arXiv subjects

Alexey Sevostyanov

Publications and source records attributed to Alexey Sevostyanov.

6 recordsLinked to original sources

arXiv2026

A tree-free approach to 3D Yang-Mills Langevin dynamic. Analytic estimates and the existence of a model for a regularity structure

Using the multi-index approach to regularity structures due to F. Otto et al., we construct a regularity structure and a model for it associated to the stochastic Langevin equation for the 3D Euclidean Yang-Mills functional. For the model we also obtain global stochastic and global pointwise weighted Besov type estimates which hold almost surely. The model is defined as a limit of a sequence of smooth models introduced with the help of a mollified noise. When the mollification is removed the sequence converges in a certain topology defined with the help of the stochastic estimates. To obtain these results we develop the multi-index approach for systems of equations with vector-valued white noises. This project is motivated by the problem for constructing 3D Euclidean Yang-Mills measure and by the earlier results of the author on the related problem of canonical quantization of the Yang-Mills field on the Minkowski space.

math.PR↗

arXiv2026

Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture

This monograph, along with a self-consistent presentation of the theory of q-W-algebras including the construction of algebraic group analogues of Slodowy slices, contains a description of q-W-algebras in terms of Zhelobenko type operators introduced in the book. This description is applied to prove the De Concini-Kac-Procesi conjecture on the dimensions of irreducible modules over quantum groups at roots of unity.

math.QA↗

arXiv2021

Towards non-perturbative quantization and the mass gap problem for the Yang-Mills Field

We reduce the problem of quantization of the Yang-Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections on $\mathbb{R}^3$. We suggest a formally self-adjoint expression for the quantized Yang-Mills Hamiltonian as an operator on the corresponding Lebesgue $L^2$-space. In the case when the Yang-Mills field is associated to the Abelian group $U(1)$ we define the probability measure which depends on two real parameters $m>0$ and $c\neq 0$. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is $\{0\}\cup[\frac12m, \infty)$, i.e. it has a gap.

hep-th↗

arXiv2021

A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence

We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\widehat{\mathfrak g}$ of level $-2h^\vee-k$, where $h^\vee$ is the dual Coxeter number of the underlying semisimple Lie algebra $\mathfrak g$ and $k\in \mathbb{C}$, and the category of finitely generated representations of the W-algebra associated to $\widehat{\mathfrak g}$ of level $k$. When $k=-h^\vee$ this yields an equivalence between a category of representations of $\widehat{\mathfrak g}$ of central charge $-h^\vee$ and the category ${\rm Coh}({\rm Op}_{^LG}(D^\times))$ of coherent sheaves on the space ${\rm Op}_{^LG}(D^\times)$ of $^LG$-opers on the punctured disc $D^\times$, where $^LG$ is the Langlands dual group to the algebraic group of adjoint type with Lie algebra $\mathfrak g$. This can be regarded as a version of the local geometric Langlands correspondence. The above mentioned equivalences generalize to the case of affine Lie algebras the Skryabin equivalence between the categories of generalized Gelfand-Graev representations of $\mathfrak g$ and the categories of representations of the corresponding finitely generated W-algebras, and Kostant's results on the classification of Whittaker modules over $\mathfrak g$.

math.RT↗

arXiv2013

The geometric meaning of Zhelobenko operators

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup of G corresponding to the nilradical n of b. We show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence N\times h -> b given by the restriction of the adjoint action. Simple geometric proofs of formulas for the "classical" counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

math.RT↗

arXiv1995

The classical r-matrix method for nonlinear sigma-model

The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax matrix.

hep-th↗