Schur indices, BPS particles, and Argyres-Douglas theories

Abstract

Abstract:We conjecture a precise relationship between the Schur limit of the superconformal index of four-dimensional N = 2 field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the special case of free field theories, N = 2 QED, and SU(2) gauge theory coupled to fundamental matter. Assuming the validity of our proposal, we compute the Schur index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras. Based on our results we propose that the chiral algebra of the generalized Argyres-Douglas theory (A k−1 , A N −1 ) with k and N coprime, is the vacuum sector of the (k, k + N ) W k minimal model, and that the Schur index is the associated vacuum character.

Superconformal index, BPS monodromy and chiral algebras

“…The case N = −1, the Schur case, is a special case of this which was already conjectured and checked in some examples in [9].…”

“…For N = −1 this map is expected to be especially simple: Tr M(q) −1 gets identified with the SCFT index in the Schur limit which is independent of p. Strong evidence for this 9 This version of the quantum KS formula holds under two genericity assumptions: 1) the only massless states are IR-free photons, and 2) massive BPS states with equal BPS phases, arg Zγ = arg Z γ , are mutually local i.e. γ, γ = 0.…”

“…In later work [8] it was found that, by considering operators contributing to the Schur limit of the superconformal index, one obtains chiral algebras in 2d, with a very specific central charge: c 2d = −12c 4d , where c 2d is the 2d central charge of the Virasoro algebra and c 4d is the c-function of the 4d SCFT. Motivated by this, in a recent work [9] it was shown that the Schur index specialization is equal to the trace of the inverse monodromy operator Tr M(q) −1 .…”

“…In order to connect the monodromy traces and usual SCFT invariants, one has to treat all BPS states of T IR on the same footing, including the massless ones which are not taken into account in the KS product M (q). This gives an explanation of the prescription suggested in [9]. Then we redefine the monodromy operator by inserting the factors which correspond to the r massless photons, seen as chargeless BPS vector multiplets,…”

“…identification was given in [9]. However, even for the Schur case we encounter a difficulty because the Tr M(q) −1 is not always well defined, as computed from the BPS spectrum.…”