Imamoglu, C. Zhang, W. Shimoda, Y. Fang, and B. Shi, "Saliency detection by forward and backward cues in deep- cnn," in 2017 IEEE International Conference on Image Processing (ICIP), 2017, pp. 430- 434. [16] L. Jiang, M. Xu, X. Wang, and L. Sigal, "Saliency- guided image translation," in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021, pp. 16 504- 16 513. [17] O. Russakovsky\*, J. Deng\*, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei- Fei, "Imagenet large scale visual recognition challenge," International Journal of Computer Vision, vol. 115, pp. 211- 252, 2015. [18] M. Muja and D. G. Lowe, "Scalable nearest neighbor algorithms for high dimensional data," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 36, no. 11, pp. 2227- 2240, 2014. [19] A. Caglayan, N. Imamoglu, A. B. Can, and R. Nakamura, "When cnns meet random rnns: Towards multilevel analysis for rgb- d object and scene recognition," Computer Vision and Image Understanding, p. 103373, 2022. [20] R. Socher, B. Huval, B. Bath, C. D. Manning, and A. Y. Ng, "Convolutional- recursive deep learning for 3d object classification," in Advances in Neural Information Processing Systems, 2012, pp. 656- 664.
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2510.26131v1
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# Exploring Object-Aware Attention Guided Frame Association for RGB-D SLAM
Ali Caglayan\*1, Nevrez Imamoglu\*1, Oguzhan Guclu\*2, Ali Osman Serhatoglu\*3 Ahmet Burak Can3, Ryosuke Nakamura1 1 National Institute of Advanced Industrial Science and Technology, Tokyo, Japan 2 Sahibinden, Istanbul, Turkiye, 3 Hacettepe University, Ankara, Turkiye {ali.caglayan, nevrez.imamoglu, r.nakamura}@aist.go.jp guclu.oguzhan@outlook.com, {aoserhatoglu, abc}@cs.hacettepe.edu.tr
## Abstract
Attention models have recently emerged as a powerful approach, demonstrating significant progress in various fields. Visualization techniques, such as class activation mapping, provide visual insights into the reasoning of convolutional neural networks (CNNs). Using network gradients, it is possible to identify regions where the network pays attention during image recognition tasks. Furthermore, these gradients can be combined with CNN features to localize more generalizable, task- specific attentive (salient) regions within scenes. However, explicit use of this gradient- based attention information integrated directly into CNN representations for semantic object understanding remains limited. Such integration is particularly beneficial for visual tasks like simultaneous localization and mapping (SLAM), where CNN representations enriched with spatially attentive object locations can enhance performance. In this work, we propose utilizing task- specific network attention for RGB- D indoor SLAM. Specifically, we integrate layer- wise attention information derived from network gradients with CNN feature representations to improve frame association performance. Experimental results indicate improved performance compared to baseline methods, particularly for large environments.
## 1 Introduction
Attention mechanisms have recently gained significant popularity in deep learning, enhancing performance in various computer vision tasks, including object detection [1] and tracking [2], image generation [3], keypoint selection [4], person re- identification [5], as well as odometry [6] and segmentation [7] in point cloud data. Deep learning methods have also become essential components in machine vision applications for autonomous systems, particularly SLAM, a crucial capability for robots and self- driving vehicles [8]. However, as emphasized by [9], there is still considerable room for improvement in deep learning- based SLAM, especially
<center>Figure 1. Overview of the RGB-D SLAM framework utilizing attention-guided deep features for enhanced frame association. </center>
in tasks involving geometric reasoning or frame association. For instance, CNN features from a pre- trained model were successfully utilized in [9] to address loop closure detection within an RGB- D SLAM framework, achieving improved performance over state- of- the- art methods on the TUM RGB- D benchmark [10].
Visualization techniques such as class activation mapping (CAM) enable the understanding of CNN decisions by highlighting image regions where the network is most attentive [11]. Gradient- based methods further enhance these visual explanations by leveraging network gradients to identify the most influential visual regions contributing to network predictions [12]. Typically, these regions correspond to high- level semantic features crucial for network decisions, making gradient- based attention methods valuable for tasks such as weakly- supervised detection and segmentation [13]. Inspired by this, recent studies utilize attention information to reduce the need for large- scale training data labeled at pixel- level, thus improving performance across various weakly- supervised visual tasks [13].
In [14], class activation mapping (CAM) modules [11] are explicitly integrated into CNNs as attention branches to directly learn and modulate network attention. Although these methods provide effective attention maps that enhance network recognition per
<--- Page 1 --->
formance, they introduce additional trainable parameters into the network. In contrast, gradient- based approaches such as Grad- CAM [12] can also obtain network attention maps without adding extra parameters. For example, inspired by [12] and [13], the method presented in [15], identifies attention regions for generalized object localization in a weakly- supervised manner. By integrating gradient information with CNN features, this approach effectively highlights attention- relevant regions for different objects, enabling better performance on various visual tasks.
Although supervised attention mechanisms have been effectively applied to various vision tasks [16, 5, 4], explicit utilization of gradient- based attention information, beyond visualization, to enrich CNN representations with object semantics remains relatively limited, especially in complex tasks such as SLAM. In fact, gradient- based attention obtained from network layers (without additional training or fine- tuning) could potentially guide CNN features toward more effective representation of object semantics. This approach can suppress irrelevant regions and emphasize distinctive objects, enhancing scene understanding. Such integration is particularly valuable for visual tasks like RGB- D SLAM, as demonstrated by [9], where CNN representations of spatially attentive object regions significantly improved frame association performance.
In this work, we propose to explicitly leverage task- specific network attention to enhance RGB- D indoor SLAM performance (see Figure 1). Specifically, we integrate CNN semantic layer representations with gradient- based, layer- wise attention maps generated by an ImageNet- pretrained network [17] as in [15]. These attention- guided representations emphasize distinctive object- aware regions with suppressed background, enabling more robust frame associations for improved loop closure detection compared to the RGB- D SLAM approach proposed in [9]. Although our attention- based approach currently focuses on frame association using color images, it can potentially be extended to other tasks, such as motion estimation or efficient keyframe/keypoint selection. Experimental results demonstrate promising initial improvements in mapping performance through this attention- enhanced representation approach.
## 2 Proposed Method
### 2.1 SLAM Framework
The SLAM system in [9] is a graph- based framework that utilizes feature- based odometry estimation and a deep feature indexing mechanism for loop closure detection. The system builds a pose graph by inserting nodes for each incoming frame and estimates odometry and loop closures through feature- based matching and deep feature indexing, respectively.
For odometry estimation, the transformation between consecutive frames is computed by detecting and matching keypoints, then applying RANSAC to estimate robust transformations. Loop closure detection, on the other hand, employs a deep feature- based mechanism integrated with task- specific network attention (see Section 2.2). Unlike [9], we propose an enhanced approach where CNN layer representations are modulated by gradient- based attention maps, effectively highlighting objects of interest and suppressing background noise. Specifically, deep features extracted from semantic layers are modulated using network gradients to encode object- aware attention information. These attention- guided features are subsequently passed through random recursive neural networks (RNNs) to produce compact, semantic- rich representations for indexing (see Figure 2).
Deep features extracted from keyframes are indexed into a priority search k- means tree [18]. During the loop closure search, the indexed deep features are queried, and candidate matches are identified based on feature similarity. An adaptive thresholding step is then applied to eliminate outliers. Finally, each candidate frame goes through a motion estimation procedure (the same as in the odometry estimation step) relative to the current frame, and loop closures are determined based on the quality of the resulting transformations.
The loop closure search process is crucial for map accuracy, as incorrect loop closure detection can lead to graph optimization failure, resulting in an inaccurately constructed map. Our proposed integration of gradient- based attention into CNN features provides a more robust frame representation, resulting in improved scene understanding and more accurate loop closures (e.g., up to 10 to 20 cm in large environments of the TUM RGB- D benchmark [10]).
### 2.2 Attention Guided CNN Features
The proposed attention- guided deep feature extraction module (Figure 2) provides semantically rich representations tailored for improved RGB- D loop closure detection. Specifically, we leverage a task- specific salient object detection approach that combines forward and backward features from an ImageNet- pretrained VGG network, as introduced in [15]. In our approach, deep representations from selected CNN layers (i.e., block 5, see Figure 2) are modulated using gradient- based, layer- wise attention maps. These gradients highlight object- aware regions, effectively suppressing irrelevant background information. This process enables the extraction of more discriminative CNN features for improved scene representation [19].
Unlike methods such as Grad- CAM [12] or distinct class saliency [13], which initialize gradients by setting a specific class to 1 and others to 0; our approach follows [15] and directly utilizes the actual class prediction scores from the softmax output of the network. These
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<center>Figure 2. Detailed view of the proposed attention-guided, object-aware feature extraction process. </center>
prediction scores are used as initial gradients for backpropagation to compute object saliency values, capturing the attentive regions for all objects at the desired network layer \(\mathbf{L}_l\) , independent of specific class labels. The gradients of the predicted class scores at a selected layer are formulated as in Eq. 1:
\[\mathbf{G}_l = \frac{\partial\mathbf{S}}{\partial\mathbf{L}_l} \quad (1)\]
where \(\mathbf{G}_l\) represents the gradient of object scores \(\mathbf{S}\) with respect to the feature activations at \(\mathbf{L}_l\) [13]. During backpropagation, we employ partially guided backpropagation between separated blocks at max- pooling layers for computational efficiency. Specifically, negative gradients are suppressed only at these transitions, unlike the method in [13], which sets all negative gradients to 0 across all layers. Once the gradient \(\mathbf{G}_l\) is obtained, we compute the attention- guided feature representation \(\mathbf{F}_l\) as follows:
\[\mathbf{F}_l = \delta (\mathbf{L}_l,\mathbf{G}_l) \quad (2)\]
where \(\delta\) represents the fusion function that combines the feed- forward CNN layer features \(\mathbf{L}_l\) with gradient- derived attention maps \(\mathbf{G}_l\) , highlighting the most salient object regions. For a given layer \(l\) , we explore multiple fusion strategies to integrate object attention features \((\mathbf{G}_l)\) with forward activations \((\mathbf{L}_l)\) , effectively suppressing background clutter. These strategies include (i) directly applying the normalized gradient tensor (Eq.3, Eq.4) and (ii) generating a global object saliency map by summing the gradient tensor across channels (Eq.5, Eq.6). We denote these attention strategies as direct attention modulation (DAM), exponential attention modulation (EAM), global attention fusion (GAF), and exponential global attention (EGA), corresponding to the following formulations in Eq. 3, 4, 5, and 6, respectively.
\[\begin{array}{rl} & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N(\mathbf{G})}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N(\mathbf{G})}}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}} \end{array} \quad (6)\]
Here, \(\odot\) denotes the Hadamard product, and \(N(.)\) represents the normalization function, which scales \(\mathbf{G}\) to the range [0,1] to serve as an attention mask for \(\mathbf{L}\) . Unlike [15], where gradients are normalized for general feature enhancement, we normalize gradients specifically to suppress activations related to background clutter, ensuring a stronger focus on salient objects. This approach produces attention- guided features where activations corresponding to object regions remain dominant, improving representation quality for scene understanding.
### 2.3 Random RNN for Feature Encoding
After obtaining object attention- guided CNN features from block 5 (L5 following [19]), the next step is to encode these representations into a more compact space. Directly using these high- dimensional features for frame- to- frame comparison can degrade SLAM performance due to the curse of dimensionality. To address this, we employ RNNs [20] to pool the features into a lower- dimensional, compact, and separable representation, as in [9]. Unlike [9], we first apply average pooling before reshaping the CNN activations. To adapt high- dimensional VGG L5 features, we merge every two activation maps by averaging pixels, reducing the feature size to \(7 \times 7 \times 256\) . We then reshape the activations to \(14 \times 14 \times 64\) for RNN processing. RNNs recursively merge adjacent vectors into parent
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Table 1. Accuracy comparison of attentionguided models against the baseline [9], measured in RMS-ATE (m), on the fr1 (small) and fr2 (large) sequences.
<table><tr><td></td><td>baseline [9]</td><td>GAF</td><td>EAM</td><td>EGA</td><td>DAM</td></tr><tr><td>360</td><td>0.056</td><td>0.054</td><td>0.056</td><td>0.051</td><td>0.053</td></tr><tr><td>desk</td><td>0.020</td><td>0.020</td><td>0.019</td><td>0.020</td><td>0.020</td></tr><tr><td>desk2</td><td>0.030</td><td>0.030</td><td>0.028</td><td>0.031</td><td>0.028</td></tr><tr><td>floor</td><td>0.029</td><td>0.029</td><td>0.030</td><td>0.029</td><td>0.029</td></tr><tr><td>plant</td><td>0.035</td><td>0.036</td><td>0.035</td><td>0.036</td><td>0.038</td></tr><tr><td>room</td><td>0.047</td><td>0.049</td><td>0.050</td><td>0.049</td><td>0.049</td></tr><tr><td>teddy</td><td>0.038</td><td>0.040</td><td>0.039</td><td>0.038</td><td>0.039</td></tr><tr><td>average</td><td>0.0364</td><td>0.0369</td><td>0.0367</td><td>0.0363</td><td>0.0366</td></tr><tr><td>large_no_loop</td><td>0.355</td><td>0.242</td><td>0.179</td><td>0.139</td><td>0.137</td></tr><tr><td>large_with_loop</td><td>0.357</td><td>0.342</td><td>0.348</td><td>0.353</td><td>0.357</td></tr><tr><td>pioneer_360</td><td>0.150</td><td>0.137</td><td>0.152</td><td>0.160</td><td>0.150</td></tr><tr><td>pioneer_slam</td><td>0.428</td><td>0.398</td><td>0.417</td><td>0.395</td><td>0.355</td></tr><tr><td>pioneer_slam2</td><td>0.160</td><td>0.163</td><td>0.166</td><td>0.164</td><td>0.158</td></tr><tr><td>pioneer_slam3</td><td>0.282</td><td>0.265</td><td>0.267</td><td>0.264</td><td>0.271</td></tr><tr><td>average</td><td>0.289</td><td>0.258</td><td>0.255</td><td>0.246</td><td>0.238</td></tr></table>
vectors using tied weights and a tanh activation function [20]. We employ the one- level structured RNN from [19], where each RNN outputs a \(k\) - dimensional feature vector ( \(k = 64\) ). Following [9], we use 16 RNNs, producing a final 1024- dimensional feature vector ( \(64 \times 16 = 1024\) ).
## 3 Experiments
We evaluated the performance of the proposed approach on the popular TUM RGB- D dataset [10], using the fr1 and fr2 sequences to assess performance in both medium- and large- scale indoor environments. The fr2 sequences, recorded in a large industrial halls with more challenging conditions, provide a more rigorous evaluation than the fr1 sequences.
Table 1 presents the RMS- ATE (root mean square of absolute trajectory error in meters) for different attention fusion strategies compared to the baseline [9]. On the fr1 sequences, object- attentive features do not show a significant improvement over the baseline. This is likely because the small- scale sequences contain fewer distinctive objects, limiting the advantage of semantic attention. When the scene is centered around a single object, low- level features may provide more reliable frame associations than high- level object- aware attention. Moreover, if the sequence of sample data is around one particular object, it is neither easy nor feasible for the network to distinguish foreground object and background clutter using the proposed object attentive gradients. Consequently, attention- guided features offer no clear benefit in these cases. However, both the baseline and attention- based models achieve high accuracy, with errors close to the ground truth, indicating that attention integration does not negatively impact performance in small- scale settings.
In contrast, the fr2 sequences show a clear per formance gain with object- attentive features, supporting the idea that attention- based SLAM can enhance large- scale mapping by prioritizing object regions over background clutter. As seen in Table 1, all attention- based models significantly reduce RMS- ATE compared to the baseline. The observed drift errors range between 10 cm and 35 cm, which is acceptable for these highly challenging large- scale sequences. These improvements demonstrate that attention- guided feature representations can generalize well to complex, real- world environments, making them promising for large- scale autonomous navigation tasks. Our ablative study on different attention fusion strategies confirms that the direct attention modulation (DAM) method consistently outperforms other approaches, yielding the best accuracy across most sequences. Figure 3 visualizes sample estimated trajectories using DAM- based object attention on fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3. The proposed model effectively minimizes RMS- ATE errors, producing trajectory maps closely aligned with ground truth results. The results show that leveraging object attention in SLAM can reduce cumulative drift and improve long- term trajectory consistency, particularly in environments with rich semantic content.
<center>Figure 3. Comparison of estimated trajectories using the DAM attention model against ground truth for the fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3 sequences. </center>
## 4 Conclusion
We proposed a gradient- based object- attentive approach for loop closure detection in RGB- D SLAM, integrating attention- guided features by modulating CNN representations with object- attentive gradients. To our knowledge, this is the first attempt to incorporate attention mechanisms in a SLAM system this way. Experimental results demonstrate the effectiveness of our approach, particularly in large- scale environments. The strong performance on the fr2 sequences suggests that attention- guided features could also be beneficial for outdoor mapping applications. Future work includes using eye- fixation trained networks, exploring attention- based keypoint detection and keyframe selection, and extending the method to a multi- modal RGB- D setting for enhanced performance.
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## References
[1] W. Wang, Q. Lai, H. Fu, J. Shen, H. Ling, and R. Yang, "Salient object detection in the deep learning era: An in- depth survey," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 6, pp. 3239- 3259, 2022. [2] Z. Zhou, W. Pei, X. Li, H. Wang, F. Zheng, and Z. He, "Saliency- associated object tracking," in Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 9866- 9875. [3] Y. Zhang, N. Wu, C. Z. Lin, G. Wetzstein, and Q. Sun, "Gazefusion: Saliency- guided image generation," ACM Transactions on Applied Perception, vol. 21, no. 4, pp. 1- 19, 2024. [4] G. Tinchev, A. Penate- Sanchez, and M. Fallon, "Skd: Keypoint detection for point clouds using saliency estimation," IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3785- 3792, 2021. [5] X. Ren, D. Zhang, X. Bao, and Y. Zhang, "S2- net: Semantic and salient attention network for person re- identification," IEEE Transactions on Multimedia, pp. 1- 1, 2022. [6] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Attention- guided lidar segmentation and odometry using image- to- point cloud saliency transfer," Multimedia Systems, vol. 30, no. 4, p. 188, 2024. [7] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Salidar: Saliency knowledge transfer learning for 3d point cloud understanding," in BMVC, 2022, p. 584. [8] R. Mur- Artal and J. D. Tardos, "Orb- slam2: an open- source slam system for monocular, stereo and rgb- d cameras," IEEE Transactions on Robotics, vol. 33, no. 5, pp. 1255- 1262, 2017. [9] O. Guclu, A. Caglayan, and A. B. Can, "Rgb- d indoor mapping using deep features," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2019. [10] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers, "A benchmark for the evaluation of rgb- d slam systems," in Proc. of the International Conference on Intelligent Robot Systems (IROS), October 2012.
[11] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, "Learning deep features for discriminative localization," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016. [12] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, "Grad- cam: Visual explanations from deep networks via gradient- based localization," in Proceedings of the IEEE International Conference on Computer Vision (ICCV), Oct 2017. [13] W. Shimoda and K. Yanai, "Distinct class- specific saliency maps for weakly supervised semantic segmentation," in Proceedings of the European Conference on Computer Vision (ECCV) Workshops, October 2016. [14] H. Fukui, T. Hirakawa, T. Yamashita, and H. Fujiyoshi, "Attention branch network: Learning of attention mechanism for visual explanation," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019. [15] N. Imamoglu, C. Zhang, W. Shimoda, Y. Fang, and B. Shi, "Saliency detection by forward and backward cues in deep- cnn," in 2017 IEEE International Conference on Image Processing (ICIP), 2017, pp. 430- 434. [16] L. Jiang, M. Xu, X. Wang, and L. Sigal, "Saliency- guided image translation," in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021, pp. 16 504- 16 513. [17] O. Russakovsky\*, J. Deng\*, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei- Fei, "Imagenet large scale visual recognition challenge," International Journal of Computer Vision, vol. 115, pp. 211- 252, 2015. [18] M. Muja and D. G. Lowe, "Scalable nearest neighbor algorithms for high dimensional data," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 36, no. 11, pp. 2227- 2240, 2014. [19] A. Caglayan, N. Imamoglu, A. B. Can, and R. Nakamura, "When cnns meet random rnns: Towards multilevel analysis for rgb- d object and scene recognition," Computer Vision and Image Understanding, p. 103373, 2022. [20] R. Socher, B. Huval, B. Bath, C. D. Manning, and A. Y. Ng, "Convolutional- recursive deep learning for 3d object classification," in Advances in Neural Information Processing Systems, 2012, pp. 656- 664.
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2510.26131v1
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# Exploring Object-Aware Attention Guided Frame Association for RGB-D SLAM
Ali Caglayan\*1, Nevrez Imamoglu\*1, Oguzhan Guclu\*2, Ali Osman Serhatoglu\*3 Ahmet Burak Can3, Ryosuke Nakamura1 1 National Institute of Advanced Industrial Science and Technology, Tokyo, Japan 2 Sahibinden, Istanbul, Turkiye, 3 Hacettepe University, Ankara, Turkiye {ali.caglayan, nevrez.imamoglu, r.nakamura}@aist.go.jp guclu.oguzhan@outlook.com, {aoserhatoglu, abc}@cs.hacettepe.edu.tr
## Abstract
Attention models have recently emerged as a powerful approach, demonstrating significant progress in various fields. Visualization techniques, such as class activation mapping, provide visual insights into the reasoning of convolutional neural networks (CNNs). Using network gradients, it is possible to identify regions where the network pays attention during image recognition tasks. Furthermore, these gradients can be combined with CNN features to localize more generalizable, task- specific attentive (salient) regions within scenes. However, explicit use of this gradient- based attention information integrated directly into CNN representations for semantic object understanding remains limited. Such integration is particularly beneficial for visual tasks like simultaneous localization and mapping (SLAM), where CNN representations enriched with spatially attentive object locations can enhance performance. In this work, we propose utilizing task- specific network attention for RGB- D indoor SLAM. Specifically, we integrate layer- wise attention information derived from network gradients with CNN feature representations to improve frame association performance. Experimental results indicate improved performance compared to baseline methods, particularly for large environments.
## 1 Introduction
Attention mechanisms have recently gained significant popularity in deep learning, enhancing performance in various computer vision tasks, including object detection [1] and tracking [2], image generation [3], keypoint selection [4], person re- identification [5], as well as odometry [6] and segmentation [7] in point cloud data. Deep learning methods have also become essential components in machine vision applications for autonomous systems, particularly SLAM, a crucial capability for robots and self- driving vehicles [8]. However, as emphasized by [9], there is still considerable room for improvement in deep learning- based SLAM, especially
<center>Figure 1. Overview of the RGB-D SLAM framework utilizing attention-guided deep features for enhanced frame association. </center>
in tasks involving geometric reasoning or frame association. For instance, CNN features from a pre- trained model were successfully utilized in [9] to address loop closure detection within an RGB- D SLAM framework, achieving improved performance over state- of- the- art methods on the TUM RGB- D benchmark [10].
Visualization techniques such as class activation mapping (CAM) enable the understanding of CNN decisions by highlighting image regions where the network is most attentive [11]. Gradient- based methods further enhance these visual explanations by leveraging network gradients to identify the most influential visual regions contributing to network predictions [12]. Typically, these regions correspond to high- level semantic features crucial for network decisions, making gradient- based attention methods valuable for tasks such as weakly- supervised detection and segmentation [13]. Inspired by this, recent studies utilize attention information to reduce the need for large- scale training data labeled at pixel- level, thus improving performance across various weakly- supervised visual tasks [13].
In [14], class activation mapping (CAM) modules [11] are explicitly integrated into CNNs as attention branches to directly learn and modulate network attention. Although these methods provide effective attention maps that enhance network recognition per
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formance, they introduce additional trainable parameters into the network. In contrast, gradient- based approaches such as Grad- CAM [12] can also obtain network attention maps without adding extra parameters. For example, inspired by [12] and [13], the method presented in [15], identifies attention regions for generalized object localization in a weakly- supervised manner. By integrating gradient information with CNN features, this approach effectively highlights attention- relevant regions for different objects, enabling better performance on various visual tasks.
Although supervised attention mechanisms have been effectively applied to various vision tasks [16, 5, 4], explicit utilization of gradient- based attention information, beyond visualization, to enrich CNN representations with object semantics remains relatively limited, especially in complex tasks such as SLAM. In fact, gradient- based attention obtained from network layers (without additional training or fine- tuning) could potentially guide CNN features toward more effective representation of object semantics. This approach can suppress irrelevant regions and emphasize distinctive objects, enhancing scene understanding. Such integration is particularly valuable for visual tasks like RGB- D SLAM, as demonstrated by [9], where CNN representations of spatially attentive object regions significantly improved frame association performance.
In this work, we propose to explicitly leverage task- specific network attention to enhance RGB- D indoor SLAM performance (see Figure 1). Specifically, we integrate CNN semantic layer representations with gradient- based, layer- wise attention maps generated by an ImageNet- pretrained network [17] as in [15]. These attention- guided representations emphasize distinctive object- aware regions with suppressed background, enabling more robust frame associations for improved loop closure detection compared to the RGB- D SLAM approach proposed in [9]. Although our attention- based approach currently focuses on frame association using color images, it can potentially be extended to other tasks, such as motion estimation or efficient keyframe/keypoint selection. Experimental results demonstrate promising initial improvements in mapping performance through this attention- enhanced representation approach.
## 2 Proposed Method
### 2.1 SLAM Framework
The SLAM system in [9] is a graph- based framework that utilizes feature- based odometry estimation and a deep feature indexing mechanism for loop closure detection. The system builds a pose graph by inserting nodes for each incoming frame and estimates odometry and loop closures through feature- based matching and deep feature indexing, respectively.
For odometry estimation, the transformation between consecutive frames is computed by detecting and matching keypoints, then applying RANSAC to estimate robust transformations. Loop closure detection, on the other hand, employs a deep feature- based mechanism integrated with task- specific network attention (see Section 2.2). Unlike [9], we propose an enhanced approach where CNN layer representations are modulated by gradient- based attention maps, effectively highlighting objects of interest and suppressing background noise. Specifically, deep features extracted from semantic layers are modulated using network gradients to encode object- aware attention information. These attention- guided features are subsequently passed through random recursive neural networks (RNNs) to produce compact, semantic- rich representations for indexing (see Figure 2).
Deep features extracted from keyframes are indexed into a priority search k- means tree [18]. During the loop closure search, the indexed deep features are queried, and candidate matches are identified based on feature similarity. An adaptive thresholding step is then applied to eliminate outliers. Finally, each candidate frame goes through a motion estimation procedure (the same as in the odometry estimation step) relative to the current frame, and loop closures are determined based on the quality of the resulting transformations.
The loop closure search process is crucial for map accuracy, as incorrect loop closure detection can lead to graph optimization failure, resulting in an inaccurately constructed map. Our proposed integration of gradient- based attention into CNN features provides a more robust frame representation, resulting in improved scene understanding and more accurate loop closures (e.g., up to 10 to 20 cm in large environments of the TUM RGB- D benchmark [10]).
### 2.2 Attention Guided CNN Features
The proposed attention- guided deep feature extraction module (Figure 2) provides semantically rich representations tailored for improved RGB- D loop closure detection. Specifically, we leverage a task- specific salient object detection approach that combines forward and backward features from an ImageNet- pretrained VGG network, as introduced in [15]. In our approach, deep representations from selected CNN layers (i.e., block 5, see Figure 2) are modulated using gradient- based, layer- wise attention maps. These gradients highlight object- aware regions, effectively suppressing irrelevant background information. This process enables the extraction of more discriminative CNN features for improved scene representation [19].
Unlike methods such as Grad- CAM [12] or distinct class saliency [13], which initialize gradients by setting a specific class to 1 and others to 0; our approach follows [15] and directly utilizes the actual class prediction scores from the softmax output of the network. These
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<center>Figure 2. Detailed view of the proposed attention-guided, object-aware feature extraction process. </center>
prediction scores are used as initial gradients for backpropagation to compute object saliency values, capturing the attentive regions for all objects at the desired network layer \(\mathbf{L}_l\) , independent of specific class labels. The gradients of the predicted class scores at a selected layer are formulated as in Eq. 1:
\[\mathbf{G}_l = \frac{\partial\mathbf{S}}{\partial\mathbf{L}_l} \quad (1)\]
where \(\mathbf{G}_l\) represents the gradient of object scores \(\mathbf{S}\) with respect to the feature activations at \(\mathbf{L}_l\) [13]. During backpropagation, we employ partially guided backpropagation between separated blocks at max- pooling layers for computational efficiency. Specifically, negative gradients are suppressed only at these transitions, unlike the method in [13], which sets all negative gradients to 0 across all layers. Once the gradient \(\mathbf{G}_l\) is obtained, we compute the attention- guided feature representation \(\mathbf{F}_l\) as follows:
\[\mathbf{F}_l = \delta (\mathbf{L}_l,\mathbf{G}_l) \quad (2)\]
where \(\delta\) represents the fusion function that combines the feed- forward CNN layer features \(\mathbf{L}_l\) with gradient- derived attention maps \(\mathbf{G}_l\) , highlighting the most salient object regions. For a given layer \(l\) , we explore multiple fusion strategies to integrate object attention features \((\mathbf{G}_l)\) with forward activations \((\mathbf{L}_l)\) , effectively suppressing background clutter. These strategies include (i) directly applying the normalized gradient tensor (Eq.3, Eq.4) and (ii) generating a global object saliency map by summing the gradient tensor across channels (Eq.5, Eq.6). We denote these attention strategies as direct attention modulation (DAM), exponential attention modulation (EAM), global attention fusion (GAF), and exponential global attention (EGA), corresponding to the following formulations in Eq. 3, 4, 5, and 6, respectively.
\[\begin{array}{rl} & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N(\mathbf{G})}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N(\mathbf{G})}}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}\\ & {\delta (\mathbf{L},\mathbf{G}) = \mathbf{L}\odot \mathbf{e}^{N\left(\sum_{i}N(\mathbf{G}_{ij})\right)}} \end{array} \quad (6)\]
Here, \(\odot\) denotes the Hadamard product, and \(N(.)\) represents the normalization function, which scales \(\mathbf{G}\) to the range [0,1] to serve as an attention mask for \(\mathbf{L}\) . Unlike [15], where gradients are normalized for general feature enhancement, we normalize gradients specifically to suppress activations related to background clutter, ensuring a stronger focus on salient objects. This approach produces attention- guided features where activations corresponding to object regions remain dominant, improving representation quality for scene understanding.
### 2.3 Random RNN for Feature Encoding
After obtaining object attention- guided CNN features from block 5 (L5 following [19]), the next step is to encode these representations into a more compact space. Directly using these high- dimensional features for frame- to- frame comparison can degrade SLAM performance due to the curse of dimensionality. To address this, we employ RNNs [20] to pool the features into a lower- dimensional, compact, and separable representation, as in [9]. Unlike [9], we first apply average pooling before reshaping the CNN activations. To adapt high- dimensional VGG L5 features, we merge every two activation maps by averaging pixels, reducing the feature size to \(7 \times 7 \times 256\) . We then reshape the activations to \(14 \times 14 \times 64\) for RNN processing. RNNs recursively merge adjacent vectors into parent
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Table 1. Accuracy comparison of attentionguided models against the baseline [9], measured in RMS-ATE (m), on the fr1 (small) and fr2 (large) sequences.
<table><tr><td></td><td>baseline [9]</td><td>GAF</td><td>EAM</td><td>EGA</td><td>DAM</td></tr><tr><td>360</td><td>0.056</td><td>0.054</td><td>0.056</td><td>0.051</td><td>0.053</td></tr><tr><td>desk</td><td>0.020</td><td>0.020</td><td>0.019</td><td>0.020</td><td>0.020</td></tr><tr><td>desk2</td><td>0.030</td><td>0.030</td><td>0.028</td><td>0.031</td><td>0.028</td></tr><tr><td>floor</td><td>0.029</td><td>0.029</td><td>0.030</td><td>0.029</td><td>0.029</td></tr><tr><td>plant</td><td>0.035</td><td>0.036</td><td>0.035</td><td>0.036</td><td>0.038</td></tr><tr><td>room</td><td>0.047</td><td>0.049</td><td>0.050</td><td>0.049</td><td>0.049</td></tr><tr><td>teddy</td><td>0.038</td><td>0.040</td><td>0.039</td><td>0.038</td><td>0.039</td></tr><tr><td>average</td><td>0.0364</td><td>0.0369</td><td>0.0367</td><td>0.0363</td><td>0.0366</td></tr><tr><td>large_no_loop</td><td>0.355</td><td>0.242</td><td>0.179</td><td>0.139</td><td>0.137</td></tr><tr><td>large_with_loop</td><td>0.357</td><td>0.342</td><td>0.348</td><td>0.353</td><td>0.357</td></tr><tr><td>pioneer_360</td><td>0.150</td><td>0.137</td><td>0.152</td><td>0.160</td><td>0.150</td></tr><tr><td>pioneer_slam</td><td>0.428</td><td>0.398</td><td>0.417</td><td>0.395</td><td>0.355</td></tr><tr><td>pioneer_slam2</td><td>0.160</td><td>0.163</td><td>0.166</td><td>0.164</td><td>0.158</td></tr><tr><td>pioneer_slam3</td><td>0.282</td><td>0.265</td><td>0.267</td><td>0.264</td><td>0.271</td></tr><tr><td>average</td><td>0.289</td><td>0.258</td><td>0.255</td><td>0.246</td><td>0.238</td></tr></table>
vectors using tied weights and a tanh activation function [20]. We employ the one- level structured RNN from [19], where each RNN outputs a \(k\) - dimensional feature vector ( \(k = 64\) ). Following [9], we use 16 RNNs, producing a final 1024- dimensional feature vector ( \(64 \times 16 = 1024\) ).
## 3 Experiments
We evaluated the performance of the proposed approach on the popular TUM RGB- D dataset [10], using the fr1 and fr2 sequences to assess performance in both medium- and large- scale indoor environments. The fr2 sequences, recorded in a large industrial halls with more challenging conditions, provide a more rigorous evaluation than the fr1 sequences.
Table 1 presents the RMS- ATE (root mean square of absolute trajectory error in meters) for different attention fusion strategies compared to the baseline [9]. On the fr1 sequences, object- attentive features do not show a significant improvement over the baseline. This is likely because the small- scale sequences contain fewer distinctive objects, limiting the advantage of semantic attention. When the scene is centered around a single object, low- level features may provide more reliable frame associations than high- level object- aware attention. Moreover, if the sequence of sample data is around one particular object, it is neither easy nor feasible for the network to distinguish foreground object and background clutter using the proposed object attentive gradients. Consequently, attention- guided features offer no clear benefit in these cases. However, both the baseline and attention- based models achieve high accuracy, with errors close to the ground truth, indicating that attention integration does not negatively impact performance in small- scale settings.
In contrast, the fr2 sequences show a clear per formance gain with object- attentive features, supporting the idea that attention- based SLAM can enhance large- scale mapping by prioritizing object regions over background clutter. As seen in Table 1, all attention- based models significantly reduce RMS- ATE compared to the baseline. The observed drift errors range between 10 cm and 35 cm, which is acceptable for these highly challenging large- scale sequences. These improvements demonstrate that attention- guided feature representations can generalize well to complex, real- world environments, making them promising for large- scale autonomous navigation tasks. Our ablative study on different attention fusion strategies confirms that the direct attention modulation (DAM) method consistently outperforms other approaches, yielding the best accuracy across most sequences. Figure 3 visualizes sample estimated trajectories using DAM- based object attention on fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3. The proposed model effectively minimizes RMS- ATE errors, producing trajectory maps closely aligned with ground truth results. The results show that leveraging object attention in SLAM can reduce cumulative drift and improve long- term trajectory consistency, particularly in environments with rich semantic content.
<center>Figure 3. Comparison of estimated trajectories using the DAM attention model against ground truth for the fr1_plant, fr2_pioneer_slam, and fr2_pioneer_slam3 sequences. </center>
## 4 Conclusion
We proposed a gradient- based object- attentive approach for loop closure detection in RGB- D SLAM, integrating attention- guided features by modulating CNN representations with object- attentive gradients. To our knowledge, this is the first attempt to incorporate attention mechanisms in a SLAM system this way. Experimental results demonstrate the effectiveness of our approach, particularly in large- scale environments. The strong performance on the fr2 sequences suggests that attention- guided features could also be beneficial for outdoor mapping applications. Future work includes using eye- fixation trained networks, exploring attention- based keypoint detection and keyframe selection, and extending the method to a multi- modal RGB- D setting for enhanced performance.
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## References
[1] W. Wang, Q. Lai, H. Fu, J. Shen, H. Ling, and R. Yang, "Salient object detection in the deep learning era: An in- depth survey," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 6, pp. 3239- 3259, 2022. [2] Z. Zhou, W. Pei, X. Li, H. Wang, F. Zheng, and Z. He, "Saliency- associated object tracking," in Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 9866- 9875. [3] Y. Zhang, N. Wu, C. Z. Lin, G. Wetzstein, and Q. Sun, "Gazefusion: Saliency- guided image generation," ACM Transactions on Applied Perception, vol. 21, no. 4, pp. 1- 19, 2024. [4] G. Tinchev, A. Penate- Sanchez, and M. Fallon, "Skd: Keypoint detection for point clouds using saliency estimation," IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3785- 3792, 2021. [5] X. Ren, D. Zhang, X. Bao, and Y. Zhang, "S2- net: Semantic and salient attention network for person re- identification," IEEE Transactions on Multimedia, pp. 1- 1, 2022. [6] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Attention- guided lidar segmentation and odometry using image- to- point cloud saliency transfer," Multimedia Systems, vol. 30, no. 4, p. 188, 2024. [7] G. Ding, N. Imamoglu, A. Caglayan, M. Murakawa, and R. Nakamura, "Salidar: Saliency knowledge transfer learning for 3d point cloud understanding," in BMVC, 2022, p. 584. [8] R. Mur- Artal and J. D. Tardos, "Orb- slam2: an open- source slam system for monocular, stereo and rgb- d cameras," IEEE Transactions on Robotics, vol. 33, no. 5, pp. 1255- 1262, 2017. [9] O. Guclu, A. Caglayan, and A. B. Can, "Rgb- d indoor mapping using deep features," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2019. [10] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers, "A benchmark for the evaluation of rgb- d slam systems," in Proc. of the International Conference on Intelligent Robot Systems (IROS), October 2012.
[11] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, "Learning deep features for discriminative localization," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016. [12] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, "Grad- cam: Visual explanations from deep networks via gradient- based localization," in Proceedings of the IEEE International Conference on Computer Vision (ICCV), Oct 2017. [13] W. Shimoda and K. Yanai, "Distinct class- specific saliency maps for weakly supervised semantic segmentation," in Proceedings of the European Conference on Computer Vision (ECCV) Workshops, October 2016. [14] H. Fukui, T. Hirakawa, T. Yamashita, and H. Fujiyoshi, "Attention branch network: Learning of attention mechanism for visual explanation," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019. [15] N. Imamoglu, C. Zhang, W. Shimoda, Y. Fang, and B. Shi, "Saliency detection by forward and backward cues in deep- cnn," in 2017 IEEE International Conference on Image Processing (ICIP), 2017, pp. 430- 434. [16] L. Jiang, M. Xu, X. Wang, and L. Sigal, "Saliency- guided image translation," in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021, pp. 16 504- 16 513. [17] O. Russakovsky\*, J. Deng\*, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei- Fei, "Imagenet large scale visual recognition challenge," International Journal of Computer Vision, vol. 115, pp. 211- 252, 2015. [18] M. Muja and D. G. Lowe, "Scalable nearest neighbor algorithms for high dimensional data," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 36, no. 11, pp. 2227- 2240, 2014. [19] A. Caglayan, N. Imamoglu, A. B. Can, and R. Nakamura, "When cnns meet random rnns: Towards multilevel analysis for rgb- d object and scene recognition," Computer Vision and Image Understanding, p. 103373, 2022. [20] R. Socher, B. Huval, B. Bath, C. D. Manning, and A. Y. Ng, "Convolutional- recursive deep learning for 3d object classification," in Advances in Neural Information Processing Systems, 2012, pp. 656- 664.
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2510.25700v1
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# Timelike Holographic Complexity
Mohsen Alishahiha School of Quantum Physics and Matter Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395- 5531, Tehran, Iran
Motivated by the pseudo- entropy program, we investigate timelike subregion complexity within the holographic "Complexity=Volume" framework, extending the usual spatial constructions to Lorentzian boundary intervals. For hyperbolic timelike regions in AdS geometries, we compute the corresponding bulk volumes and demonstrate that, despite the Lorentzian embedding, the resulting subregion complexity remains purely real. We further generalize our analysis to AdS black brane geometries, where the extremal surfaces can either be constant- time hypersurfaces or penetrate the horizon. In all cases, the computed complexity exhibits the same universal UV divergences as in the spacelike case but shows no imaginary contribution, underscoring its causal and geometric origin. This stands in sharp contrast with the complex- valued pseudo- entropy and suggests that holographic complexity preserves a genuinely geometric and real character even under Lorentzian continuation.
## I. INTRODUCTION
Quantum entanglement has long been recognized as a fundamental probe of quantum correlations and spacetime structure in holographic theories [1- 3]. Traditionally, entanglement entropy is defined for spatial subregions at a fixed time slice, capturing the pattern of quantum correlations among spatially separated degrees of freedom in the boundary theory. However, recent developments have extended this concept beyond equal- time slices, leading to a novel quantity known as pseudo- entropy [4- 6]. This extension enables the study of entanglement- like measures for timelike subregions and for transitions between quantum states.
Pseudo- entropy generalizes entanglement entropy to the case of nontrivial quantum state transitions. Given an initial and a final state, \(|\psi_i\rangle\) and \(|\psi_f\rangle\) , one defines a transition matrix
\[\tau = \frac{|\psi_i\rangle\langle\psi_f|}{\langle\psi_i|\psi_f\rangle}, \quad (1)\]
and the pseudo- entropy of a subsystem \(A\) (of a bipartite system \(A\cup B\) ) as
\[S_A^{\mathrm{pseudo}} = -\mathrm{Tr}\tau_A\log \tau_A,\qquad \tau_A = \mathrm{Tr}_B\tau . \quad (2)\]
Unlike conventional entanglement entropy, pseudo- entropy can acquire complex values due to the non- Hermitian nature of \(\tau\) . Its real part quantifies a generalized correlation between \(|\psi_i\rangle\) and \(|\psi_f\rangle\) , while its imaginary part encodes a phase- like structure associated with temporal or causal transitions. The appearance of such complex contributions points toward a richer analytic structure underlying holographic information measures.
In the context of AdS/CFT correspondence [7], pseudo- entropy naturally motivates the study of timelike
extremal surfaces anchored on timelike boundary intervals [8, 9]. These surfaces extend the Ryu- Takayanagi (RT) construction [1, 2] into Lorentzian regions of AdS, where the induced metric on the extremal surface may change signature. Such generalizations have attracted increasing interest in recent years (see e.g. [10- 16]). The corresponding holographic entanglement entropy for timelike subregions generally acquires an imaginary part, which is interpreted as the bulk dual of the complex- valued pseudo- entropy. This complexification hints that other quantum information measures, such as quantum complexity, might also admit timelike or complexified extensions.
Holographic quantum complexity provides a geometric measure of how difficult it is to prepare a particular boundary state from a simple reference state using a minimal set of quantum operations. The two best- known proposals are the "Complexity=Volume" (CV) conjecture [17, 18]—further extended to subregion complexity in [19] (see also [20, 21])—and the "Complexity=Action" (CA) conjecture [22, 23], whose subregion version has been explored in [24- 26]. These frameworks have deepened our understanding of information storage and state preparation in holographic field theories, particularly for spatially defined subsystems and global states.
Despite this progress, the generalization of holographic complexity to timelike subregions has remained largely unexplored. Motivated by the pseudo- entropy framework, we introduce and analyze a timelike version of subregion complexity within the CV proposal. This involves evaluating the bulk volume enclosed by timelike extremal surfaces—extending the original construction of [19] into Lorentzian regions of AdS. Conceptually, this provides a natural holographic dual for a measure of "temporal circuit depth" or the complexity associated with time evolution between quantum configurations in the boundary theory.
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In this work, we focus first on timelike hyperbolic subregions in \(\mathrm{AdS}_{d + 2}\) , building on the explicit extremal surfaces found in [8, 9]. We compute the corresponding bulk volumes and analyze their divergence structures. Remarkably, we find that unlike the case of pseudo- entropy—timelike subregion complexity remains purely real. This indicates that holographic complexity retains a purely geometric interpretation, even in the presence of Lorentzian embeddings, and may therefore probe aspects of spacetime geometry inaccessible to entanglement- based measures.
We then extend our analysis to thermal states described by AdS black brane geometries, where the presence of horizons renders the extremal surfaces more intricate. We show that, depending on the causal nature of the subregion, the extremal surfaces can penetrate the horizon, and the resulting complexity exhibits both universal UV divergences and finite, horizon- dependent corrections.
The remainder of this letter is organized as follows. In Sec. II, we compute the timelike subregion complexity for hyperbolic regions in pure AdS. In Sec. III, we generalize the construction to AdS black hole backgrounds and analyze the resulting behavior near and beyond the horizon. Sec. IV is devoted to concluding remarks and possible extensions.
## II. TIMELIKE SUBREGION AND HOLOGRAPHIC VOLUME
Following [8, 9], we consider a timelike boundary subregion embedded in an \(\mathrm{AdS}_{d + 2}\) spacetime with \(d \geq 1\) . The bulk geometry is described by the metric
\[ds^{2} = \frac{R^{2}}{r^{2}}\left(dr^{2} + dy^{2} - d\xi^{2} + \xi^{2}dX_{d - 1}^{2}\right), \quad (3)\]
where \(y\) is a fixed spectator coordinate, \(X_{d - 1}\) denotes the \((d - 1)\) - dimensional transverse space, and \(R\) is the AdS radius. The boundary subsystem is chosen as a timelike hyperbolic region \(\xi^{2} \leq \frac{T^{2}}{r^{2}}\) , with \(T\) the boundary time interval. The corresponding area functional takes the form
\[A = H_{d - 1}R^{d}\int dr\frac{\xi^{d - 1}\sqrt{1 - \xi^{r2}}}{r^{d}}, \quad (4)\]
where \(H_{d - 1} = \mathrm{Vol}(X_{d - 1})\) is the transverse volume. Extremizing this functional yields two branches of extremal surfaces [8, 9]
\[\xi^{2} = r^{2} + \frac{T^{2}}{4},\qquad \xi^{2} = r^{2} - \frac{T^{2}}{4}, \quad (5)\]
corresponding to spacelike and timelike surfaces, respectively. The timelike branch exists only for \(r \geq \frac{T}{2}\) , defining a causal cutoff in the bulk geometry.
These two branches together constitute the full extremal surface configuration that defines the timelike
subregion problem. As demonstrated in [8, 9], while the area of the spacelike branch remains real, the timelike branch introduces an imaginary contribution to the entanglement entropy. This motivates studying its analog in holographic complexity, where one expects both real and potentially complex geometric contributions.
In the subregion complexity CV prescription [19], holographic complexity is proportional to the bulk volume enclosed by the extremal surfaces:
\[V = H_{d - 1}R^{d + 1}\int_{\xi \leq f(r)}dr d\xi \frac{\xi^{d - 1}}{r^{d + 1}}, \quad (6)\]
where \(f(r)\) is the extremal profile defined in Eq. (5). In this case, the relevant volume is the region enclosed between the two branches, as illustrated in Figure 1. Performing the \(\xi\) integration then yields the volume difference between the branches, given by
\[V = \frac{2H_{d - 1}R^{d + 1}}{d}\left[\int_{\epsilon}^{\infty}dr\frac{(r^{2} + \frac{T^{2}}{4})^{\frac{d}{2}}}{r^{d + 1}} -\int_{\frac{T}{2}}^{\infty}dr\frac{(r^{2} - \frac{T^{2}}{4})^{\frac{d}{2}}}{r^{d + 1}}\right],\]
where \(\epsilon\) is a UV cutoff near the boundary \(r = 0\) . The second integral starts at \(r = T / 2\) since the timelike branch exists only for \(r \geq T / 2\) . The factor of 2 arises from the symmetry of the volume under consideration.
<center>Figure 1. The colored region between the spacelike (green) and timelike (red) extremal surfaces represents the bulk volume associated with the timelike subregion complexity. </center>
Following [19], the timelike subregion complexity is defined as
\[\mathcal{C}_{\mathrm{T}} = \frac{V}{GR}, \quad (8)\]
where \(G\) is the \((d + 1)\) - dimensional Newton constant. It is clear from Eq. (7) that both integrals individually exhibit IR divergences at \(r \to \infty\) , but these cancel in the difference, leaving a finite volume. The UV divergences at \(r \to 0\) are regularized by \(\epsilon\) .
Although Eq. (7) can be expressed in terms of hypergeometric functions for general \(d\) , it is more instructive to present explicit results for lower dimensions. Evaluating
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the integrals for \(d = 1,2,3,4\) gives
\[\begin{array}{r l} & {\mathrm{AdS}_{3}:\quad C_{\mathrm{T}} = \frac{R}{G}\frac{T}{2\epsilon},}\\ & {\mathrm{AdS}_{4}:\quad C_{\mathrm{T}} = \frac{V_{1}R^{2}}{2G}\left(\frac{T^{2}}{8\epsilon^{2}} +\log \frac{T}{2\epsilon} +\frac{1}{2}\right),}\\ & {\mathrm{AdS}_{5}:\quad C_{\mathrm{T}} = \frac{V_{2}R^{3}}{3G}\left(\frac{T^{3}}{24\epsilon^{3}} +\frac{3T}{4\epsilon}\right),}\\ & {\mathrm{AdS}_{6}:\quad C_{\mathrm{T}} = \frac{V_{3}R^{4}}{4G}\left(\frac{T^{4}}{64\epsilon^{4}} +\frac{T^{2}}{4\epsilon^{2}} +\log \frac{T}{2\epsilon} +\frac{3}{4}\right).} \end{array} \quad (12)\]
The divergence structure mirrors that of the spacelike CV proposal: power- law UV divergences corresponding to short- distance correlations, and universal logarithmic terms appearing in even dimensions. The leading divergence scales with the volume of the boundary subregion, confirming that the dominant contribution to holographic complexity arises from the near- boundary geometry.
An important distinction emerges when comparing with the entanglement entropy of timelike regions: the complexity expressions in Eq. (9) are purely real. While the timelike branch of the extremal surface leads to an imaginary contribution in entanglement entropy due to the analytic continuation across the light cone, in the complexity- volume relation the integration is performed over a strictly Lorentzian region with \(r \geq T / 2\) , yielding a real- valued geometric volume.
To gain a better intuition for the quantity we are computing, let us focus on the three- dimensional case. The corresponding extremal surfaces in Poincaré and global coordinates are depicted in Fig. 2 (see also [8]). In the context of entanglement entropy, when one transitions to global coordinates, the timelike branch becomes apparent, giving rise to the imaginary contribution in the pseudo- entropy. For complexity, if one ignores the timelike surface, the relevant bulk volume would appear to be that enclosed by the spacelike RT curve and the Poincaré horizon in the global coordinate patch. However, once the timelike extremal surface is included, the proper volume is the region enclosed between the spacelike and timelike branches—illustrated by the shaded region in Fig. 2.
One might naturally wonder whether this configuration implies a hidden causal connection between the two branches, absent in the usual spacelike case, which could in principle lead to an imaginary contribution to the complexity. At present, however, it remains unclear whether such a causal volume genuinely exists 1.
From a technical perspective, for odd \(d\) both limits of the timelike integral acquire phase factors that cancel, leaving a purely real result. Hence, timelike complexity measures only the accessible Lorentzian bulk volume,
<center>Figure 2. Extremal surfaces for a timelike entangling region (blue interval) in Poincaré (left) and global (right) coordinates. In the global case, in addition to the spacelike extremal surface (green), there exists a timelike one (red). The timelike subregion complexity corresponds to the bulk volume enclosed between these two surfaces, illustrated by the shaded (colored) region. This figure essentially reproduces Fig. 3 of [8] for clarity and comparison. Here, \((r, z)\) and \((\tau , \rho)\) denote the Poincaré and global coordinate systems, respectively. </center>
while pseudo- entropy captures phase information related to temporal correlations.
Comparing our results with those for spacelike subregions obtained in [19], we find that although the overall divergence structure of timelike subregion complexity closely parallels the spacelike case, the two are not related by a simple Wick rotation. Both share the same hierarchy of UV divergences—power- law terms reflecting short- distance correlations and logarithmic terms in even dimensions—but their finite parts differ in both magnitude and interpretation. In particular, for odd- dimensional AdS backgrounds, timelike complexity lacks the universal constant term that characterizes its spacelike counterpart. This absence suggests that timelike complexity has a fundamentally distinct geometric origin, one not obtainable by analytic continuation of the spatial result. Rather, it arises directly from the Lorentzian causal structure of the extremal hypersurface.
To further clarify this point, consider repeating the extremization procedure for a spacelike subregion possessing spherical symmetry at the boundary. The corresponding area functional admits two distinct extremal solutions:
\[\xi (r) = \sqrt{\ell^2 - r^2},\qquad \xi (r) = i\sqrt{\ell^2 + r^2}, \quad (13)\]
where \(\ell\) is the radius of the boundary entangling region.
The first branch corresponds to the well- known RT surface [1], which provides the minimal- area prescription for holographic entanglement entropy and forms the basis for computing holographic subregion complexity [19]. The second branch, however, is purely imaginary and has traditionally been excluded because it does not correspond to a real Lorentzian extremal surface in the AdS bulk. In the usual treatment, this complex embedding is interpreted as unphysical, as it extends beyond the domain of
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real- valued minimal surfaces anchored on the boundary.
From the present perspective, however, this "discarded" branch carries deep physical significance. Upon analytic continuation, it reproduces the timelike extremal branch encountered in the study of timelike entanglement entropy and timelike subregion complexity. The imaginary embedding coordinate \(\xi = i\sqrt{\ell^2 + r^2}\) corresponds to a continuation of the minimal surface across the light cone, thereby encoding the causal extension of the entangling region into the timelike domain. Its contribution is responsible for the appearance of the imaginary component in pseudo- entropy, reflecting phase information associated with non- unitary or time- reflected evolutions of the boundary state.
In this sense, the two branches- one real and one imaginary- are not independent but rather analytic continuations of a unified geometric configuration. The real branch represents the standard RT surface probing spatial entanglement, while the imaginary branch extends the geometry into a "shadow" region of AdS, corresponding to the temporal or causally continued part of the boundary subregion. This complexified structure naturally bridges spacelike and timelike holographic quantities: the real branch encodes spatial correlations, whereas the imaginary branch captures the causal and temporal entanglement information underlying pseudo- entropy.
Within the framework of holographic complexity for spacelike subregions, only the real Lorentzian volume contributes directly to the CV proposal, providing a real- valued geometric measure of computational depth. However, the presence of an imaginary extremal branch suggests that a more general, complexified notion of holographic complexity may exist- one that unifies spatial and temporal information within a single analytic structure. Such an extension could naturally connect holographic complexity, pseudo- entropy, and causal holographic information through complex AdS embeddings, offering a richer picture of quantum information geometry in Lorentzian spacetimes. Nonetheless, as our explicit calculations show, even for the timelike subsystem- where the extremal configuration includes a timelike branch- the resulting complexity remains entirely real, reflecting the geometric rather than entropic nature of the CV prescription.
## III. TIMELIKE SUBREGION COMPLEXITY FOR ADS BLACK HOLES
It is natural to extend our discussion of timelike subregion complexity to thermal states, which are holographically dual to AdS black hole or black brane geometries. In this context, the thermal nature of the boundary state is encoded in the presence of a horizon, introducing a new geometric scale that influences the causal structure of the extremal surfaces. For concreteness, we consider a \((d + 2)\) - dimensional AdS black hole with metric
\[ds^{2} = \frac{R^{2}}{r^{2}}\left(-f(r)dt^{2} + \frac{dr^{2}}{f(r)} +dx^{2} + dY_{d - 1}^{2}\right), \quad (14)\]
where
\[f(r) = 1 - \frac{r^{d + 1}}{r_{h}^{d + 1}}, \quad (15)\]
and \(r_h\) denotes the horizon radius. The conformal boundary lies at \(r = 0\) .
We consider a timelike subregion in the form of a strip extending along the \(y_i\) directions but bounded in time by the finite interval \(t \in [- T / 2, T / 2]\) at fixed spatial coordinate \(x\) . Although, unlike the hyperbolic case discussed in the previous section, this setup lacks a clear interpretation of bulk extremal surface, we can still follow the general extremization procedure for the area functional associated with a timelike embedding \(t = t(r)\) [9]. The induced metric on the hypersurface is
\[ds_{\mathrm{ind}}^{2} = \frac{R^{2}}{r^{2}}\left[\left(\frac{1}{f(r)} -f(r)t'(r)^{2}\right)dr^{2} + dY_{d - 1}^{2}\right], \quad (16)\]
where \(t'(r) = dt / dr\) and \(dY_{d - 1}^{2}\) denotes the \((d - 1)\) - dimensional transverse section. The corresponding area functional is
\[A = \mathcal{R}_{d - 1}R^{d}\int dr\frac{\sqrt{\frac{1}{f(r)}} - f(r)t'(r)^{2}}{r^{d}}, \quad (17)\]
with \(\mathcal{R}_{d - 1} = \mathrm{Vol}(Y_{d - 1})\) .
Extremizing this functional yields a conserved quantity,
\[\frac{-f(r)t'(r)}{r^d\sqrt{\frac{1}{f(r)} - f(r)t'(r)^2}} = \mathrm{constant}, \quad (18)\]
which can be interpreted as an "energy" parameter for the surface profile \(t(r)\) . For real- valued \(t'(r)\) this constant must be real; however, for timelike subregions this reality condition is subtle, and in general no globally real solution exists unless the extremal surface extends beyond the horizon.
Besides the trivial solution \(t'(r) = 0\) , the presence of a horizon allows for an extremal surface that penetrates it, with the turning point \(r_0\) lying behind the horizon \((r_0 > r_h)\) . This possibility is not merely formal—it closely parallels the behavior of extremal surfaces in time- dependent backgrounds such as Vaidya geometries [27], where the surface crosses horizons to capture causal correlations. Even though our setup is static, the subregion itself is intrinsically time- dependent, which justifies such an extension. Under this assumption, the extremal surface satisfies
\[\frac{f(r)t'(r)}{\sqrt{\frac{1}{f(r)} - f(r)t'(r)^2}} = \sqrt{\bar{f}(r_0)}\left(\frac{r}{r_0}\right)^d, \quad (19)\]
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where
\[\tilde{f} (r_0) = \left(\frac{r_0}{r_h}\right)^{d + 1} - 1, \quad (20)\]
for \(r_0 > r_h\) . Solving for \(t'(r)\) gives
\[t'(r) = -\frac{\left(\frac{r}{r_0}\right)^d\sqrt{\tilde{f}(r_0)}}{f(r)\sqrt{f(r) + \left(\frac{r}{r_0}\right)^{2d}\tilde{f}(r_0)}}. \quad (21)\]
The CV- type volume enclosed by this extremal surface is
\[V = 2\mathcal{R}_{d - 1}R^{d + 1}\int dr\frac{t(r)}{r^{d + 1}}, \quad (22)\]
where the factor of two reflects time- reflection symmetry. Integrating by parts and using \(t(0) = T / 2\) , we obtain
\[\mathcal{C}_{\mathrm{T}} = \frac{\mathcal{R}_{d - 1}R^{d}T}{dG\epsilon^{d}} +\frac{2\mathcal{R}_{d - 1}R^{d}}{dG}\int_{\epsilon}^{r_0}dr\frac{t'(r)}{r^{d}}, \quad (23)\]
with \(\epsilon\) a UV cutoff. Substituting (21) yields 2
\[\begin{array}{l}{\mathcal{C}_{\mathrm{T}} = \frac{\mathcal{R}_{d - 1}R^{d}T}{dG\epsilon^{d}}}\\ {-\frac{2\mathcal{R}_{d - 1}R^{d}\sqrt{\tilde{f}(r_{0})}}{dG r_{0}^{d}}\int_{\epsilon}^{r_{0}}\frac{dr}{f(r)\sqrt{f(r) + \left(\frac{r}{r_{0}}\right)^{2d}\tilde{f}(r_{0})}}.} \end{array} \quad (24)\]
The leading divergence is proportional to the boundary volume of the subsystem, as expected from the near- boundary AdS structure. Importantly, the total timelike complexity remains real, even when the extremal surface penetrates the horizon. This behavior contrasts with that of timelike entanglement entropy, which typically acquires imaginary contributions associated with pseudo- entropy.
Although we have so far ignored the trivial \(t'(r) = 0\) configuration—corresponding to a constant- time slice extending from the boundary to the horizon—it is plausible that, under certain conditions, a transition could occur between this static branch and the horizon- penetrating one. Such a transition would resemble phase changes between distinct extremal surfaces in the usual entanglement entropy context.
## IV. CONCLUSION
In this work, we have introduced and systematically analyzed the notion of timelike subregion complexity within the holographic CV framework. Motivated by the pseudo- entropy program, we have extended the study of holographic complexity from spatial to timelike boundary regions and evaluated the corresponding bulk volumes for both pure AdS and AdS black brane geometries.
For timelike hyperbolic regions in pure AdS \(d + 2\) , we showed that the bulk volume enclosed between the spacelike and timelike extremal surfaces remains purely real, despite the Lorentzian nature of the embedding. This stands in sharp contrast with the case of pseudo- entropy, where analytic continuation across the light cone generically produces an imaginary component. The absence of any imaginary part in the complexity indicates that the CV functional retains a genuinely geometric interpretation even when extended to timelike configurations. In this sense, timelike complexity measures the accessible Lorentzian volume bounded by causally admissible surfaces, providing a probe of spacetime geometry that is insensitive to analytic continuation or phase structure.
We then extended our analysis to thermal states dual to AdS black branes, where the causal structure and the presence of a horizon introduce new subtleties. In this case, extremal surfaces can penetrate the horizon, depending on the causal character of the boundary subregion, while a trivial solution corresponding to a constant- time hypersurface also exists, extending smoothly from the boundary to the horizon. Remarkably, even when the extremal surface crosses the horizon, the resulting CV volume—and hence the subregion complexity—remains entirely real. This reinforces the interpretation of timelike complexity as a measure of accessible bulk volume determined by causal reach rather than analytic continuation. It is also plausible that a transition could occur between the trivial and horizon- penetrating branches, signaling a new type of geometric phase structure.
The absence of any imaginary contribution in \(\mathcal{C}_{\mathrm{T}}\) suggests that holographic complexity possesses a fundamentally different analytic structure from that of pseudo- entropy. Whereas pseudo- entropy can reflect features of transition amplitudes—such as phase information—complexity appears to probe the underlying geometry of information itself. Its purely real character indicates that holographic complexity encodes the intrinsic "depth" or cost of traversing the bulk spacetime, independent of any temporal analytic continuation.
This purely real nature is, in fact, not unexpected. Although the construction originates from the transition matrix (1), which is not Hermitian, it remains a well- posed and meaningful question to ask how "distant" the final state \(|\psi_f\rangle\) is from the initial state \(|\psi_i\rangle\) . In this sense, \(\mathcal{C}_{\mathrm{T}}\) can be interpreted as quantifying the minimal number of effective "gates" or operations required to transform \(|\psi_i\rangle\) into \(|\psi_f\rangle\) restricted to a subregion—a quantity that is naturally real.
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Of course, an important open question remains—namely, how to compute timelike complexity directly from the boundary field theory. Even for spatial subregions, a precise field-theoretic definition of subregion complexity is still lacking, with partial insights coming from circuit complexity, path- integral optimization, or tensor- network approaches. Extending these frameworks to Lorentzian or timelike subregions poses additional conceptual and technical challenges, as one must incorporate causal and non- unitary aspects of boundary dynamics. A consistent field- theoretic formulation of timelike complexity would thus not only clarify the operational meaning of our holographic construction but also reveal how temporal and causal structures are encoded in the boundary state space.
We note also that working with timelike entangling regions reveals an intriguing new feature. By relaxing the physical boundary conditions on the entangling region, additional extremal surfaces can emerge, though their physical interpretation remains uncertain. In particular, we could identify a Lorentzian branch with an induced metric of Lorentzian signature, corresponding to an extremal surface that originates from a point in the bulk and terminates at the horizon. Such configurations suffer from infrared divergences, which typically arise from the near- horizon region of the geometry where the redshift factor becomes large. In this regime, the integrand of the corresponding volume or area functional develops a logarithmic divergence, signaling that the extremal surface probes deep into the infrared part of the bulk spacetime. Physically, this behavior reflects the accumulation of contributions from regions close to the horizon, where the proper volume effectively becomes infinite due to the warping of the metric. Understanding the precise regularization and physical interpretation of these IR divergences—whether they correspond to long- time or large- scale phenomena in the dual field theory—remains an open and interesting question.
Very recently, the paper [28] (see also [29]) appeared, in which the authors investigated the dynamics of spatial subregion complexity under time evolution and uncovered a series of striking behaviors. They showed that for subsystems larger than half the total system, the complexity grows linearly for exponentially long times, whereas for smaller subsystems the growth halts and
eventually reverses, leading to a sharp transition at the half- size threshold. At finite temperature, additional critical subsystem sizes emerge, determining whether the complexity saturates instantaneously or continues to grow.
Although their setup focuses on spatial subregions evolving in time, it shares deep conceptual parallels with our Lorentzian analysis. Both frameworks reveal that subregion complexity is highly sensitive to causal structure and the presence of horizons or effective causal barriers. In particular, the sharp transitions identified in [28] may correspond, in our timelike setup, to transitions between distinct extremal branches—such as the constant- time and horizon- penetrating surfaces. Moreover, the finite- temperature critical behavior observed in their study closely parallels our finding that timelike extremal surfaces exhibit qualitatively different regimes depending on whether the turning point lies inside the horizon or corresponds to a static configuration, hinting at analogous transition phenomena in the timelike CV framework.
A natural extension of our work, beyond generalizations to other geometries, would be to formulate and compute the timelike version of the CA conjecture on a Wheeler- DeWitt patch anchored to a timelike interval. Such an analysis could clarify the role of analytic continuation and illuminate potential phase ambiguities in Lorentzian holographic complexity. We hope to return to these questions in future work.
## ACKNOWLEDGEMENTS
I would like to thank Andreas Karch, Souvik Banerjee and Tadashi Takayanagi for their valuable comments. I am also grateful to Mohammad Javad Vasli for many insightful discussions on various aspects of holographic complexity. I would further like to thank the CERN Department of Theoretical Physics for their warm hospitality during the course of this work. This research was supported by the Iran National Science Foundation (INSF) under Project No. 4023620. I also acknowledge the assistance of ChatGPT for editorial help in refining and polishing the manuscript.
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2510.25700v1
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# Timelike Holographic Complexity
Mohsen Alishahiha School of Quantum Physics and Matter Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395- 5531, Tehran, Iran
Motivated by the pseudo- entropy program, we investigate timelike subregion complexity within the holographic "Complexity=Volume" framework, extending the usual spatial constructions to Lorentzian boundary intervals. For hyperbolic timelike regions in AdS geometries, we compute the corresponding bulk volumes and demonstrate that, despite the Lorentzian embedding, the resulting subregion complexity remains purely real. We further generalize our analysis to AdS black brane geometries, where the extremal surfaces can either be constant- time hypersurfaces or penetrate the horizon. In all cases, the computed complexity exhibits the same universal UV divergences as in the spacelike case but shows no imaginary contribution, underscoring its causal and geometric origin. This stands in sharp contrast with the complex- valued pseudo- entropy and suggests that holographic complexity preserves a genuinely geometric and real character even under Lorentzian continuation.
## I. INTRODUCTION
Quantum entanglement has long been recognized as a fundamental probe of quantum correlations and spacetime structure in holographic theories [1- 3]. Traditionally, entanglement entropy is defined for spatial subregions at a fixed time slice, capturing the pattern of quantum correlations among spatially separated degrees of freedom in the boundary theory. However, recent developments have extended this concept beyond equal- time slices, leading to a novel quantity known as pseudo- entropy [4- 6]. This extension enables the study of entanglement- like measures for timelike subregions and for transitions between quantum states.
Pseudo- entropy generalizes entanglement entropy to the case of nontrivial quantum state transitions. Given an initial and a final state, \(|\psi_i\rangle\) and \(|\psi_f\rangle\) , one defines a transition matrix
\[\tau = \frac{|\psi_i\rangle\langle\psi_f|}{\langle\psi_i|\psi_f\rangle}, \quad (1)\]
and the pseudo- entropy of a subsystem \(A\) (of a bipartite system \(A\cup B\) ) as
\[S_A^{\mathrm{pseudo}} = -\mathrm{Tr}\tau_A\log \tau_A,\qquad \tau_A = \mathrm{Tr}_B\tau . \quad (2)\]
Unlike conventional entanglement entropy, pseudo- entropy can acquire complex values due to the non- Hermitian nature of \(\tau\) . Its real part quantifies a generalized correlation between \(|\psi_i\rangle\) and \(|\psi_f\rangle\) , while its imaginary part encodes a phase- like structure associated with temporal or causal transitions. The appearance of such complex contributions points toward a richer analytic structure underlying holographic information measures.
In the context of AdS/CFT correspondence [7], pseudo- entropy naturally motivates the study of timelike
extremal surfaces anchored on timelike boundary intervals [8, 9]. These surfaces extend the Ryu- Takayanagi (RT) construction [1, 2] into Lorentzian regions of AdS, where the induced metric on the extremal surface may change signature. Such generalizations have attracted increasing interest in recent years (see e.g. [10- 16]). The corresponding holographic entanglement entropy for timelike subregions generally acquires an imaginary part, which is interpreted as the bulk dual of the complex- valued pseudo- entropy. This complexification hints that other quantum information measures, such as quantum complexity, might also admit timelike or complexified extensions.
Holographic quantum complexity provides a geometric measure of how difficult it is to prepare a particular boundary state from a simple reference state using a minimal set of quantum operations. The two best- known proposals are the "Complexity=Volume" (CV) conjecture [17, 18]—further extended to subregion complexity in [19] (see also [20, 21])—and the "Complexity=Action" (CA) conjecture [22, 23], whose subregion version has been explored in [24- 26]. These frameworks have deepened our understanding of information storage and state preparation in holographic field theories, particularly for spatially defined subsystems and global states.
Despite this progress, the generalization of holographic complexity to timelike subregions has remained largely unexplored. Motivated by the pseudo- entropy framework, we introduce and analyze a timelike version of subregion complexity within the CV proposal. This involves evaluating the bulk volume enclosed by timelike extremal surfaces—extending the original construction of [19] into Lorentzian regions of AdS. Conceptually, this provides a natural holographic dual for a measure of "temporal circuit depth" or the complexity associated with time evolution between quantum configurations in the boundary theory.
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In this work, we focus first on timelike hyperbolic subregions in \(\mathrm{AdS}_{d + 2}\) , building on the explicit extremal surfaces found in [8, 9]. We compute the corresponding bulk volumes and analyze their divergence structures. Remarkably, we find that unlike the case of pseudo- entropy—timelike subregion complexity remains purely real. This indicates that holographic complexity retains a purely geometric interpretation, even in the presence of Lorentzian embeddings, and may therefore probe aspects of spacetime geometry inaccessible to entanglement- based measures.
We then extend our analysis to thermal states described by AdS black brane geometries, where the presence of horizons renders the extremal surfaces more intricate. We show that, depending on the causal nature of the subregion, the extremal surfaces can penetrate the horizon, and the resulting complexity exhibits both universal UV divergences and finite, horizon- dependent corrections.
The remainder of this letter is organized as follows. In Sec. II, we compute the timelike subregion complexity for hyperbolic regions in pure AdS. In Sec. III, we generalize the construction to AdS black hole backgrounds and analyze the resulting behavior near and beyond the horizon. Sec. IV is devoted to concluding remarks and possible extensions.
## II. TIMELIKE SUBREGION AND HOLOGRAPHIC VOLUME
Following [8, 9], we consider a timelike boundary subregion embedded in an \(\mathrm{AdS}_{d + 2}\) spacetime with \(d \geq 1\) . The bulk geometry is described by the metric
\[ds^{2} = \frac{R^{2}}{r^{2}}\left(dr^{2} + dy^{2} - d\xi^{2} + \xi^{2}dX_{d - 1}^{2}\right), \quad (3)\]
where \(y\) is a fixed spectator coordinate, \(X_{d - 1}\) denotes the \((d - 1)\) - dimensional transverse space, and \(R\) is the AdS radius. The boundary subsystem is chosen as a timelike hyperbolic region \(\xi^{2} \leq \frac{T^{2}}{r^{2}}\) , with \(T\) the boundary time interval. The corresponding area functional takes the form
\[A = H_{d - 1}R^{d}\int dr\frac{\xi^{d - 1}\sqrt{1 - \xi^{r2}}}{r^{d}}, \quad (4)\]
where \(H_{d - 1} = \mathrm{Vol}(X_{d - 1})\) is the transverse volume. Extremizing this functional yields two branches of extremal surfaces [8, 9]
\[\xi^{2} = r^{2} + \frac{T^{2}}{4},\qquad \xi^{2} = r^{2} - \frac{T^{2}}{4}, \quad (5)\]
corresponding to spacelike and timelike surfaces, respectively. The timelike branch exists only for \(r \geq \frac{T}{2}\) , defining a causal cutoff in the bulk geometry.
These two branches together constitute the full extremal surface configuration that defines the timelike
subregion problem. As demonstrated in [8, 9], while the area of the spacelike branch remains real, the timelike branch introduces an imaginary contribution to the entanglement entropy. This motivates studying its analog in holographic complexity, where one expects both real and potentially complex geometric contributions.
In the subregion complexity CV prescription [19], holographic complexity is proportional to the bulk volume enclosed by the extremal surfaces:
\[V = H_{d - 1}R^{d + 1}\int_{\xi \leq f(r)}dr d\xi \frac{\xi^{d - 1}}{r^{d + 1}}, \quad (6)\]
where \(f(r)\) is the extremal profile defined in Eq. (5). In this case, the relevant volume is the region enclosed between the two branches, as illustrated in Figure 1. Performing the \(\xi\) integration then yields the volume difference between the branches, given by
\[V = \frac{2H_{d - 1}R^{d + 1}}{d}\left[\int_{\epsilon}^{\infty}dr\frac{(r^{2} + \frac{T^{2}}{4})^{\frac{d}{2}}}{r^{d + 1}} -\int_{\frac{T}{2}}^{\infty}dr\frac{(r^{2} - \frac{T^{2}}{4})^{\frac{d}{2}}}{r^{d + 1}}\right],\]
where \(\epsilon\) is a UV cutoff near the boundary \(r = 0\) . The second integral starts at \(r = T / 2\) since the timelike branch exists only for \(r \geq T / 2\) . The factor of 2 arises from the symmetry of the volume under consideration.
<center>Figure 1. The colored region between the spacelike (green) and timelike (red) extremal surfaces represents the bulk volume associated with the timelike subregion complexity. </center>
Following [19], the timelike subregion complexity is defined as
\[\mathcal{C}_{\mathrm{T}} = \frac{V}{GR}, \quad (8)\]
where \(G\) is the \((d + 1)\) - dimensional Newton constant. It is clear from Eq. (7) that both integrals individually exhibit IR divergences at \(r \to \infty\) , but these cancel in the difference, leaving a finite volume. The UV divergences at \(r \to 0\) are regularized by \(\epsilon\) .
Although Eq. (7) can be expressed in terms of hypergeometric functions for general \(d\) , it is more instructive to present explicit results for lower dimensions. Evaluating
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the integrals for \(d = 1,2,3,4\) gives
\[\begin{array}{r l} & {\mathrm{AdS}_{3}:\quad C_{\mathrm{T}} = \frac{R}{G}\frac{T}{2\epsilon},}\\ & {\mathrm{AdS}_{4}:\quad C_{\mathrm{T}} = \frac{V_{1}R^{2}}{2G}\left(\frac{T^{2}}{8\epsilon^{2}} +\log \frac{T}{2\epsilon} +\frac{1}{2}\right),}\\ & {\mathrm{AdS}_{5}:\quad C_{\mathrm{T}} = \frac{V_{2}R^{3}}{3G}\left(\frac{T^{3}}{24\epsilon^{3}} +\frac{3T}{4\epsilon}\right),}\\ & {\mathrm{AdS}_{6}:\quad C_{\mathrm{T}} = \frac{V_{3}R^{4}}{4G}\left(\frac{T^{4}}{64\epsilon^{4}} +\frac{T^{2}}{4\epsilon^{2}} +\log \frac{T}{2\epsilon} +\frac{3}{4}\right).} \end{array} \quad (12)\]
The divergence structure mirrors that of the spacelike CV proposal: power- law UV divergences corresponding to short- distance correlations, and universal logarithmic terms appearing in even dimensions. The leading divergence scales with the volume of the boundary subregion, confirming that the dominant contribution to holographic complexity arises from the near- boundary geometry.
An important distinction emerges when comparing with the entanglement entropy of timelike regions: the complexity expressions in Eq. (9) are purely real. While the timelike branch of the extremal surface leads to an imaginary contribution in entanglement entropy due to the analytic continuation across the light cone, in the complexity- volume relation the integration is performed over a strictly Lorentzian region with \(r \geq T / 2\) , yielding a real- valued geometric volume.
To gain a better intuition for the quantity we are computing, let us focus on the three- dimensional case. The corresponding extremal surfaces in Poincaré and global coordinates are depicted in Fig. 2 (see also [8]). In the context of entanglement entropy, when one transitions to global coordinates, the timelike branch becomes apparent, giving rise to the imaginary contribution in the pseudo- entropy. For complexity, if one ignores the timelike surface, the relevant bulk volume would appear to be that enclosed by the spacelike RT curve and the Poincaré horizon in the global coordinate patch. However, once the timelike extremal surface is included, the proper volume is the region enclosed between the spacelike and timelike branches—illustrated by the shaded region in Fig. 2.
One might naturally wonder whether this configuration implies a hidden causal connection between the two branches, absent in the usual spacelike case, which could in principle lead to an imaginary contribution to the complexity. At present, however, it remains unclear whether such a causal volume genuinely exists 1.
From a technical perspective, for odd \(d\) both limits of the timelike integral acquire phase factors that cancel, leaving a purely real result. Hence, timelike complexity measures only the accessible Lorentzian bulk volume,
<center>Figure 2. Extremal surfaces for a timelike entangling region (blue interval) in Poincaré (left) and global (right) coordinates. In the global case, in addition to the spacelike extremal surface (green), there exists a timelike one (red). The timelike subregion complexity corresponds to the bulk volume enclosed between these two surfaces, illustrated by the shaded (colored) region. This figure essentially reproduces Fig. 3 of [8] for clarity and comparison. Here, \((r, z)\) and \((\tau , \rho)\) denote the Poincaré and global coordinate systems, respectively. </center>
while pseudo- entropy captures phase information related to temporal correlations.
Comparing our results with those for spacelike subregions obtained in [19], we find that although the overall divergence structure of timelike subregion complexity closely parallels the spacelike case, the two are not related by a simple Wick rotation. Both share the same hierarchy of UV divergences—power- law terms reflecting short- distance correlations and logarithmic terms in even dimensions—but their finite parts differ in both magnitude and interpretation. In particular, for odd- dimensional AdS backgrounds, timelike complexity lacks the universal constant term that characterizes its spacelike counterpart. This absence suggests that timelike complexity has a fundamentally distinct geometric origin, one not obtainable by analytic continuation of the spatial result. Rather, it arises directly from the Lorentzian causal structure of the extremal hypersurface.
To further clarify this point, consider repeating the extremization procedure for a spacelike subregion possessing spherical symmetry at the boundary. The corresponding area functional admits two distinct extremal solutions:
\[\xi (r) = \sqrt{\ell^2 - r^2},\qquad \xi (r) = i\sqrt{\ell^2 + r^2}, \quad (13)\]
where \(\ell\) is the radius of the boundary entangling region.
The first branch corresponds to the well- known RT surface [1], which provides the minimal- area prescription for holographic entanglement entropy and forms the basis for computing holographic subregion complexity [19]. The second branch, however, is purely imaginary and has traditionally been excluded because it does not correspond to a real Lorentzian extremal surface in the AdS bulk. In the usual treatment, this complex embedding is interpreted as unphysical, as it extends beyond the domain of
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real- valued minimal surfaces anchored on the boundary.
From the present perspective, however, this "discarded" branch carries deep physical significance. Upon analytic continuation, it reproduces the timelike extremal branch encountered in the study of timelike entanglement entropy and timelike subregion complexity. The imaginary embedding coordinate \(\xi = i\sqrt{\ell^2 + r^2}\) corresponds to a continuation of the minimal surface across the light cone, thereby encoding the causal extension of the entangling region into the timelike domain. Its contribution is responsible for the appearance of the imaginary component in pseudo- entropy, reflecting phase information associated with non- unitary or time- reflected evolutions of the boundary state.
In this sense, the two branches- one real and one imaginary- are not independent but rather analytic continuations of a unified geometric configuration. The real branch represents the standard RT surface probing spatial entanglement, while the imaginary branch extends the geometry into a "shadow" region of AdS, corresponding to the temporal or causally continued part of the boundary subregion. This complexified structure naturally bridges spacelike and timelike holographic quantities: the real branch encodes spatial correlations, whereas the imaginary branch captures the causal and temporal entanglement information underlying pseudo- entropy.
Within the framework of holographic complexity for spacelike subregions, only the real Lorentzian volume contributes directly to the CV proposal, providing a real- valued geometric measure of computational depth. However, the presence of an imaginary extremal branch suggests that a more general, complexified notion of holographic complexity may exist- one that unifies spatial and temporal information within a single analytic structure. Such an extension could naturally connect holographic complexity, pseudo- entropy, and causal holographic information through complex AdS embeddings, offering a richer picture of quantum information geometry in Lorentzian spacetimes. Nonetheless, as our explicit calculations show, even for the timelike subsystem- where the extremal configuration includes a timelike branch- the resulting complexity remains entirely real, reflecting the geometric rather than entropic nature of the CV prescription.
## III. TIMELIKE SUBREGION COMPLEXITY FOR ADS BLACK HOLES
It is natural to extend our discussion of timelike subregion complexity to thermal states, which are holographically dual to AdS black hole or black brane geometries. In this context, the thermal nature of the boundary state is encoded in the presence of a horizon, introducing a new geometric scale that influences the causal structure of the extremal surfaces. For concreteness, we consider a \((d + 2)\) - dimensional AdS black hole with metric
\[ds^{2} = \frac{R^{2}}{r^{2}}\left(-f(r)dt^{2} + \frac{dr^{2}}{f(r)} +dx^{2} + dY_{d - 1}^{2}\right), \quad (14)\]
where
\[f(r) = 1 - \frac{r^{d + 1}}{r_{h}^{d + 1}}, \quad (15)\]
and \(r_h\) denotes the horizon radius. The conformal boundary lies at \(r = 0\) .
We consider a timelike subregion in the form of a strip extending along the \(y_i\) directions but bounded in time by the finite interval \(t \in [- T / 2, T / 2]\) at fixed spatial coordinate \(x\) . Although, unlike the hyperbolic case discussed in the previous section, this setup lacks a clear interpretation of bulk extremal surface, we can still follow the general extremization procedure for the area functional associated with a timelike embedding \(t = t(r)\) [9]. The induced metric on the hypersurface is
\[ds_{\mathrm{ind}}^{2} = \frac{R^{2}}{r^{2}}\left[\left(\frac{1}{f(r)} -f(r)t'(r)^{2}\right)dr^{2} + dY_{d - 1}^{2}\right], \quad (16)\]
where \(t'(r) = dt / dr\) and \(dY_{d - 1}^{2}\) denotes the \((d - 1)\) - dimensional transverse section. The corresponding area functional is
\[A = \mathcal{R}_{d - 1}R^{d}\int dr\frac{\sqrt{\frac{1}{f(r)}} - f(r)t'(r)^{2}}{r^{d}}, \quad (17)\]
with \(\mathcal{R}_{d - 1} = \mathrm{Vol}(Y_{d - 1})\) .
Extremizing this functional yields a conserved quantity,
\[\frac{-f(r)t'(r)}{r^d\sqrt{\frac{1}{f(r)} - f(r)t'(r)^2}} = \mathrm{constant}, \quad (18)\]
which can be interpreted as an "energy" parameter for the surface profile \(t(r)\) . For real- valued \(t'(r)\) this constant must be real; however, for timelike subregions this reality condition is subtle, and in general no globally real solution exists unless the extremal surface extends beyond the horizon.
Besides the trivial solution \(t'(r) = 0\) , the presence of a horizon allows for an extremal surface that penetrates it, with the turning point \(r_0\) lying behind the horizon \((r_0 > r_h)\) . This possibility is not merely formal—it closely parallels the behavior of extremal surfaces in time- dependent backgrounds such as Vaidya geometries [27], where the surface crosses horizons to capture causal correlations. Even though our setup is static, the subregion itself is intrinsically time- dependent, which justifies such an extension. Under this assumption, the extremal surface satisfies
\[\frac{f(r)t'(r)}{\sqrt{\frac{1}{f(r)} - f(r)t'(r)^2}} = \sqrt{\bar{f}(r_0)}\left(\frac{r}{r_0}\right)^d, \quad (19)\]
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where
\[\tilde{f} (r_0) = \left(\frac{r_0}{r_h}\right)^{d + 1} - 1, \quad (20)\]
for \(r_0 > r_h\) . Solving for \(t'(r)\) gives
\[t'(r) = -\frac{\left(\frac{r}{r_0}\right)^d\sqrt{\tilde{f}(r_0)}}{f(r)\sqrt{f(r) + \left(\frac{r}{r_0}\right)^{2d}\tilde{f}(r_0)}}. \quad (21)\]
The CV- type volume enclosed by this extremal surface is
\[V = 2\mathcal{R}_{d - 1}R^{d + 1}\int dr\frac{t(r)}{r^{d + 1}}, \quad (22)\]
where the factor of two reflects time- reflection symmetry. Integrating by parts and using \(t(0) = T / 2\) , we obtain
\[\mathcal{C}_{\mathrm{T}} = \frac{\mathcal{R}_{d - 1}R^{d}T}{dG\epsilon^{d}} +\frac{2\mathcal{R}_{d - 1}R^{d}}{dG}\int_{\epsilon}^{r_0}dr\frac{t'(r)}{r^{d}}, \quad (23)\]
with \(\epsilon\) a UV cutoff. Substituting (21) yields 2
\[\begin{array}{l}{\mathcal{C}_{\mathrm{T}} = \frac{\mathcal{R}_{d - 1}R^{d}T}{dG\epsilon^{d}}}\\ {-\frac{2\mathcal{R}_{d - 1}R^{d}\sqrt{\tilde{f}(r_{0})}}{dG r_{0}^{d}}\int_{\epsilon}^{r_{0}}\frac{dr}{f(r)\sqrt{f(r) + \left(\frac{r}{r_{0}}\right)^{2d}\tilde{f}(r_{0})}}.} \end{array} \quad (24)\]
The leading divergence is proportional to the boundary volume of the subsystem, as expected from the near- boundary AdS structure. Importantly, the total timelike complexity remains real, even when the extremal surface penetrates the horizon. This behavior contrasts with that of timelike entanglement entropy, which typically acquires imaginary contributions associated with pseudo- entropy.
Although we have so far ignored the trivial \(t'(r) = 0\) configuration—corresponding to a constant- time slice extending from the boundary to the horizon—it is plausible that, under certain conditions, a transition could occur between this static branch and the horizon- penetrating one. Such a transition would resemble phase changes between distinct extremal surfaces in the usual entanglement entropy context.
## IV. CONCLUSION
In this work, we have introduced and systematically analyzed the notion of timelike subregion complexity within the holographic CV framework. Motivated by the pseudo- entropy program, we have extended the study of holographic complexity from spatial to timelike boundary regions and evaluated the corresponding bulk volumes for both pure AdS and AdS black brane geometries.
For timelike hyperbolic regions in pure AdS \(d + 2\) , we showed that the bulk volume enclosed between the spacelike and timelike extremal surfaces remains purely real, despite the Lorentzian nature of the embedding. This stands in sharp contrast with the case of pseudo- entropy, where analytic continuation across the light cone generically produces an imaginary component. The absence of any imaginary part in the complexity indicates that the CV functional retains a genuinely geometric interpretation even when extended to timelike configurations. In this sense, timelike complexity measures the accessible Lorentzian volume bounded by causally admissible surfaces, providing a probe of spacetime geometry that is insensitive to analytic continuation or phase structure.
We then extended our analysis to thermal states dual to AdS black branes, where the causal structure and the presence of a horizon introduce new subtleties. In this case, extremal surfaces can penetrate the horizon, depending on the causal character of the boundary subregion, while a trivial solution corresponding to a constant- time hypersurface also exists, extending smoothly from the boundary to the horizon. Remarkably, even when the extremal surface crosses the horizon, the resulting CV volume—and hence the subregion complexity—remains entirely real. This reinforces the interpretation of timelike complexity as a measure of accessible bulk volume determined by causal reach rather than analytic continuation. It is also plausible that a transition could occur between the trivial and horizon- penetrating branches, signaling a new type of geometric phase structure.
The absence of any imaginary contribution in \(\mathcal{C}_{\mathrm{T}}\) suggests that holographic complexity possesses a fundamentally different analytic structure from that of pseudo- entropy. Whereas pseudo- entropy can reflect features of transition amplitudes—such as phase information—complexity appears to probe the underlying geometry of information itself. Its purely real character indicates that holographic complexity encodes the intrinsic "depth" or cost of traversing the bulk spacetime, independent of any temporal analytic continuation.
This purely real nature is, in fact, not unexpected. Although the construction originates from the transition matrix (1), which is not Hermitian, it remains a well- posed and meaningful question to ask how "distant" the final state \(|\psi_f\rangle\) is from the initial state \(|\psi_i\rangle\) . In this sense, \(\mathcal{C}_{\mathrm{T}}\) can be interpreted as quantifying the minimal number of effective "gates" or operations required to transform \(|\psi_i\rangle\) into \(|\psi_f\rangle\) restricted to a subregion—a quantity that is naturally real.
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Of course, an important open question remains—namely, how to compute timelike complexity directly from the boundary field theory. Even for spatial subregions, a precise field-theoretic definition of subregion complexity is still lacking, with partial insights coming from circuit complexity, path- integral optimization, or tensor- network approaches. Extending these frameworks to Lorentzian or timelike subregions poses additional conceptual and technical challenges, as one must incorporate causal and non- unitary aspects of boundary dynamics. A consistent field- theoretic formulation of timelike complexity would thus not only clarify the operational meaning of our holographic construction but also reveal how temporal and causal structures are encoded in the boundary state space.
We note also that working with timelike entangling regions reveals an intriguing new feature. By relaxing the physical boundary conditions on the entangling region, additional extremal surfaces can emerge, though their physical interpretation remains uncertain. In particular, we could identify a Lorentzian branch with an induced metric of Lorentzian signature, corresponding to an extremal surface that originates from a point in the bulk and terminates at the horizon. Such configurations suffer from infrared divergences, which typically arise from the near- horizon region of the geometry where the redshift factor becomes large. In this regime, the integrand of the corresponding volume or area functional develops a logarithmic divergence, signaling that the extremal surface probes deep into the infrared part of the bulk spacetime. Physically, this behavior reflects the accumulation of contributions from regions close to the horizon, where the proper volume effectively becomes infinite due to the warping of the metric. Understanding the precise regularization and physical interpretation of these IR divergences—whether they correspond to long- time or large- scale phenomena in the dual field theory—remains an open and interesting question.
Very recently, the paper [28] (see also [29]) appeared, in which the authors investigated the dynamics of spatial subregion complexity under time evolution and uncovered a series of striking behaviors. They showed that for subsystems larger than half the total system, the complexity grows linearly for exponentially long times, whereas for smaller subsystems the growth halts and
eventually reverses, leading to a sharp transition at the half- size threshold. At finite temperature, additional critical subsystem sizes emerge, determining whether the complexity saturates instantaneously or continues to grow.
Although their setup focuses on spatial subregions evolving in time, it shares deep conceptual parallels with our Lorentzian analysis. Both frameworks reveal that subregion complexity is highly sensitive to causal structure and the presence of horizons or effective causal barriers. In particular, the sharp transitions identified in [28] may correspond, in our timelike setup, to transitions between distinct extremal branches—such as the constant- time and horizon- penetrating surfaces. Moreover, the finite- temperature critical behavior observed in their study closely parallels our finding that timelike extremal surfaces exhibit qualitatively different regimes depending on whether the turning point lies inside the horizon or corresponds to a static configuration, hinting at analogous transition phenomena in the timelike CV framework.
A natural extension of our work, beyond generalizations to other geometries, would be to formulate and compute the timelike version of the CA conjecture on a Wheeler- DeWitt patch anchored to a timelike interval. Such an analysis could clarify the role of analytic continuation and illuminate potential phase ambiguities in Lorentzian holographic complexity. We hope to return to these questions in future work.
## ACKNOWLEDGEMENTS
I would like to thank Andreas Karch, Souvik Banerjee and Tadashi Takayanagi for their valuable comments. I am also grateful to Mohammad Javad Vasli for many insightful discussions on various aspects of holographic complexity. I would further like to thank the CERN Department of Theoretical Physics for their warm hospitality during the course of this work. This research was supported by the Iran National Science Foundation (INSF) under Project No. 4023620. I also acknowledge the assistance of ChatGPT for editorial help in refining and polishing the manuscript.
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2510.25950v1
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# LETTER TO THE EDITOR
# Lyman-\(\alpha\) radiation pressure regulates star formation efficiency
Daniele Manzoni \(^{a}\) and Andrea Ferrara \(^{a}\)
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Received 8 September 2025 / Accepted 27 October 2025
## ABSTRACT
Order- unity star formation efficiencies (SFE) in early galaxies may explain the overabundance of bright galaxies observed by JWST at high redshift. Here we show that Lyman- \(\alpha\) (Lyα) radiation pressure limits the gas mass converted into stars, particularly in primordial environments. We develop a shell model including Lyα feedback, and validate it with one- dimensional hydrodynamical simulations. To account for Lyα resonant scattering, we adopt the most recent force multiplier fits, including the effect of Lyα photon destruction by dust grains. We find that, independently of their gas surface density \(\Sigma_{g}\) , clouds are disrupted on a timescale shorter than a free- fall time, and even before supernova explosions if \(\Sigma_{g} \gtrsim 10^{3} M_{\odot} \mathrm{pc}^{- 2}\) . At \(\log (Z / Z_{\odot}) = - 2\) , relevant for high- redshift galaxies, the SFE is \(0.01 \lesssim \xi_{\ast} \lesssim 0.66\) for \(10^{3} \lesssim \Sigma_{g} [M_{\odot} \mathrm{pc}^{- 2}] \lesssim 10^{5}\) . The SFE is even lower for decreasing metallicity. Near- unity SFEs are possible only for extreme surface densities, \(\Sigma_{g} \gtrsim 10^{5} M_{\odot} \mathrm{pc}^{- 2}\) , and near- solar metallicities. We conclude that Lyα radiation pressure severely limits a possible extremely efficient, feedback- free phase of star formation in dense, metal- poor clouds.
Key words. Galaxies: star formation, high redshift – ISM: clouds – Methods: analytical, numerical
## 1. Introduction
The James Webb Space Telescope (JWST) has opened exciting new avenues for understanding the galaxy formation process (Naidu et al. 2022; Roberts- Borsani et al. 2022; Castellano et al. 2023; Adams et al. 2023; Robertson et al. 2023, 2024). Among JWST's unexpected findings is a striking overabundance of bright \((M_{\mathrm{UV}} < - 20)\) , very blue (UV slope \(\beta \lesssim - 2\) ) galaxies at redshift \(z \gtrsim 10\) (Harikane et al. 2023; McLeod et al. 2024). The abundance of these "blue monsters" challenges our current understanding of galaxy formation (Mason et al. 2023; Mirocha & Furlanetto 2023).
UV luminosity functions derived from JWST observations can be reconciled with current galaxy formation models if early galaxies suffer minimal dust obscuration (Ferrara et al. 2023). This condition is achieved when the galaxy luminosity exceeds the Eddington limit, and radiation- driven outflows expel dust from the galaxy (Ziparo et al. 2023; Ferrara 2024).
A straightforward alternative involves increasing the star formation rate by requiring remarkably high star formation efficiencies (order- unity) in early galaxies (Dekel et al. 2023; Li et al. 2023). This condition could be met if star formation takes place in molecular clouds undergoing feedback- free bursts (Dekel et al. 2023). If the free- fall time of a star- forming cloud is shorter than the delay time of supernova explosions \((\approx 1 \mathrm{Myr})\) , star formation can proceed unimpeded during this phase, converting nearly all gas mass into stars. Analytical estimates and numerical simulations (Dekel et al. 2023; Menon et al. 2023) show that neither photoionisation feedback nor radiation pressure on dust appears to significantly limit SFE.
However, additional feedback could be provided by Lyman- \(\alpha\) (Lyα) radiation pressure (Nebrin et al. 2024; Ferrara 2024). Lyα is a resonant line, and multiple scatterings of Lyα photons with neutral hydrogen can impart significant momentum to the gas. The strength of Lyα feedback can be characterised by the force multiplier \(M_{F}\) , defined as \(M_{F} \equiv \dot{p}_{\alpha} / (L_{\alpha} / c)\) (Di
jkstra & Loeb 2008), where \(\dot{p}_{\alpha}\) represents the force from Lyα multiple scattering, and \(L_{\alpha} / c\) is the force in the single- scattering limit. The force multiplier roughly scales as \(M_{F} \propto \tau^{1 / 3}\) (Neufeld 1990), where \(\tau\) is the total Lyα optical depth at line centre, with additional dependencies on geometry, dust content and gas velocity (Nebrin et al. 2024; Smith et al. 2025).
For typical feedback- free clouds at redshift \(z \gtrsim 10\) , with optical depths \(\log \tau_{0} \gtrsim 10\) and metallicities \(Z \gtrsim 0.02 Z_{\odot}\) (Dekel et al. 2023), we expect \(M_{F} \sim 100\) (Tomaselli & Ferrara 2021; Nebrin et al. 2024). Analytical estimates further suggest that the force from Lyα photons is \(\sim 20\) times stronger than the force from photoionisation and UV radiation pressure onto dust (Tomaselli & Ferrara 2021).
The role of Lyα feedback has been investigated in various contexts. Using analytical estimates, Abe & Yajima (2018) derived the critical star formation efficiency, \(\hat{\epsilon}_{*}\) , above which Lyα feedback can evacuate the gas and suppress star formation. For typical feedback- free clouds, their findings suggest \(\hat{\epsilon}_{*} \sim 0.6\) .
Kimm et al. (2018) used 3D RHD simulations of a metal- poor dwarf galaxy, including a subgrid model for Lyα momentum transfer, to study Lyα radiation pressure feedback. They found that Lyα radiation pressure regulates star- forming cloud dynamics before supernovae, reducing star formation efficiency and star cluster numbers by factors of two and five, respectively.
Recently, Nebrin et al. (2024) and Smith et al. (2025) derived analytical solutions of Lyα radiative transfer in the diffusion approximation for a uniform cloud, incorporating previously neglected physics such as velocity gradients, Lyα photon destruction, and recoil. The authors also provide convenient fits for the force multiplier, which could significantly improve existing subgrid models.
Ideally, to fully assess the impact of Lyα scattering on gas dynamics—and consequently on star formation—3D RHD simulations with on- the- fly Lyα radiative transfer should be performed. However, the computational cost would be prohibitive
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with standard Monte Carlo radiative transfer methods. Instead, the resonant discrete diffusion Monte Carlo algorithm (rDDMC) offers a feasible solution (Smith et al. 2018), making 3D Lyα RHD simulations achievable in the near future. To date, the only Lyα RHD simulation has been carried out in spherical symmetry by Smith et al. (2017), who investigated the dynamical impact of Lyα pressure on galactic winds, using both stars and black holes as sources. Their results confirmed that Lyα radiation pressure plays a crucial role in driving galactic winds and leaves observable imprints. In this Letter, we investigate the maximum SFE allowed by Lyα feedback using a shell expansion model. Lyα radiation pressure is accounted for adopting force multiplier fits provided in Nebrin et al. (2024).
The paper is structured as follows. In Sect. 2 we introduce the shell model, which we validate with 1D hydrodynamical simulations. In Sect. 3 we derive the SFE versus cloud surface density for different gas metallicities. Section 4 provides a critical discussion; conclusions are drawn in Sect. 5.
## 2. Lyα-driven shells
Consider a uniform spherical giant molecular cloud (GMC) with mass \(M_{c}\) and density \(\rho\) . The virial parameter \(\alpha_{\mathrm{vir}}\) is defined as
\[\alpha_{\mathrm{vir}} = \frac{5\sigma^2R_c}{3GM_c}. \quad (1)\]
In virial equilibrium, \(\alpha_{\mathrm{vir}} = 5 / 3\) , and we can derive from Eq. 1 the expressions for the cloud radius \(R_{c} = \sigma t_{\mathrm{ff}}\) , free- fall time \(t_{\mathrm{ff}} = (3 / 4\pi G\rho)^{1 / 2}\) and 1D velocity dispersion \(\sigma = (GM_{c} / t_{\mathrm{ff}})^{1 / 3}\) . The gas surface density is defined as \(\Sigma_{g} = M_{c} / \pi R_{c}^{2}\) .
Once a star cluster of mass \(M_{*}\) has formed at the centre of the cloud, massive stars producing H- ionising photons drive the formation and expansion of an H II region.
Shell models have already been applied to the study of GMC disruption by H II region expansion and radiation pressure (Krumholz & Matzner 2009; Fall et al. 2010; Murray et al. 2010; Kim et al. 2016). Here we include Lyα radiation pressure, neglected in previous works. We then perform 1D hydrodynamical simulations in spherical symmetry to validate the shell model approach (see Appendix A for the methods).
The time evolution of the shell radius \(R_{s}(t)\) is described by the momentum equation,
\[\frac{\mathrm{d}}{\mathrm{d}t} [M_{s}R_{s}] = M_{F}(R_{s})\frac{L_{\alpha}}{c} -\frac{GM_{s}(M_{*} + M_{s} / 2)}{R_{s}^{2}}, \quad (2)\]
where \(M_{s} = M(< R_{s}) = 4\pi (1 - \epsilon_{*})\rho R_{s}^{3} / 3\) is the gas mass accumulated in the shell. The final outcome of the shell motion can be either cloud disruption or recollapse, depending on the balance of the forces on the right- hand side of Eq. 2. The second term is the gravity force, \(F_{g}\) , which includes the shell self- gravity. The first term is the Lyα force, \(F_{\alpha}\) , which is the novel ingredient of our shell model. The strength of Lyα feedback is determined by the Lyα luminosity \(L_{\alpha}\) and by the force multiplier \(M_{F}\) .
The Lyα luminosity is related to the ionisation rate \(\dot{N}_{\gamma}\) by \(L_{\alpha} = (2 / 3)E_{\alpha}\dot{N}_{\gamma}\) , where \(E_{\alpha} = 10.2 \mathrm{eV}\) is the energy of Lyα photons. For a burst of star formation the ionisation rate is \(\dot{N}_{\gamma} = 10^{46.88}(M_{*} / M_{\odot})\mathrm{s}^{- 1}\) , assuming a fixed \(Z / Z_{\odot} = 1 / 50\) (consistent with the value measured in early galaxies) and a 1- 100 \(M_{\odot}\) Salpeter IMF (Schaerer 2003). The corresponding Lyα luminosity is \(L_{\alpha} = 8.3\times 10^{35}(M_{*}/M_{\odot})\mathrm{erg}\mathrm{s}^{- 1} = l_{\alpha}(M_{*}/M_{\odot})\) , where we have defined the Lyα luminosity per unit stellar mass as \(l_{\alpha}\) .
The force multiplier \(M_{F}\) depends on the Lyα optical depth at line centre \(\tau\) and scales as \(M_{F}\propto \tau^{1 / 3}\) in static, dust- free media. To account for Lyα destruction by dust, we adopt the fits to \(M_{F}\) provided in Nebrin et al. (2024). The force multiplier in dusty media depends on the gas- to- dust ratio \(D\) . Here we follow the standard assumption from galaxy formation simulations, \(D / D_{\mathrm{MW}} = Z / Z_{\odot}\) (Hopkins et al. 2023), where the Milky Way value is \(D_{\mathrm{MW}} = 1 / 162\) .
The shell optical depth can be written as \(\tau (R_{s}) = (1 - \epsilon_{*})\rho /(m_{p}\sigma_{\alpha})R_{s}\) , considering pure hydrogen gas. The Lyα cross section at line centre is \(\sigma_{\alpha} = 5.88\times 10^{- 13}(T / 100 \mathrm{K})^{- 1 / 2}\) . With this setup, the only model free parameter is the SFE \(\epsilon_{*} = M_{*} / M_{c}\) . We determine its maximum possible value in Sect. 3.
For illustration, we plot the predicted evolution of the shell radius and velocity from the shell model in Fig. 1 by solid lines. Filled circles mark the disruption time when \(R_{s} = R_{c}\) . We present the results for \(\Sigma_{g} = 10^{4}M_{\odot} \mathrm{pc}^{- 2}\) , \(\log (Z / Z_{\odot}) = - 2\) and \(\epsilon_{*} = 1\%\) , \(5\%\) , \(10\%\) , and \(30\%\) . Hydrodynamical simulation results are shown for comparison by dashed lines. For a detailed discussion of the simulations, see Appendix A.
The SFE controls both the disruption timescale and the terminal shell velocity. A larger SFE produces faster disruption and higher final velocities.
The shell solution agrees well with the simulations. This consistency holds across the full range of metallicities, surface densities, and SFEs explored. Therefore, we can rely on Eq. 2 solutions to estimate the maximum SFE instead of running full simulations.
## 3. Maximum SFE allowed by Lyα feedback
The simplest approach would be to deduce the SFE by requiring that Lyα and gravitational forces balance when the shell reaches the cloud radius. Setting the left- hand side of Eq. 2 to zero (i.e. \(F_{\mathrm{tot}} = F_{\alpha} - F_{g} = 0\) ), and further imposing \(R_{s} = R_{c}\) , we find (see also Kim et al. 2016; Abe & Yajima 2018; Nebrin et al. 2024):
\[\frac{\hat{\epsilon}_{*}}{1 - \hat{\epsilon}_{*}^{2}} = \frac{\Sigma_{g}}{\Sigma_{\mathrm{crit}}}, \quad (3)\]
where the critical surface density is given by
\[\Sigma_{\mathrm{crit}} = \frac{M_{F}l_{\alpha}}{\pi GcM_{\odot}} = 316M_{F}(\Sigma_{g},Z)M_{\odot}\mathrm{pc}^{-2}. \quad (4)\]
This solution is straightforward; however, it provides no timescale information. Moreover, if the cloud takes longer than a free- fall time to be disrupted, additional star formation could occur. This would lead to an underestimate of the final SFE.
To proceed, we solve Eq. 2 numerically. To derive the maximum SFE \(\hat{\epsilon}_{*}\) , a cloud disruption criterion must be specified. We first compute the shell evolution for a given SFE, up to the final time \(t_{d} = t(R_{s} = R_{c})\) . We then check if one of the two conditions is satisfied: (a) \(F_{\mathrm{tot}} = 0\) , or (b) \(\dot{R}_{s} = v_{\mathrm{esc}} = [(1 + \epsilon_{*})GM_{c} / R_{c}]^{1 / 2}\) . If neither of the two conditions is satisfied, we iterate by adjusting \(\epsilon_{*}\) .
Condition (a) requires that when Lyα pressure balances gravity, the collapse of the gas halts and star formation ceases. In contrast, the stricter requirement \(\dot{R}_{s} = v_{\mathrm{esc}}\) in condition (b) ensures that the cloud is totally dispersed, preventing any future collapse. This condition demands stronger feedback, since gravity can already be balanced by Lyα radiation pressure even when the shell velocity remains below \(v_{\mathrm{esc}}\) . Consequently, the values of \(\hat{\epsilon}_{*}\) from condition (b) are always larger than those from condition (a).
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<center>Fig. 1. Left: Shell radius as a function of time normalised to the free-fall time for \(\epsilon_{*} = 1\%\) (green), \(5\%\) (water green), \(10\%\) (blue) and \(30\%\) (purple). The cloud surface density and metallicity are \(\Sigma_{g} = 10^{4} M_{\odot} \mathrm{pc}^{-2}\) and \(\log (Z / Z_{\odot}) = -2\) , respectively. Both the shell solution (Eq. 2, solid line) and the simulation predictions (dashed) are shown. Right: Same for the shell velocity. </center>
Figure 2 shows the resulting maximum SFE for both cases as a function of surface density, for metallicities \(\log (Z / Z_{\odot}) = - 6, - 4, - 2, 0\) . If condition (a) is applied the SFE reduces to Eq. 3, once the dependence of the critical surface density on gas surface density and metallicity is included. For comparison, we also show the maximum SFE obtained from Eq. 3, for a fixed \(\Sigma_{\mathrm{crit}} = 2000 M_{\odot} \mathrm{pc}^{- 2}\) (see, e.g., Somerville et al. 2025). For reference, we find \(\Sigma_{\mathrm{crit}} = 1.4 - 1.7 \times 10^{5} M_{\odot} \mathrm{pc}^{- 2}\) for \(\log (Z / Z_{\odot}) = - 2\) across our gas surface density range. The final time \(t_{d}\) — when the shell radius reaches \(R_{c}\) — is shown in units of the cloud free- fall time and in Myr in the central and right panels of Fig. 2.
Lyα strongly limits the SFE, even for the densest clouds. At \(\log (Z / Z_{\odot}) = - 2\) , typical of high- redshift galaxy metallicities, the SFE is \(0.01 \lesssim \hat{\epsilon}_{*} \lesssim 0.66\) for \(10^{3} \lesssim \Sigma_{g} [M_{\odot} \mathrm{pc}^{- 2}] \lesssim 10^{5}\) . For very metal- poor GMCs, \(\log (Z / Z_{\odot}) \leq - 4\) , the SFE is always \(\hat{\epsilon}_{*} \lesssim 0.34\) . Near- unity SFEs are possible only for extreme surface densities, \(\Sigma_{g} = 10^{5} M_{\odot} \mathrm{pc}^{- 2}\) , and near- solar metallicities. We have quoted here the less restrictive values from condition (b).
We note that the cloud disruption timescale is always \(t_{d} \lesssim t_{\mathrm{ff}}\) , independent of density. 3D RHD simulations of GMCs that include stellar wind and radiative (UV, optical and IR) feedback show that the stellar mass is assembled over \(\sim 3 - 4 t_{\mathrm{ff}}\) (Hopkins et al. 2023; Menon et al. 2023). In these simulations, the star formation rate declines after reaching its peak, and in our model the central cluster has already formed. Given the suppressed star formation rate and the short timescale, almost no additional stellar mass forms as the shell expands. Hence, the derived \(\hat{\epsilon}_{*}\) values represent a reliable upper limit.
## 4. Discussion
Key to our results is the value of the force multiplier \(M_{F}\) . In our model, we neglected some physical effects that could limit Lyα feedback, such as velocity gradients, the Lyα source extension and turbulence.
All the gas in the shell has the same velocity, and therefore we should deal with a bulk velocity. Significant suppression of the force multiplier is expected only at velocities \(v \sim 500 \mathrm{km} \mathrm{s}^{- 1} (N_{\mathrm{HI}} / 10^{20} \mathrm{cm}^{- 2})^{1 / 2}\) (Tomaselli & Ferrara 2021) for
clouds with \(\mathrm{HI}\) column density \(N_{\mathrm{HI}}\) . Within our parameter range, \(21 \lesssim \log N_{\mathrm{HI}} \lesssim 25\) , strong suppression of the force multiplier is expected at velocities around \(1500 \mathrm{km} \mathrm{s}^{- 1}\) for the least dense clouds. However, since the simulated shell velocities remain within \(0 - 1000 \mathrm{km} \mathrm{s}^{- 1}\) , the velocity dependence of the force multiplier can be neglected, particularly for the massive clouds that are the main focus of this study.
The spatial extent of the Lyα source can also limit feedback, particularly in dusty media (Nebrin et al. 2024). 3D RHD simulations of GMCs show that UV radiation pressure can be reduced by flux cancellation (Menon et al. 2023). We model this effect for Lyα pressure in Appendix B. Source extension lowers the force multiplier and enhances SFE, especially at \(\log (Z / Z_{\odot}) = 0\) . For \(\log (Z / Z_{\odot}) \lesssim - 2\) , its impact is negligible when \(R_{*} / R_{c} \lesssim 0.25\) .
In turbulent media, Lyα photons escape more easily through low- density channels. Fluctuating velocity gradients introduce large Doppler shifts, which further aid photon escape. Nebrin et al. (2024) showed that the suppression of \(M_{F}\) scales as \(M^{- 8 / 9}\) , where \(M = \sigma / c_{s}\) is the Mach number. For \(M = 10\) and \(\log (Z / Z_{\odot}) = - 2\) , the force multiplier is suppressed by a factor of \(\sim 3\) , yielding \(\Sigma_{\mathrm{crit}} = 4.9 - 6.7 \times 10^{4} M_{\odot} \mathrm{pc}^{- 2}\) . The qualitative behaviour of the SFE is unchanged: order- unity SFE occurs only at \(\Sigma_{g} \gtrsim \Sigma_{\mathrm{crit}}\) .
We found that Lyα feedback disrupts clouds on short timescales, \(t_{d} \lesssim t_{\mathrm{ff}}\) . For \(\Sigma_{g} \gtrsim 10^{3} M_{\odot} \mathrm{pc}^{- 2}\) , the free- fall time is \(< 1\) Myr. This is comparable to, or even shorter than, the delay to the first supernova explosions. Thus, Lyα radiation pressure acts as an efficient pre- supernova feedback channel and prevents a feedback- free phase.
Our model assumes that a stellar cluster of mass \(M_{*} = \epsilon_{*} M_{c}\) forms at the cloud centre. In reality, Lyα radiation pressure operates as soon as the first massive stars form. Lyα- driven shells create low- density ionised bubbles around individual stars. Their expansion and overlap suppress further star formation and reduce the star formation rate until the final SFE is reached. A more detailed investigation of this process is deferred to a companion paper (Ferrara et al. 2025).
We note that in our model the shell is driven solely by Lyα radiation pressure. In reality, several additional feedback chan
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<center>Fig. 2. Left: Maximum SFE as a function of surface density for metallicity \(\log (Z / Z_{\odot}) = -6\) (blue), \(-4\) (purple), \(-2\) (red), 0 (orange). Solid or dashed curves show where the zero-force \((F_{\mathrm{tot}} = 0)\) or the escape velocity \((\hat{R}_{\mathrm{s}} = v_{\mathrm{esc}})\) conditions are satisfied. The dotted curve shows the time-independent solution (Eq. 3) for the reference value \(\Sigma_{\mathrm{crit}} = 2000 M_{\odot} \mathrm{pc}^{-2}\) used by Somerville et al. (2025). Coloured ticks mark the maximum SFE across the surface density range for each metallicity. Middle: Cloud disruption time in units of the free-fall time as a function of the cloud surface density for various metallicities as indicated in the colour bar. Solid or dashed curves show where the zero-force \((F_{\mathrm{tot}} = 0)\) or the escape velocity \((\hat{R}_{\mathrm{s}} = v_{\mathrm{esc}})\) conditions are satisfied. Right: As the middle panel, with \(t_d\) in units of Myr. </center>
nels operate. The swept- up gas in the shell is neutral, and lies outside the Strömgren radius \(R_{s}\) . The H II region thus provides a kick- start to the shell expansion. Photoionisation and radiation pressure on dust also contribute. The latter is dominated by the more numerous non- ionising photons and becomes more important as \(Z\) increases. Finally, stellar winds from massive stars, neglected here, inject yet further energy. Therefore, our results represent generous upper limits to the actual SFE, since they neglect both the impact of early Lyα feedback on star formation and the contribution of other feedback channels.
## 5. Summary
We have combined a shell model and radiation hydrodynamic simulations to study Lyα radiation pressure feedback in GMCs. We derived the momentum equation for a shell including gravity and Lyα force. We validated the solution with 1D simulations in spherical symmetry, finding good agreement. From these models we deduced the upper limit on SFE set by Lyα feedback, requiring a total vanishing force or shell velocity equal to the cloud escape velocity at the cloud boundary.
We found that Lyα radiation pressure can strongly limit the SFE achievable in molecular clouds. Once a central star cluster forms, the Lyα- driven shell reaches the cloud boundary in \(\lesssim t_{\mathrm{ff}}\) at any surface density. This is shorter than the delay to the first supernova explosions for \(\Sigma_{g} \gtrsim 10^{3} M_{\odot} \mathrm{pc}^{- 2}\) . Thus, Lyα radiation pressure prevents a feedback- free phase of star formation.
At \(\log (Z / Z_{\odot}) = - 2\) , relevant for high- redshift galaxies, the SFE is \(0.01 \lesssim \hat{\epsilon}_{\mathrm{s}} \lesssim 0.66\) for \(10^{3} \lesssim \Sigma_{g} [M_{\odot} \mathrm{pc}^{- 2}] \lesssim 10^{5}\) . For very metal- poor GMCs, \(\log (Z / Z_{\odot}) \lesssim - 4\) , the SFE is always \(\hat{\epsilon}_{\mathrm{s}} \lesssim 0.34\) . Near- unity SFEs are possible only for extreme surface densities, \(\Sigma_{g} = 10^{5} M_{\odot} \mathrm{pc}^{- 2}\) , and near- solar metallicities. Our results are likely to overestimate the SFE, since they neglect both the impact of early Lyα feedback on star formation and the contribution of other feedback channels.
Acknowledgements. We thank the referee, M. Krumholz, for constructive comments. We also thank A. Smith for useful discussions.
## References
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## Appendix A: Hydrodynamical simulations
To validate the shell model, we perform 1D spherically symmetric hydrodynamical simulations adopting a finite- volume formulation. We use a fixed uniform grid with 1000 cells with boundaries \((0.01,1)\times R_{c}\) . We apply free boundary conditions and compute the intercell fluxes with an HLLC Riemann solver. We enforce density and pressure floors to avoid complete gas depletion from the cells. As an initial condition, we adopt a uniform cloud in pressure equilibrium with an ambient gas of density \(\rho_{\mathrm{amb}} = 10^{- 3}\rho\) . The gas is almost isothermal (polytropic index \(\gamma \approx 1\) ) at \(T = 30\mathrm{K}\) .
The fluid state vector \(\mathbf{Q}_i = (m_i,p_i,E_i)\) encoding the mass, momentum and energy of each cell \(i\) is evolved according to
\[\begin{array}{l}\frac{\mathrm{d}\mathbf{Q}_i}{\mathrm{d}t} = -\int_{\partial V_i}\mathbf{F}(\mathbf{U})\mathrm{d}\mathbf{n} + \mathbf{S}_i,\\ \displaystyle \mathbf{F}(\mathbf{U}) = \left( \begin{array}{c}\rho \\ \rho v \\ (\rho e + P)v \end{array} \right),\quad \mathbf{S}_i = \left( \begin{array}{c}0 \\ F_g + F_\alpha \\ v(F_g + F_\alpha) \end{array} \right). \end{array} \quad (A.1)\]
The gravity force in each cell with centre of mass \(R_{i}\) is \(F_{g} = GM(< R_{i})m_{i} / R_{i}^{2}\) . To compute the Lyα force we first compute the Lyα optical depth at cell boundaries, \(\tau_{i\pm 1 / 2} = \tau (R_{i\pm 1 / 2})\) . The net Lyα force on the cell is then given by
\[F_{\alpha} = [M_{F}(\tau_{i + 1 / 2}) - M_{F}(\tau_{i - 1 / 2})]\frac{L_{\alpha}}{c}. \quad (A.3)\]
Here we note that the force multipliers in Nebrin et al. (2024) were derived under the assumption of a uniform cloud. However, as soon as the shell forms, this approximation breaks down and in spherical symmetry an arbitrary density profile is not equivalent to a homogeneous representation in optical- depth space. Nevertheless, adopting Eq. A.3 is consistent within a factor of \(\lesssim 2\) (Lao & Smith 2020).
As an illustration we discuss the model with \(\Sigma_{g} = 10^{4}M_{\odot}\mathrm{pc}^{- 2}\) , \(\log (Z / Z_{\odot}) = - 2\) , and \(\epsilon_{s} = 30\%\) . Figure A.1 shows the number density and velocity profiles for different snapshots, with time in units of the free- fall time indicated by the colour bar. A thin shell forms and increases its density as it expands. We identify the shell as the densest cell at each time.
The lower panel of Fig. A.1 shows the shell velocity over time (dashed black line). The velocity peak is slightly offset with the shell position, coinciding with low- density gas just behind it. As a result the curve does not coincide with the velocity peaks. This suggests that a fraction of Lyα photons originates from fast- moving ionised gas just behind the shell. These photons are Doppler- shifted out of resonance with the slowly moving neutral gas beyond the shell and with the gas in the shell moving in the opposite direction. However, as discussed in Sect. 4, only velocities of order \(v\sim 500(N_{\mathrm{HI}} / 10^{20}\mathrm{cm}^{- 2})^{1 / 2}\mathrm{km}\mathrm{s}^{- 1}\) can substantially suppress the force multiplier. This behaviour was derived by Tomaselli & Ferrara (2021) for Doppler- shifted Lyα photons interacting with static gas. The velocity shifts in our models are too weak to affect Lyα radiation pressure, especially in the densest clouds.
The shell accelerates quickly at early times and then slows down due to mass accumulation and gravity. With this parameter set the cloud is disrupted on a short timescale, \(t_{d} = t(R_{s} = R_{c}) = 0.24t_{\mathrm{ff}}\) .
## Appendix B: Impact of extended sources
We now examine how source extension affects the force multiplier and, consequently, the maximum star formation efficiency.
<center>Fig. A.1. Upper: Number density profiles for different snapshots of the simulation. Time in free-fall time units is colour-coded. The surface density of the cloud is \(\Sigma_{g} = 10^{4}M_{\odot}\mathrm{pc}^{-2}\) , with metallicity \(\log (Z / Z_{\odot}) = -2\) and \(\mathrm{SFE}\epsilon = 30\%\) . Lower: Same for the velocity profiles. The black dashed line shows the time evolution of the shell velocity, corresponding to the velocity of the cell with highest density. </center>
For a uniform, static cloud of total optical depth \(\tau\) , the force multiplier is
\[M_{F}(\tau) = N(a_{v}\tau)^{1 / 3},\]
where \(a_{v} = 4.7\times 10^{- 3}(T / 100\mathrm{K})^{- 1 / 2}\) is the Voigt parameter. The constant \(N\) encodes the effect of source geometry: for a point source \(N = 3.51\) , while for a source uniformly distributed throughout the cloud it decreases by a factor of \(\sim 7\) to \(N = 0.51\) due to flux cancellation. Additional suppression arises from Lyα photon destruction by dust, which is more significant for extended sources. These effects are included in the fitting relations of Nebrin et al. (2024), to which we refer for further details.
In our shell model, we can account for both point- like and extended components of the source. For a uniform source of radius
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\(R_{*}\) , stars located within the shell act as point sources, contributing a luminosity
\[L_{\alpha}^{\mathrm{point}} = \left\{ \begin{array}{ll}L_{\alpha}(R_{*} / R_{c})^{3}, & R_{*}< R_{*},\\ L_{\alpha}, & R_{*} \geq R_{*} \end{array} \right. \quad (B.1)\]
while the remaining luminosity is uniformly distributed,
\[L_{\alpha}^{\mathrm{uni}} = L_{\alpha} - L_{\alpha}^{\mathrm{point}}.\]
The total Lyα force on the shell is then
\[F_{\alpha} = M_{F}^{\mathrm{point}}\frac{L_{\alpha}^{\mathrm{point}}}{c} + M_{F}^{\mathrm{uni}}\frac{L_{\alpha}^{\mathrm{uni}}}{c}. \quad (B.2)\]
We evaluate the resulting maximum SFE for source extensions \(R_{*} / R_{c} = 0, 0.1, 0.25, 0.5, 0.75\) , and 1 (Fig. B.1). The results are shown only for \(\log (Z / Z_{\odot}) = - 2\) and 0, as the influence of source extension increases with metallicity. The escape- velocity condition is adopted and the results are similar for the zero- force case. For \(\log (Z / Z_{\odot}) = - 2\) , the SFE remains identical to the point- source case for \(R_{*} / R_{c} \leq 0.25\) , with noticeable enhancement only for \(R_{*} \gtrsim 0.5\) . At solar metallicity, the effect becomes significant for \(R_{*} \gtrsim 0.1\) , reflecting stronger attenuation of the force multiplier through combined flux cancellation and Lyα destruction by dust.
<center>Fig. B.1. Maximum SFE as a function of cloud surface density for different source extensions: \(R_{*} = 0.1\) (cyan), 0.25 (light blue), 0.5 (yellow), 0.75 (orange), and 1 (red), shown with dashed lines. The point-source case is shown in solid blue. For reference, the SFE corresponding to \(\Sigma_{\mathrm{crit}} = 2000 \mathrm{M}_{\odot} \mathrm{pc}^{-2}\) is indicated by a grey dotted line. </center>
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