Gauge Theories and Macdonald Polynomials

Abstract

We study the N=2 four-dimensional superconformal index in various interesting limits, such that only states annihilated by more than one supercharge contribute. Extrapolating from the SU(2) generalized quivers, which have a Lagrangian description, we conjecture explicit formulae for all A-type quivers of class S, which in general do not have one. We test our proposals against several expected dualities. The index can always be interpreted as a correlator in a two-dimensional topological theory, which we identify in each limit as a certain deformation of two-dimensional Yang-Mills theory. The structure constants of the topological algebra are diagonal in the basis of Macdonald polynomials of the holonomies.Comment: 57 pages, 6 figures; v2: references added and minor improvements; v3: typos correcte

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

“…The high temperature limit, when both basic parameters p and q tend to 1, maps indices to the hyperbolic integrals describing three-dimensional partition functions [24], while the low temperature limit, when p and q go to zero, degenerates the indices to Hilbert series counting the gauge invariant operators [25,26]. The intermediate case, when only one of JHEP07(2017)041 the base parameters p or q goes to zero, reduces to the Macdonald theory -a relation which was noticed first in [27] and later developed in detail in [28].…”

“…Applying the same transformation as above leads, in complete analogy, to (4.27) and 3 k=1 α k = 3 i=1 ω i . We have also performed the calculation for all the other classical groups, with the result being 28) where d G is the dimension of the group. This is again in full agreement with [9].…”

Compactifications of 5d SCFTs with a twist

“…In section 4 we use the known Hall-Littlewood index for class S theories [22][23][24] and properties of the compactification to conjecture an expression for the Hall-Littlewood index for the theories we presented. This is then checked for the cases where the conjectured 4d theory is known.…”

“…Therefore, as a simple test, we shall compromise on examining the N = 2 case, which is the first new case. 3 Also we shall compute the index in the simplified Hall-Littlewood limit [22] (see also appendix A). We expand the index in a power series in τ , evaluating to order τ 5 .…”

“…As the Higgs branch is invariant under quantum corrections the 4d Higgs branch should be identical to a subspace of the original invariant under the twist. For the theory at hand, the Hall-Littlewood index, which is made from operators belonging to the Hall-Littlewood chiral ring, should be identical to the Hilbert series of the Higgs branch [22,37]. Thus symmetric combinations of Higgs branch operators of the 5d theory should descend to Higgs branch operators of the 4d theory, and thus to part of the Hall-Littlewood chiral ring.…”