Nekrasov prepotential with fundamental matter from the quantum spin chain

TL;DR

This paper verifies the conjectured relation between Nekrasov functions and quantum integrable systems by checking the case of $SU(N_c)$ gauge theories with fundamental matter, demonstrating the equivalence of quantum periods from different integrable models.

Contribution

It extends previous checks of Nekrasov functions to include gauge theories with fundamental hypermultiplets, establishing the connection with the Baxter equation of the spin chain and the Gaudin system.

Findings

Baxter equation reproduces quantum periods for $SU(N_c)$ with matter.

Confirmed the Nekrasov function and integrable system correspondence in new case.

Showed equivalence between spin chain and Gaudin system quantum periods.

Abstract

Nekrasov functions were conjectured in \cite{Mironov:2009uv} to be related to exact Bohr-Sommerfeld periods of quantum integrable systems. This statement was thoroughly checked for the case of the pure $SU(N_c)$ gauge theory in \cite{Mironov:2009dv} and \cite{Popolitov:2010bz}. Here we successfully perform a set of checks in the case of gauge group $SU(N_c)$ with additional $N_f$ fundamental hypermultiplets. We show that the Baxter equation for the spin chain gives the same quantum periods as the one for the Gaudin system in this case.