Abstract
The counting of BPS states in four-dimensional \mathcal{N}=1𝒩=1 theories has attracted a lot of attention in recent years. For superconformal theories, these states are in one-to-one correspondence with local operators in various short representations. The generating function for this counting problem has branch cuts and hence several Cardy-like limits, which are analogous to high-temperature limits. Particularly interesting is the second sheet, which has been shown to capture the microstates and phases of supersymmetric black holes in AdS_55. Here we present a 3d Effective Field Theory (EFT) approach to the high-temperature limit on the second sheet. We use the EFT to derive the behavior of the index at orders \beta^{-2},\beta^{-1},\beta^0β−2,β−1,β0. We also make a conjecture for O(\beta)O(β), where we argue that the expansion truncates up to exponentially small corrections. An important point is the existence of vector multiplet zero modes, unaccompanied by massless matter fields. The runaway of Affleck-Harvey-Witten is however avoided by a non-perturbative confinement mechanism. This confinement mechanism guarantees that our results are robust.
Highlights
It is a standard fact that every 4d N = 1 theory with a continuous R-symmetry can be studied on the spatial manifold S3 while preserving four supercharges [1,2,3,4], see the reviews [5, 6]
The potential grows like V eff(u) ∼ u2/β2 around the origin which in the effective field theory on S1 × 3 we interpret by saying that the Chern-Simons term lifts the Coulomb branch. This renders the Effective Field Theory (EFT) approach more robust on the second sheet than on the first sheet, where the existence of a minimum at the origin relies on some additional assumptions which we reviewed above
The effective field theory techniques we use allow us to establish that there is a local minimum of V eff(u) at u = 0 and they allow us to make predictions for the corresponding contributions to the index at each order in β
Summary
The effective field theory techniques we use allow us to establish that there is a local minimum of V eff(u) at u = 0 and they allow us to make predictions for the corresponding contributions to the index at each order in β These facts are model independent and presumably hold for non-Lagrangian theories. Since the zero modes are trivially gapped, all we need to do is to recompute the integrals (2.4)–(2.7) below, and this will automatically provide a prediction for the β−2, β−1, β0 terms This is conceptually different from the first sheet, where there is typically a massless theory at the origin and the β0 term in the asymptotic expansion has to be studied on a case-by-case basis. As this paper was being completed, the preprint [67] discussing related topics appeared
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