# BPS state counting, stability structures and derived algebraic geometry

**Summer School, University of Hamburg, 31.8.–4.9.2009**

## Abstracts

### Loeser: Introduction to the motivic Milnor fiber

Talk 1: After a review of the topological Milnor fiber, we shall present the construction of the motivic Milnor fiber and its relation with Hodge spectrum.Talk 2: We shall construct the motivic convolution operator and state motivic Thom-Sebastiani. We shall present three different generalizations of motivic Thom-Sebastiani obtained in joint work with Guibert and Merle.

### Macri: Derived categories and quivers

We will discuss some examples for Stellari's talk by using (and introducing) quiver representations.### Neitzke

Talk 1: What is a BPS state?Abstract: I will describe the notion of "BPS state" as it is understood by physicists in terms of supersymmetric quantum field theory, how generalized Donaldson-Thomas invariants are physically defined in terms of BPS states, and what "wall-crossing" means in this context.

Talk 2: Hitchin systems and quantum field theory.

Abstract: I will describe the general relation between supersymmetric quantum field theories and integrable systems, and a specific example of such a quantum field theory which corresponds to a Hitchin system.

Talk 3: BPS states and wall-crossing in Hitchin systems.

Abstract: I will give a description of the hyperkahler structure on the Hitchin system, in which the BPS states of the corresponding quantum field theory play a crucial role. By investigating this description closely we will see the wall-crossing formula of Kontsevich and Soibelman emerge naturally.

### Reineke

Talk 1: Poisson automorphisms and quiver moduliThe definition of quiver representations, their moduli spaces, and their Hall algebras is reviewed. Formulas for the cohomology of quiver moduli are derived from identities in the Hall algebra. Then a map from part of the Hall algebra to a group of Poisson automorphisms of a formal power series ring is constructed, expressing certain identities in this group via quiver moduli.

Talk 2: Functional equations for quiver moduli and integrality

The study of the cohomology of quiver moduli is continued, deriving functional equations relating their Euler characteristic. Via some elementary number theory, this is used to derived integrality properties of some wall-crossing DT-invariants of Kontsevich and Soibelman.

### Soibelman: Lecture course on motivic Donaldson-Thomas invariants

Abstract: Aim of my lectures is to explain our joint work with Maxim Kontsevich in which we suggested a general approach to Donaldson-Thomas type invariants ("count of BPS states and refined BPS states" in the language of physics). It is based on the ideas of non-commutative derived algebraic geometry and motivic integration.Talk 1: Motivation, A-infinity and Calabi-Yau categories with ind-constructible structure. Stability conditions and the counting problem.

Talk 2: Motivic Hall algebras, orientation data, motivic weights and motivic Donaldson-Thomas invariants.

Talk 3: Quasi-classical limit and integrality conjecture, numerical DT-invariants and stability data on graded Lie algebras and wall-crossing formulas.

Talk 4: Cohomological Hall algebras and motivic DT-invariants. If time permits, I am going to discuss the relation to complex integrable systems, cluster transformations and tropical geometry.