Equivariant holomorphic anomaly equation

TL;DR

This paper proves holomorphic anomaly equations for equivariant Gromov-Witten theories of local P2 and P3, extending previous work by removing equivariant variable specializations and generalizing to full equivariant settings.

Contribution

It extends prior results by establishing holomorphic anomaly equations in full equivariant settings for local P2 and P3, and generalizes to equivariant formal quintic theory.

Findings

Proved holomorphic anomaly equations for local P2 and P3.

Extended results to full equivariant settings.

Generalized to equivariant formal quintic theory.

Abstract

In [16] the fundamental relationship between stable quotient invariants and the B-model for local P2 in all genera was studied under some specialization of equivariant variables. We generalize the argument of [16] to full equivariant settings without the specialization. Our main results are the proof of holomorphic anomaly equations for the equivariant Gromov-Witten theories of local P2 and local P3. We also state the generalization to full equivariant formal quintic theory of the result in [17].