Factorisation and holomorphic blocks in 4d

TL;DR

This work uncovers a unifying block decomposition for supersymmetric partition functions across 3d and 4d theories on Seifert-like manifolds, showing that anomaly cancellation is the key to factorising Coulomb branch integrands into holomorphic blocks. By recasting lens space and twisted indices in terms of holomorphic block squares and identifying explicit block expressions (in terms of Theta functions, elliptic Gamma functions and elliptic hypergeometric series), it demonstrates how 3d and 4d partition functions emerge from gluing these blocks via appropriate pairings or r-pairings. The results extend known 3d factorisations to lens spaces, S^2_A x S^1, and S^2 x T^2, and introduce 4d holomorphic blocks that generalise the 3d blocks, with concrete constructions for SQED and SQCD. The formalism also clarifies the role of anomalies as obstructions to factorisation and provides a framework for deriving block identities and contour prescriptions in higher-dimensional supersymmetric theories.

Abstract

We study N=1 theories on Hermitian manifolds of the form M^4=S^1xM^3 with M^3 a U(1) fibration over S^2, and their 3d N=2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D^2xT^2 and D^2xS^1. We prove that when the 4d and 3d anomalies are cancelled the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D^2xT^2 and D^2xS^1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.