Factorisation and holomorphic blocks in 4d

TL;DR

This paper investigates the factorization of partition functions of N=1 theories on specific 4-manifolds into holomorphic blocks, revealing a deep structure relating 4d and 3d supersymmetric theories through anomaly cancellation and manifold decomposition.

Contribution

It demonstrates that under anomaly cancellation, 4d and 3d partition functions factorize into sums of products of holomorphic blocks, linking geometric decomposition with quantum field theory partition functions.

Findings

Partition functions factorize into holomorphic blocks.

Factorization holds when anomalies are canceled.

Explicit evaluation of Coulomb branch integrals confirms the structure.

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.