Quantum integrability and quantum Schubert calculus

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.