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Abstract: The superconformal index of 4-dimensional N=4 SYM theory with gauge group U(N) can be expressed as a unitary matrix integral with a double-trace potential. Work by Gaiotto and Lee, and by Imamura and collaborators suggests that the superconformal index of the U(N) gauge theory should be expressible as a convergent series whose terms are indices of associated U(k) gauge theories realized as the worldvolume theories of stacks of k giant-graviton branes in the holographic dual of the U(N) theory. From matrix integral manipulations, Murthy derives a different convergent expansion for the index which holds very generally, but whose physical interpretation is a priori unclear. The connection between the two expansions is also a priori unknown. In this talk, I will explain the relation between these two expansions and provide an algorithm for determining the terms of the Gaiotto-Lee expansion starting from Murthy's. In the process of deriving this relation, we also uncover an interesting identity involving unitary matrix integrals which leads to a conjecture about what physical interpretation the terms in Murthy's expansion might have. This part of the talk is based on 2302.04887 (https://arxiv.org/abs/2302.04887).
If time permits, I would like to also briefly explain the resolution of a puzzle surrounding the strong-coupling phase of the Gross-Witten-Wadia unitary matrix integral. The non-perturbative corrections in the strong-coupling phase have historically been confusing since the standard eigenvalue tunneling calculation leads to exponentially enhanced (rather than suppressed) contributions, which means a different type of instanton is responsible for the corrections. I will briefly explain how the ghost instanton of Marino, Schiappa, and Schwick (anti-eigenvalue tunneling) leads to the desired corrections. This part of the talk is based on 2308.06320 (https://arxiv.org/abs/2308.06320) in collaboration with Raghu Mahajan and Chitraang Murdia.
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