arXiv subjects

Anton A. Gerasimov

Publications and source records attributed to Anton A. Gerasimov.

14 recordsLinked to original sources

arXiv2025

Global $GL_2$ Hecke-Baxter operator

We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group $GL_2$. This is an element of a global Hecke algebra associated with the double coset space $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$. Eigenvalues of the global Hecke-Baxter operator acting on the $GL_2$-Eisenstein series are given by the corresponding global $L$-factors. This construction generalizes our previous construction of the Hecke-Baxter operators over local completions $\mathbb{R}$ and $\mathbb{Q}_p$ of the number field $\mathbb{Q}$. Presumably, zeroes of the corresponding global $L$-factors should be subjected to an arithmetic version of the Bethe ansatz equations.

math.RT↗

arXiv2025

The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator: principal series representations

Previously introduced the $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator is a one-parameter family of elements in the commutative spherical Hecke algebra $\mathcal{H}(GL_{\ell+1}(\mathbb{R}),O_{\ell+1})$. Its action on spherical vectors in spherical principle series representations of $GL_{\ell+1}(\mathbb{R})$ is given by multiplication by the Archimedean $L$-factors associated to these representations. In this note we propose an extension of the construction to other (non-spherical) $GL_{\ell+1}(\mathbb{R})$ principle series representations providing a relevant generalization of the notions of spherical vector, commutative spherical Hecke algebra and the Hecke-Baxter operator to the general case. Action of the introduced Hecke-Baxter operator on the generalized spherical vectors is given by multiplication by the Archiemdean $L$-factor associated to the corresponding principle series representation of $GL_{\ell+1}(\mathbb{R})$.

math.RT↗

arXiv2022

Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group $G$ with a maximal torus $H\subset G$, Tits defined an extension $W_G^T$ of the corresponding Weyl group $W_G$. The extended group is supplied with an embedding into the normalizer $N_G(H)$, such that $W_G^T$ together with $H$ generate $N_G(H)$. In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form $G(\mathbb{R})\subset G$, which leads to the simple topological description of $W^T_G$. We also consider a variation of the Tits construction associated with compact real form $U$ of $G$. In this case we define an extension $W_G^U$ of the Weyl group $W_G$, naturally embedded into the group extension $\widetilde{U}:=U\rtimesΓ$ of the compact real form $U$ by the Galois group $Γ={\rm Gal}(\mathbb{C}/\mathbb{R})$. Generators of $W^U_G$ are squared to identity as in the Weyl group $W_G$. However, the non-trivial action of $Γ$ by outer automorphisms requires $W^U_G$ to be a non-trivial extension of $W_G$. This gives a specific presentation of the maximal torus normalizer of the group extension $\widetilde{U}$. Finally, we describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.

math.RT↗

arXiv2016

On spectral cover equations in Simpson integrable systems

Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact Kähler manifold $X$. We propose a description of the corresponding spectral cover of $X$ as the fiberwise projective dual to a hypersurface in the projectivization $\mathbb{P}(\mathcal{T}_{X} \oplus \mathcal{O}_X)$ of the tangent bundle $\mathcal{T}_X$ to $X$. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.

math.AG↗

arXiv2012

Representation Theory over Tropical Semifield and Langlands Correspondence

Recently we propose a class of infinite-dimensional integral representations of classical gl(n+1)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by U(n+1)-equivariant symplectic volumes of appropriate Kahler U(n+1)-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids GL(n+1,R), R being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over Q_p by taking the formal limit p->1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field Q_1. The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa.

math.AG↗

arXiv2011

On topological field theory representation of higher analogs of classical special functions

Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions and Gamma-function were identified with correlation functions in topological field theories on a two-dimensional disk. Mirror symmetry of the underlying topological field theory leads to a dual finite-dimensional integral representations reproducing classical integral representations for the corresponding special functions. The mirror symmetry interchanging infinite- and finite-dimensional integral representations provides an incarnation of the local Archimedean Langlands duality on the level of classical special functions. In this note we provide some directions to higher-dimensional generalizations of our previous results. In the first part we consider topological field theory representations of multiple local L-factors introduced by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the second part we are dealing with generalizations based on consideration of topological Yang-Mills theories on non-compact four-dimensional manifolds. Presumably, in analogy with the mirror duality in two-dimensions, S-dual description should be instrumental for deriving integral representations for a particular class of quantum field theory correlation functions and thus providing a new interesting class of special functions supplied with canonical integral representations.

hep-th↗

arXiv2011

On universal Baxter operator for classical groups

The universal Baxter operator is an element of the Archimedean spherical Hecke algebra H(G,K), K be a maximal compact subgroup of a Lie group G. It has a defining property to act in spherical principle series representations of G via multiplication on the corresponding local Archimedean L-factors. Recently such operators were introduced for G=GL_{\ell+1}(R) as generalizations of the Baxter operators arising in the theory of quantum Toda chains. In this note we provide universal Baxter operators for classical groups SO_{2\ell}, Sp_{2\ell} using the results of Piatetski-Shapiro and Rallis on integral representations of local Archimedean L-factors.

math.RT↗

arXiv2007

Two-dimensional Gauge Theories and Quantum Integrable Systems

In this paper the relation between 2d topological gauge theories and Bethe Ansatz equations is reviewed. In addition we present some new results and clarifications. We hope the relations discussed here are particular examples of more general relations between quantum topological fields theories in dimensions $d\leq 4$ and quantum integrable systems.

hep-th↗

arXiv2006

Higgs Bundles, Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the $N$-particle sector. This implies the full equivalence between the above gauge theory and the $N$-particle sub-sector of the quantum theory of Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the $G/G$ gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed.

hep-th↗

arXiv2004

Towards Integrability of Topological Strings I: Three-forms on Calabi-Yau manifolds

The precise relation between Kodaira-Spencer path integral and a particular wave function in seven dimensional quadratic field theory is established. The special properties of three-forms in 6d, as well as Hitchin's action functional, play an important role. The latter defines a quantum field theory similar to Polyakov's formulation of 2d gravity; the curious analogy with world-sheet action of bosonic string is also pointed out.

hep-th↗

arXiv2001

On non-abelian structures in open string field theory

Multi-brane backgrounds are studied in the framework of the background independent open string field theory. A simple description of the non-abelian degrees of freedom is given. Algebra of the differential operators acting on the space of functions on the space-time provides a natural tool for the discussion of this phenomena.

hep-th↗

arXiv2001

On Unification of RR Couplings

We consider the couplings of RR fields with open string sector for $Dp$-${\overline{Dp}}$ backgrounds of various $p$. Proposed approach, based on the approximation of the open string algebra by the algebra of differential operators, provides the unified description of these couplings and their interrelations.

hep-th↗

arXiv2000

Stringy Higgs Mechanism and the Fate of Open Strings

We propose a refinement of the physical picture describing different vacua in bosonic string theory. The vacua with closed strings and open strings are connected by the string field theory version of the Higgs mechanism, generalizing the Higgs mechanism of an abelian gauge field interacting with a complex scalar. In accordance with Sen's conjecture, the condensation of the tachyon is an essential part of the story. We consider this phenomenon from the point of view of both a world-sheet sigma-model and the target-space theory. In the Appendix the relevant remarks regarding the choice of the coordinates in the background independent open string field theory are given.

hep-th↗

arXiv2000

On Exact Tachyon Potential in Open String Field Theory

In these notes we revisit the tachyon lagrangian in the open string field theory using background independent approach of Witten from 1992. We claim that the tree level lagrangian (up to second order in derivatives and modulo some class of field redefinitions) is given by $L = e^{-T} (\partial T)^2 + (1+T)e^{-T}$. Upon obvious change of variables this leads to the potential energy $-ϕ^2 \log {ϕ^2 \over e}$ with canonical kinetic term. This lagrangian may be also obtained from the effective tachyon lagrangian of the p-adic strings in the limit $p\to 1$. Applications to the problem of tachyon condensation are discussed.

hep-th↗