Superconformal index of low-rank gauge theories via the Bethe Ansatz

Abstract

We study the Bethe Ansatz formula for the superconformal index, in the case of 4d $\mathcal{N}=4$ super-Yang-Mills with gauge group $SU(N)$. We observe that not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the index, and thus formulate "reduced BAEs" such that all and only their solutions contribute. We then propose, sharpening a conjecture of Arabi Ardehali et al. [arXiv:1912.04169], that there is a one-to-one correspondence between branches of solutions to the reduced BAEs and vacua of the 4d $\mathcal{N}=1^{\ast }$ theory. We test the proposal in the case of $SU(2)$ and $SU(3)$. In the case of $SU(3)$, we confirm that there is a continuous family of solutions, whose contribution to the index is non-vanishing.

24 pages; v2: refs added, typos corrected, slightly clarified relation with the N=1* theory

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Cited by in corpus (10)

• A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index
• The exact Schur index in closed form
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS${}_5$
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• Expanding on the Cardy-like limit of the superconformal index of 4d $\mathcal{N}=1$ ABCD SCFTs
• Kerr-Newman black holes from $\mathcal{N}=1^{\ast }$
• The topologically twisted index of $\mathcal{N}=4$ SU($N$) Super-Yang-Mills theory and a black hole Farey tail
• Global variants of $\mathcal{N}=1^{\ast }$ theories and Calogero-Moser systems