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# On the 4d superconformal index near roots of unity: Bulk and Localized contributions.

Nov 2021

This paper studies one particular approach to the expansion near roots of unity of the superconformal index of 4d $SU(N)$ $\mathcal{N}=4$ SYM. In such a limit, middle-dimensional walls of non-analyticity emerge in the complex analytic extension of the integrand. These walls intersect the integration contour at infinitesimal vicinities and come from both, the vector and chiral multiplet contributions. We will call these intersections $bits$ and the complementary region $bulk$. For $N=2$ the index can be conveniently divided in three integrals, two of them over regions covering the vector and chiral bits, the third one over the bulk. At very leading order in an expansion near roots of unity, these three sub-integrals match the contributions of the three Bethe roots that localize the $SU(2)$ superconformal index. The dominating contribution comes from the integral over vector bits, the next subleading one comes from the integral over chiral bits, and the next-to-next subleading one comes from the integral over part of the bulk. Some observations are made suggesting that the expansion near roots of unity could be a useful tool to complete the localization formulas that have been recently argued to compute the $SU(N)$ index for $N\,\geq\,3\,$.

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