Instantons, Topological Strings and Enumerative Geometry

TL;DR

This paper explores the deep connections between instanton counting in supersymmetric gauge theories, topological string theory, and enumerative invariants, revealing insights into black hole entropy and wall-crossing phenomena.

Contribution

It provides a detailed analysis of how instanton counting relates to enumerative invariants and black hole entropy across various dimensions within topological string theory.

Findings

Instanton counting relates to black hole entropy calculations.

Wall-crossing phenomena affect enumerative invariants like Donaldson-Thomas and Gromov-Witten.

Moduli spaces of sheaves and their Euler characteristics are key to understanding these connections.

Abstract

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four and two dimensions which naturally arise in the context of topological string theory on certain non-compact threefolds. We describe how the instanton counting in these gauge theories are related to the computation of the entropy of supersymmetric black holes, and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.