Daily Papers

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 show that curve enumeration invariants of complex threefolds with nef anti-canonical bundle are determined by their values on local curves. This implies the MNOP conjecture of Maulik, Nekrasov, Okounkov, and Pandharipande relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants, for all complex threefolds with nef anti-canonical bundle (in particular, all Calabi--Yau threefolds) and primary insertions (no descendents), given its known validity for local curves due to Bryan, Okounkov, and Pandharipande. The main new technical ingredient in our work is a generic transversality result for holomorphic curves in complex manifolds. Due to the rigidity of complex structures, this result is necessarily weaker than the corresponding generic transversality property for holomorphic curves in almost complex manifolds. Despite this weaker nature, it is enough to obtain our main result by following the proof of the Gopakumar--Vafa integrality conjecture by Ionel and Parker.

We present an algorithm that constructs Calabi-Yau threefold orientifolds and their F-theory uplifts to elliptically-fibered Calabi-Yau fourfolds, embedded in toric varieties at codimension one and two respectively. The resulting Calabi-Yau fourfolds arise from triangulations of 6d reflexive polytopes -- which our method constructs from orientifold data -- and are smooth away from isolated terminal singularities. For many of our fourfolds, the construction of the mirror manifold is immediate, enabling the computation of fourfold periods, and thus the seven-brane superpotential. We present multiple examples that demonstrate these capabilities. Our algorithms work with CYTools and are available through a GitHub repository.

In the present paper we study the singularity structure of the exterior field of neutron stars with the aid of the four-parameter exact solution of the Einstein-Maxwell equations. The complete analysis of this problem in the generic case becomes possible due to the implementation of the novel analytical approach to the resolution of the singularity condition, and it shows the absence of the ring singularities off the symmetry axis in the positive mass case, as well as the possibility of the removal of the ring singularity by a strong magnetic field in the negative mass case. The solution takes an extraordinarily simple form in the equatorial plane, very similar to that of the Kerr solution, which makes it most suitable for astrophysical applications as the simplest model of a rotating magnetized deformed mass. It also provides a nontrivial example confirming a recent claim that the varphi component of the electromagnetic four-potential has features inconsistent with the intrinsic properties of the electrovac metric, while the magnetic field is represented correctly by the t component of the dual electromagnetic four-potential.

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy can be circumvented by finding a height vector in the strict interior of the intersection of the secondary cones associated with each 2-face triangulation. We demonstrate that such triangulations are generated with orders of magnitude fewer operations than the naive approach of generating all FRSTs and selecting only those differing on 2-faces. Similar methods are also presented to directly generate (the support of) the secondary subfan of all fine triangulations, relevant for random sampling of FRSTs.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Fine, regular, and star triangulations (FRSTs) of four-dimensional reflexive polytopes give rise to toric varieties, within which generic anticanonical hypersurfaces yield smooth Calabi-Yau threefolds. We introduce CYTransformer, a deep learning model based on the transformer architecture, to automate the generation of FRSTs. We demonstrate that CYTransformer efficiently and unbiasedly samples FRSTs for polytopes across a range of sizes, and can self-improve through retraining on its own output. These results lay the foundation for AICY: a community-driven platform designed to combine self-improving machine learning models with a continuously expanding database to explore and catalog the Calabi-Yau landscape.

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.

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.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

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.

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.

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.

The dynamics of Einstein-conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman-Robertson--Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (`Small Bang'), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (`Big Bang'). Thus, in the present EccH model there is a new kind of physical spacetime singularity (`Small Bang'). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout-Englert-Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.

We first prove a rigidity result for uniformly rotating vortex patch solutions to Euler equations in bounded simply-connected planar domains. Namely, if a bounded simply-connected domain with real-analytic boundary contains a nontrivial vortex patch that remains strictly inside the domain while undergoing a uniform rigid rotation, then the ambient domain must be a disc centered at the rotation center. The proof combines the rotating-frame formulation with harmonic unique continuation and the generalized Schwarz's reflection principle across analytic boundary arcs. These tools imply that, unless the outer domain is rotationally invariant, the velocity must vanish in an open region surrounding the patch, contradicting the nonzero circulation generated by the patch. In the same spirit, we show that only circular domains possess a circular point vortex orbit. As a second result, we construct quasi-periodic vortex patch solutions to the Euler equations in domains that are small perturbations of the unit disc D. The amplitudes of the solutions are linked to the size of the domain deformation, allowing the classical Rankine vortices bcdotD, bin(0,1), to be regarded as equilibrium states. Our analysis is perturbative with respect to the construction developed in the undeformed case, and the resulting quasi-periodic solutions exist for most values of the inner radius b. This provides the first construction of quasi-periodic vortex patch solutions to the Euler equations in non-radial domains. Moreover, combined with the rigidity result established above, this shows that, in the periodic case, these solutions cannot arise from a purely rigid rotation.

In this paper, we first study the consequence of spacetime translations and Lorentz transformations on Celestial CFT OPEs. Working with the light transforms of the operators belonging to the modified Mellin basis, we found that the leading order singularity in the OPE of such operators could be fixed purely using Poincaré symmetries owing to the non-trivial action of the translations on these operators. The OPE coefficient is then fixed using the soft limit of the correlation functions. We check that this singular structure obtained from symmetries is consistent with the OPE limit of three-point functions. This approach could potentially be useful for studying Celestial CFT without adverting to bulk physics. As another goal, we explore the significance of light transformation in Carrollian CFTs. In the special cases we considered, we show that light transformation equips us with a map between two branches of Carroll CFT in d=3 dimension at the level of correlation functions in the near coincident limit.

We study singularities of height functions on singular surfaces in R^3 parameterized by smooth map-germs A-equivalent to H_k in Mond's classification, and the versality of the family of the height functions. We also study relations of the singularities of the height functions with the parabolic locus of these singular surfaces.

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.

Spacetime singularities displayed by classical black hole solutions are expected to be cured by high-energy effects, beyond the regime of validity of General Relativity. On the other hand, there are well-known examples of classical theories allowing for nonsingular black hole solutions. The most prominent is obtained by complementing the gravitational sector of General Relativity with a minimally coupled nonlinear-electrodynamics sector. It has recently been shown that purely electric/purely magnetic nonsingular black holes in an Einstein-nonlinear-electrodynamics setup are always plagued by a gradient instability. In this work, we derive the quadratic action for linear perturbations of static, spherically symmetric dyonic solutions of Einstein-nonlinear-electrodynamics theories with a general dependence on both invariants. We show that regularity of the center, absence of ghosts and angular gradient stability cannot hold simultaneously: arbitrarily close to a regular center, at least one photon polarization has a negative squared angular propagation speed. We further show that regularity drives one electromagnetic kinetic eigenvalue to zero at the center, which may limit perturbative control near the core. Evading the obstruction requires extending the theory, for example by introducing additional degrees of freedom.

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 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.

We introduce a classification conjecture for kappa-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of Z_2^2times O_3-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

A classical problem in convex and discrete geometry asks for the convex polyhedron of greatest volume whose vertices are chosen from the unit sphere S^2. For a prescribed number N of vertices, the problem is known only in a small number of cases. In this paper we resolve the next outstanding case, N=9. We prove that every convex polyhedron with at most nine vertices on S^2 has volume at most 32sqrt{3-3}, with equality, up to rotation, precisely for a triaugmented triangular prism of an explicitly determined shape. The proof combines combinatorial and geometric reductions with sharp volume estimates. By a theorem of Berman and Hanes (Mathematische Annalen, 1970), a volume maximizer must be simplicial, reducing the 2,606 combinatorial types of 9-vertex polyhedra to 50. We prove that a maximizer cannot have a trivalent vertex, leaving only five combinatorial types, which are treated using geometric and combinatorial arguments. In particular, we determine the exact maximizer within the triaugmented triangular prism class, and characterize the equality case.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We also extend Ballas-Cooper-Leitner's definition of generalized cusp to allow for virtually solvable fundamental group, and we produce the first such example with non-virtually nilpotent fundamental group. Along with a companion paper, this allows to build strictly convex cusps and generalized cusps whose fundamental group is any finitely generated virtually nilpotent group. This also has interesting consequences for the theory of relatively Anosov representations.

In this paper, we study 6-dimensional GKM manifolds with 4 fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex projective space C P^3 with standard complex structure (P2) blow up of S^6 at a fixed point, diffeomorphic to C P^3 (P3) C P^3 as the homogeneous space Sp(2)/(U(1) times Sp(1)) with non-standard almost complex structure (Q1) complex quadric Q_3 with standard complex structure (Q2) blow up of S^6 along isotropy 2-sphere, diffeomorphic to Q_3 (S) S^2 times S^4, obtained as equivariant gluing along orbits of two S^6's

We consider Proca field perturbations in a five-dimensional Schwarzschild-anti-de Sitter (Schwarzschild-AdS_{5}) black hole geometry. Using the vector spherical harmonic (VSH) method, we show that the Proca field decomposes into scalar-type and vector-type components according to their tensorial behavior on the three-sphere. Two degrees of freedom of the field are described by scalar-type components, which are coupled due to the mass term, while the remaining two degrees of freedom are described by a vector-type component, which decouples completely. Motivated by the Frolov-Krtouš-Kubizňák-Santos (FKKS) ansatz in the limit of zero spin, we use a field transformation to decouple the scalar-type components at the expense of introducing a complex separation parameter β. This parameter can be determined analytically, and its values correspond to two distinct polarizations of the scalar-type sector: "electromagnetic" and "non-electromagnetic", denoted by β_{+} and β_{-}, respectively. In the scalar-type sector, the radial differential equation for each polarization is a Fuchsian differential equation with five singularities, whereas in the vector-type sector, the radial equation has four singularities. By means of the isomonodromy method, we reformulate the boundary value problem in terms of the initial conditions of the Painlevé VI τ function and, using a series expansion of the τ function, we compute the scalar-type and vector-type quasinormal modes (QNMs) in the small horizon limit. Our results are in overall very good agreement with those obtained via the numerical integration method. This shows that the isomonodromy method is a reliable method to compute quasinormal modes in the small horizon limit with high accuracy.

This review discusses the recently proposed fuzzy sphere regularization for studying 2+1D critical phenomena, particularly three-dimensional (3D) conformal field theory (CFT). The fuzzy sphere scheme not only offers remarkable efficiency in extracting extensive CFT data at low computational cost but also reveals unexpected connections among 3D CFT (critical phenomena), noncommutative geometry, and the quantum Hall effect. We introduce the fundamental ideas of fuzzy sphere regularization, emphasizing its role in demonstrating the state-operator correspondence of 3D CFTs on the S^2 times R geometry. Additionally, we review key developments in this approach across various directions and outline potential future applications.

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 complete theory of null Cartan normal helices in Minkowski space-time E^4_1 is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit C-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in E^4_1 is established.

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 study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions 3leq nleq 14. We show that this dimensional threshold n_*=14 is sharp by constructing in ngeq 15 nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, n_*^{LCF}=18, and the metrics with umbilic boundary, n_*^{umb} =20. In all three settings we also prove that the Weyl--umbilicity vanishing conjecture holds for 4leq nleq n_* and construct counterexamples in n>n_* by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe--Escobar equation, outside the conformal class of the round hemisphere.

We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by g(t) satisfies in dimension ngeq 3 $g(t)=c_n t |log t|^{-1{n - 2}} (1 + o(1)), tto 0,$ and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are Z/2-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin^--structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.

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.

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.

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

Single-minus tree-level n-gluon scattering amplitudes are reconsidered. Often presumed to vanish, they are shown here to be nonvanishing for certain "half-collinear" configurations existing in Klein space or for complexified momenta. We derive a piecewise-constant closed-form expression for the decay of a single minus-helicity gluon into n-1 plus-helicity gluons as a function of their momenta. This formula nontrivially satisfies multiple consistency conditions including Weinberg's soft theorem.

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

The tensor rank of a bilinear map is the least number of multiplications any bilinear algorithm needs to compute it; for the multiplication of an algebra it measures how cheaply the algebra can be multiplied at all. For the even-dimensional real normed division algebras it is 3 for the complex numbers and 8 for the quaternions, both classical, while for the octonions O only a range was known: at least 15 (Fiduccia and Zalcstein, 1977) and at most 30 (Cariow and Cariowa). We prove $18 le R_{R}(T_{O}) le 25. The lower bound peels the eight slices of T_{O} down to two and bounds the rank of the surviving pencil through the octonion norm. Nothing in it is special to dimension 8: the same steps give R_{R}(T_A) \ge 5{2}n - 2 for every real normed division algebra A of even dimension n, sharp for C and H and the best bound we know for O. The upper bound is a separate construction, an explicit rank-25 decomposition certified by a Krawczyk argument, in exact rational arithmetic, to sit within 10^{-6} of an exact one. The same two arguments pin down the rank of a smaller three-slice quaternion tensor τ, giving R_{R}(τ) = 7$. The Lean 4 kernel checks the lower bounds and the Krawczyk existence principle; the accompanying scripts check the certificate's finitely many exact-rational inequalities.

In the 3-3-1 model with right-handed neutrinos, three triplets of scalars engender the correct sequence of symmetry breaking, SU(3)_C times SU(3)_L times U(1)_X rightarrow SU(3)_C times SU(2)_L times U(1)_Y rightarrow SU(3)_C times U(1)_{EM}, generating mass for all fermions, except neutrinos. Tiny neutrino masses may be achieved by adding one sextet of scalars to the original scalar content. As consequence, it emerges a very complex scalar sector, involving terms that violate lepton number explicitly, too. The main obstacle to the development of the phenomenology of such scenario is the knowledge of its spectrum of scalars since, now, there are 15 massive scalar particles on it. The proposal of this work is to do an exhaustive analysis of such scalar sector with lepton number being explicitly violated at low, electroweak and high energy scales by means of trilinear terms in the potential. The first case can be addressed analytically and, as a nice result, we have observed that the scalar content of such case is split into two categories: One belonging to the 331 energy scale and the other belonging to the EWSB energy scale, with the last recovering the well known THDM+triplet. For the other cases, the scalar sector can be addressed only numerically. Hence, we proposed a very general approach for the numerical study of the potential, avoiding simplifications that can make us reach conclusions without foundation. We show that, in the case of lepton number being explicitly violated at electroweak scale, it is possible to recover the same physics of the THDM+triplet, as the previous case. Among all the possibilities, we call the attention to one special case which generates the 3HDM+triplet scenario. For the last case, when lepton number is violated at high energy scale, the sextet become very massive and decouples from the original scalar content of the 3-3-1 model.

For any given dimension d, all reflexive d-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of (d+1)-tuples of integers (weights), or combinations of k-tuples of weights with k<d+1. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge numbers.

We study the pseudospectrum of the Kerr black hole for gravitational perturbations with spin weight s=2, the case of greatest interest in gravitational-wave astronomy. In the hyperboloidal framework with two-dimensional Chebyshev collocation, we formulate the quasinormal mode problem as a non-self-adjoint eigenvalue problem. For sneq0 the fields Ψ^{(s)} and Ψ^{(-s)} are independent, so the conserved current method yields no single-field energy norm. We overcome this with the Teukolsky--Starobinsky identities, which relate Ψ^{(-2)} to a fourth-order operator on Ψ^{(2)} and yield an H^2 energy norm after symmetrization. With the H^1 norm known for s=0, this suggests the pattern H^{|s|}, verified for s=0 and s=2. All results use an ω-independent two-field hierarchy. We find that the pseudospectrum tightens by 1.1 to 3.5 decades at each step of L^2to H^1to H^2, yet stays large in every norm. The s=+2 and s=-2 operators are isospectral but not isopseudospectral: at twelve of fourteen probes the ψ_0 resolvent exceeds the ψ_4 one by 0.19 to 0.82 decades in the H^2 norm, while at the two probes furthest along the QNM line the ordering reverses by 0.18 to 0.19 decades. The gravitational pseudospectrum exceeds the scalar one by 0.5 to 5.2 decades in the window minimum and by 5.9 decades in L^2 at ωM=0.47-0.17,i, while along the damping direction the s=2 resolvent is non-monotonic and the s=0 resolvent smooth. QNM spectral instability is therefore a universal feature of Kerr, and is stronger for gravitational perturbations.

In this paper we study the behavior of geodesics on cones over arbitrary C^3-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for cone generatrices. This investigation is inspired by our results on billiards inside cones over manifolds where similar results hold true.

For every integer Nge4, we construct a fixed smooth, non-locally-conformally-flat metric on the n-dimensional unit sphere S^n for which the constant Q_{2N}-curvature equation admits an L^infty-unbounded sequence of positive solutions. The construction applies for nge2N+20 when N=4,5, nge2N+19 when 6le Nle8, nge2N+18 when 9le Nle17, and nge2N+17 when Nge18. For each L in N cup {0}, the metric can be chosen arbitrarily close to the round metric in the C^L norm. Together with the known cases N=1,2,3, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total Q_{2N}-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying Ric=(n-1)g. Building on Juhl's formulas, we identify this Hessian, for every NinN and n>2N, as a degree-N polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real n, this matrix changes from negative definite to indefinite at n_N=2N+a_*+c_*N^{-1}+O(N^{-2}) as Ntoinfty, with a_*approx16.201871 and c_*approx13.512880, explaining the eventual bound nge2N+17.

We develop a direct paradifferential approach to nonlinear graph integral operators arising in contour dynamics. The main methodological result is an abstract structure-preserving paralinearization theorem, valid in arbitrary dimension, for a class of homogeneous kernels whose nonlinear dependence on the graph is expressed through normalized finite differences. The theorem covers both critical principal-value kernels and subcritical locally integrable kernels, gives explicit formulas for the principal symbols, and separates the finite-order symbolic contribution generated by the homogeneous singularity from order-zero far-field terms and arbitrarily smoothing remainders. A structural feature of the reduction is the cancellation of the intermediate symbolic order; at the critical endpoint the construction also preserves the exact skew-adjoint structure of the original singular integral. As a main application, we consider the three-dimensional two-phase free-boundary Euler equations with constant interfacial background vorticity. We first derive an autonomous contour-dynamics equation in canonical surface variables and then apply the abstract theorem directly to the singular integral operators generated by the Birkhoff--Rott formulation. This yields the complete paradifferential structure of the system without taking the Dirichlet--Neumann reduction as a starting point. After introducing an Alinhac good unknown and diagonalizing the resulting system, we obtain local well-posedness for small Sobolev perturbations in the stable Kelvin--Rayleigh--Taylor regime, uniformly on compact subsets of the stable parameter set.

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.

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 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 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.

Motivated by the peculiar features of the trans-Neptunian object (TNO) Máni and the discovery of ring systems around Centaurs and trans-Neptunian bodies, we investigate the stability of particles around a small irregular body hosting a deep equatorial crater. This study is the third contribution in a sequence devoted to the dynamics around non-spherically symmetric bodies. Using Máni as a reference model, whose crater depth exceeds 10% of its radius, we map the system stability through complementary methods: Poincaré surfaces of section (PSS), survival maps, and the finite-time Lyapunov exponent (FTLE). In the nominal case, the 1!:!1, 2!:!1, 3!:!1, and 4!:!1 spin-orbit resonances (SORs) are consistently identified by all techniques. The prominent 3!:!1 SOR exhibits a bifurcated structure at high eccentricities, matching structural transitions observed at lower Jacobi constant values in the PSSs. Increasing the rotation rate (λ) and crater mass ratio (μ) enlarges resonance widths and promotes overlap, making the 2!:!1 and 4!:!1 SORs progressively dominant. Lower rotation rates allow stable trajectories to persist at higher eccentricities, whereas the 1!:!1 SOR is highly sensitive to stronger perturbations and disappears in the most extreme cases. In contrast to mass-anomaly models, which can clear the region interior to the 2!:!1 SOR, the crater configuration considered here preserves stable regions in this vicinity, suggesting alternative scenarios for the formation and maintenance of debris structures around irregular minor bodies.

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.

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.

Symmetry has long provided the organizing principle for understanding how materials respond to external perturbations. Here we show that what is permitted by symmetry need not be what actually emerges from the underlying system, and this distinction becomes particularly consequential for symmetry-related gapless Dirac fermions such as in altermagnets. A perturbation that shifts the Dirac points drives a Lifshitz reconstruction and generates an intrinsically nonanalytic effective potential, giving rise to undiscovered singular responses. In Dirac altermagnets, this nonanalyticity acquires a magnetic manifestation through C-paired spin-polarized Dirac cones: breaking the pairing symmetry C lifts their degeneracy and converts the associated Lifshitz reconstruction into an itinerant magnetization. As a consequence, piezomagnetism, the induction of a net magnetization by a mechanical strain, becomes nonlinear and nonanalytic, with Mproptosgn(varepsilon)varepsilon^2, whereas the linear response is absent despite being allowed by symmetry. Accordingly, the leading response cannot be inferred from the conventional symmetry-based paradigm of material response built upon the cornerstones of Neumann's principle and analytic Landau expansion. First-principles calculations identify the monolayer Fe_2S as a candidate material realization for such singular magnetic responses. These results suggest quantum singularities as a new degree of freedom for engineering material responses.

In this note we construct large ensembles of supersymmetry breaking solutions arising in the context of flux compactifications of type IIB string theory. This class of solutions was previously proposed in arXiv:hep-th/0402135 for which we provide the first explicit examples in Calabi-Yau orientifold compactifications with discrete fluxes below their respective tadpole constraint. As a proof of concept, we study the degree 18 hypersurface in weighted projective space CP_{1,1,1,6,9}. Furthermore, we look at 10 additional orientifolds with h^{1,2}=2,3. We find several flux vacua with hierarchical suppression of the vacuum energy with respect to the gravitino mass. These solutions provide a crucial stepping stone for the construction of explicit de Sitter vacua in string theory. Lastly, we also report the difference in the distribution of W_0 between supersymmetric and non-supersymmetric minima.

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from oriented matroid theory which we show are both necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope and invariant under the polytope's orientation-preserving symmetries (SL(d,Z) ltimes Z^d). When implemented with a certain masking procedure, one can also guarantee that every rollout produces a fine triangulation (in 2D). On unseen polygons with N_pts leq 40, dualGNN is the most uniform FRT sampler we tested, and even a model trained on a single polygon generalizes well to other polygons. The model is small (sim92k parameters), trains in sim7.5 hours on a single consumer GPU, and runs without modification on an M1 MacBook Pro. We apply dualGNN to string theory, uniformly sampling Calabi-Yau threefolds at h^{1,1}=86 and consistent with uniformity at h^{1,1}=128. This is an order of magnitude beyond previous learned methods with a model sim1000times smaller. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN .

We develop a modular wall crossing theory for quasimaps to a projective variety, allowing independent variation of the boundary coefficients and the quasimap weight. Building on the K-stability of quasimaps introduced by Hashizume and the first author, we construct projective moduli spaces in the stable, Calabi--Yau, and log Fano regimes, together with wall crossing morphisms. A central construction is the moduli theory of boundary polarized Calabi--Yau quasimaps, which retains an ample polarization at the numerically trivial locus and allows comparison with suitable perturbations toward the stable and log Fano regions. The stable theory applies in arbitrary genus, while the comparisons through the Calabi--Yau locus concern genus zero. For degree-one boundary divisors, the resulting framework relates weighted stable maps and quasimaps to Hassett spaces and GIT quotients of weighted points on mathbb P^1. In a companion paper, we apply this framework to give a modular interpolation between Miranda's GIT compactification of rational elliptic surfaces and the Baily--Borel compactification of an eight-dimensional ball quotient.

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.

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 investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, (varepsilon,d,σ_0), a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity ν=pm1/2 and the Berry phase γ=π for loops encircling the EL, while ν=0 and γ=0 for those do not encircle the EL. Through matrix perturbation theory, we prove that the ε-pseudospectrum contour size scales as ε^{1/q} at an EP, where q is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear ε scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.

In our prior work toward Bartnik's static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm large classes of boundary hypersurfaces are static regular. In this note, we further improve some of those prior results. Specifically, we show that any hypersurface in an open and dense subfamily of a certain general smooth one-sided family of hypersurfaces (not necessarily a foliation) is static regular. The proof uses some of our new arguments motivated from studying the conjecture for boundary data near an arbitrary static vacuum metric.

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of E(p, q)-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces R^{p,q}. They cover, for instance, E(3)-equivariance on R^3 and Poincaré-equivariance on Minkowski spacetime R^{1,3}. Our approach is based on an implicit parametrization of O(p,q)-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

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.

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.

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).

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally projecting a convex set onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge-Ampere equation.

We propose a 1+1 dimensional CFT dual structure for quantum gravity and matter on the extended 2+1 dimensional BTZ black hole, realized as a quotient of the Poincare patch of AdS_3. The quotient spacetime includes regions beyond the singularity, "whiskers", containing timelike and lightlike closed curves, which at first sight seem unphysical. The spacetime includes the usual AdS-asymptotic boundaries outside the horizons as well as boundary components inside the whiskers. We show that local boundary correlators with some endpoints in the whisker regions: (i) are a protected class of amplitudes, dominated by effective field theory even when the associated Witten diagrams appear to traverse the singularity, (ii) describe well-defined diffeomorphism-invariant quantum gravity amplitudes in BTZ, (iii) sharply probe some of the physics inside the horizon but outside the singularity, and (iv) are equivalent to correlators of specific non-local CFT operators in the standard thermofield entangled state of two CFTs. In this sense, the whisker regions can be considered as purely auxiliary spacetimes in which these useful non-local CFT correlators can be rendered as local boundary correlators, and their diagnostic value more readily understood. Our results follow by first performing a novel reanalysis of the Rindler view of standard AdS/CFT duality on the Poincare patch of AdS, followed by exploiting the simple quotient structure of BTZ which turns the Rindler horizon into the BTZ black hole horizon. While most of our checks are within gravitational effective field theory, we arrive at a fully non-perturbative CFT proposal to probe the UV-sensitive approach to the singularity.

Fix a strip width \(ρ>0\), a Diophantine frequency \(α\), an integer \(N\ge2\), and \(λ>1\). For the family \[ V_{\varepsilon,δ}(x) =2λ\cos(2πN x)+\varepsilonδ(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(ρ\) perturbations such that every \(δ\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(δ)>0\) for which \[ gathered Leb\left\{ E\inΣ(V_{\varepsilon,δ},α): ω(α,E;V_{\varepsilon,δ})=s \right\}>0, 0<\varepsilon<\varepsilon_0(δ), \qquad s=1,\ldots,N. gathered \] The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves \(s\) pairs of simple zeros on a positive-measure spectral set.

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn--Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of 10^{-9} and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between 0.966 and 0.976, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within 0.07^circ, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

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 study holomorphic foliations on normal crossings varieties arising as semistable degenerations. We do so by we exploring the notion of foliated d-semistability using the language of logarithmic structures in the sense of Fontaine-Illusie. First, we identify both local and global obstructions to d-semistability. In order to analyze the existence of smoothings, we develop a logarithmic deformation theory of foliations and show that the corresponding moduli functor admits a versal hull.

We establish a charge-regular formulation of separated Kerr--Newman perturbations and an explicit map from master solutions to physical radiative fields. We derive regular angular and radial first-order systems and their second-order reductions, with finite-charge deformations of the spin pm1 and pm2 Teukolsky equations. The field and boundary normalizations recover the standard Kerr conventions while accounting for the nonuniform zero-charge limit at infinity. For physical Einstein--Maxwell perturbations with the usual peeling behavior, we obtain explicit outgoing gravitational and electromagnetic amplitudes and energy fluxes at nonzero real frequency. The resulting finite-charge radiation-composition ratios depend only on the regular angular functions, independently of the radial excitation.

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).

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.

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.

By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip-Fisher-Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively s-curved magnetic system on a closed real-analytic manifold has only finitely many closed totally s-magnetic hypersurfaces, unless the magnetic 2-form is trivial and the underlying metric is hyperbolic.

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A,B,C,D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.

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.

In this paper, we consider a critical Grushin-type problem with double potentials. By applying the reduction argument and local Pohozaev identities, we construct a new family of solutions to this problem, which are concentrated at points lying on the top and the bottom circles of a cylinder.

In this review, we discuss the relevance and impact of studying Calabi-Yau threefolds in the context of global model building in string phenomenology. First, taking a phenomenologist-friendly approach, we review how the topologies of the various divisors and curves of the compactifying CY threefolds play a crucial role for generating the various ``suitable" classes of effective scalar potentials, within the framework of the popular moduli stabilization schemes such as KKLT and LVS. Subsequently, we discuss the impact of the specifics of the CY threefold geometries in the minimal LVS inflationary models such as fibre inflation, in particular, along the challenges such as the inflaton field-range bound. In this regard, we discuss a multi-field approach in which several fibre moduli assist to drive successful inflation having a sufficient number of efolds, without getting close to their individual Kähler cone boundaries.

The static tidal Love numbers of black holes vanish in general relativity (GR), but are generically non-zero, and in fact 'run' logarithmically with scale, in more general theories or beyond the static limit. We expose a model-agnostic framework for the running of Love numbers at the dynamical level, valid for any tidal perturbation on any static, spherically symmetric, asymptotically flat background, based on the near-zone expansion in powers of the frequency ω and on the monodromy at spatial infinity. A master recurrence formula determines the monodromy generator to all orders in ω^2, whose three independent entries fully encode the running of the Love numbers. This structure explains the quadratic nature of the renormalization group equation, here proved to all orders and for any type of Love number. We discuss the scheme dependence of the logarithmic coefficients, which is made transparent by the monodromy form. In particular, this clarifies that the anomalous exponents of the tidal field, the so-called 'renormalized angular momentum' of the Mano-Suzuki-Takasugi formalism, are scheme-invariant and directly computable on any spacetime, without need of explicit solutions. We discuss several applications to black holes, including the derivation of a closed-form formula for the leading dynamical running of axial Love numbers of Reissner-Nordström black holes, proving in particular that it vanishes in the extremal limit. For generic black holes beyond GR, we prove that dynamical running probes deformations of the metric at lower multipoles than the static one, and we illustrate the power of the method by deriving some exact results for the Hayward and Bardeen regular black holes.

We establish the existence of compactly supported solutions of the inviscid incompressible 2D Boussinesq equation with C^{1,frac{4{3}}-1-varepsilon}cap L^{2} force that develop a singularity in finite time. Importantly, the force preserves this regularity at the blow-up time. Moreover, the forces in the vorticity and density equations have compact support. The mechanism behind the blow-up is an accumulated hysteresis effect on the vorticity caused by an infinite chain of "degenerate" pendula and flickering density.

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.

We establish a unified framework for understanding the results on the Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes--Consani (2023), and Connes--Consani--Moscovici (2025+) from the perspective of the screw function introduced in Suzuki (2023). An advantage of the approach via the screw function is that it provides a method to study the Weil quadratic form, which is originally defined in terms of distributions, by means of continuous functions. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function can be obtained as the limit, as a to infty, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval [-a,a]. All these results are obtained without assuming the Riemann Hypothesis. This conjecture may be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the Riemann zeta function.

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.

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.

The paper studies the quantum action for the five-dimensional real φ^3-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The explicit form of the first three coefficients for the renormalization constants is presented. The absence of non-local singular contributions and partial results for the fourth correction are discussed.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

A set of infinitesimal {rm Virasoro_{,L}}rm Virasoro_{,R} diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of `Wald-Zoupas' counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges c_L=c_R=12J. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.

We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity k within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" ρ(k) (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that ρ(k) declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts ρ(k) by a factor of sim3 and produces or exacerbates a clear peak at k=6. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.

In this paper, we investigate the quantum behaviour near the classical singularity of a black hole. We describe the black-hole interior by the Kantowski-Sachs metric, with a string propagating in this region. This approach is motivated by the fact that, near the classical singularity, spacetime curvature reaches Planck or string scales, necessitating a description in terms of string-theoretic degrees of freedom. The string dynamics is formulated via the Polyakov action, and the combined system is canonically quantized to derive the corresponding Wheeler-DeWitt equation. Although the equation is separable in the radial and azimuthal string sectors, the remaining geometric and string degrees of freedom are nontrivially coupled. To address this, we employ a perturbative approach and obtain the wave function up to first order. At the zeroth-order, the angular part is described by the general Heun function, while the geometric part is expressed in terms of the Bessel function. The first-order wave function is governed by a source term containing the zeroth-order solution, and we obtain its explicit expression in terms of the complementary function and the particular integral. We confirm regularity of the wave function at both zeroth and first order, at the singularity as well as at the horizon. Regularity at the horizon imposes a discretization condition on one eigenvalue, while another eigenvalue is quantized due to the azimuthal periodicity of the wave function. Importantly, both zeroth- and first-order solutions satisfy the DeWitt boundary condition, implying resolution of the black hole singularity within this quantum framework.

We present deep observations in targeted regions of the string landscape through a combination of analytic and dedicated numerical methods. Specifically, we devise an algorithm designed for the systematic construction of Type IIB flux vacua in finite regions of moduli space. Our algorithm is universally applicable across Calabi-Yau orientifold compactifications and can be used to enumerate flux vacua in a region given sufficient computational efforts. As a concrete example, we apply our methods to a two-modulus Calabi-Yau threefold, demonstrating that systematic enumeration is feasible and revealing intricate structures in vacuum distributions. Our results highlight local deviations from statistical expectations, providing insights into vacuum densities, superpotential distributions, and moduli mass hierarchies. This approach opens pathways for precise, data-driven mappings of the string landscape, complementing analytic studies and advancing the understanding of the distribution of flux vacua. This allows us to obtain different types of solutions with hierarchical suppressions, e.g.~vacua with small values of the Gukov-Vafa-Witten superpotential |W_0|. We find an example with |W_0| = 5.547 times 10^{-5} at large complex structure, without light directions and the use of non-perturbative effects.

For a real quadratic field Q(d), we study the norm-form energy N = S_ζ^2 - d cdot S_L^2, where S_ζ and S_L are Lorentzian-weighted zero sums with w(ρ) = 2/(1/4 + γ^2). We prove three main results. (1) Spacelike spectral data: N < 0 unconditionally for all squarefree d > 1, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level M, dens{N > 0} leq 2|f_{S_L^{(M)}}|_infty cdot (W_1(ζ)/d + ε_M), established unconditionally via Jacobi--Anger resonance analysis. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice Λ_infty has finite rank (verified for M leq 20, where rank = 0), the sharp asymptotic dens{N > 0} = C(d)/d + o(1/d) holds. For d = 5, C(5) = 2,f_{S_L}(0)cdotE[|S_ζ|] = 0.1191; the constant depends on d through the zeros of L(s,χ_d), and C(d) = O(1/log d) as d to infty.

In this paper, we sharpen results obtained by the author in 2023. The new results reduce the Mathieu Conjecture on SU(N) (formulated for all compact connected Lie groups by O. Mathieu in 1997) to a conjecture involving only functions on R^ntimes (S^1)^m with n,m non-negative integers instead of involving functions on R^ntimes (S^1setminus{1})^m. The proofs rely on a more recent work of the author (2024) and a specific KAK decomposition. Finally, with these results we can also improve the results on the groups Sp(N) and G_2 in the latter paper, since they relied on the construction introduced in the 2023 paper.

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

We present the analytic evaluation of the gravitational energy and of the angular momentum flux with tidal effects for inspiraling compact binaries, at next-to-next-to-leading post-Newtoian (2PN) order, within the effective field theory diagrammatic approach. We first compute the stress-energy tensor for a binary system, that requires the evaluation of two-point Feynman integrals, up to two loops. Then, we extract the multipole moments of the system, which we present for generic orbits in center-of-mass coordinates, and which are needed for the evaluation of the total gravitational energy and the angular momentum flux, for generic orbits. Finally, we provide the expression of gauge invariant quantities such as the fluxes, and the mode amplitudes and phase of the emitted gravitational wave, for circular orbits. Our findings are useful to update earlier theoretical studies as well as related phenomenological analyses, and waveform models

We study complex saddle points of the worldline path integral for massive charged particles in two-dimensional maximally symmetric backgrounds threaded by homogeneous electric fields. Within the worldline formalism, we show that the semiclassical saddles generically require complexifying both the target spacetime and the proper-time parameter. The resulting solutions comprise two slices of a single complex family, distinguished by the direction of proper-time evolution. The one evolving in imaginary proper time carries an action that typically controls the exponential suppression of non-perturbative effects, while that evolving in real proper time may encode additional complex phases responsible for quantum interference between saddles. Treating this family as a whole, we give a prescription for constructing and classifying all topologically inequivalent instanton sectors, and we find that the worldline action depends only on their topological data, namely winding numbers around the compact directions and certain isolated punctures. Moreover, in Anti-de Sitter spacetime, the qualitative structure of the solutions is governed by the dimensionless ratio R |eE|/m (R being the curvature radius), which distinguishes sub-extremal, extremal, and super-extremal regimes. By contrast, no analogous distinction occurs in de Sitter---where the instantons are insensitive to this ratio---or in Minkowski spacetime, which lacks a curvature scale altogether. We also recast the action as a sum over residues of the relevant (non-conserved) conjugate momentum, making transparent which analytic data control each contribution. Finally, we discuss the subtleties that arise when solutions built in various local patches are lifted to global (Anti-)de Sitter space. These impose further necessary conditions on which saddles can in principle contribute to the one-loop path integral.

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in GI. In GI, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szego-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L'Hopital's rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. {For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner.} The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results.

In this paper, we show that for any finite subgroup Γ< O(4) acting freely on S^3, there exists a 4-dimensional complete Riemannian manifold (M,g) with {rm Ric}_g geq 0 , such that the asymptotic cone of (M,g) is C(S_δ^3 /Γ) for some δ= δ(Γ) >0. This answers a question of Bruè-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of 4-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bruè-Pigati-Semola [arXiv:2405.03839], one can classify the 4-dimensional non-collapsed tangent cone in the topological sense.

Solving the strong CP problem with spontaneous CP violation (SCPV) in SUSY requires that the constant term of the superpotential and the SUSY-breaking F-term be real to high accuracy. When both are generated dynamically, by gaugino condensation and by dynamical SUSY breaking (DSB), their phases need not vanish even if every theta-angle vanishes and every coupling is real; e.g., the multivalued non-perturbative superpotential may sit on a branch with a complex phase. In fact, gaugino condensation, in pure SUSY Yang-Mills theory and in its Affleck-Dine-Seiberg generalization, is safe only for N_c - N_f in {1,2} (N_c = 2 for pure SYM); otherwise the phase of the constant term reaches the gluino mass through anomaly mediation. The SUSY-breaking F-term is the main source of the MSSM soft masses, including the gluino mass, so its phase reaches theta-bar as well. In the framework of global SUSY, the 3-2 model introduces no phase. The ISS model is safe for non-degenerate quark masses in that sector; in the degenerate (or nearly degenerate) limit, however, a complex vacuum orientation can generate an O(1) phase. The IYIT model can generate a purely imaginary F-term: for roughly half of the real Yukawa parameter space, the meson VEV is purely imaginary and CP is spontaneously broken. In all three, the vacuum of the field coupled to the messengers is aligned with the constant term, so a complex constant term generates a strong CP phase through a second route, separate from anomaly mediation.

We compute the elliptic genera of two-dimensional N=(2,2) and N=(0,2) gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi-Yau, N=(2,2) SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric N=(0,2) model.

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of R^n, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a W^{2,infty} critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n-p_0, for some p_0 in (2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

We identify seven new 192-periodic infinite families of elements in the 2-primary stable homotopy groups of spheres. Although their Hurewicz image is trivial for topological modular forms, they remain nontrivial after T(2)- as well as K(2)-localization. We also obtain new information about 2-torsion and 2-divisibility of some of the previously known 192-periodic infinite families in the stable stems.

Ap\'{e}ry-type (inverse) binomial series have appeared prominently in the calculations of Feynman integrals in recent years. In our previous work, we showed that a few large classes of the non-alternating Ap\'ery-type (inverse) central binomial series can be evaluated using colored multiple zeta values of level four (i.e., special values of multiple polylogarithms at fourth roots of unity) by expressing them in terms of iterated integrals. In this sequel, we shall prove that for several classes of the alternating versions we need to raise the level to eight. Our main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of π. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this work, we are interested in determining all the rational spherical triangles. We introduce a conjecture on the solutions to a trigonometric Diophantine equation. An implication of the conjecture is that the only rational spherical triangles are the ones given by Little and Coxeter. We prove some partial results towards the conjecture.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a continuous piecewise linear function with exactly \(n\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(n\) when \(F\) has exactly \(n\) fold points. The conjecture was confirmed for \(n=1\) and \(n=2\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \(n\ge3\) remained open. In this paper, we show that the maximal number of limit cycles is at least \(|3n-4|\) for \(n\in\mathbb N^+\), thereby disproving Tonnelier's conjecture for \(n\ge3\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \(2^{28(n+1)^2}\).

We develop a novel systematic method for constructing exact, static, and asymptotically flat black-hole solutions in shift-symmetric and Z_2-symmetric beyond Horndeski theories. In contrast to previous studies, our approach does not rely on the restrictive reciprocal condition g_{tt}g_{rr}=-1, allowing for more general geometries characterized by two independent metric functions. The resulting solution space contains two qualitatively distinct types of hair originating from different branches of the theory. One branch is characterized by khronometric hair, while the other by genuine scalar hair. In both, the scalar profile defines a nontrivial preferred foliation, but the shift-symmetry Noether current and the associated propagating degree of freedom are trivial for the former and nonvanishing for the latter. Focusing mainly on the second branch, we derive several distinct families of exact black-hole solutions. In general, these solutions are singular and exhibit primary scalar hair associated with an independent scalar charge. Nevertheless, we demonstrate that they can be regularized to yield regular black-hole or solitonic configurations, for which the scalar hair becomes secondary. We further investigate the stability properties of these solutions under linear perturbations by analyzing the axial sector and the monopole (ell=0) mode of the even-parity sector. For representative solutions with scalar hair, the perturbative analysis provides evidence for linear stability under the modes considered.

Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture that fuses multiple TNNs into a larger network, intended to solve a broader class of problems than a single TNN. I evaluate this architecture, referred to as a "stacked tensorial neural network" (STNN), on a parametric PDE with three independent variables and three parameters. The three parameters correspond to one PDE coefficient and two quantities describing the domain geometry. The STNN provides an accurate reduced-order description of the solution manifold over a wide range of parameters. There is also evidence of meaningful generalization to parameter values outside its training data. Finally, while the STNN architecture is relatively simple and problem agnostic, it can be regularized to incorporate problem-specific features like symmetries and physical modeling assumptions.

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.

Recent years have witnessed considerable achievements in facial avatar reconstruction with neural volume rendering. Despite notable advancements, the reconstruction of complex and dynamic head movements from monocular videos still suffers from capturing and restoring fine-grained details. In this work, we propose a novel approach, named Tri^2-plane, for monocular photo-realistic volumetric head avatar reconstructions. Distinct from the existing works that rely on a single tri-plane deformation field for dynamic facial modeling, the proposed Tri^2-plane leverages the principle of feature pyramids and three top-to-down lateral connections tri-planes for details improvement. It samples and renders facial details at multiple scales, transitioning from the entire face to specific local regions and then to even more refined sub-regions. Moreover, we incorporate a camera-based geometry-aware sliding window method as an augmentation in training, which improves the robustness beyond the canonical space, with a particular improvement in cross-identity generation capabilities. Experimental outcomes indicate that the Tri^2-plane not only surpasses existing methodologies but also achieves superior performance across quantitative and qualitative assessments. The project website is: https://songluchuan.github.io/Tri2Plane.github.io/.

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.

We systematically explore the space of renormalization group flows of four-dimensional N=1 superconformal field theories (SCFTs) triggered by relevant deformations, as well as by coupling to free chiral multiplets with relevant operators. In this way, we classify all possible fixed point SCFTs that can be obtained from certain rank 1 and 2 supersymmetric gauge theories with small amount of matter multiplets, identifying 7,346 inequivalent fixed points which pass a series of non-trivial consistency checks. This set of fixed points exhibits interesting statistical behaviors, including a narrow distribution of central charges (a, c), a correlation between the number of relevant operators and the ratio a/c, and trends in the lightest operator dimension versus a/c. The ratio a/c of this set is distributed between 0.7228 and 1.2100, where the upper bound is larger than that of previously known interacting SCFTs. Moreover, we find a plethora of highly non-perturbative phenomena, such as (super)symmetry enhancements, operator decoupling, non-commuting renormalization group flows, and dualities. We especially identify amongst these fixed points a new SCFT that has smaller central charges (a, c) = (633{2000},683{2000}) than that of the deformed minimal Argyres-Douglas theory, as well as novel Lagrangian duals for certain N=1 deformed Argyres-Douglas theories. We provide a website https://qft.kaist.ac.kr/landscape to navigate through our set of fixed points.

With the aim of testing massive gravity in the context of black hole physics, we investigate the gravitational radiation emitted by a massive particle plunging into a Schwarzschild black hole from slightly below the innermost stable circular orbit. To do so, we first construct the quasinormal and quasibound resonance spectra of the spin-2 massive field for odd and even parity. Then, we compute the waveforms produced by the plunging particle and study their spectral content. This allows us to highlight and interpret important phenomena in the plunge regime, including (i) the excitation of quasibound states, with particular emphasis on the amplification and slow decay of the post-ringdown phase of the even-parity dipolar mode due to harmonic resonance; (ii) during the adiabatic phase, the waveform emitted by the plunging particle is very well described by the waveform emitted by the particle living on the innermost stable circular orbit, and (iii) the regularized waveforms and their unregularized counterparts constructed from the quasinormal mode spectrum are in excellent agreement. Finally, we construct, for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summing over partial waveforms.

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.

In this paper, we establish effective equidistribution of transverse intersection points between properly immersed totally geodesic submanifolds of complementary dimensions in a finite-volume hyperbolic manifold with respect to the hyperbolic volume measure, as the volume of the submanifolds tends to infinity.