Modular factorization of superconformal indices

Superconformal indices of four-dimensional N = 1 gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an SL(3, Z) and SL(2, Z) proportional to Z2 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 function. This leads to our conjecture for the modular factorization of superconformal lens indices of general N = 1 gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED and sketch the extension of our arguments to more general gauge theories. Assuming the validity of the conjecture, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with SL(3, Z). Finally, we use this framework to propose a formula for the general lens space index.

Automated summary

The superconformal indices of four-dimensional N = 1 gauge theories have been studied and their modular properties have been investigated. It has been shown that the indices can be factorized into holomorphic blocks, which are related to the underlying geometry's Heegaard splitting. By extending the transformations to a more general setting, it is concluded that the indices can be factorized into holomorphic blocks parametrized by modular (congruence sub)groups.