Abstract
We study \( \mathcal{N}=1 \) theories on Hermitian manifolds of the form M 4 = S 1 × M 3 with M 3 a U(1) fibration over S 2, and their 3d \( \mathcal{N}=2 \) reductions. These manifolds admit an Heegaard-like decomposition in solid tori D 2 × T 2 and D 2 × S 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 2 × T 2 and D 2 × S 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.
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Nieri, F., Pasquetti, S. Factorisation and holomorphic blocks in 4d. J. High Energ. Phys. 2015, 155 (2015). https://doi.org/10.1007/JHEP11(2015)155
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DOI: https://doi.org/10.1007/JHEP11(2015)155