Factorisation and holomorphic blocks in 4d

Abstract: We study N=1 theories on Hermitian manifolds of the form M4 = S1× M3 with M3 a U(1) fibration over S2, and their 3d N=2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D2× T2 and D2× S1. 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 D2× T2 and D2× S1 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.