First Order String Theory and the Kodaira-Spencer Equations. II

TL;DR

This paper investigates how the first-order bosonic string theory's correlation functions impose the Kodaira-Spencer equations on target-space Beltrami differentials, linking conformal invariance with complex structure deformations.

Contribution

It explicitly derives the Kodaira-Spencer equations from correlation function divergences in first-order string theory, connecting world-sheet conformal invariance with target-space complex structure conditions.

Findings

Correlation functions' divergences lead to Kodaira-Spencer equations.

Consistency with conformal invariance requires target-space Beltrami differentials to satisfy these equations.

Explicit checks performed for three-point and four-point correlators.

Abstract

The first-order bosonic string theory, perturbed by primary operator, corresponding to the deformation of target-space complex structure is considered. We compute the correlation functions in this theory and study their divergencies. It is found, that consistency of these correlation functions with the world-sheet conformal invariance requires the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. This statement is checked explicitly for the three-point and four-point correlators, containing one probe operator. We discuss the origin of these divergences and their relation with beta-functions or effective action and polyvertex structures in BRST approach.