Schedule

All lectures in HG G43 [HWZ]

Monday [07.01.2019]

13:30-14:30 Caldararu I
14:45-15:45 Tu I
16:30-17:30 Sheridan I

Tuesday [08.01.2019]

10:30-11:30 Shadrin
13:30-14:30 Sheridan II
15:00-16:00 Caldararu II
16:30-17:30 Tu II

Wednesday [09.01.2019]

10:30-11:30 Ganatra I
13:30-14:30 Sheridan III
15:00-16:00 Caldararu III
16:30-17:30 Ruan

Thursday [10.01.2019]

15:00-16:00 Ganatra II
16:30-17:30 Tu III

Titles and Abstracts

A. Caldararu

Talk I: Introduction to Costello's invariants

Abstract: I will give an overview of the main moving parts involved in the definition of Costello's invariants, under one simplifying assumption. Main topics: (1) the Weyl algebra, Fock module, and BV algebra associated to a representation of the category of annuli; (2) solutions of the Maurer-Cartan equation in BV algebras; (3) the BV algebra structure on singular chains on moduli spaces of curves and the distinguished solution of the MC equation (string vertices); (4) the abstract categorical Gromov-Witten potential as an element of a Fock space; (5) using a splitting of the Hodge-de Rham spectral sequence to convert the abstract GW potential to a concrete one.

Talk II: Costello's invariants (the real deal)

Abstract: In the previous lecture I used a fundamental, unrealistic simplification of Costello's theory, namely I pretended that the TCFT associated to a cyclic A_infty algebra could be defined on chains with no inputs, only outputs. In the second lecture I will remove this simplifying assumption, using a certain Koszul resolution also suggested by Costello (unpublished). Along the way I will give a completely precise definition of string vertices (down to signs!) which is amenable to computer calculations. This approach allows us to give a new, precise definition of the concrete categorical GW potential, but several non-explicit choices need to be made. If time allows, I will discuss how we hope to use the Givental group action on the space of splittings to remove these non-explicit choices to get a computable categorical GW potential.

Talk III: The elliptic curve example

Abstract: We put together all the moving parts we have developed in the previous lectures in order to calculate the g=1, n=1 B-model GW invariant of the family of elliptic curves, using the Hodge-de Rham splitting obtained from monodromy around the cusp point. If time allows we will also discuss similar calculations for the category of A_n singularities arising from the category of matrix factorizations.

S. Ganatra

Title: Open-closed maps and Calabi-Yau structures I and II

Abstract: Open-closed TFTs are classified (according to Costello, Kontsevich-Vlassopoulous, Lurie) by A-infinity categories equipped with Calabi-Yau (CY) structures. There are two types of CY structures of relevance, "smooth CY" structures classify TFTs with at least one output and "proper CY" structures classify TFTs with at least one input. We will review this formalism, show how a version of the open-closed map allows one to equip Fukaya categories with a geometrically meaningful CY structure of either type (under suitable hypotheses), and describe some initial applications.

Y. Ruan

Title: The structure of higher genus Gromov-Witten theory of quintic 3-folds

Abstract: One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten theory of compact Calabi-Yau 3-fold. There have been a collection of remarkable conjectures from physics for the so-called 14 one-parameter models, the simplest compact Calabi-Yau 3-folds similar to the quintic 3-folds. The backbone of this collection of conjectures are four structural conjectures: (1) Yamaguchi-Yau finite generation; (2) Holomorphic anomaly equation; (3) Orbifold regularity and (4) Conifold gap condition. In the talk, I will present a proof of first three conjectures for quintic 3-folds. This is a joint work with F. Janda and S. Guo. Our proof is based on certain localization formula from log GLSM theory developed by Q. Chen, F. Janda and myself.

S. Shadrin

Title: Cohomological field theories and BV algebras

Abstract: I'll discuss how a version of the Givental group action on cohomological field theories emerges from the structure of a differential graded BV algebra with a homotopically trivial BV operator. There are several different approaches to this general statement, but they all work so far only in genus 0, and I'll try to indicate the principle difficulties of their generalisations to higher genera. It also has a surprising and a very deep generalisation, where all the involved structures become non-commutative. If I'll have sufficient time, I'll talk about it as well. The talk is based on my joint papers with Anton Khoroshkin and Nikita Markarian and Vladimir Dotsenko and Bruno Vallette.

N. Sheridan

Talk I: Hodge-theoretic mirror symmetry

Talk II: VHS from categories

Talk III: Open-closed maps

Abstracts for I-III: I will explain in detail how to compute genus-zero Gromov-Witten invariants of a Calabi-Yau manifold from its Fukaya category. The main steps are to show that the periodic cyclic homology of the Fukaya category carries a natural structure of a Variation of Hodge Structures (VHS); and to identify this structure with the quantum VHS, from which the rational Gromov-Witten invariants can be extracted. This is based on joint work with Ganatra and Perutz.

J. Tu

Talk I: Categorical Saito Theory I

Abstract: We present the construction of an explicit cyclic minimal model for the category of matrix factorizations associated with a potential function W on C^N with an isolated singularity at origin. The key observation is to use Kontsevich's deformation quantization technique. As an application of this new technology, we prove the analogue of Caldararu's conjecture for matrix factorizations.

Talk II: Categorical Saito Theory II

Abstract: The proof of the analogue of Caldararu's conjecture for matrix factorizations follows from a comparison result between Variation of Semi-infinite Hodge Structures. In this talk, we describe a proof of this comparison using Tsygan formality map which looks like a B-model open-closed map. If time permits, we shall also present some recent progress on categorical Saito theory of equivariant matrix factorizations (specifically orbifold LG-models).

Talk III: Givental's formula and Costello's higher genus categorical invariants

Abstract: We describe some work in progress to use Givental's formula to explicitly describe Costello's higher genus categorical Gromov-Witten invariants. Almost no axioms of GW theory are known to hold for Costello's categorical invariants, except the dimension axiom. Nevertheless, we argue that, in the categorical context, the holomorphic anomaly equation is a formal consequence of the Givental's formula.