arXiv subjects

Roman Travkin

Publications and source records attributed to Roman Travkin.

At least 19 recordsLinked to original sources

arXiv2026

Relative Langlands duality and Koszul duality

Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$.

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arXiv2024

Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

We establish an $S$-duality converse to the one studied by the 1st, 2nd and 4th authors; this is also a case of a twisted version of the relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh.. Namely, we prove that the $S$-dual of $\text{SO}(2n+1)\times \text{Sp}(2n)$ acting on the tensor product of their tautological representations is the symplectic mirabolic space $\text{Sp}(2n)\times\text{Sp}(2n)$ acting on the product $T^* \text{Sp}(2n)$ and the tautological representations of $\text{Sp}(2n)$. (Note that due to the anomaly, the dual of the second factor $\text{Sp}(2n)$ is the metaplectic dual, i.e. $\text{Sp}(2n)$). We also formulate the corresponding global conjecture, which describes explicitly the categorical theta-correspondence on the Langlands dual side.

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arXiv2024

Gaiotto Conjecture for $\mathrm{Rep}_q(\mathrm{F}(4))$

This paper is a part of the series proving the Gaiotto conjecture for basic classical quantum supergroups. The previous part arXiv:2107.02653 [math.RT] , arXiv:2306.09556 [math.RT], proved the Gaiotto conjecture for the general linear quantum supergroups $U_q(\mathfrak{gl}(N|M))$. Here we deal with the exceptional quantum supergroup $U_q(\mathfrak{f}(4))$.

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arXiv2023

Lagrangian subvarieties of hyperspherical varieties

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $\Lambda_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $\Lambda_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}\Lambda_{\mathscr X}$ and $\mathrm{Irr}\Lambda_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

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arXiv2023

Twisted Gaiotto equivalence for $\rm{GL}(M|N)$

In $\rm{GL}_N$, a series of subgroups indexed by $0\leq M\leq N-1$ were noticed by H. Jacquet-I. Piatetski Shapiro-J. Shalika, J. Cogdell, and D. Gaiotto. It was conjectured by D. Gaiotto that the categories of twisted D-modules on the affine Grassmannian of $\rm{GL}_N$ with equivariant structure with respect to these subgroups are equivalent to the categories of representations of quantum supergroups $U_q(\mathfrak{gl}(M|N))$. When $M=0$ and $M=N-1$, this conjecture was proved due to D. Gaitsgory and A. Braverman-M. Finkelberg-R. Travkin, respectively. In this paper, we prove the other cases. We adapt the global method originating from arXiv:math/0611323 and arXiv:0705.4571. In order to compare the global definition of Gaiotto category with the local definition, we generalize the local-global comparison theorem of arXiv:1811.02468 to a general setting.

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arXiv2022

The canonical global quantization of symplectic varieties in characteristic $p$

Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[η] \in H^0(X, Ω^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In this paper we construct a functorial in $(X, [η])$ formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves on $X$. We also construct its natural extension to a quasi-coherent sheaf of categories $\mathrm{QCoh}_h$ on the product $X^{(1)} \times {\mathbb S}$ of the Frobenius twist of $X$ and the projective line ${\mathbb S}=\mathbb P^1$, viewed as the one-point compactification of $\mathrm{Spec}\ \! k[h]$. Its global sections over $X^{(1)} \times \{0\}$ is the category of quasi-coherent sheaves on $X$. If $X$ is affine, $\mathrm{QCoh}_h$, restricted to $X^{(1)}\times \mathrm{Spf} \ \! k[[h]]$, is equivalent to the category of modules over the distinguished "Frobenius-constant" quantization of $(X,[η])$ defined by Bezrukavnikov and Kaledin.

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arXiv2022

Orthosymplectic Satake equivalence

This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

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arXiv2022

Orthosymplectic Satake equivalence, II

This is a companion paper of arXiv:1909.11492 and arXiv:1912.01930. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted $Sp(2n,{\mathbb C}[\![t]\!])$-equivariant $D$-modules on the so called mirabolic affine Grassmannian of $Sp(2n)$. We also discuss (conjectural) extension of this equivalence to the case of quantum supergroups and to some exceptional supergroups.

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arXiv2022

Untwisted Gaiotto equivalence

This is a successive paper of arXiv:1909.11492. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), \chi_{M, N})$-equivariant D-modules on $Gr_{N}$. We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ as a category of D-modules on the mirabolic subgroup $Mir_L(F)$ with certain equivariant conditions for any $L$ bigger than $N$ and $M$.

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arXiv2022

Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)

We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying certain anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our construction goes through certain "universal" ring object in the twisted derived Satake category of the symplectic group $Sp(2n)$. The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf introduced by S.Lysenko on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We also discuss applications of our construction to a potential mathematical construction of $S$-duality for super-symmetric boundary conditions in 4-dimensional gauge theory and to (some extension of) the conjectures of D.Ben-Zvi, Y.Sakellaridis and A.Venkatesh.

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arXiv2021

Mirabolic Robinson-Schensted-Knuth correspondence

The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of $GL(V)$ arising from $Fl(V)\times Fl(V)\times V$. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V)\times V$.

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arXiv2021

Gaiotto conjecture for $Rep_q(GL(N-1|N))$

We prove D.Gaiotto's conjecture about geometric Satake equivalence for quantum supergroup $U_q({\mathfrak{gl}}(N-1|N))$ for generic $q$. The equivalence goes through the category of factorizable sheaves.

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arXiv2021

A colimit presentation of $\mathcal{D}(G(K))$ via the Bott-Samelson hypercover

Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the $\infty$-category of D-modules on the loop group of $G$ is equivalent to the monoidal colimit of the $\infty$-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'

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arXiv2021

Homotopical presentations of braid groups via reduced lifts

In 1997, Deligne showed that the reduced lift presentation of a finite type generalized braid group remains correct if it is (suitably) interpreted as a presentation of a topological monoid. In this expository paper, we point out that Deligne's argument does not require the 'finite type' hypothesis, so it gives a different proof of a theorem proved by Dobrinskaya in 2006. We also review how to use this result to construct an action of the braid group on the finite or affine Hecke $\infty$-category via intertwining functors.

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arXiv2020

The affine Hecke category is a monoidal colimit

Let $G$ be a semisimple simply-connected algebraic group over an algebraically closed field of characteristic zero. We prove that the affine Hecke category associated to the loop group of $G$ is equivalent to the colimit, evaluated in the $\infty$-category of monoidal stable $\infty$-categories, of the finite type Hecke subcategories associated to standard parahoric subgroups. The main ingredient is an inductive characterization of colimits indexed by (sufficiently nice) bistratified categories. Our method is very general and can be used to prove a number of analogous 'colimit theorems,' e.g. for D-modules on the loop group.

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arXiv2019

Resolutions with conical slices and descent for the Brauer group classes of certain central reductions of differential operators in characteristic $p$

For a smooth variety $X$ over an algebraically closed field of characteristic $p$, to a differential 1-form $α$ on the Frobenius twist $X^{(1)}$ one can associate an Azumaya algebra $\mathcal D_{X,α}$, defined as a certain central reduction of the algebra $\mathcal D_X$ of "crystalline differential operators" on $X$. For a resolution of singularities $π:X\to Y$ of an affine variety $Y$, we study for which $α$ does the class $[\mathcal D_{X,α}]$ in the Brauer group $\mathrm{Br}(X^{(1)})$ descend to $Y^{(1)}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of non-commutative resolutions of $Y$. We prove that the classes $[\mathcal D_{X,α}]$ descend étale locally for all $α$ if $\mathcal O_Y\simeq π_* \mathcal O_X$ and $R^{1,2}π_*\mathcal O_X =0$. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[\mathcal D_{X,α}]$ descend to $Y^{(1)}$ globally for all $α$. Finally we give some examples, in particular we show that Slodowy slices, Nakajima quiver varieties and hypertoric varieties are resolutions with conical slices.

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arXiv2019

Mirabolic Satake equivalence and supergroups

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $GL_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

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arXiv2016

Moduli spaces, indecomposable objects and potentials over a finite field

Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give formulas for an exponential sum over the set of absolutely indecomposable objects and a stacky sum over the set of all objects of the category, respectively, in terms of the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible etale local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

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