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Nekrasov prepotential with fundamental matter from the quantum spin chain

Yegor Zenkevich

TL;DR

This work tests the Nekrasov–Seiberg correspondence for $SU(N_c)$ gauge theories with $N_f$ fundamental hypermultiplets by solving the Baxter equation of the XXX spin chain and comparing quantum deformed Seiberg–Witten periods to the Nekrasov prepotential. The authors derive a perturbative solution in $\hbar$, construct a differential operator $\hat{\mathcal{O}}$ that maps classical periods to their quantum-corrected counterparts, and validate the matching of $A$- and $B$-periods against $\mathcal{F}_{Nek}$ up to $O(\hbar^2)$ and $\Lambda^{2N_c-N_f}$, including explicit one-instanton checks in several cases. They also compute the exact one-loop prepotential for $SU(2)$ with up to four hypermultiplets, showing that the B-periods are invariant under different placements of masses between the Baxter equation's two exponential terms and agree with perturbative Nekrasov results to all orders in $\hbar$. Together, these results support the quantum–integrable systems interpretation of the Nekrasov prepotential in the presence of fundamental matter and encourage further generalization beyond the cases studied. The approach offers a concrete bridge between spin-chain Baxter equations and the Seiberg–Witten framework, with potential implications for the AGT correspondence and exact quantization.

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.

Nekrasov prepotential with fundamental matter from the quantum spin chain

TL;DR

This work tests the Nekrasov–Seiberg correspondence for gauge theories with fundamental hypermultiplets by solving the Baxter equation of the XXX spin chain and comparing quantum deformed Seiberg–Witten periods to the Nekrasov prepotential. The authors derive a perturbative solution in , construct a differential operator that maps classical periods to their quantum-corrected counterparts, and validate the matching of - and -periods against up to and , including explicit one-instanton checks in several cases. They also compute the exact one-loop prepotential for with up to four hypermultiplets, showing that the B-periods are invariant under different placements of masses between the Baxter equation's two exponential terms and agree with perturbative Nekrasov results to all orders in . Together, these results support the quantum–integrable systems interpretation of the Nekrasov prepotential in the presence of fundamental matter and encourage further generalization beyond the cases studied. The approach offers a concrete bridge between spin-chain Baxter equations and the Seiberg–Witten framework, with potential implications for the AGT correspondence and exact quantization.

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 gauge theory in \cite{Mironov:2009dv} and \cite{Popolitov:2010bz}. Here we successfully perform a set of checks in the case of gauge group with additional 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.

Paper Structure

This paper contains 15 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Solving the Baxter equation
  3. The differential operator
  4. A-periods to the order $\Lambda^{2N_c-N_f}$
  5. Nekrasov functions
  6. Checks up to $\hbar^2$
  7. Zero instanton approximation
  8. One instanton approximation
  9. Exact one-loop part in the $SU(2)$ case
  10. Pure gauge theory
  11. One hypermultiplet
  12. Two hypermultiplets
  13. Three hypermultiplets
  14. Four hypermultiplets
  15. Conclusions

Figures (1)

  • Figure 1: The B-contour $\mathcal{C}$ consisting of two stretches $\mathcal{C}_1$ and $\mathcal{C}_2$ encircles the cut shown in the complex $\xi$ plane.