Shift versus Extension in Refined Partition Functions

TL;DR

This paper investigates how shifting mass parameters in N=2 gauge theory affects the partition function's symmetry and the applicability of the holomorphic anomaly equation, revealing that a specific shift restores symmetry and simplifies the analysis.

Contribution

It demonstrates that a particular shift of mass parameters restores Z_2 symmetry and removes the extension in the holomorphic anomaly equation for the partition function.

Findings

Shift of mass parameters restores Z_2 symmetry.

The extension in the holomorphic anomaly equation is removed.

Connections to other theoretical contexts are discussed.

Abstract

We have recently shown that the global behavior of the partition function of N=2 gauge theory in the general Omega-background is captured by special geometry in the guise of the (extended) holomorphic anomaly equation. We here analyze the fate of our results under the shift of the mass parameters of the gauge theory. The preferred value of the shift, noted previously in other contexts, restores the Z_2 symmetry of the instanton partition function under inversion of the Omega-background, and removes the extension. We comment on various connections.