Mirror Symmetry and Spin Curves
April 28 - 30, 2014
Abstracts


Alexander Buryak
Double ramification cycles and integrable hierarchies.
I will present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. It is based on an integration over double ramification cycles and is motivated by the Symplectic Field Theory. We conjecture that for a cohomological field theory formed by Witten’s r-spin classes the construction gives the r-KdV hierarchy. The conjecture is verified in the cases r=2,3

Hiroshi Iritani
Gamma Conjectures for Fano Manifolds
The Gamma class is a characteristic class of a complex manifold defined by Euler's Gamma function. For Fano manifolds we associate an "asymptotic class" defined by the asymptotic behaviour of solution of quantum differential equation. We say that a Fano manifold satisfies the Gamma conjecture if the asymptotic class coincides the Gamma class. We discuss several examples where this conjecture holds, and explain relationships to mirror symmetry and Dubrovin's conjecture. This is based on a joint work with Sergey Galkin and Vasily Golyshev.

Tyler Jarvis
A Mathematical Approach to the Gauged Linear Sigma Model
I will describe recent joint work with Huijun Fan and Yongbin Ruan on the construction of a moduli space and a virtual cycle for the gauged Witten equation. This work generalizes the so-called "FJRW theory" to infinite (reductive) groups. As a special case we get the hybrid model studied by Clader, but our theory describes a much more general situation, including complete intersections of toric varieties and even general non-Abelian groups. We conjecture that this construction will also provide a geometric interpretation of the Landau-Gizburg/Calabi-Yau correspondence.

Maxim Kazarian
Topological recursion for enumeration of dessins d'enfant
Grothendieck's dessins d'enfant serve to encode Belyi functions, that is, coverings of the Riemann sphere with three ramification points. We consider the generating function for the numbers of such coverings with prescribed ramification profile over one point and arbitrary ramification types over two others. It proves out that this function possesses nice integrable properties: it satisfies Virasoro constraints, cut-and-join type evolution equation, KP hierarchy, and topological recursion. Similar integrable properties hold for the case when the ramification profile over one of the two remaining points is (222...2). The topological recursion for this case has been established by M. Mulase with coauthors earlier. The talk is based on a joint work with P.Zograf.

Jérémy Guéré
Genus-zero quantum invariants of chain polynomials and mirror symmetry
This talk will be about the quantum invariants for singularities, which were introduced by Fan, Jarvis, and Ruan in 2007. The main obstacle is the computation of the virtual class; this poses problems even in genus zero, where the concavity hypothesis may still fail. In 2011, Polishchuk and Vaintrob have given an algebraic construction of the virtual class. I will present a comprehensive computation of every genus-zero quantum invariants for a range of singularities called chain polynomials (with maximal symmetries). These singularities do not respect concavity and their quantum invariants involve also some broad states (the counterpart in quantum singularity theory to primitive cohomology insertions in Gromov-Witten theory). We then get back to FJRW invariants by answering in this precise case the natural question about the compatibility between FJR and PV constructions. We finally get a mirror symmetry theorem for chain singularities.

Jun Li
Toward an effective theory of GW invariants of quintic CY threefolds
TBA

Motohico Mulase
Quantum curves: what they are, and what they do
A quantum curve is a family of ordinary differential equations that controls the mirror B-model of an A-model counting problem. For example, the quantum curve for the Witten-Kontsevich theory of intersection numbers on the moduli of stable curves is the Airy differential equation, and the one for Catalan numbers is the Hermite differential equation. Infinite-order differential equations appear for the case of r-spin intersection numbers and Gromov-Witten invariants of the projective line. Quantum curves are also constructed for the spectral curves of Hitchin fibrations. In this talk, starting with presenting concrete mathematical examples, a glimpse of this rich subject is presented.

Nicolas Orantin
Topological recursion, Frobenius manifold and higher genus Gromov-Witten invariants
The topological recursion method is a formalism developed in the context of random matrix theories in order to solve an associated problem of combinatorics consisting in the enumeration of discrete surfaces. This inductive procedure allows to enumerate such surfaces of arbitrary topology out of the only genus 0 data. This theory has further been formalized out of the context of random matrices and mysteriously solved many problem of enumerative geometry using a universal inductive procedure. In this talk, I will present this topological recursion procedure and explain the reason why it solves many problems of enumerative geometry at once. I will show that, given a semi-simple Frobenius manifold, one can identify the formula of the ancestor formal Gromov-Witten potential derived by Givental with the correlation functions computed by a local version of the topological recursion. The role of mirror symmetry will be explained and exemplified in the computation of the Gromov-Witten invariants of the projective line. Based on joint works with Chekhov, Dunin-Barkowski, Eynard, Norbury, Shadrin and Spitz.

Clélia Pech
On Landau-Ginzburg models for cominuscule homogeneous spaces.
In this talk, I will explain how to construct mirrors for some cominuscule homogeneous spaces, including Lagrangian Grassmannians and quadrics. These mirrors stem from a general Lie-theoretic construction for homogeneous spaces by K. Rietsch. Here I will give explicit expressions for the superpotentials, in "natural" coordinates related to the cohomology of the variety. I will also explain the cluster algebra structure which exists on the mirrors, and give applications to quantum cohomology (relations, J-function). This is joint work with respectively K. Rietsch, and K. Rietsch and L. Williams.

Alexander Polishchuk
Dimension axiom in the algebraic approach to Fan-Jarvis-Ruan theory
Fan-Jarvis-Ruan theory is an analog of Gromov-Witten theory in which the role of the target space is played by a quasihomogeneous polynomial with isolated singularity. In the algebraic approach to this theory the correlators are constructed as maps induced on Hochschild homology by the Fourier-Mukai functors associated with some matrix factorizations. In this approach it is not clear why the obtained cohomology classes on moduli spaces of curves live in the dimensions predicted by the analytic approach of Fan-Jarvis-Ruan. I will explain the proof of this property for homogeneous polynomials.

Leonid Positselski
Matrix factorizations and derived categories of the second kind
The homotopy category of matrix factorizations (of a section of a line bundle on an algebraic variety) is naturally triangulated, but the conventional construction of the derived category is not applicable, as matrix factorizations have no cohomology modules or sheaves. I will explain the constructions of the absolute derived and coderived categories that can be applied instead, and discuss the triangulated categories that one obtains in this way, and the natural functors and equivalences between them. One of such equivalences, called the covariant Serre-Grothendieck duality, is particularly unfamiliar and interesting. I will describe its behavior with respect to the conventional and extraordinary inverse image functors.

Loek Spitz
An approach to the r-ELSV conjecture using the Givental-CEO-recursion equivalence.
In my talk, I will introduce the r-ELSV conjecture, due to D. Zvonkine, which generalizes the ELSV formula to the world of r-spin curves. That is, it equates certain generalizations of ordinary single Hurwitz numbers to some intersection numbers on the moduli space of spin curves. The conjecture is already quite old, but has resisted attempts to prove it. Recently, a new strategy has appeared that might prove useful. Thus, I will introductuce the Givental-CEO-recursion equivalence, and I will show why one could hope that this equivalence could be used to prove the r-ELSV conjecture. In fact, using this equivalence, one can show that the r-ELSV conjecture is equivalent to the so-called r-Bouchard-Marino conjecture, which introduces a whole new way one might try to go about proving the r-ELSV conjecture.

Dimitri Zvonkine
Cohomological relations on Mbar_{g,n} via 3-spin structures
We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n}. This family contains all relations known to this day and is expected to be complete and optimal. The construction uses the Frobenius manifold of the A_2 singularity, the 3-spin Witten class and the Givental-Teleman classification of semi-simple cohomological field theories. Joint with R. Pandharipande and A. Pixton.