Microstate counting via Bethe Ansätze in the 4d $$ \mathcal{N} $$ = 1 superconformal index

Abstract

We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe-Ansatz approach recently applied by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large N limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of N D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and Y p,q quiver gauge theories. For generic quiver gauge theories we find a particular correction to the superconformal index in the Cardy limit that happens to vanish in the case of N = 4 supersymmetric Yang-Mills. We estimate how such correction affects the entropy of the would be dual electrically charged rotating AdS 5 black holes.

Microscopic entropy of rotating electrically charged AdS4 black holes from field theory localization

“…Going beyond the Cardy-like limit, however, seems more formidable. The absence of a Bethe Ansatz description for the superconformal index seems to block a route that was successfully taken in the 4d context [42,43]. It would be interesting to consider near-BPS configurations by turning on temperature or slightly violating the BPS constraint (4.54), in the same spirit as [50].…”

“…Several groups have now independently reproduced the entropy function by studying the partition function or the superconformal index of N " 4 SYM [34][35][36] by slightly reinterpreting the original work in the superconformal index [26]. Further extensions for the general growth of the N " 1 superconformal index [37][38][39][40][41] including via the Bethe-Ansatz approach [42,43] have been achieved. After this important progress, the generalizations to other dimensions were also considered.…”

Supersymmetric phases of 4d $$ \mathcal{N} $$ = 4 SYM at large N

“…Following the insightful observation of [3], this puzzle has been revisited recently [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] The emergent picture shows that there is actually an exponential growth of states contained in the index, which is captured by the saddle-point estimate if one allows the potentials σ, τ , ϕ to take values in the complex plane away from the pure imaginary values assumed in [2]. 1 There are essentially three strands of analysis contained in the recent progress:…”

Black hole entropy function for toric theories via Bethe Ansatz

“…Note added: When we were finalizing our paper the preprint [45], which has a significant overlap with our work, has been posted on arXiv. The authors' conclusions disagree with ours at various points.…”