Cohomological vertex operators

TL;DR

This paper explores the mathematical structure of vertex operators in string theory on Calabi-Yau manifolds, relating them to Ext sheaf cohomology and homological extensions of B-branes.

Contribution

It establishes a precise correspondence between string vertex operators and Ext sheaf cohomology groups, and defines correlation functions for these operators in the derived category framework.

Findings

Vertex operators correspond to elements of Ext sheaf cohomology groups.

Correlation functions for these vertex operators are explicitly defined.

Strings with different ghost numbers relate to obstructions in extensions.

Abstract

Given a Calabi-Yau manifold and considering the $B$-branes on it as objects in the derived category of coherent sheaves, we identify the vertex operators for strings between two branes with elements of the cohomology groups of Ext sheaves. We define the correlation functions for these general vertex operators. Strings stretching between two coherent sheaves are studied as homological extensions of the corresponding branes. In this context, we relate strings between different pairs of branes when there are maps between these branes. We also interpret some strings with ghost number $k$ as obstructions for lifts or extensions of strings with ghost number $k-1$.