Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds

TL;DR

This paper constructs holomorphic generating functions for invariants counting semistable coherent sheaves on Calabi-Yau 3-folds, revealing a unique, PDE-satisfying structure linked to string theory and mirror symmetry.

Contribution

It introduces a method to combine conjectural invariants into holomorphic functions that satisfy a PDE, suggesting a new geometric and physical framework.

Findings

Holomorphic generating functions are essentially uniquely determined by continuity and holomorphicity.

These functions satisfy a specific partial differential equation.

The structure implies a flat connection in an infinite-dimensional Lie algebra context.

Abstract

Let X be a Calabi-Yau 3-fold, T=D^b(coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions Z on T. It is conjectured that one can define rational numbers J^a(Z) for Z in Stab(T) and a in the numerical Grothendieck group K(T) generalizing Donaldson-Thomas invariants, which `count' Z-semistable (complexes of) coherent sheaves on X in class a, and whose transformation law under change of Z is known. This paper explains how to combine such invariants J^a(Z), if they exist, into a family of holomorphic generating functions F^a:Stab(T) --> C. Surprisingly, requiring the F^a to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.