arXiv subjects

Daniil Klyuev

Publications and source records attributed to Daniil Klyuev.

12 recordsLinked to original sources

arXiv2026

Residue construction of quantized Coulomb branches

We describe the image of a quantized BFN Coulomb branch $\mathcal{A}_{G,N}^{\hbar=1}$ under localization (abelianization) map for any $G,N$. In most cases, such as quiver gauge theories without loops, this description works for any flavors, sometimes we need to take generic or formal flavor parameters. The answer is given by a roots and residue condition similar to Ginzburg---Kapranov---Vasserot construction of DAHA arXiv:alg-geom/9512017. As a corollary, for any quantized conical Coulomb branch (and zero or small flavors) we provide a mathematical construction of the sphere trace introduced by Gaiotto and Okazaki arXiv:1911.11126.

math.RT↗

arXiv2026

Positive Traces on Certain ${\rm SL}(2)$ Coulomb Branches

For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $ρ$ of $\mathcal{A}$, there is a notion of a positive trace. When we have a three-dimensional $\mathcal{N}=4$ gauge theory or four-dimensional $\mathcal{N}=2$ gauge theory compactified on a circle, classification of positive traces on its Coulomb branch $\mathcal{A}$ can give a better understanding of this theory. We classify positive traces on $\mathcal{A}$ in two cases. The first case is when $\mathcal{A}$ is a quantization of a Kleinian singularity of type $D$, with certain restriction on the quantization parameter. The second case is when $\mathcal{A}=K^{{\rm SL}(2,\mathbb{C}[[t]])\rtimes \mathbb{C}_q^{\times}}({\rm Gr}_{{\rm PGL}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure ${\rm SL}(2)$ and ${\rm PGL}(2)$ gauge theories.

hep-th↗

arXiv2025

Positive traces on quantized abelian Coulomb branches

Let $A=A_{G,N}^{\hbar=1}$ be a quantized Coulomb branch with an antilinear automorphism $ρ$. A map $T\colon A\to\mathbb{C}$ is called a positive trace if $T(aρ(a))>0$ for all nonzero $a\in A$. Positive traces on Coulomb branches appear in the study of supersymmetric gauge theories. We classify positive traces on all abelian Coulomb branches, meaning $G=(\mathbb{C}^{\times})^d$ is a torus.

math.RT↗

arXiv2025

Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras

Let $\mathcal{A}$ be a quantized ($K$-theoretic) BFN Coulomb branch with $G=\mathbb{C}^*$ and any $N$, that is, $\mathcal{A}$ is a generalized Weyl or $q$-Weyl algebra. Let $M$ be an $\mathcal{A}$-$\overline{\mathcal{A}}$ bimodule. Choosing an automorphism $ρ$ of $\mathcal{A}$ we can define the notion of an invariant Hermitian form: $(au,v)=(u,vρ(a))$ for all $a\in \mathcal{A}$ and $u,v\in M$. We obtain a classification of invariant positive definite forms on $M$ in the case when $M$ is Harish-Chandra in the sense of Losev and quantization parameter is generic.

math.RT↗

arXiv2024

A different approach to positive traces on generalized q-Weyl algebras

Positive twisted traces are mathematical objects that could be useful in computing certain parameters of superconformal field theories. The case when $\mathcal{A}$ is a $q$-Weyl algebra and $ρ$ is a certain antilinear automorphism of $\mathcal{A}$ was considered in arXiv:2105.12652. Here we consider more general choices of $ρ$. In particular, we show that for $ρ$ corresponding to a standard Schur index of a four-dimensional gauge theory a positive trace is unique.

math.RT↗

arXiv2023

Analytic Langlands correspondence for $\operatorname{PGL}_2(\mathbb{C})$ on a genus one curve with parabolic structures

Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan. On one side of this correspondence there are certain operators on $L^2(\operatorname{Bun}_G)$, called Hecke operators, where $\operatorname{Bun}_G$ is the variety of stable $G$-bundles on $X$ and $L^2(\operatorname{Bun}_G)$ is a Hilbert space of square-integrable half-densities. The compactness conjecture says that Hecke operators are bounded and, moreover, compact. In arXiv:2106.05243 Etingof, Frenkel and Kazhdan prove this and other conjectures in the case of $G=\operatorname{PGL}_2$ and $X=\mathbb{P}^1$ with parabolic structures. We investigate the case of $G=\operatorname{PGL}_2$ and genus one curve over complex numbers with parabolic structures. We obtain an explicit formula for Hecke operators and prove the compactness conjecture in this case.

math.RT↗

arXiv2023

On the Analytic Langlands Corrrespondence for $\operatorname{PGL}_2$ in Genus 0 with Wild Ramification

The analytic Langlands correspondence was developed by Etingof, Frenkel and Kazhdan in arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243, arXiv:2311.03743. For a curve $X$ and a group $G$ over a local field $F$, in the tamely ramified setting one considers the variety $\operatorname{Bun}_G$ of stable $G$-bundles on $X$ with Borel reduction at a finite subset $S\subset X$ of points. On one side of this conjectural correspondence there are Hecke operators on $L^2(\operatorname{Bun}_G)$, the Hilbert space of square-integrable half-densities on $\operatorname{Bun}_G$; on the other side there are certain opers with regular singularities at $S$. In this paper we prove the main conjectures of analytic Langlands correspondence in the case $G = \operatorname{PGL}_2$, $X=\mathbb{P}^1_{\mathbb{C}}$ with wild ramification, i.e. when several points in $S$ are collided together.

math.AG↗

arXiv2023

Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A

Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.

math.QA↗

arXiv2022

Multiplication Kernels for the Analytic Langlands Program in Genus Zero

We provide an explicit proof of a recent result of Gaiotto arXiv:2110.02255 which gives an explicit formula for a so-called "multiplication kernel'' $K_3(x, y, z; t)$ intertwining the action of Hecke operators and Gaudin operators in three sets of variables. This function $K_3$ arises naturally in the context of the analytic formulation of the geometric Langlands program in the genus-zero case arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. We also discuss how the kernel $K_3$ relates to other objects typically considered in the analytic Langlands program.

math.RT↗

arXiv2022

Twisted Traces and Positive Forms on Generalized $q$-Weyl Algebras

Let ${\mathcal A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ)$, where $P$ is a Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on ${\mathcal A}$ is called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in {\mathbb C}$ with $|s|=1$ and all $a,b\in {\mathcal A}$. In this paper we classify positive definite invariant Hermitian forms on generalized $q$-Weyl algebras.

math.RT↗

arXiv2021

Deformations of pairs of Kleinian singularities

Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger.

math.RT↗

arXiv2020

On unitarizable Harish-Chandra bimodules for deformations of Kleinian singularities

The notion of a Harish-Chandra bimodule, i.e. finitely generated $U(\mathfrak{g})$-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [Ivan Losev, Dimensions of irreducible modules over W-algebras and Goldie ranks. arXiv:1209.1083]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, i.e. the algebra itself, for a deformation of Kleinian singularity of type $A$ is unitarizable. We obtain a partial classification of unitarizable regular bimodules.

math.RT↗