Daily Papers

We show how aspects of the R-charge of N=2 CFTs in four dimensions are encoded in the q-deformed Kontsevich-Soibelman monodromy operator, built from their dyon spectra. In particular, the monodromy operator should have finite order if the R-charges are rational. We verify this for a number of examples including those arising from pairs of ADE singularities on a Calabi-Yau threefold (some of which are dual to 6d (2,0) ADE theories suitably fibered over the plane). In these cases we find that our monodromy maps to that of the Y-systems, studied by Zamolodchikov in the context of TBA. Moreover we find that the trace of the (fractional) q-deformed KS monodromy is given by the characters of 2d conformal field theories associated to the corresponding TBA (i.e. integrable deformations of the generalized parafermionic systems). The Verlinde algebra gets realized through evaluation of line operators at the loci of the associated hyperKahler manifold fixed under R-symmetry action. Moreover, we propose how the TBA system arises as part of the N=2 theory in 4 dimensions. Finally, we initiate a classification of N=2 superconformal theories in 4 dimensions based on their quiver data and find that this classification problem is mapped to the classification of N=2 theories in 2 dimensions, and use this to classify all the 4d, N=2 theories with up to 3 generators for BPS states.

It was demonstrated recently that the W_{1+infty} algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized widetilde W algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the widetilde W algebras studied earlier. We suggest a definition of the generalized widetilde W algebra as differential operators in variables p_k basing on the matrix realization of the W_{1+infty} algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W_{1+infty} algebra than is contained in its commutative subalgebras. The positive integer rays are associated with widetilde W algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric tau-functions corresponding to the completed cycles.

Every four-dimensional {cal N}=2 superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any {cal N}=2 SCFT should obey a finite order modular differential equation. By way of the "high temperature" limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the "Deligne-Cvitanović exceptional series" of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class {cal S} theories, and {cal N}=4 super Yang-Mills with su(n) gauge group for small-to-moderate values of n.

We continue our studies of the correspondence between 4d N=2 SCFTs and 2d W-algebras. The purpose of this paper is to study the relationship between 2d lisse W-algebras and their 4d SCFT partners. The lisse W-algebra is the W-algebra whose associated Zhu's C_2 algebra is finite dimensional. As the associated variety of Zhu's C_2 algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of the Higgs branch on the 4d side. We classify 4d N=2 SCFTs which do not admit Higgs branch, then these theories would give lisse W-algebras through the 4d/2d correspondence. In particular, we predict the existence of a large class of new non-admissible lisse W-algebras, which have not been studied before. The 4d theories corresponding to lisse W-algebra can appear in the Higgs branches of generic 4d N=2 SCFTs, therefore they are crucial to understand the Higgs branches of N=2 SCFTs.

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky--Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator \[ Y=\partial_y^2-ω^2y^2+1/4-c^2{y^2} \] is of Laguerre type, while the parabolic integral \(W=L_2\) is its algebraic Heun partner. With \(Z=[Y,W]\), the defining relations are \[ [Y,Z]=16ω^2W-2bY,\qquad [W,Z]=6Y^2-4HY+2bW+8ω^2(1-c^2), \] where \(H\) is central. This gives a direct superintegrable realization of the Laguerre--Heun algebra.

This paper is devoted to the classification and studying properties of complex unital 3-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type (2, 1) and two algebras for type (1, 2). For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree 2. Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting Z-graded Lie algebras, providing their dimensions and Levi decompositions.

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type J on a punctured sphere. We denote the AD theories as (J^b[k],Y), where J^b[k] and Y represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where J^b[k] has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra W^{k_{2d}}(J,Y), where k_{2d}=-h+ b{b+k} with h being the dual Coxeter number of J. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b=h. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.

Four-dimensional N=2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S. The class S duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Is there more to Dirac's gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac's space-time algebra to Clifford algebra Cl(0,6). The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the extended vierbein field and the extended weak gauge field is allowed. In conjunction with the composite Higgs model, we examine the vierbein field as a Cooper-pair-like fermion-antifermion condensation. Quantum gravity is realized indirectly via the quantized standard model spinor fields which underlie the composite space-time metric. We propose that the fundamental energy scales of the universe including the Planck scale are emergent and resulted from quantum condensations, thus possibly addressing the cosmological constant problem through an unconventional multi-scale renormalization procedure for multiplications of divergent Feynman integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle-antiparticle interchange, leading to a Majorana-type mass that conserves lepton number. Additionally, with reshuffling the traditional quark-lepton pairing pattern of three generations of fermions, we explore a three-Higgs-doublet model with Higgs VEVs 246 GeV, 42 GeV and 2.5 GeV which could explain the mass hierarchies of fermions.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

We study the N=2 four-dimensional superconformal index in various interesting limits, such that only states annihilated by more than one supercharge contribute. Extrapolating from the SU(2) generalized quivers, which have a Lagrangian description, we conjecture explicit formulae for all A-type quivers of class S, which in general do not have one. We test our proposals against several expected dualities. The index can always be interpreted as a correlator in a two-dimensional topological theory, which we identify in each limit as a certain deformation of two-dimensional Yang-Mills theory. The structure constants of the topological algebra are diagonal in the basis of Macdonald polynomials of the holonomies.

The Hopf fibration is an important object in mathematics and physics. A landmark discovery in topology and a fundamental object in the theory of Lie groups, the Hopf fibration has a wide variety of physical applications including magnetic monopoles, rigid body mechanics, and quantum information theory. This expository article presents an introduction to the Hopf fibration that is accessible to undergraduate students. We use the algebra of quaternions to illustrate algebraic and geometric properties of the Hopf fibration and the connection with rotations of 3-space that is the basis for natural applications to physics and engineering.

This thesis is divided into three parts. The first part deals with cylindric plane partitions. The second with lambda-determinants and the third with commutators in semi-circular systems. For more detailed abstract please see inside. Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of section one is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result is a (q,t)-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result is an explicit combinatorial interpretation of the Macdonald weight occurring in the (q,t)-analog using the non-intersecting lattice path model for cylindric plane partitions. Alternating sign matrices were discovered by Robbins and Rumsey whilst studying λ-determinants. In the second part of this thesis we prove a multi-parameter generalization of the λ-determinant, generalizing a recent result by di Francesco. Like the original λ-determinant, our formula exhibits the Laurent phenomenon. Semicircular systems were first introduced by Voiculescu as a part of his study of von Neumann algebras. In the third part of this thesis we study certain commutator subalgebras of the semicircular system. We find a projection matrix with an interesting self-similar structure. Making use of our projection formula we given an alternative, elementary proof that the semicircular system is a factor.

For any {rm E}-rigid presentation e, we construct an orthogonal projection functor to {rm rep}(e^perp) left adjoint to the natural embedding. We establish a bijection between presentations in {rm rep}(e^perp) and presentations compatible with e. For quivers with potentials, we show that {rm rep}(e^perp) forms a module category of another quiver with potential. We derive mutation formulas for the delta-vectors of positive and negative complements and the dimension vectors of simple modules in {rm rep}(e^perp), enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

Let V be a highest weight module over a Kac-Moody algebra g, and let conv V denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv V, i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN 2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans. Amer. Math. Soc. 2017] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous work of Satake [Ann. of Math. 1960], Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and Casselman [Austral. Math. Soc. 1997]. For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type. We further answer a question of Michel Brion, by showing that the localization of conv V along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

A large family of 4d N=2 SCFT's was introduced in arXiv:1210.2886. Its elements D_p(G) are labelled by a positive integer p\in N and a simply-laced Lie group G; their flavor symmetry is at least G. In the present paper we study their physics in detail. We also analyze the properties of the theories obtained by gauging the diagonal symmetry of a collection of D_{p_i}(G) models. In all cases the computation of the physical quantities reduces to simple Lie-theoretical questions. To make the analysis more functorial, we replace the notion of the BPS-quiver of the N=2 QFT by the more intrinsic concept of its META-quiver. In particular: 1) We compute the SCFT central charges a, c, k, and flavor group F for all D_p(G) models. 2) We identify the subclass of D_p(G) theories which correspond to previously known SCFT's (linear SU and SO-USp quiver theories, Argyres-Douglas models, superconformal gaugings of Minahan-Nemeshanski E_r models, etc.), as well as to non-trivial IR fixed points of known theories. The D_p(E_r) SCFT's with pgeq 3 cannot be constructed by any traditional method. 3) We investigate the finite BPS chambers of some of the models. 4) As a by product, we prove three conjectures by Xie and Zhao, and provide new checks of the Argyres-Seiberg duality.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

Among the simple Lie algebras, D_4 is distinguished as the unique one whose group of outer-automorphisms is bigger than Z_2. We study the compactifications of the D_4 (2,0) Theory on a punctured Riemann surface, C, with outer-automorphism twists around cycles of C lying in Z_3subset Aut(D_4)= S_3. The resulting 4D N=2 SCFTs have a number of new and interesting properties. As byproduct, we discover a new rank-1 N=2 SCFT with flavour symmetry group SU(4).

We study an infinite family of strongly coupled four-dimensional mathcal N=2 superconformal field theories (SCFTs) distinguished by a trivial Higgs branch, known as the (A_2,D_{3m+1}) Argyre-Douglas theories. We propose that their associated vertex operator algebras (VOAs) are the doublet algebras mathcal A(4m+2), an infinite family of non-rational vertex operator superalgebras with a remarkably simple strong generating set of only three fields. We provide several highly nontrivial checks of this proposal. In particular, we reproduce the four-dimensional conformal anomalies a and c from the VOA and analytically prove the exact equality between the Schur index of the SCFT and the supercharacter of the VOA. A key ingredient is a diagonal-gauging realization of the (A_2,D_{3m+1}) theories in terms of two simpler building blocks, which makes the Schur-index computation tractable. Remarkably, we find that the resulting Schur index of the (A_2,D_{3m+1}) theory coincides with that of mathcal N=4 SU(2) Super-Yang-Mills theory, up to an overall prefactor and an appropriate identification of fugacities. We also discuss a generalization to the two-parameter family (A_{2s},D_{(2s+1)m+1}), whose members likewise have trivial Higgs branches and admit diagonal-gauging realizations. A particularly interesting subfamily is (A_{2s},D_{2s+2}), for which the four-dimensional conformal anomalies coincide, a=c. Our results reveal a systematic connection between Higgsless SCFTs, diagonal gauging, and strongly finite but non-rational vertex operator algebras.

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green matrix entries g(x,y). We also have already seen that that the determinants of L or g are both the product of the h(x). In the case where h(x) is the parity of dimension, the sum of the energy values was the standard Euler characteristic and the determinant was a unit. If h(x) was the unit in the ring then L,g are integral quadratic forms which are isospectral and inverse matrices of each other. We prove here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries. The quadratic energy expression is Wu characteristic in the case when h is dimension parity. For general h, the quadratic energy expression resembles an Ising Heisenberg type interaction. The conjugate of g is the inverse of L if h takes unit values in a normed ring or in the group of unitary operators in an operator algebra.

We present, for the first time, a novel theoretical approach to address the problem of correspondence free multivector cloud registration in conformal geometric algebra. Such formalism achieves several favorable properties. Primarily, it forms an orthogonal automorphism that extends beyond the typical vector space to the entire conformal geometric algebra while respecting the multivector grading. Concretely, the registration can be viewed as an orthogonal transformation (\it i.e., scale, translation, rotation) belonging to SO(4,1) - group of special orthogonal transformations in conformal geometric algebra. We will show that such formalism is able to: (i) perform the registration without directly accessing the input multivectors. Instead, we use primitives or geometric objects provided by the conformal model - the multivectors, (ii) the geometric objects are obtained by solving a multilinear eigenvalue problem to find sets of eigenmultivectors. In this way, we can explicitly avoid solving the correspondences in the registration process. Most importantly, this offers rotation and translation equivariant properties between the input multivectors and the eigenmultivectors. Experimental evaluation is conducted in datasets commonly used in point cloud registration, to testify the usefulness of the approach with emphasis to ambiguities arising from high levels of noise. The code is available at https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was submitted to the International Journal of Computer Vision and is currently under review.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A_N, A_M) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

An influential conjecture by Witten states that there is an instanton Floer homology of four-manifolds with corners that in certain situations is isomorphic to Khovanov homology of a given knot K. The Floer chain complex is generated by Nahm pole solutions of the Kapustin-Witten equations on R^3 times R^+_y with an additional monopole-like singular behaviour along the knot K inside the three-dimensional boundary at y=0. The Floer differential is given by counting solutions of the Haydys-Witten equations that interpolate between Kapustin-Witten solutions along an additional flow direction R_s. This article investigates solutions of a decoupled version of the Kapustin-Witten and Haydys-Witten equations on R_s times R^3 times R^+_y, which in contrast to the full equations exhibit a Hermitian Yang-Mills structure and can be viewed as a lift of the extended Bogomolny equations (EBE) from three to five dimensions. Inspired by Gaiotto-Witten's approach of adiabatically braiding EBE-solutions to obtain generators of the Floer homology, we propose that there is an equivalence between adiabatic solutions of the decoupled Haydys-Witten equations and non-vertical paths in the moduli space of EBE-solutions fibered over the space of monopole positions. Moreover, we argue that the Grothendieck-Springer resolution of the Lie algebra of the gauge group provides a finite-dimensional model of this moduli space of monopole solutions. These considerations suggest an intriguing similarity between Haydys-Witten instanton Floer homology and symplectic Khovanov homology and provide a novel approach towards a proof of Witten's gauge-theoretic interpretations of Khovanov homology.

A universal class of light-ray operators formed from null integrals of the stress tensor is constructed in generic interacting Lorentzian conformal field theories in four spacetime dimensions. This class of light-ray operators generates the wedge algebra of {rm w}_{1+infty}, which was recently identified among the asymptotic symmetries of asymptotically flat spacetimes. In four-dimensional conformal field theories with an additional spin-one conserved current, a second universal class of light-ray operators is constructed and shown to generate the ''S algebra,'' the gauge-theoretic analog of {rm w}_{1+infty}. Finally, a precise relation is established between the one-point functions of these light-ray operators in scalar states and the universal soft factors in the infinite tower of soft graviton theorems. The results presented in this paper will be accompanied by detailed calculations and proofs in a longer forthcoming work.

We identify the free half shuffle algebra of Schützenberger (1958) with an algebra of real-valued functionals on paths, where the half shuffle emulates integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities of Dzhumadil'daev (2007). We use these identities to prove that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas. Moreover, we consider minimal sets of iterated integrals defined through the recursive application of the half shuffle on Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture. Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem.

Daniel McNulty et.al (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the Lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908-25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any complex Hadamard matrix of order-six whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of complex Hadamard submatrices of order-two of any H2-reducible matrix is not 10,...,16,18. We also put forward some possible directions for further development.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

The infinite tower of positive-helicity soft gravitons in any minimally coupled, tree-level, asymptotically flat four-dimensional (4D) gravity was recently shown to generate a w_{1+infty} asymptotic symmetry algebra. It is natural to ask whether this classical algebra acquires quantum corrections at loop level. We explore this in quantum self-dual gravity, whose amplitudes acquire known one-loop exact all-plus helicity quantum corrections. We show using collinear splitting formulae that, remarkably, the w_{1+infty} algebra persists in quantum self-dual gravity without corrections.

We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or L_{infty} algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic L_{infty} algebra of the bulk theory is equivalent, up to homotopy, to a cyclic L_{infty} algebra on the boundary. The resulting action is the `on-shell action' conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic L_{infty} algebras.

The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with the dot action. A similar result holds between unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties. A recent result by Abreu-Nigro enabled us to prove these results by showing the modular law for the geometrical objects, and this is indeed done by Precup-Sommers and Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular law through GKM theory.

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of `trinion' singularities which exhibit these properties. In Type IIB, they give rise to 4d N=2 SCFTs that we call D_p^b(G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d N=4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as `ugly' components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver `bad'. We propose a way to redeem the badness of these quivers using a class S realization. We also discover new S-dualities between different D_p^b(G)-trinions. For instance, a certain E_8 gauging of the E_8 Minahan-Nemeschansky theory is S-dual to an E_8-shaped Lagrangian quiver SCFT.

We develop a technique for counting the number of stress tensor multiplets in a 4D N=2 SCFT. This provides a simple diagnostic for when an isolated (non-Lagrangian) SCFT is a product of two (or more) such theories. In class-S, the basic building blocks are the isolated SCFTs arising from the compactification of a 6D (2,0) theory on a 3-punctured sphere ("fixture"). We apply our technique to determine when a fixture is a product SCFT. The answer is that this phenomenon is surprisingly rare. In the low-rank A_{N-1}, D_N theories and the E_6 theory studied by the first author and his collaborators, it occurs less than 1% of the time. Of the 2979 fixtures in the (untwisted and twisted) E_6 theory, only 23 are product SCFTs. Of these, 22 were known to the original authors. The new one is a product of the {(E_7)}_8 Minahan-Nemeschansky theory and a new rank-2 SCFT.

We consider the anti-symmetrization of the half-shuffle on words, which we call the 'area' operator, since it corresponds to taking the signed area of elements of the iterated-integral signature. The tensor algebra is a so-called Tortkara algebra under this operator. We show that the iterated application of the area operator is sufficient to recover the iterated-integral signature of a path. Just as the "information" the second level adds to the first one is known to be equivalent to the area between components of the path, this means that all the information added by subsequent levels is equivalent to iterated areas. On the way to this main result, we characterize (homogeneous) generating sets of the shuffle algebra. We finally discuss compatibility between the area operator and discrete integration and stochastic integration and conclude with some results on the linear span of the areas of areas.

Let G be a Lie group equipped with commuting left and right linear representations on a vector space V, and let GJoin V be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group GJoin V from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their mathfrak g- and V-components, where mathfrak g is the Lie algebra of G. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on mathbb R^r, G_r^2. For the matrix model, the conjugation representation decomposes Mat(3) into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a (1+1)-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.

I propose a Hamiltonian framework for chiral gauge theories (CGT) based on a Euclidean formulation which uses 2n dimensional chiral fermions on the boundary of a 2n+1 dimensional disk. In the original Euclidean formulation, boundary gauge fields were extended into the bulk using 2n + 1 dimensional gauge field equations of motion (EOM) which creates a bottleneck for constructing a Hamiltonian. I present an alternate proposal for extending the gauge fields into the bulk that is compatible with both a Hamiltonian framework and a Euclidean path integral. Applying this to Abelian gauge fields produces exact expressions of interior fields as a functional of the boundary fields, which can be directly used in a Hamiltonian formulation. Euclidean analysis of the new gauge field extension shows that it can preserve the non-perturbative content of the original construction, including the behavior of topological charge, associated chiral fermion zero modes and an absence of the strong CP problem when applied to the Standard Model. This construction opens up a route to a Hamiltonian treatment and future quantum simulation of CGTs on a disk boundary.

In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proof of the results concerning the compactification and the renormalization. This proof involves finitely many calculations done with exact integer arithmetic.

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being d-complicial (dgeq 1), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal A_{infty}-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian d-truncated differential graded categories, generalizing results of Enomoto.

In this paper we discuss a higher spin extension of the holographic symmetry algebra for graviton and gluon. Our primary observation is that in the presence of higher spin particles the soft symmetry algebra has a subalgebra isomorphic to w_{infty} which is generated by the conformally soft higher spin particles. This w_{infty} subalgebra does not commute with the w_{1+infty} subalegbra generated by the conformally soft gravitons. The same thing holds for the colored higher spin particles. One gets a subalgebra isomorphic to the S-algebra which is generated by the conformally soft colored higher spin particles. We further verify the soft algebra for colored higher spin particles using the (tree-level) 4-point MHV amplitude of the higher spin Yang-Mills theory constructed in arXiv:2210.07130. At the end we also discuss the higher spin extension of the deformed holographic symmetry algebra for non-zero cosmological constant as constructed in arXiv:2312.00876.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

We classify equivariant C^*-actions on moduli spaces of Higgs bundles corresponding to the Painlevé equations. Using this, we compute the Floer-theoretic filtrations on the cohomology of these spaces, introduced by Ritter and the second author in arXiv:2304.13026. We compare it with the ``P=W'' and the filtration obtained by multiplicities of the irreducible components of the nilpotent cone, ultimately deducing that the Floer-theoretic filtration coincides with the multiplicity filtration, for all 2-dimensional Higgs moduli.

We investigate the unflavoured Schur indices of class mathcal S theories of modest rank, and in the case of N=4 super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their modular properties. We find closed form expressions for many of these theories in terms of quasimodular forms of level one or two, with the curious feature that in general they are sums of quasimodular forms of different weights. For type a_1 theories, the index can be fixed by taking a simple Ansatz for the family of quasimodular forms appearing in the expansion of this type and demanding that the result be sufficiently regular as qto0. For higher rank cases, an equally simple construction is lacking, but we nevertheless find that these indices can be expressed in terms of mixed-weight quasimodular forms.

We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds X. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi--Yau fourfold Y=Tot(K_X) and formulate the corresponding colored DT4 vertex. A K_X-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of X is canonically isomorphic to the constrained Hilbert functor of subschemes of Y lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

We introduce a Heegaard-Floer homology functor from the category of oriented links in closed 3-manifolds and oriented surface cobordisms in 4-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. Some of the basic properties of this functor are also explored.

In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation p^2 = 2(E-V(q)) defines a genus-g Riemann surface X, whose periods are naturally organized by its Jacobian variety J(X) = C^{g}/Λ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in SL(2g,Z) but not in the symplectic group Sp(2g,Z) once ggeq2, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.

Motivated by a variant of Atiyah-Floer conjecture proposed in L2 and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) \(Y\), perturbed by closed 2-forms of the form \(r*d\ff+w\), where \(r\geq 1\), where \(\ff\) is a harmonic Morse function with certain linear growth at the ends of \(Y\), and \(w\) is a certain closed 2-form.

Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate quiver mutation -- an operation that transforms one quiver (or directed multigraph) into another -- which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? In this paper, we use graph neural networks and AI explainability techniques to independently discover mutation equivalence criteria for quivers of type D. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type D, adding to the growing evidence that modern machine learning models are capable of learning abstract and parsimonious rules from mathematical data.

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types F_4, E_6, E_7, E_8 and G_2, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

This paper gives an explicit isomorphic mapping from the 240 real R^{8} roots of the E_8 Gosset 4_{21} 8-polytope to two golden ratio scaled copies of the 120 root H_4 600-cell quaternion 4-polytope using a traceless 8times8 rotation matrix U with palindromic characteristic polynomial coefficients and a unitary form e^{i\mathbb{U}}. It also shows the inverse map from a single H_4 600-cell to E_8 using a 4Dhookrightarrow8D chiral leftleftrightarrowright mapping function, φ scaling, and U^{-1}. This approach shows that there are actually four copies of each 600-cell living within E_8 in the form of chiral H_{4L}oplusφH_{4L}oplusH_{4R}oplusφH_{4R} roots. In addition, it demonstrates a quaternion Weyl orbit construction of H_4-based 4-polytopes that provides an explicit mapping between E_8 and four copies of the tri-rectified Coxeter-Dynkin diagram of H_4, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E_8-based Grand Unified Theories or GUTs.

A well-established organisational principle for Argyres--Douglas-type N=2 superconformal field theories in four dimensions is to characterise such theories by the data defining a(n irregular) Hitchin system on CP^1. The dictionary between Hitchin system data and various features of the corresponding SCFT has been studied extensively, but the overall structure of the resulting space of SCFTs still appears quite complicated. In this work, we systematically delineate a variety of simplifications that arise within this class of constructions due to several large classes of isomorphisms between SCFTs associated with inequivalent Hitchin system data (and their exactly marginal gaugings). We restrict to the most studied class of theories, namely the type A theories without outer automorphism twists.

Let Msubset B(H) be a semifinite von Neumann algebra, where B(H) denotes the algebra of all bounded linear operators on a Hilbert space H, and let τ be a fixed faithful normal semifinite trace on M.Let E_τ be the fully symmetric space associated with a fully symmetric Banach function space E on [0,infty).Using a complex interpolation argument based on the three-lines theorem on a strip, we show that for positive operators a,bin E_τ and tin[0,1], $ |a^t b^{1-t}+b^t a^{1-t}|_{E_τ}le 2^{max{2|t-1/2|-1/2,;0}};|a+b|_{E_τ}. In particular, we obtain the sharp constant 1 for t\in[1/4,3/4]: |a^t b^{1-t}+b^t a^{1-t}|_{E_τ}le |a+b|_{E_τ}. $ This extends the work of Kittaneh--Ricard in Linear Algebra Appl. 710 (2025), 356--362 and covers the results of Liu--He--Zhao in Acta Math. Sci. Ser. B (Engl. Ed.) 46 (2026), 62--68

In arXiv:2008.04330 it was shown that supertranslation and SL(2,mathbb C) current algebra symmetries, corresponding to leading and subleading soft graviton theorems, are enough to determine the tree level MHV graviton scattering amplitudes. In this note we clarify the role of subsubleading soft graviton theorem in this context.

In this article we determine the maximal possible order of the automorphism group of the form ag + b, where a and b are integers, of a complex three and four-dimensional family of compact Riemann surfaces of genus g, appearing for all genus. In addition, we construct and describe explicit complex three and four-dimensional families possessing these maximal number of automorphisms.

We show how the Schur-Weyl duality that exists between the partition algebra and the symmetric group results in a stronger theoretical foundation for characterising all of the possible permutation equivariant neural networks whose layers are some tensor power of the permutation representation M_n of the symmetric group S_n. In doing so, we unify two separate bodies of literature, and we correct some of the major results that are now widely quoted by the machine learning community. In particular, we find a basis of matrices for the learnable, linear, permutation equivariant layer functions between such tensor power spaces in the standard basis of M_n by using an elegant graphical representation of a basis of set partitions for the partition algebra and its related vector spaces. Also, we show how we can calculate the number of weights that must appear in these layer functions by looking at certain paths through the McKay quiver for M_n. Finally, we describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

The Weak Gravity Conjecture has recently been re-formulated in terms of a particle with non-negative self-binding energy. Because of the dual conformal field theory (CFT) formulation in the anti-de Sitter space the conformal dimension Delta (Q) of the lowest-dimension operator with charge Q under some global U(1) symmetry must be a convex function of Q. This property has been conjectured to hold for any (unitary) conformal field theory and generalized to larger global symmetry groups. Here we refine and further test the convex charge conjecture via semiclassical computations for fixed charge sectors of different theories in different dimensions. We analyze the convexity properties of the leading and next-to-leading order terms stemming from the semiclassical computation, de facto, extending previous tests beyond the leading perturbative contributions and to arbitrary charges. In particular, the leading contribution is sufficient to test convexity in the semiclassical computations. We also consider intriguing cases in which the models feature a transition from real to complex conformal dimensions either as a function of the charge or number of matter fields. As a relevant example of the first kind, we investigate the O(N) model in 4+epsilon dimensions. As an example of the second type we consider the U(N)times U(M) model in 4-epsilon dimensions. Both models display a rich dynamics where, by changing the number of matter fields and/or charge, one can achieve dramatically different physical regimes. We discover that whenever a complex conformal dimension appears, the real part satisfies the convexity property.

We study 4D N=2 superconformal field theories that arise as the compactification of the six-dimensional (2,0) theory of type E_6 on a punctured Riemann surface in the presence of Z_2 outer-automorphism twists. We explicitly carry out the classification of these theories in terms of three-punctured spheres and cylinders, and provide tables of properties of the Z_2-twisted punctures. An expression is given for the superconformal index of a fixture with twisted punctures of type E_6, which we use to check our identifications. Several of our fixtures have Higgs branches which are isomorphic to instanton moduli spaces, and we find that S-dualities involving these fixtures imply interesting isomorphisms between hyperKähler quotients of these spaces. Additionally, we find families of fixtures for which the Sommers-Achar group, which was previously a Coulomb branch concept, acts non-trivially on the Higgs branch operators.

We compute the Cuntz semigroup of a unital graph C*-algebra by combining Hong and Szymański's realisation of the entire ideal lattice of a graph C*-algebra, Ara, Moreno and Pardo's computation of the Murray-von Neumann semigroup of a unital graph C*-algebra and various Cuntz semigroup techniques. We also investigate certain properties of such Cuntz semigroups, and demonstrate that at the level of the Cuntz semigroup, there is no obstruction to the conjecture that the nuclear dimension of any graph C*-algebra is at most one.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. We introduce the moduli space of multi-curves and show how it leads to invariants. Our construction is based on an idea of Witten. In the special case that each connected component of the Lagrangian submanifold has the rational homology of a sphere we define rational numbers F_{g,h} for each genus g and h boundary components.

To each four dimensional Ngeq 2 supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of N=4 super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the S-duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of N=4 sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.

Crystal symmetry plays a fundamental role in determining its physical, chemical, and electronic properties such as electrical and thermal conductivity, optical and polarization behavior, and mechanical strength. Almost all known crystalline materials have internal symmetry. However, this is often inadequately addressed by existing generative models, making the consistent generation of stable and symmetrically valid crystal structures a significant challenge. We introduce WyFormer, a generative model that directly tackles this by formally conditioning on space group symmetry. It achieves this by using Wyckoff positions as the basis for an elegant, compressed, and discrete structure representation. To model the distribution, we develop a permutation-invariant autoregressive model based on the Transformer encoder and an absence of positional encoding. Extensive experimentation demonstrates WyFormer's compelling combination of attributes: it achieves best-in-class symmetry-conditioned generation, incorporates a physics-motivated inductive bias, produces structures with competitive stability, predicts material properties with competitive accuracy even without atomic coordinates, and exhibits unparalleled inference speed.

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book, showing that (A1) Albert algebras are exceptional and in particular that a central simple Jordan algebra over a field is exceptional if and only if it is an Albert algebra; (A2) A regular lattice in a real Albert algebra is also an Albert algebra; (A3) a Freudenthal algebra over a field is split by an extension of degree dividing 6; and (A4) a Freudenthal subalgebra of rank 9 in an Albert algebra can be used to describe the Albert algebra as a Tits construction.

We develop a bottom-up framework to reconstruct the asymptotic spectrum of light towers in four-dimensional N=1 effective field theories directly from their Kähler potential. Our construction uses the Integral Scaling Relation, which relates the mass scales of towers becoming light at infinite distance to the tensions of EFT strings, together with the Emergent String Conjecture applied recursively upon decompactification. These conditions organize the tower scaling vectors into a lattice generated by the EFT-string vectors and select the globally consistent tower polytopes. When applied to Kähler potentials arising from string compactifications, our algorithm precisely reproduces the known arrangements of towers and duality frames. We then classify the admissible polytopes for asymptotic Kähler potentials Ksim-log P(s), with P(s) a homogeneous polynomial of degree at most seven, and show that certain apparently consistent Kähler potentials are incompatible with the assumed quantum-gravity constraints. For general polynomials, additional restrictions arise from gluing the tower arrangements across different growth sectors. Remarkably, every tower polytope allowed by our reconstruction can be obtained as a concrete slice of the polytope associated with M-theory on a Joyce G_2-manifold. This provides evidence for a form of string universality in which EFT strings act as the fundamental building blocks of the UV tower structure in the EFT perturbative regimes.

Consider a standard graded artinian k-algebra B and an extension of B by a new variable, A=Botimes_k k[x]/(x^d) for some dgeq 1. We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions Botimes_k k[x]/(x^d) to have the weak or the strong Lefschetz property.

Among a huge variety of known two-dimensional materials, some of them have anisotropic crystal structures; examples include so different systems as a few-layer black phoshphorus (phosphorene), beryllium nitride BeN_4, van der Waals magnet CrSBr, rhenium dichalgogenides ReX_2. As a consequence, their optical and electronic properties turn out to be highly anisotropic as well. In some cases, the anisotropy results not just in a smooth renormalization of observable properties in comparison with the isotropic case but in the appearance of dramatically new physics. The examples are hyperbolic plasmons and excitons, strongly anisotropic ordering of adatoms at the surface of two-dimensional or van der Waals materials, essential change of transport and superconducting properties. Here, we present a systematic review of electronic structure, transport and optical properties of several representative groups of anisotropic two-dimensional materials including semiconductors, anisotropic Dirac and semi-Dirac materials, as well as superconductors.

Let λ_{mathbb K}(m) denote the maximal absolute projection constant among m-dimensional Banach spaces over mathbb K=mathbb R or mathbb C. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for λ_{mathbb R}(m). Numerical evidence suggests that these bounds may be sharp in dimensions 6 and 8. We also discuss weighted spherical (2,2)-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions 4 and 5 indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an EDR. We mention three criteria: (1) a Hermite ring R is an EDR iff for all pairs (a,c)in R^2, the product homomorphism U(R/Rac)times Ubigl(R/Rc(1-a)bigr)to U(R/Rc) between groups of units is surjective; (2) a reduced Hermite ring R is an EDR iff it is a pre-Schreier ring and for each ain R, every zero determinant unimodular 2times 2 matrix with entries in R/Ra lifts to a zero determinant matrix with entries in R; (3) a Bézout domain R is an EDD iff for all triples (a,b,c)in R^3 there exists a unimodular pair (e,f)in R^2 such that (a,e) and (be+af,1-a-bc) are unimodular pairs. We use these criteria to show that each Bézout ring R that is an (SU)_2 ring (as introduced by Lorenzini) such that for each nonzero ain R there exists no nontrivial self-dual projective R/Ra-module of rank 1 generated by 2 elements (e.g., all its elements are squares), is an EDR.

We study the superconformal indices of 4d theories coming from 6d N=(2,0) theory of type Γon a Riemann surface, with the action of the outer-automorphism σin the trace. We find that the indices are given by the partition function of a deformed 2d Yang-Mills on the Riemann surface with gauge group G which is S-dual to the subgroup of Γfixed by σ. In the 2-parameter deformed version, we find that it is governed not by Macdonald polynomials of type G, but by Macdonald polynomials associated to twisted affine root systems.

The origin of a chemical reaction between two reactant atoms is associated to the activation energy, with the assumption that, high-energy collisions between these atoms, are the ones that overcome the activation energy. Here, we (i) show that a stronger attractive van der Waals (vdW) and electron-ion Coulomb interactions between two polarized atoms are responsible to initiate a chemical reaction, either before or after the collision. We derive this stronger vdW attraction formula exactly using the quasi one-dimensional Drude model within the ionization energy theory and the energy-level spacing renormalization group method. Along the way, we (ii) expose the precise physical mechanism responsible for the existence of a stronger vdW interaction for both long and short distances, and also show how to technically avoid the electron-electron Coulomb repulsion between polarized electrons from these two reactant atoms. Finally, we properly and correctly associate the existence of this stronger attraction to Ramachandran's 'normal limits' (distance shorter than what is allowed by the standard vdW bond) between chemically nonbonded atoms.

This paper presents a systematic cataloging of the generators of celestial symmetries on phase space. Starting from the celestial OPEs, we first show how to extract a representation of the general-spin analog of the wedge subalgebra of w_{1+infty} on the phase space of massless matter fields of arbitrary helicity. These generators can be expressed as light-sheet operators that are quadratic in the matter fields at future or past null infinity. We next show how to extend these symmetries beyond the wedge. Doing so requires us to augment the quadratic operators with: 1) linear terms corresponding to primary descendants of the negative helicity gauge fields the matter modes couple to, and 2) a tower of higher-particle composite operator contributions. These modes can be realized as light-ray operators supported on generators of null infinity, but local on the celestial sphere. Finally, we construct a representation of the celestial symmetries that captures how the positive helicity gauge fields transform. We close by discussing how these celestial symmetries inform our choice of detector operators.

We completely classify the atomic summands in a graph product (M,varphi) = *_{v in G} (M_v,varphi_v) of von Neumann algebras with faithful normal states. Each type I factor summand (N,psi) is a tensor product of type I factor summands (N_v,psi_v) in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the (N_v,psi_v)'s using explicit polynomials associated to the graph.

We derive sufficient conditions for exact functors on locally finite abelian categories to preserve Loewy diagrams of objects. We apply our results to determine sufficient conditions for induction functors associated to simple current extensions of vertex algebras to preserve Loewy diagrams.

We determine the 4-point correlation function and amplitude in planar, maximally supersymmetric Yang-Mills theory to 12 loops. We find that the recently-introduced 'double-triangle' rule in fact implies the previously described square and pentagon rules; and when applied to 12 loops, it fully determines the 11-loop correlator and fixes all but 3 of the (22,024,902) 12-loop coefficients; these remaining coefficients can be subsequently fixed using the '(single-)triangle' rule. Not only do we confirm the Catalan conjecture for anti-prism graphs, but we discover evidence for a greatly generalized Catalan conjecture for the coefficients of all polygon-framed fishnet graphs. We provide all contributions through 12 loops as ancillary files to this work.

S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.

Gauging fermion parity and summing over spin structures are subtly distinct operations. We introduce 'bosonisation cohomology' groups H_B^{d+2}(X) to capture this difference, for theories in spacetime dimension d equipped with maps to some X. Non-trivial classes in H_B^{d+2}(X) contain theories for which (-1)^F is anomaly-free, but spin structure summation is anomalous. We formulate a sequence of cobordism groups whose failure to be exact is measured by H_B^{d+2}(X), and from here we compute it for X=pt. The result is non-trivial only in dimensions din 4Z+2, being due to the presence of gravitational anomalies. The first few are H_B^4=Z_2, probed by a theory of 8 Majorana-Weyl fermions in d=2, then H_B^8=Z_8, H_B^{12}=Z_{16}times Z_2. We rigorously derive a general formula extending this to every spacetime dimension. Along the way, we compile many general facts about (fermionic and bosonic) anomaly polynomials, and about spin and pin^- (co)bordism generators, that we hope might serve as a useful reference for physicists working with these objects. We briefly discuss some physics applications, including how the H_B^{12} class is trivialised in supergravity. Despite the name, and notation, we make no claim that H_B^bullet(X) actually defines a cohomology theory (in the Eilenberg-Steenrod sense).

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

The composition gamma circ eta of Jordan curves gamma and eta in universal Teichm\"uller space is defined through the composition h_gamma circ h_eta of their conformal weldings. We show that whenever gamma and eta are curves of finite Loewner energy I^L, the energy of the composition satisfies $I^L(gamma circ eta) lesssim_K I^L(gamma) + I^L(eta), with an explicit constant in terms of the quasiconformal K of \gamma and \eta. We also study the asymptotic growth rate of the Loewner energy under n self-compositions \gamma^n := \gamma \circ \cdots \circ \gamma, showing limsup_{n rightarrow infty} 1{n}log I^L(gamma^n) lesssim_K 1, again with explicit constant. Our approach is to define a new conformally-covariant rooted welding functional W_h(y), and show W_h(y) \asymp_K I^L(\gamma) when h is a welding of \gamma and y is any root (a point in the domain of h). In the course of our arguments we also give several new expressions for the Loewner energy, including generalized formulas in terms of the Riemann maps f and g for \gamma which hold irrespective of the placement of \gamma on the Riemann sphere, the normalization of f and g, and what disks D, D^c \subset \mathbb{C} serve as domains. An additional corollary is that I^L(\gamma) is bounded above by a constant only depending on the Weil--Petersson distance from \gamma$ to the circle.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Appell functions are a large class of multi-variable quasi-elliptic functions, which are also instances of higher depth mock modular forms. A central aspect of these functions is their non-holomorphic modular completion. Our previous work developed Appell functions for a general positive definite lattice Λ, and expressed the non-holomorphic completion in terms of the generalized error functions M_P, which are integrals over a P-dimensional hyperplane in Λotimes C. In this sequel, we express M_P as a P-dimensional iterated integral of variables w_jin H, such that the completion of the Appell function takes the form of an iterated Eichler integral of modular forms. We apply this to the case of the root lattice of the A_N Lie algebra, which is of particular interest because of their appearance in partition functions of topological twisted Yang-Mills theories.

At leading order in the genus expansion the ETH matrix model for DSSYK reproduces its correlators by construction, while its higher-genus corrections are conjectured to capture higher-topology contributions in the dual sine-dilaton gravity -- a correspondence established so far only for the disk and the wormhole. At fixed genus the correlators are built from discrete volumes N_{g,n}, polynomial in q-deformed zeta values ζ_q(2k) with q=e^{-λ}, λ being the DSSYK coupling. These lie in the ring of quasimodular forms generated by the Eisenstein series E_2,E_4,E_6, whose S-duality yields an exact closed form for the leading non-perturbative correction as λto0, controlled by widetilde q=e^{-4π^2/λ}. Known at disk level, this scale is shown here to govern the fixed-genus, higher-boundary amplitudes as well. We show that the term linear in widetilde q, at leading order in λ, is captured entirely by the q-deformed Weil--Petersson volumes, and reduces to a finite sum of intersection numbers of κ-classes on the moduli space mathcal M_{g,n} of stable curves, computable without repeating the topological recursion that produced the N_{g,n}. We tabulate it for every (g,n) whose q-deformed volume is known in closed form, and extend it to (3,1),(3,2),(4,1), where none is available. The construction is not restricted to leading order: we work out O(widetilde q^2) for the same cases.

In an earlier paper [arXiv:2511.03669] we extracted the OPE of celestial CFT operator duals of positive helicity graviton and scalar particles from the Mellin transformed relevant MHV amplitudes of conformal gravity, realised as the bosonic subsector of the Berkovits-Witten theory. A soft theorem analysis of bulk MHV amplitudes established that this conformal gravity exhibits a chiral bms_4 symmetry on the celestial sphere with the associated sl(2,R) current algebra, which acquires a non-trivial central extension, unlike the Einstein gravity. Here we construct a 2d chiral CFT free-field realisation of the relevant chiral bms_4 algebra in terms of three free scalars (ϕ_i) and three (β_i,γ_i) ghost pairs, and propose vertex operators for the positive-helicity graviton primary G^{++}_Δ(z,z) as well as the scalar primary Φ_Δ(z,z), and compute their OPEs. These OPEs reproduce exactly those obtained from the bulk conformal gravity MHV amplitudes, providing a concrete celestial dual description of its MHV sector.

This work investigates the q-deformation of (r,s)-Airy structures and their realization via q-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order q-WKB solution for the matrix systems associated with the q-quantized curve E_q(x,y)=0. We demonstrate that the resulting non-perturbative connected q-amplitudes satisfy a set of shifted q-loop equations, which can be interpreted as the Ward identities of a q-deformed W(gl_r) algebra. Our main result provides a rigorous classification of admissible (r,s,q) pairs and q-Casimir configurations that satisfy the q-topological type property. This ensures that the semi-classical expansion is uniquely governed by the q-topological recursion, offering new insights into the q-quantization of mirror curves and their underlying algebraic structures.

We study theories of type D_4 in class-S, with nonabelian outer-automorphism twists around various cycles of the punctured Riemann surface C. We propose an extension of previous formulæ for the superconformal index to cover this case and classify the SCFTs corresponding to fixtures (3-punctured spheres). We then go on to study families of SCFTs corresponding to once-punctured tori and 4-punctured spheres. These exhibit new behaviours, not seen in previous investigations. In particular, the generic theory with 4 punctures on the sphere from non-commuting Z_2 twisted sectors has six distinct weakly-coupled descriptions.

We consider four dimensional CHL models with sixteen spacetime supersymmetries obtained from orbifolds of type IIA superstring on K3 x T^2 by a Z_N symmetry acting (possibly) non-geometrically on K3. We show that most of these models (in particular, for geometric symmetries) are self-dual under a weak-strong duality acting on the heterotic axio-dilaton modulus S by a "Fricke involution" S --> -1/NS. This is a novel symmetry of CHL models that lies outside of the standard SL(2,Z)-symmetry of the parent theory, heterotic strings on T^6. For self-dual models this implies that the lattice of purely electric charges is N-modular, i.e. isometric to its dual up to a rescaling of its quadratic form by N. We verify this prediction by determining the lattices of electric and magnetic charges in all relevant examples. We also calculate certain BPS-saturated couplings and verify that they are invariant under the Fricke S-duality. For CHL models that are not self-dual, the strong coupling limit is dual to type IIA compactified on T^6/Z_N, for some Z_N-symmetry preserving half of the spacetime supersymmetries.

We study two-point functions of chiral primary operators in N=4 Super Yang-Mills in the presence of a half-BPS surface defect. Such a defect preserves only the diagonal combination of transverse rotations and of a global U(1) R-symmetry, so that the correlator depends on one additional cross-ratio and splits into sectors of definite angular charge. We solve the superconformal Ward identities and construct the bulk and defect superconformal blocks in all sectors, including the charged ones, which have no analogue for defects preserving transverse rotations. The solution contains a holomorphic sector, which we fix non-perturbatively by mapping it to a chiral algebra correlator. We extend this map to defects breaking transverse rotations, where it requires a redefinition of the modes of the two-dimensional current algebra, and use it to fix the charged sectors as well. We then use the defect dispersion relation to reconstruct the full correlator of the dimension-two chiral primary at leading order at large N and strong coupling. In the charged sectors long multiplets are absent and the answer is conjecturally exact in both N and the coupling; expanding it at weak coupling, we find perfect agreement with known results. From these correlators we extract infinitely many new pieces of defect CFT data. Finally, we reproduce the strong-coupling answer by an independent Witten diagram bootstrap on the rotation-invariant locus.

We derive the tree-level interaction vertices of the linear sigma model with quarks in the presence of global rotation. Starting from the scalar and fermionic field solutions in a rotating background, we construct the corresponding spectral representations and use them to evaluate the mesonic self-interactions and Yukawa couplings of the model. We show that rotation modifies the usual momentum-space structure of the interaction vertices: conservation of longitudinal momentum is accompanied by angular-momentum selection rules, while the transverse dynamics is encoded in rotation-induced radial form factors involving products of Bessel functions. The resulting expressions provide a set of interaction rules adapted to the symmetries and finite geometry of the rotating system. Although derived within the linear sigma model with quarks, the structures obtained for scalar self-interactions and scalar-fermion couplings can be extended to a broader class of quantum field theories under global rotation. These results provide the building blocks for perturbative calculations and contribute toward a microscopic quantum-field-theoretical description of interacting matter in rotating environments.

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {Haldane bundle dataset}, which consists of synthetically generated complex line bundles on the 2-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT. Extremal surfaces in AdS quantify the "geometric" entanglement between different regions in physical space for the dual CFT. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS_5timesS^5, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 SYM. Making this proposal precise is subtle for a number of reasons, the most obvious being that from the bulk one usually has access to gauge-invariant quantities only, while a split into subgroups is inherently gauge variant. We study N=4 SYM on the Coulomb branch, where some of the issues can be mitigated and the proposal can be sharpened. Continuing back to the original AdS_5timesS^5 geometry, we obtain a modified proposal, based on the relation of the internal space to the R-symmetry group.

Quantum materials hosting Weyl fermions have opened a new era of research in condensed matter physics. First proposed in 1929 in particle physics, Weyl fermions have yet to be observed as elementary particles. In 2015, Weyl fermions were detected as collective electronic excitations in the strong spin-orbit coupled material tantalum arsenide, TaAs. This discovery was followed by a flurry of experimental and theoretical explorations of Weyl phenomena in materials. Weyl materials naturally lend themselves to the exploration of the topological index associated with Weyl fermions and their divergent Berry curvature field, as well as the topological bulk-boundary correspondence giving rise to protected conducting surface states. Here, we review the broader class of Weyl topological phenomena in materials, starting with the observation of emergent Weyl fermions in the bulk and of Fermi arc states on the surface of the TaAs family of crystals by photoemission spectroscopy. We then discuss some of the exotic optical and magnetic responses observed in these materials, as well as the progress in developing some of the related chiral materials. We discuss the conceptual development of high-fold chiral fermions, which generalize Weyl fermions, and we review the observation of high-fold chiral fermion phases by taking the rhodium silicide, RhSi, family of crystals as a prime example. Lastly, we discuss recent advances in Weyl-line phases in magnetic topological materials. With this Review, we aim to provide an introduction to the basic concepts underlying Weyl physics in condensed matter, and to representative materials and their electronic structures and topology as revealed by spectroscopic studies. We hope this work serves as a guide for future theoretical and experimental explorations of chiral fermions and related topological quantum systems with potentially enhanced functionalities.

In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near the Planck scale. The key idea is to investigate the Betti numbers of coarse-grained path integral histories, regularized in terms of dynamical triangulations, as a function of the coarse-graining scale. In two dimensions our analysis exhibits the well-known fractal structure of Euclidean quantum gravity.

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik--Zamolodchikov equation to the quantum 6j-symbols of the algebra U_q(g): U_{q} = U cdot Φ^{KZ}_{Q} . We prove this relation in the case of the sl_2 Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the sl_N algebra for representations [2]otimes [1] otimes [1], [2]otimes [2] otimes [1], the non-trivial cases with multiplicities --- [2,1]^{otimes 3}, [2,2]^{otimes 3}, and in the case of the so_5 Lie algebra for the tensor cube of the fundamental representation.

In an unpublished preprint batanin, Batanin conjectures that it is possible to take `slices' of a globular operad, thereby isolating the algebraic structure in each dimension. It was further hypothesised that the slices of a globular operad for some theory of higher category contain essential information about those higher categories, namely whether or not they are equivalent to the fully weak variety. In this paper, we use the theory of presentations for globular operads developed in Me to provide a concrete definition of slices, and calculate the slices for several key theories of n-category.

Gross, Mansour, and Tucker introduced the partial-duality polynomial of a ribbon graph [Distributions, European J. Combin. 86, 1--20, 2020], the generating function enumerating partial duals by the Euler genus. Chmutov and Vignes-Tourneret wondered if this polynomial and its conjectured properties would hold for general delta-matroids, which are combinatorial abstractions of ribbon graphs. Yan and Jin contributed to this inquiry by identifying a subset of delta-matroids-specifically, even normal binary ones-whose twist polynomials are characterized by a singular term. Building upon this foundation, the current paper expands the scope of the investigation to encompass even non-binary delta-matroids, revealing that none of them have width-changing twists.

The AKSZ formalism is a construction of topological field theories where the target spaces are differential graded symplectic manifolds. In this paper, we describe an analogue of the AKSZ formalism where the target spaces are differential graded contact manifolds. We show that the space of fields inherits a weak contact structure, and we construct a solution to the analogue of the classical master equation, defined via the Jacobi bracket. In the n=1 case, we recover the Jacobi sigma model, and in the n=2 case, we obtain three-dimensional topological field theories associated to Courant-Jacobi algebroids.

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

A covariant formalism is used in order to examine the status of Maxwell equations and to unify the concept of balances, for all chemical engineering applications in relation with electrodynamics. The resulting formal structure serves as a discussion in studying the determinism of electromagnetic systems, and examining the theoretical foundations for a general classification of chemical engineering applications when non-conservative forces are concerned. A strategy for modeling such applications is then sketched and the balances for the main conserved and non-conserved extensive quantities are then summarized in their covariant and classical forms.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

For a graph G with n vertices and m edges, Lin et al. Lin define the atom--bond sum-connectivity (ABS) matrix of G such that the (i,j)^{th} entry is \[ 1 - \frac{2{d_i + d_j}} \] if vertex v_i is adjacent to the vertex v_j, and 0 otherwise. In this article, we determine the characteristic polynomial of the ABS matrix for certain specific classes of graphs. Furthermore, we compute the ABS eigenvalues and the ABS energy for these classes.

We study a set of four-dimensional N=2 superconformal field theories (SCFTs) widehatΓ(G) labeled by a pair of simply-laced Lie groups Γ and G. They are constructed out of gauging a number of D_p(G) and (G, G) conformal matter SCFTs; therefore they do not have Lagrangian descriptions in general. For Γ= D_4, E_6, E_7, E_8 and some special choices of G, the resulting theories have identical central charges (a=c) without taking any large N limit. Moreover, we find that the Schur indices for such theories can be written in terms of that of N=4 super Yang--Mills theory upon rescaling fugacities. Especially, we find that the Schur index of D_4(SU(N)) theory for N odd is written in terms of MacMahon's generalized sum-of-divisor function, which is quasi-modular. For generic choices of Γ and G, it can be regarded as a generalization of the affine quiver gauge theory obtained from D3-branes probing an ALE singularity of type Γ. We also comment on a tantalizing connection regarding the theories labeled by Γ in the Deligne--Cvitanović exceptional series.

In this article, we introduce and study singular foliations of b^k-type. These singular foliations formalize the properties of vector fields that are tangent to order k along a submanifold W subset M. Our first result is a classification of these foliations, relating them to geometric structures defined in a formal neighborhood of the submanifold, such as jets of distributions that are involutive up to order k-1. When W is a hypersurface, singular foliations of b^k-type are Lie algebroids. In this particular case, they are generalizations of the b^k-tangent bundles introduced by Scott. Indeed, they are always locally isomorphic to b^k-tangent bundles, but globally such an isomorphism is obstructed by a holonomy invariant. Our second main result is a Riemann-Hilbert-style classification of singular foliations of b^k-type in terms of holonomy representations. In this paper, we study singular foliations of b^k-type from several different perspectives. In particular: (1) We study the problem of extending a k-th-order foliation to a (k+1)-th order foliation and prove that this is obstructed by a characteristic class. (2) When W is a hypersurface, we give a detailed study of algebroid differential forms and extend Scott's calculation of the cohomology. (3) We study algebroid symplectic forms in terms of the geometric structures induced on W. In particular, we find that there is a close relationship between the above obstruction class for extensions and the symplectic variation of the symplectic foliation induced on W.

Reduction in effective spacetime dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wavefunction factors depend nontrivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a "covert" pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios.

This is the compendium of the cluster algebra and quiver package for sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. In this compendium, we include the relevant theory to introduce the reader to cluster algebras assuming no prior background; this exposition has been written to be accessible to an interested undergraduate. Throughout this compendium, we include examples that the user can run in the sage notebook or command line, and then close with a detailed description of the data structures and methods in this package.

Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the tableaux method to classify cluster variables in Grassmannian cluster algebras C[Gr(k,n)] up to (k,n)=(3,12), (4,10), or (4,12) up to a certain number of columns of tableaux, using HPC clusters. These datasets are made available on GitHub. Supervised and unsupervised machine learning methods are used to analyse this data and identify structures associated to tableaux corresponding to cluster variables. Conjectures are raised associated to the enumeration of tableaux at each rank and the tableaux structure which creates a cluster variable, with the aid of machine learning.

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)toLearn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto Ay^A. Using the fact that (Poly,otimes) is monoidal closed, we show that a map Ato B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [Ay^A,By^B]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p in Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.

We study a kinematical consequence of the Hopf-algebraic momentum composition law in κ-Minkowski spacetime. The same curved momentum space can be described in different coordinates. In the bicrossproduct basis the ordered-plane-wave labels are the translation-generator eigenvalues, so the relevant map is one-to-one. In the classical basis, instead, the translation eigenvalues P_μ are nonlinearly related to the ordered-plane-wave labels p_μ. This relation can fail to be globally one-to-one in a high-momentum region. When a given classical-basis four-momentum admits more than one real auxiliary preimage, the branch-sensitive quantity P_+equiv P_0+P_4=κe^{p_0/κ} enters the coproduct and resolves the branches in two-particle states. Imposing the vanishing total-momentum constraint therefore gives branch-dependent κ-deformed back-to-back momentum correlations. In a single-branch regime this is just a deformed correlated product, while in a multibranch regime a state specified only by P_μ can be expanded into distinct auxiliary branches. If P_μ are taken as the directly meaningful momenta, the physical content is the resulting deformed correlation pattern. If the auxiliary variables p_μ are assigned operational meaning, the same constrained state can be interpreted as a superposition over different auxiliary branches. We also compare this structure with standard regular self-adjoint nonrelativistic minimal-length models and find no analogous smooth local two-real-branch inversion on their physical domains.

We consider the superconformal index of class S theories of type D, which arise by compactification of the (2,0) D_n theories on a punctured Riemann surface C. We also allow for the presence of twist lines on C associated to the Z_2 outer automorphism of D_n. For a two-parameter slice (p=0, q, t) in the space of superconformal fugacities, we determine the 2d TQFT that computes the index.

The study of twisted two-dimensional (2D) materials, where twisting layers create moiré superlattices, has opened new opportunities for investigating topological phases and strongly correlated physics. While systems such as twisted bilayer graphene (TBG) and twisted transition metal dichalcogenides (TMDs) have been extensively studied, the broader potential of a seemingly infinite set of other twistable 2D materials remains largely unexplored. In this paper, we define "theoretically twistable materials" as single- or multi-layer structures that allow for the construction of simple continuum models of their moiré structures. This excludes, for example, materials with a "spaghetti" of bands or those with numerous crossing points at the Fermi level, for which theoretical moiré modeling is unfeasible. We present a high-throughput algorithm that systematically searches for theoretically twistable semimetals and insulators based on the Topological 2D Materials Database. By analyzing key electronic properties, we identify thousands of new candidate materials that could host rich topological and strongly correlated phenomena when twisted. We propose representative twistable materials for realizing different types of moiré systems, including materials with different Bravais lattices, valleys, and strength of spin-orbital coupling. We provide examples of crystal growth for several of these materials and showcase twisted bilayer band structures along with simplified twisted continuum models. Our results significantly broaden the scope of moiré heterostructures and provide a valuable resource for future experimental and theoretical studies on novel moiré systems.

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions 0, 1, and 2. This work is primarily expository, revisiting proofs from Pont1 with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of Pont2 regarding π_2^S is incorrect.

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically studied via the Nekrasov-Shatashvili functions. In this paper, we explore another aspect of the relation between N=2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlev\'e equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg-Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an O(2) matrix model. We then show that these eigenfunctions are computed by surface defects in SU(2) super Yang-Mills in the self-dual phase of the Omega-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Let L subset X be a compact embedded Lagrangian in a compact symplectic manifold. We present the moduli spaces of holomorphic maps of arbitrary genus with boundary on L as a global Kuranishi chart, generalising the work of Abouzaid-McLean-Smith and Hirschi-Swaminathan. We use this to define an open-closed Deligne-Mumford theory whose open genus zero part is the Fukaya A_infty algebra associated to L, and whose closed part gives the Gromov--Witten theory of X. Combined with results of Costello, this has applications in obtaining Gromov--Witten invariants from the Fukaya category.

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric mathcal D-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic mathcal D-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric D-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.