Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

Abstract

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

Large N superconformal indices for 3d holographic SCFTs

“…These developments have sparked a flurry of activity centered at understanding the SCI of N = 4 SYM theory and its generalizations to other N = 1 holographic SCFTs that provide the microscopic origin of the entropy of dual AdS 5 black holes. A remarkable feature of the SCI in 4d N = 1 holographic SCFTs is that it allows for an all order expansion in a Cardy-like limit in which the size of the S 1 is taken to be much smaller than the radius of the S 3 , see [14,15] as well as [16][17][18][19][20][21][22][23] for further discussion on the 4d SCI in this Cardy-like limit. In the large N limit, on the other hand, the exact evaluation of the 4d SCI becomes much more involved due to multiple competing saddles; see [24][25][26][27][28][29][30][31][32][33][34][35][36][37] for a selection of recent results on various aspects of the SCI in this large N limit.…”

“…The generic Bethe potential (2.31) and the generic BA formula for the TTI (2.27) for the ABJM theory can be read off from the ones for the N 0,1,0 theory, (3.1) and (3.2), simply by setting r 1 = r 2 = 0. One can check that the resulting expressions match the Bethe potential 16 and the TTI of the ABJM theory in [27] respectively after the identifications…”

“…) 16 To match the Bethe potentials up to gauge holonomy independent terms, we have used the inversion formula (A.4) under the assumption 0…”

Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography

“…1 Indeed, (1.5) maps into the formula given there if one makes the replacement φ I Q I → φQ R , where Q R denotes the R-charge and φ = r I φ I is the R-symmetry chemical potential. 2 One way to prove (1.5) is therefore to extend the three-dimensional effective field theory approach of [28,29] to the flavoured case: the twisted supersymmetric reduction on a small Euclidean time circle discussed there can also be performed in the presence of background vector multiplets coupling to flavour currents; this leads to additional supersymmetric Chern-Simons contact terms in three dimensions [44,45]. Here we choose a different route and present a quicker, though more formal, way to reach the same result, which extends the equivariant integration of the anomaly polynomial presented in [30] (see also [46]) to the flavoured case.…”

Anomaly matching across dimensions and supersymmetric Cardy formulae

“…See [49] for a review. A supersymmetric back- 14 See also [45] for related discussions. 15 Note that the superconformal index I and the partition function Z S 3 ×S 1 differs by a contribution which is known as the Casimir energy [46,47] log Z S 3 ×S 1 = −βEC + log I .…”