Shift versus Extension in Refined Partition Functions

TL;DR

The paper investigates how refined partition functions in the general $\Omega$-background are governed by special geometry through holomorphic anomaly equations, including the extended version. A mass shift $m \to m + \frac{\epsilon_1+\epsilon_2}{2}$ is shown to restore $Z_2$ symmetry and remove the extension, with a one-parameter family $Z_\xi$ revealing that $\xi=0$ yields a symmetric, standard holomorphic anomaly, while $\xi eq0$ preserves the extended structure. The amplitudes obey the extended holomorphic anomaly for all $\xi$, with boundary data from monopole/dyon points fixing the leading behavior, and the symmetry restoration (notably for $N_f=2$ at $\xi=0$) provides a compelling consistency check. A string-theoretic interpretation via geometric engineering links the shift to a shifted Kähler modulus and an open–closed string background, suggesting that the open-closed wavefunction with parameter $\xi$ is equivalent to a closed-string wavefunction with mass shifted by $\delta m \propto \xi$, thereby connecting refined amplitudes, the refined topological vertex, and brane backgrounds in a unified framework.

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.