Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Abstract

We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3, Z) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N = 2 SCFT with E 6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N = 1 theory with enhanced E 7 flavour symmetry.

An extremization principle for the entropy of rotating BPS black holes in AdS5

“…The quantity E can be interpreted as a combination of 't Hooft anomaly polynomials that arise studying the partition function Z N =4 (∆ I , ω i ) on S 3 × S 1 or the superconformal index I(∆ I , ω i ) for N = 4 SYM [13,16]. Some explicit expressions are given in appendix C. In particular, E is formally equal to the supersymmetric Casimir energy of N = 4 SYM as a function of the chemical potentials (see for example equation (4.50) in [13] and appendix C).…”

“…However, due to a large cancellation between bosonic and fermionic states, the index is a quantity of order one for generic values of the fugacities while the entropy scales like N 2 [6]. We also know that the supersymmetric partition function on S 3 × S 1 is equal to the superconformal index only up to a multiplicative factor e −βE SUSY , where the supersymmetric Casimir energy E SUSY is of order N 2 [10][11][12][13][14][15][16]. However, it is not clear what the average energy of the vacuum should have to do with the entropy, which is the degeneracy of ground states.…”

A note on the entropy of rotating BPS AdS7 × S4 black holes

“…A choice of determination for the chemical potentials should be made when performing limits, for example low-and high-temperature, or modular transformations of the integrand of the corresponding matrix models, since these operations typically involve multi-valued functions. Examples in the analogous four-dimensional case can be found in [19][20][21][22]44]. It is then interesting to ask whether there exists a limit in the fugacities, subject to the constraint (1.6), where the quantity (1.4) dominates the partition function or the index.…”

N $$ \mathcal{N} $$ = 1 supersymmetric indices and the four-dimensional A-model

“…Incidentally, we should also note that the "S-transformed" S 3 × S 1 background, corresponding to (p 1 , p 2 ) = (0, 1), has appeared before in the literature in the form of the so-called "modified S 3 index" [65]. In our language, this simply corresponds to using the fibering operator F 2 , instead of F 1 , in the construction of the three-sphere index.…”

“…By performing an SL(2, Z) transformation, one simply obtains a different realization of the M g,p ×S 1 partition function. Related modular properties of the S 3 index have been discussed in the literature, in connection with 't Hooft anomalies [71,65]. In our formalism, these modular properties are simply explained in terms of the T 2 compactification necessary to define the four-dimensional A-model.…”