Quantum cohomology and quantum K-theory

Sophie Germain Building, Paris Diderot
15-17 January 2014

Speakers Andrea Brini, Montpellier Pierre-Emmanuel Chaput, Nancy Alessandro Chiodo, Paris Tom Coates, London (TBC) Vasily Golyshev, Moscow Vassily Gorbounov, Aberdeen Jérémy Guéré, Paris Christian Korff, Glasgow Andrew Kresch, Zurich Etienne Mann, Montpellier Clélia Pech, London Nicolas Perrin, Dusseldorf Konstanze Rietsch, London Christian Sevenheck, Mannheim Catharina Stroppel, Bonn Dimitri Zvonkine, Paris Schedule Wednesday, January 15 9h15-10h30 P.E. Chaput An introduction to quantum cohomology 10h30-11h Coffee 11h-12h15 N. Perrin An introduction to quantum K-theory 14h-14h45 K. Rietsch A survey of mirror symmetry for Grassmannians 15h-15h45 C. Pech Some aspects of mirror symmetry for quadrics 15h45-16h15 Coffee 16h15-17h C. Stroppel TBA Thursday, January 16 9h15-10h30 A. Chiodo An introduction to moduli spaces of curves 10h30-11h Coffee 11h-12h15 E. Mann An introduction to quantum D-modules 14h-14h45 C. Sevenheck Landau-Ginzburg models and quiver discriminants 15h-15h45 V. Golyshev Gamma class and Gamma conjecture 15h45-16h15 Coffee 16h15-17h A. Brini Equivariant quantum cohomology and the Toda equation Friday, January 17 9h-9h45 C. Korff Yang-Baxter algebras and equivariant quantum cohomology I 10h-10h45 V. Gorbounov Yang-Baxter algebras and equivariant quantum cohomology II 10h45-11h15 Coffee 11h15-12h A. Kresch Applications of Kiem-Li cosection localization 13h30-14h15 J. Guéré Mirror symmetry for the quantum singularity theory of chain polynomials 14h30-15h15 D. Zvonkine The Chern character of the Verlinde bundle 15h30-16h15 T. Coates Mirror Symmetry without Localization Abstracts A. Brini, Equivariant quantum cohomology and the Toda equation Since Witten's influential conjecture on the relation between the KdV hierarchy and the moduli space of stable curves, classical integrable hierarchies have been long thought to underlie the Gromov-Witten theory of complex algebraic varieties (or orbifolds) with semi-simple quantum cohomology. In this talk I will introduce a new family of integrable systems that arise as symmetry reductions of the two-dimensional Toda hierarchy and show how they are related to the equivariant orbifold quantum cohomology of the local weighted projective line. The main spinoff will be a proof of several conjectural correspondences which refine Ruan's original Crepant Resolution Conjecture for a wide class of examples. This talk is based on joint works with Cavalieri-Ross, Carlet-Romano-Rossi and Borot-Klemm. T. Coates, Mirror Symmetry without Localization I will describe joint work with Cristina Manolache. We show that, for a Fano toric variety X, the spaces of stable maps to X and quasimaps to X are "virtually birational". This implies that Gromov--Witten invariants and quasimap invariants of X coincide. It gives a new and geometric proof of mirror symmetry, for a broad class of spaces (complete intersections in Fano toric varieties), which does not rely on localization calculations in T-equivariant cohomology. V. Golyshev, Gamma class and gamma conjecture We discuss the gamma conjecture for rank 1 Fano threefolds and give a proof in one particular case, that of V_12 (joint work in progress with D. Zagier). V. Gorbounov & C. Korff, Yang-Baxter algebras and equivariant quantum cohomology I & II We provide a description of the equivariant quantum cohomology ring of the Grassmannian in terms of exactly solvable or integrable lattice models starting from special solutions of the Yang-Baxter equation. We find explicit combinatorial formulae for the weighted sums (partition functions) over special non-intersecting lattice paths and show that they are generating functions for equivariant Gromov-Witten invariants. Exploiting techniques from integrable systems, the so-called Bethe ansatz, we establish a series of new results for the Grassmannian: we relate the partition functions to affine stable Grothendieck polynomials introduced by Lam, show that the idempotents of the ring are solutions of the quantum Knizhnik-Zamolodchikov equations and conjecture an explicit determinant formulae for the basis of the Peterson algebra. The first part of the talk (CK) will be describing the integrable system, related algebraic structures and combinatorial aspects. The second part (VG) will set our results in relation with previous works by Peterson and Lam as well as recent works by Nekrasov, Shatashvili and Maulik, Okounkov on the quantum cohomology of Nakajima varieties and sketch how these results might be generalised to general partial flag varieties. J. Guéré, Mirror symmetry for the quantum singularity theory of chain polynomials We will talk about the quantum invariants for singularities, which were introduced by Fan, Jarvis, and Ruan in 2007. Our dream is to compute them in every genus and for every singularities, but it goes through the computation of a `virtual (cohomological) class', which is very hard to get without the so-called concavity hypothesis. I will present the first explicit computation of every genus-zero quantum invariants for a range of singularities called chain polynomials, which do not respect concavity. To this purpose, I developed a machinery based on my recent notion of recursive complexes, which is fun and easy to use. As a remarkable consequence, we are going to understand in which precise sense the virtual class is viewed as a partial extension of Euler class to K-theory. Last but not least, this work yields a mirror symmetry theorem for chain singularities, at the level of local systems. You could find the full article on arXiv:1307.5070. A. Kresch, Applications of Kiem-Li cosection localization We recall the construction of cosection localized virtual cycle classes of Kiem and Li and indicate some applications that this construction has yielded, e.g., in Gromov-Witten theory (work of Chang and Li and others). C. Pech, Some aspects of mirror symmetry for quadrics In 2000, Hori and Vafa conjectured a Laurent polynomial mirror for projective hypersurfaces. Unfortunately, for a quadric Q of dimension N, these Laurent polynomials do not have the expected number of critical points - namely dim H*(Q,C) - hence they do not satisfy all the properties of a mirror. Another approach is to view Q as a homogeneous space for SO(N+2). It then has a Lie-theoretic mirror, constructed by K. Rietsch in 2008, which has the correct number of critical points. In this talk based on joint work with K. Rietsch and L. Williams, I will explain how to express the mirror W in terms of natural coordinates on an affine subspace of a `mirror homogeneous space' Y embedded into projective space P(H*(Q,C)*). I will also give a mirror symmetry result for Q involving the Dubrovin connection. This will enable me to compute explicitely the constant term of the J-function of Q - that is, the generating function for 1-pointed descendent Gromov-Witten invariants on Q - using period integrals, in the spirit of work by Bertram-Ciocan-Fontanine-van Straten for the Grassmannian Gr(2,n). K. Rietsch, A survey of mirror symmetry for Grassmannians I will describe mirror dual Landau-Ginzburg (LG) models for Grassmannians and Lagrangian Grassmannians obtained in joint work with Robert Marsh and Clelia Pech, respectively. The latter LG model is a rational function on a minuscule homogeneous space for the orthogonal (or the Spin) group, while the former takes place on a Grassmannian isomorphic to the original one but more naturally thought of as Langlands dual, so that these stories are parallel. I will explain some known and conjectured results relating these LG models to the (small) quantum cohomology rings and Dubrovin connection in the A-model. C. Sevenheck, Landau-Ginzburg-models and quiver discriminants I will present a construction on potential Landau-Ginzburg-models related to some special quiver representations. I will explain some Hodge theoretic aspects of this construction and give some speculation on how this is related to the usual mirror symmetry picture for toric stacks. D. Zvonkine, The Chern character of the Verlinde bundle We give an explicit formula for the total Chern character of the Verlinde bundle over the moduli space of stable curves in terms of tautological classes. The Chern characters of the Verlinde bundles form a semi-simple CohFT, while their ranks, given by the Verlinde formula, form a semi-simple fusion algebra. According to the Teleman's classification of semisimple CohFTs, there exists an element of Givental's group that transforms the fusion algebra into the CohFT. We determine this element using the first Chern class of the Verlinde bundle on the space of smooth curves, and the projective flatness of the Hitchin connection. (Joint with A. Marian, D. Oprea, R. Pandharipande, A. Pixton.)