Abstract
We systematically analyze the large-N limit of the superconformal index of mathcal{N} = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5 theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.
Highlights
Introduction and summaryThe last couple of years have seen good progress in the study of the-BPS superconformal index of four dimensional N = 4 super Yang-Mills theory (SYM) and, more generally, the can be finite in units of N 2, which is exactly the regime of parameters of the black hole solution in supergravity
The essence of the approach of [21] is to extend the range of eigenvalues of the unitary matrix from a circle to a torus, one of whose cycles is the original circle. This prompts us to refer to this approach as that of elliptic extension. This approach allows us to find solutions of the saddle-point equations and, further, it allows us to calculate the effective action at each saddle point in a straightforward manner
We study the basic index, which may be defined for any N = 1 supersymmetric field theory with an R-symmetry, as well as the index refined by including chemical potentials for flavor symmetries, and our focus will be to extract simple universal results for generic theories
Summary
Despite the fact that the extended integrand is not a meromorphic function, for each (m, n) saddle one can show that the original contour can be deformed so as to pass through it, at leading order in the large-N expansion These remarks answer the first two of the three questions raised above, and we leave the third question for future work. Note that the equation (1.8) gives the complete perturbative expansion around each saddle γ at large N , we do not yet have the exact non-perturbative answer — which would involve making sense of the infinite sum for every value of τ With this background and context, we can describe the main results of this paper
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