Modular factorization of superconformal indices

DOI10.1007/JHEP10(2023)105zbMATH Open1527.81104arXiv2210.17551MaRDI QIDQ6061783

Author name not available (Why is that?)

Publication date: 8 December 2023

Published in: (Search for Journal in Brave)

Abstract: Superconformal indices of four-dimensional

m a t h c a l N = 1

gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an

S L (3, m a t h b b Z)

and

S L (2, m a t h b b Z) l t i m e s m a t h b b Z^{2}

transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic

G a m m a

function. This allows us to establish the ``modular factorization of superconformal lens indices of general $m a t h c a l N = 1$ gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with $S L (3, m a t h b b Z)$ . Finally, we use this framework to propose a formula for the general lens space index.

Full work available at URL: https://arxiv.org/abs/2210.17551

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