The holomorphic anomaly for open string moduli

TL;DR

This paper extends the holomorphic anomaly equations in topological string theory to include open string moduli, analyzing their dependence and boundary structure using path integral methods and moduli space compactification.

Contribution

It completes the holomorphic anomaly equations for open string moduli, generalizing BCOV analysis to open strings with boundaries and Wilson lines.

Findings

Derived the complete open string holomorphic anomaly equations.

Analyzed anti-holomorphic dependence on open and closed moduli.

Structured the anomaly equations on boundary components of moduli space.

Abstract

We complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. We obtain the complete system by standard path integral arguments generalizing the analysis of BCOV (Commun. Math. Phys. 165 (1994) 311) to strings with boundaries. We study both the anti-holomorphic dependence on open moduli and on closed moduli in presence of Wilson lines. By providing the compactification a' la Deligne-Mumford of the moduli space of Riemann surfaces with boundaries, we show that the open holomorphic anomaly equations are structured on the (real codimension one) boundary components of this space.