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Andrey Smirnov

Publications and source records attributed to Andrey Smirnov.

34 records · Page 2Linked to original sources
arXiv2020

Capped vertex with descendants for zero dimensional $A_{\infty}$ quiver varieties

In this paper, we study the capped vertex functions associated to certain zero-dimensional type-$A$ Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions. We determine the monodromy of the vertex functions and show that it coincides with the elliptic R-matrix of symplectic dual variety. We apply our results to give the vertex functions and the characters of the tautological bundles on the quiver varieties formed from arbitrary stability conditions.

math.AG↗
arXiv2020

Pursuing quantum difference equations II: 3D-mirror symmetry

Consider a pair of symplectic varieties dual with respect to 3D-mirror symmetry. The K-theoretic limit of the elliptic duality interface is an equivariant K-theory class of the product. We show that this class provides correspondences in the product mapping the K-theoretic stable envelopes to the K-theoretic stable envelopes. This construction allows us to extend the action of various representation theoretic objects on K(X), such as action of quantum groups, quantum Weyl groups, R-matrices etc., to their action on the K-theory of the variety dual to X. In particular, we relate the wall R-matrices to the R-matrices of the dual variety. As an example, we apply our results to the Hilbert scheme of n points in the complex plane. In this case we arrive at the conjectures of E.Gorsky and A.Negut.

math.AG↗
arXiv2020

Quasimaps to zero-dimensional $A_{\infty}$-quiver varieties

We consider the moduli spaces of quasimaps to zero-dimensional $A_{\infty}$ Nakajima quiver varieties. An explicit combinatorial formula for the equivariant Euler characteristic of these moduli spaces is obtained and applications to symplectic duality are discussed.

math.AG↗
arXiv2019

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and Kähler parameters of $X$, respectively, with the Kähler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rimányi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

math.AG↗
arXiv2019

Elliptic stable envelope for Hilbert scheme of points in the plane

We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the stable envelope in equivariant $K$-theory (with arbitrary slope) and equivariant cohomology.

math.AG↗
arXiv2019

Baxter Q-operator from quantum K-theory

We define and study the quantum equivariant $K$-theory of cotangent bundles over Grassmannians. For every tautological bundle in the $K$-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous $XXZ$ spin chain. In addition, we prove that each such operator corresponds to the universal elements of quantum group $\mathcal{U}_{\hbar}(\widehat{\mathfrak{sl}}_2)$. In particular, we identify the Baxter operator for the $XXZ$ spin chain with the operator of quantum multiplication by the exterior algebra tautological bundle. The explicit universal combinatorial formula for this operator is found. The relation between quantum line bundles and quantum dynamical Weyl group is shown.

math.AG↗
arXiv2019

Characters of tangent spaces at torus fixed points and $3d$-mirror symmetry

Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for the $\mathsf{K}$-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of $X$ known as vertex functions.

math.AG↗
arXiv2016

Rationality of capped descendent vertex in $K$-theory

In this paper we analyze the fundamental solution of the \textit{quantum difference equation} (qde) for the moduli space of instantons on two-dimensional projective space. The qde is a $K$-theoretic generalization of the quantum differential equation in quantum cohomology. As in the quantum cohomology case, the fundamental solution of qde provides the capping operator in $K$-theory (the rubber part of the capped vertex). We study the dependence of the capping operator on the equivariant parameters $a_i$ of the torus acting on the instanton moduli space by changing the framing. We prove that the capping operator factorizes at $a_i\to 0$. The rationality of the $K$-theoretic 1-leg capped descendent vertex follows from factorization of the capping operator as a simple corollary.

math.AG↗
arXiv2014

Polynomials associated with fixed points on the instanton moduli space

Using the Okounkov-Maulik stable map, we identify the equivariant cohomology of instanton moduli spaces with the space of polynomials on an infinite number of variables. We define the generalized Jack polynomials as the polynomials representing the classes of fixed points under this identification. Using the abelianization theorem of Shenfeld we derive the combinatorial formula for the expansion of generalized Jack polynomials in the basis of Schur polynomials.

math-ph↗
arXiv2013

On the Instanton R-matrix

A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting R-matrices are the standard rational solutions of the Yang-Baxter equation, which are well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R-matrices, depending additionally on two equivariant parameters t_1 and t_2. In this paper we derive an explicit expression for the R-matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R-matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R-matrix the well known Lehn's formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R-matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.

math.AG↗
arXiv2012

Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N)

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce the symmetrized version of the Drinfeld associator by dropping the odd terms. The symmetrized associator gives the same knot invariants, but has a simpler structure and is fully characterized by one symmetric function which we call the Drinfeld prepotential.

hep-th↗
arXiv2012

Kontsevich integral for knots and Vassiliev invariants

We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

hep-th↗
arXiv2011

Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix

In temporal gauge A_{0}=0 the 3d Chern-Simons theory acquires quadratic action and an ultralocal propagator. This directly implies a 2d R-matrix representation for the correlators of Wilson lines (knot invariants), where only the crossing points of the contours projection on the xy plane contribute. Though the theory is quadratic, P-exponents remain non-trivial operators and R-factors are easier to guess then derive. We show that the topological invariants arise if additional flag structure (xy plane and an y line in it) is introduced, R is the universal quantum R-matrix and turning points contribute the "enhancement" factors q^ρ.

hep-th↗
arXiv2009

Notes on Chern-Simons Theory in the Temporal Gauge

We analyze the perturbative series expansion of vacuum expectation values (vevs) for Wilson loop operators in Chern-Simons (CS) gauge theory in the temporal gauge $A_{0}=0$. Following J. Labastida and E. Pérez we introduce the notion of the kernels of knot polynomial invariants - the (non-invariant) vevs of the Wilson loops, arising from CS theory in the temporal gauge. A method for exact calculation of the kernels of knot polynomial invariants is presented.

hep-th↗
arXiv2009

Degenerate Sklyanin Algebras

New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for $sl(2,\mathbb{C})$. These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The $sl(N,\mathbb{C})$-case is discussed.

math-ph↗