Abstract
We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R^2 x S^1. We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.
Highlights
Supersymmetric gauge theories provide a rich source of inspiration for various branches of mathematics
We study the Nekrasov partition function of the five dimensional U(N ) gauge theory with maximal supersymmetry on R4 ×S1 in the presence of codimension two defects
The interplay between supersymmetric gauge theories and mathematics is enhanced by introducing defects that preserve some amount of supersymmetry
Summary
Supersymmetric gauge theories provide a rich source of inspiration for various branches of mathematics. We will show that the twisted chiral ring relations are equivalent to the spectral curve of an associated classical N -body integrable system, known as the complex trigonometric Ruijsenaars-Schneider (RS) system. This provides a reformulation of the equivariant quantum K-theory of the cotangent bundle to a complete flag variety, via the Nekrasov-Shatashvili correspondence [7, 8]. The Seiberg-Witten curve of the 5d U(N ) N = 2 supersymmetric gauge theory is known to correspond to the spectral curve of the N -body elliptic Ruijsenaars-Schneider system [15] This is a deformation of the trigonometric RS system by an additional complex parameter Q.
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