Letters in Mathematical Physics最新文献

Topological string theory partition function gives rise to Gromov–Witten invariants, Donaldson–Thomas invariants and 5D BPS indices. Using the remodeling conjecture, which connects Topological Recursion with topological string theory for toric Calabi–Yau threefold, we study a more direct connection for the subclass of strip geometries. In doing so, new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal x–y duality. Through these techniques, our main result in this paper is a direct derivation of all free energies from topological recursion for general strip geometries. In analyzing the expression of free energy, we shed some light on the meaning and the influence of the x–y duality in topological string theory and its interconnection to GW and DT invariants as well as the 5D BPS index.

This paper revisits the equivalence problem between algebraic quantum field theories and prefactorization algebras defined over globally hyperbolic Lorentzian manifolds. We develop a radically new approach whose main innovative features are 1.) a structural implementation of the additivity property used in earlier approaches and 2.) a reduction of the global equivalence problem to a family of simpler spacetime-wise problems. When applied to the case where the target category is a symmetric monoidal 1-category, this yields a generalization of the equivalence theorem from [Commun. Math. Phys. 377, 971 (2019)]. In the case where the target is the symmetric monoidal (infty )-category of cochain complexes, we obtain a reduction of the global (infty )-categorical equivalence problem to simpler, but still challenging, spacetime-wise problems. The latter would be solved by showing that certain functors between 1-categories exhibit (infty )-localizations; however, the available detection criteria are inconclusive in our case.

The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used in explicit computations or approximation schemes. We present a definition of the integral that uses triangulations instead. Our definition is a coordinate–free version of the standard definition of the Riemann integral on (mathbb {R}^n), and we argue that it is the natural definition in the contexts of Lie algebroids, stochastic integration, and quantum field theory, where path integrals are defined using lattices. In particular, our definition naturally incorporates the different stochastic integrals, which involve integration over Hölder continuous paths. Furthermore, our definition is well adapted to establishing integral identities from their combinatorial counterparts. Our construction is based on the observation that, in great generality, the things that are integrated are determined by cochains on the pair groupoid. Abstractly, our definition uses the van Est map to lift a differential form to the pair groupoid. Our construction suggests a generalization of the fundamental theorem of calculus which we prove: the singular cohomology and de Rham cohomology cap products of a cocycle with the fundamental class are equal.

We consider the elliptic Calogero–Inozemtsev system of (textrm{BC}_n) type with five arbitrary constants and propose R-matrix valued generalization for (2ntimes 2n) Takasaki’s Lax pair. For this purpose, we extend the Kirillov’s (textrm{B})-type associative Yang–Baxter equations to similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa–Ueno R-operator and the Komori–Hikami K-operators satisfying the reflection equation. Then, using the Felder–Pasquier construction, the answer for the Lax pair is also written in terms of the Baxter’s 8-vertex R-matrix. As a by-product of the constructed Lax pair we also propose a (textrm{BC}_n) type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.

We analyse the structure of equivalence classes of symmetric quivers whose generating series are equal. We consider such classes constructed using the basic operation of unlinking, which increases the size of a quiver. The existence and features of such classes do not depend on a particular quiver but follow from the properties of unlinking. We show that such classes include sets of quivers assembled into permutohedra, and all quivers in a given class are determined by one quiver of the largest size, which we call a universal quiver. These findings generalise the previous ones for permutohedra graphs for knots. We illustrate our results with generic examples, as well as specialisations related to the knots–quivers correspondence.

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold M as part of the construction of quantum geodesics on the algebra ({mathcal {D}}(M)) of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first-order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein–Gordon operator, with stationary modes being solutions of the Klein–Gordon equation. As an application, we describe gravatom solutions of the Klein–Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.

We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra (mathfrak {osp}(1 vert 2)). More precisely, the quantum group depends on a root of unity (q=e^{frac{2 pi sqrt{-1}}{r}}), where r is a positive integer greater than 2, and the construction applies when r is not congruent to 4 modulo 8. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When r is congruent to (pm 1) or (pm 2) modulo 8, we relate the resulting 3-manifold invariants with physicists’ (widehat{Z})-invariants associated to (mathfrak {osp}(1 vert 2)). Finally, we establish a relation between (widehat{Z})-invariants associated to (mathfrak {sl}(2)) and (mathfrak {osp}(1 vert 2)) which was conjectured in the physics literature.

We study chiral algebra in the reduction of 3D (mathcal {N} = 2 ) supersymmetric gauge theories on an interval with the ({mathcal {N}}=(0,2)) Dirichlet boundary conditions on both ends. By invoking the 3D “twisted formalism” and the 2D (beta gamma )-description, we explicitly find the perturbative (overline{Q}_+) cohomology of the reduced theory. It is shown that the vertex algebras of boundary operators are enhanced by the line operators. A full non-perturbative result is found in the abelian case, where the chiral algebra is given by the rank two Narain lattice VOA, and two more equivalent descriptions are provided. Conjectures and speculations on the non-perturbative answer in the non-abelian case are also given.

In this article, we introduce a modification of the timelike Hausdorff measure (mathcal {V}^N) defined by McCann and Sämann on Lorentzian pre-length spaces. We write the modification of (mathcal {V}^N) as (mathcal {W}^N). We establish volume comparison inequalities by causality preserving and timelike Lipschitz maps for (mathcal {V}^N) and (mathcal {W}^N), and discuss basic properties of both (mathcal {V}^N) and (mathcal {W}^N). Moreover, we show the coincidence of (mathcal {W}^N) and (mathcal {V}^N) on smooth spacetimes and some Lorentzian pre-length spaces, and construct some examples of timelike Lipschitz maps and causality preserving maps.

We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices (Erdős and Riabov in Commun Math Phys 405(12): 282, 2024) and an intricate analysis of the deterministic approximation.