Shift versus Extension in Refined Partition Functions

Understand

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.