Schedule of Workshop 1: Mirror Symmetry

Marco Aldi: A-branes and twisted products
The main goal of this talk is to describe an approach to the study of A-branes on certain nilmanifolds. If time permits, applications to homological mirror symmetry (work in progress with Oren Ben-Bassat) will be discussed.

Dennis Auroux: Mirror Symmetry for blowups
In this talk, we will report on joint work with Abouzaid and Katzarkov about mirror symmetry for blowups. Namely, we first describe how to construct a special Lagrangian torus fibration on the blowup of a toric variety X along a codimension 2 subvariety S contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model. This construction allows one to recover geometrically the predicted mirrors in various slightly unconventional settings: pairs of pants, curves of arbitrary genus, etc.

Alessio Corti: On the Gromov-Witten theory of toric stacks
I will give an update of work in progress with Coates, Iritani and Tseng.

Lothar Goettsche: Holomorphic Euler characteristics of line bundles on moduli spaces of sheaves on surfaces
This is report on joint works with Nakajima-Yoshioka and with Zagier. Let (X,H) be a polarized algebraic surface. We consider moduli spaces M of rank two H-stable torsion-free sheaves on X, and so called determinant line bundles L on them and their holomorphic Euler characteristic χ(M,L). M and χ(M,L) depend on H via a system of walls and chambers. Using the Nekrasov partition function we determine a wallcrossing formula for the χ(M,L) in terms of theta functions. In the case that X is a rational surface we use this formula together with calculations with theta functions, to obtain generating functions for the χ(M,L) as explicit rational functions. This result is related to the strange duality conjecture of Le Potier.

Manfred Herbst: Phases of N=2 theories with boundaries
We consider B-type D-branes in N=2 supersymmetric sigma models. The latter have a rich phase structure over Kaehler moduli space. In particular, non-geometric phases containing Landau-Ginzburg models come along with geometric ones, corresponding to smooth CY manifolds. We develop an effective tool to transport D-branes between different phases. In that way we can for instance relate the derived category of coherent sheaves on a CY-hypersurface to matrix factorizations of the corresponding hypersurface polynomial.

Maxim Kontsevich: Gromov-Witten invariants and homological mirror symmetry
I will explain how to extract the Gromov-Witten invariants of a compact symplectic manifold from its Landau-Ginzburg dual. The algorithm is based on B-model for Z/2 graded Calabi-Yau categories, the deformation theory of Cohomological Field Theories, and the theory of nc Hodge structures developed recently in a joint paper with L. Katzarkov and T. Pantev.

Marcos Marino: Topological strings, instantons and resurgent functions
The generating functionals of Gromov-Witten invariants at genus g can be put together in a total "free energy" which can be regarded as a formal power series in the so-called string coupling constant. In this talk I will discuss some analyticity properties of the free energy and extend old ideas on the large order behavior of perturbation theory to topological string theory. In particular, I will argue that
1) The power series in the string coupling is asymptotic, and each term diverges like (2g)! at large g and fixed complexified Kahler parameter.
2) This asymptotic series can be promoted to a so-called "trans-series expansion." In particular, it has exponentially small corrections which are not visible in perturbation theory. These corrections can be computed explicitly in some topological string models. Physically, they correspond to nonperturbative effects in the string coupling constant due to spacetime instantons.
3) The resulting topological string trans-series is an example of a "resurgent function", a concept introduced and studied by Jean Ecalle, Ovidiu Costin and others. In particular, the precise large g behavior of the original asymptotic, formal power series can be determined in terms of the spacetime instanton data. This leads to a new set of conjectures on the asymptotic behavior of generating functionals in Gromov-Witten theory.

Yan Soibelman: BPS states for Calabi-Yau categories
This is a joint work with Maxim Kontsevich devoted to the counting of stable objects in 3d Calabi-Yau categories.

Balazs Szendroi: Aspects of the partition function of the conifold
I will recall the construction of a version of the (Donaldson-Thomas) partition function of local P¹ from its quiver model. I will then discuss extensions of this picture, including a relation to quiver mutations (work of Jafferis-Moore) and a computation of a refined partition function (joint work with Bridgeland) which makes a connection to the refined topological vertex of Iqbal-Kozcaz-Vafa.