Quiver indices and Abelianization from Jeffrey-Kirwan residues

Abstract

In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations. Using supersymmetric localization, the refined Witten index can be expressed as a residue integral with a specific contour prescription, originally due to Jeffrey and Kirwan, depending on the stability parameters. On the other hand, the physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by 'single-centered invariants'. In the case of quivers without oriented loops, we show that this decomposition naturally arises from the residue formula, as a consequence of applying the Cauchy-Bose identity to the vector multiplet contributions. For quivers with loops, the same procedure produces a natural decomposition of the single-centered invariants, which remains to be elucidated. In the process, we clarify some under-appreciated aspects of the localization formula. Part of the results reported herein have been obtained by implementing the Jeffrey-Kirwan residue formula in a public Mathematica code.

Multi-centered black holes, scaling solutions and pure-Higgs indices from localization

“…Nonetheless, we shall argue that the problem separates into a product of SU(m) non-Abelian dynamics associated to m nearly coincident Cartan variables, which can be treated using the usual Jeffrey-Kirwan residue prescription, and the Abelian dynamics of the center of motion in each SU(m) factor, which can be treated as in the previous section. One way of separating these variables is to apply the Cauchy-Bose identity for each of the U(N a ) vector multiplet determinants, as explained in [35], and then recombine the corresponding sum over permutations into a product of U(m) determinants. However, it is more economical to proceed as follows, similarly to the case of of Abelian quivers with multiple cycles in §3.4.2.…”

Pure-Higgs states from the Lefschetz-Sommese theorem

“…question to ask if the pure-Higgs states could fill the gap. 5 The results of this paper are a first step towards answering that question, but we leave an investigation of the growth of states for asymptotic values of the parameters for future work.…”

“…We'll refer to (2.6) as the refined index, since it corresponds to the refined Witten index of a quiver quantum mechanics in case K is a quiver moduli space, in that setting it also goes under the name of χ-genus [5,17]. From a more general geometric perspective the refined index is closely related to the holomorphic Euler number E :…”

“…The BPS states correspond to the supersymmetric groundstates of the quiver theory and depending on the coupling can be described geometrically in terms of the Coulomb or Higgs branch [1]. Recently various powerful techniques have been developed in the study of these supersymmetric groundstates, such as the Coulomb branch formula [4,5], localization [6][7][8], mutations [9,10] and others [11,12]. States on the Coulomb branch can be matched rather directly to a gravitational description, since both pictures describe JHEP11(2020)161 spatial bound states of dyonic centers [1,13,14].…”

“…Other quivers with oriented loops and non-abelian nodes have been considered in[5,10,11,22], except for the single example in section 3.2.2 of[11] they all have a different arrow structure than the quivers we study. The quivers discussed in section 6.1 and 6.2 of[22] match to a 2+1 node quiver in the limit b → c. Although this limit violates the assumption b < c made in[22] it correctly reproduces the Coulomb branch/induced cohomology part of the result we find in this work.4 See e.g [25].…”