N=2 Supersymmetric QCD and Integrable Spin Chains: Rational Case N_f < 2N_c

TL;DR

The paper proposes that 4d N=2 SQCD with fundamental matter and N_f<2N_c can be solved by mapping its Seiberg-Witten data to an inhomogeneous sl(2) XXX spin chain of length N_c, with the spectral curve arising from the monodromy of the spin-chain Lax product. The authors formulate the deformation of the Toda-chain description to a spin-chain framework, showing that the SW curve takes the form Tr T_{N_c}(lambda)=P_{N_c}(lambda|h)+R_{N_c-1}(lambda|m) and det T_{N_c}(lambda)=Q_{2N_c}(lambda), where the inhomogeneities lambda_i and spins K_i encode the quark masses m_gamma. Through explicit low-rank examples, they demonstrate that the XXX chain reproduces the SW low-energy data and commuting Hamiltonians, establishing a concrete integrable-system realization of the SW problem in this regime and suggesting an elliptic generalization for N_f=2N_c. This work thus links 4d SUSY gauge theory with fundamental matter to 1d integrable spin chains, providing a new computational bridge between Seiberg-Witten theory and lattice integrable models.

Abstract

The form of the spectral curve for $4d$ $N=2$ supersymmetric Yang-Mills theory with matter fields in the fundamental representation of the gauge group suggests that its $1d$ integrable counterpart should be looked for among (inhomogeneous) $sl(2)$ spin chains with the length of the chain being equal to the number of colours $N_c$. For $N_f < 2N_c$ the relevant spin chain is the simplest $XXX$-model, and this identification is in agreement with the known results in Seiberg-Witten theory.