Abstract
We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to the strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.
Highlights
In this paper we report results about a relation between N=2 supersymmetric SU(2) QCD with Nf =4 hypermultiplets and asymptotic spectral solutions of stationary Schrodinger equation with the Darboux-Treibich-Verdier (DTV) potential [1,2,3,4]
The partition function of gauge theory with surface operator is computed using localization formula developed for instanton computation, we show that the instanton partition function precisely matches with the asymptotic spectral solution when parameters appearing on both sides are correctly identified, their relation is given by (3.15) and (3.31)
Among the relations we studied between supersymmetric SU(2) gauge theory models and Schrodinger equation with periodic potentials (Mathieu, Lame and Heun equations), the case studied in this paper is the richest one of this class
Summary
In this paper we report results about a relation between N=2 supersymmetric SU(2) QCD with Nf =4 hypermultiplets and asymptotic spectral solutions of stationary Schrodinger equation with the Darboux-Treibich-Verdier (DTV) potential [1,2,3,4]. This is an example of the connection between the Omega background deformed N=2 supersymmetric gauge theories and quantum integrable models discovered by Nekrasov and Shatashvili [5]. For the N=2∗ theory, q appears as the nome of the Weierstrass elliptic function; for the Nf =4 theory, q appears as the modulus of the Jacobian elliptic functions
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