# Quantum integrability and quantum Schubert calculus

Starts:

12:00

Ends:

12:30

## Location

Kavli Royal Society Centre, Chicheley Hall, Newport Pagnell, Buckinghamshire, MK16 9JJ

## Overview

Theo Murphy international scientific meeting organised by Dr Christian Korff and Professor Vassily Gorbounov.

Depiction of the quantum Euler class (a quantity in quantum Schubert calculus) in terms of domain wall boundaries (blue lines) of the Ising model on the torus (a quantity in quantum integrable systems). Credit: Christian Korff

There is a new direction in Schubert calculus, which links the Yang-Baxter equation, the central equation in quantum integrable systems, to problems in representation theory that have their origin in enumerative geometry. This will be a cross-disciplinary meeting that brings together researchers of different career stages and from different areas in mathematics for a mix of talks and focused discussion rounds.

The speaker abstracts and biographies can be found below. Recorded audio of the presentations will be available on this page after the meeting has taken place.

Enquiries: contact the Scientific Programmes team

## Schedule of talks

### 11 June

#### 12:00-17:00

Session 1

6 talks Show detail Hide detail

#### Chairs

Professor Vassily Gorbounov, University of Aberdeen, UK

#### 13:30-14:15Enumerative symplectic duality

Professor Andrei Okounkov, Columbia University, USA

#### Abstract

This will be a report on a joint work in progress with Mina Aganagic. Its goal is to prove the equality of curve counts in symplectically dual geometries, whenever both counts can be defined using present-day technology.

#### 14:15-15:00Lessons on integrability from little string theory

Professor Mina Aganagic, University of California, Berkeley, USA

#### Abstract

A class of integrable lattice models turns out to emerge from geometry of holomorphic symplectic varieties. The physics origin of this is a certain “little” string theory in six dimensions. The six dimensional string theory gives a unified framework for studying integrable lattice models by providing relations between different approaches: the ones due to Okounkov and Maulik, due to Nekrasov and Shatashvili, and that of Costello with Witten and Yamazaki. (This is based on joint works with Andrei Okounkov, and work to appear with Nikita Nekrasov and Samson Shatashvili).

#### 15:30-16:15Super-spin-chains and gauge theories

Professor Nikita Nekrasov, The Simons Center for Geometry and Physics, Stony Brook University, USA

#### Abstract

Bethe/gauge correspondence relates quantum integrable systems to supersymmetric gauge theories. One of the mathematical consequences of this relation is the identification of the quantum cohomology ring of certain varieties with the Bethe subalgebras of quantum algebras. In this talk the two dimensional gauge theories corresponding to the Yangians of super-algebras of A type will be described. In a parallel development Nikita Nekrasov will present the four dimensional gauge theories with the surface defects supporting the (conjecturally) super-Kac-Moody chiral algebras.

#### 16:15-17:00Mirror symmetry constructions for G/P

Professor Konstanze Rietsch, King's College London, UK

#### Abstract

This talk will review some aspects of mirror symmetry for homogeneous spaces G/P and its interaction with the representation theory of G. For a cominuscule homogeneous space G/P there is an expression for the mirror LG model W in terms of coordinates which, by the geometric Satake correspondence in representation theory, are naturally identified with cohomology classes of G/P (joint works with Marsh, Pech, and Williams). A relationship between the critical points of W and its tropicalisation, and representation theory (work of Judd) will also be discussed.

### 12 June

#### 09:00-12:30

Session 2

5 talks Show detail Hide detail

#### Chairs

Dr Laurent Manivel, CNRS, Paul Sabatier University, France

#### 09:00-09:45Hypergeometric and q-hypergeometric solutions of quantum differential equations

Professor Alexander Varchenko, University of North Carolina at Chapel Hill, USA

#### Abstract

The hypergeometric and q-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^*F_\lambda$ of a partial flag variety will be discussed.  The leading term of the asymptotics of the q-hypergeometric solutions will be expressed in terms of the equivariant gamma class of $T^*F_\lambda$. That statement is analogous to the statement of the gamma conjecture for Fano varieties by Galkin, Golyshev, and Iritani.

#### 09:45-10:30Schubert puzzles and quantum integrability

Professor Paul Zinn-Justin, University of Melbourne, Australia

#### Abstract

Paul Zinn-Justin will discuss the geometric meaning of ‘puzzles’ in the context of Schubert calculus, and how the underlying quantum integrability allows to compute structure constants of the equivariant K-theory of d-step flag varieties and their cotangent bundles for d<=4 (joint work with A Knutson). If time permits, Paul will mention applications to the theory of symmetric polynomials.

#### 11:00-11:45Mirror symmetry for flag varieties via Langlands duality

Professor Thomas Lam, University of Michigan, USA

#### Abstract

Thomas Lam’s group proves Rietsch's mirror conjecture for minuscule flag varieties. The mirror theorem asserts that two systems of differential equations coincide: one arising from quantum cohomology, and the other from a Landau-Ginzburg model. The idea of the proof is to recognise the former as a Galois object and the latter as an automorphic object, and apply the (ramified) geometric Langlands correspondence. Some surprising connections to Kloosterman sums and sheaves will appear. If time permits, Thomas will speculate on the case of Fano varieties. This is joint work with Nicolas Templier.

#### 11:45-12:30Quantum Hitchin system for the Minahan-Nemeschansky E6 theory

Dr Lotte Hollands, Heriot-Watt, UK

#### Abstract

According to Nekrasov and Shatashvili the Coulomb vacua of a large class of four-dimensional N=2 theories, subjected to the Omega background in two of the four dimensions, correspond to the eigenstates of a quantisation of a Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is defined in terms of a system of Darboux coordinates on the corresponding moduli space of flat connections. In this talk Lotte Hollands will introduce such a system of Darboux coordinates on the moduli space of SL(3) flat connections on the three-punctured sphere through a procedure called abelianization. She will describe the spectral problem characterising the corresponding quantum Hitchin system, and show how this leads to a characterisation of the Coulomb vacua of the Minahan-Nemeschansky E6 theory in the Omega-background. This talk is based on work in progress with Andrew Neitzke.

Lunch

#### 13:30-17:00

Session 3

5 talks Show detail Hide detail

#### Chairs

Professor Eric Vasserot, Institute of Mathematics of Jussieu - Paris Left Bank, France

#### 13:30-14:15Motivic Chern classes, Hecke algebras, and stable envelopes

Professor Leonardo Mihalcea, Virginia Tech University, USA

#### Abstract

The Chern-Schwartz-MacPherson (CSM) class of a compact (complex) variety X is a homology class which provides an analogue of the total Chern class of the tangent bundle of X, for X singular. Its K-theoretic version, the motivic Chern class, is a class with good functorial properties, and for smooth X it normalizes to the Hirzebruch's lambda-y class of the cotangent bundle of X. One can associate these classes to any constructible subset of X, and in the talk I will discuss how one can use the Demazure-Lusztig operators in the Hecke algebra to calculate the motivic Chern classes for Schubert cells in generalized flag manifolds X=G/B. I will also discuss relations to K theoretic envelopes of Maulik and Okounkov, and a conjectural positivity property. This recovers and extends beyond Lie type A recent results obtained by Fejer, Rimanyi and Weber, using localization techniques. The talk is based on ongoing joint work with Paolo Aluffi, Changjian Su, and Jorg Schurmann.

#### 14:15-15:00Ext operators as W-algebra intertwiners: AGT with matter for general surfaces

Professor Andrei Negut, Massachusetts Institute of Technology, USA

#### Abstract

Andrei Negut will summarise an algebro-geometric proof, which works for a fairly general smooth projective surface, of the following fact: the Ext operator that Carlsson-Okounkov associate to moduli spaces of sheaves on a surface is a W-algebra intertwiner (as predicted by the work of Nekrasov and Alday-Gaiotto-Tachikawa).

#### 15:30-16:15On quantum cohomology and derived category of isotropic Grassmannians

Dr Maxim Smirnov, University of Augsburg, Germany

#### Abstract

Dubrovin’s conjecture (ICM 1998) predicts an intriguing relation between the quantum cohomology ring of a smooth projective variety X and its derived category of coherent sheaves. Namely, the generic semisimplicity of quantum cohomology of X should be equivalent to the existence of a full exceptional collection in the derived category of coherent sheaves on X. This poster will present results on the semisimplicity of the big quantum cohomology of symplectic isotropic Grassmannians IG(2,2n), its relation to unfoldings of singularities of type A_n, and Lefschetz exceptional collections in their derived categories. Based on a joint work with JA Cruz Morales, A Kuznetsov, A Mellit, and N Perrin.

#### 16:15-17:00Helix structures in quantum cohomology of Grassmannians

Dr Giordano Cotti, Max-Planck-Institut für Mathematik, Germany

#### Abstract

In occasion of the 1998 ICM in Berlin, B Dubrovin conjectured an intriguing connection between the enumerative geometry of a Fano manifold X with algebro-geometric properties of exceptional collections in the derived category D^b(X). Under the assumption of semisimplicity of the quantum cohomology of X, the conjecture prescribes an explicit form for local invariants of QH*(X), the so-called “monodromy data”, in terms of Gram matrices and characteristic classes of objects of exceptional collections. In this talk, a refinement of this conjecture will be presented, and particular attention will be given to the case of complex Grassmannians. At points of small quantum cohomology, these varieties manifest a coalescence phenomenon, whose occurrence and frequency is surprisingly subordinate to the distribution of prime numbers. A priori, the analytical description of these Frobenius structures cannot be obtained from an immediate application of the theory of isomonodromy deformations. Giordano Cotti will show how, under minimal conditions, the classical theory of M Jimbo, T Miwa, K Ueno (1981) can be extended to describe isomonodromy deformations at a coalescing irregular singularity. Furthermore, a property of quasi-periodicity of Stokes matrices associated to the points of small Quantum Cohomology of complex Grassmannians will be discussed. Based on joint works with B Dubrovin and D Guzzetti.

### 13 June

#### 09:00-12:30

Session 4

5 talks Show detail Hide detail

#### Chairs

Professor Andrei Okounkov, Columbia University, USA

#### 09:00-09:45Cherednik operators at infinity

Professor Maxim Nazarov, University of York, UK

#### Abstract

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. Maxim Nazarov introduces the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. The groups compute the limits of their operators when N tends to infinity. These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

#### 09:45-10:30Elliptic stable envelopes and finite-dimensional representation of elliptic quantum group of type $gl_N$

Professor Hitoshi Konno, Tokyo University of Marine Science and Technology, Japan

#### Abstract

This work presents a construction of elliptic stable envelopes for partial flag variety as elliptic weight functions of type $gl_N$ using representation theory of the elliptic quantum group $U_{q,p}(¥widehat{sl}_N)$. The work also presents a finite-dimensional representation of $U_{q,p}(¥widehat{sl}_N)$ on the Gelfand-Tsetlin basis. The result is described in a combinatorial way in terms of the partitions of $[1,n]$, and gives an elliptic and dynamical analogue of the geometric representation of the quantum affine algebra $U_q(¥widehat{sl}_N)$ on equivariant K-theory constructed by Ginzburg-Vasserot and Nakajima. The elliptic stable envelopes allows us to obtain a geometric interpretation of our result as a representation of $U_{q,p}(\widehat{sl}_N)$ on equivariant elliptic cohomology.

#### 11:00-11:45Quantum cohomology of some varieties with an action of an algebraic group

Dr Clelia Pech, University of Kent, UK

#### Abstract

In this talk Clelia Pech will describe a family of algebraic varieties with actions of Lie groups which are closely related to homogeneous spaces. After describing the geometry and the orbit structure of these varieties, Clelia will explain how to understand rational curves on these varieties, as well their quantum cohomology ring. This is joint work with R Gonzales, N Perrin, and A Samokhin, see arXiv:1803.05063.

#### 12:30-13:30

Lunch and meeting close

Quantum integrability and quantum Schubert calculus

11 - 13 June 2018

Kavli Royal Society Centre, Chicheley Hall Newport Pagnell Buckinghamshire MK16 9JJ