Defects and quantum Seiberg-Witten geometry

Abstract

Abstract:We study the Nekrasov partition function of the five dimensional U(N ) gauge theory with maximal supersymmetry on R 4 ×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 × 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.

Mirror theories of 3d $$ \mathcal{N} $$ = 2 SQCD

“…m j denote of course real masses for the SU(N + k) symmetry in theory (A) and FI parameters in theory (B). Note that (4.47) was proven in [23] for a general value of m A . 5 Equation (4.47) implies the equality between the following two parition functions (as in (4.47)):…”

“…Exploiting the fact that (2.28) is true for generic (complex) values of m A (as was proven in [23]), we can immediately derive the equality of S 3 b partition functions for our deformed T (SU(N )) theory and its mirror since this simply follows from a specialization of (2.28): on one side the monopole superpotential breaks the topological symmetries and H − C to the diagonal subgroup, therefore all the ξ i parameters and m A should be identified. This sets the real parts of all ξ i and m A to a single parameter which we shall denote by ξ.…”

“…This forces us to mix the R-symmetry with a certain combination of the Cartan components of the (now broken) SU(N ) Higgs branch (HB) symmetry. Specifically, the generator which replaces 23) and the trial R-symmetry becomes…”

Supersymmetric Casimir energy and the anomaly polynomial

“…In the case of 6d N = (2, 0) SCFTs, the relevant calculations in the "chiral algebra" limit are presented in [71]. For general parameters, it may also be possible to extend the 5d partition function computations of section 3 to include defects using results from [72,73]. The 4d N = 2 superconformal index in the presence of various kinds of defects has also been computed in [54,[74][75][76], which may provide a starting point.…”

Refined 3d-3d correspondence

“…. , h N } are not in general the eigenvalues of H. The holonomy matrix H can be identified with the Lax matrix of the complex N -body trigonometric Ruijsenaars-Schneider model [24,25] and therefore methods from classical JHEP04(2017)170 integrable systems are very useful. In particular, a convenient set of invariant functions on M is obtained by expanding the Lax determinants…”