Abstract
We compute the Coulomb branch partition function of the 4d \( \mathcal{N} \) = 2 vector multiplet on closed simply-connected quasi-toric manifolds B. This includes a large class of theories, localising to either instantons or anti-instantons at the torus fixed points (including Donaldson-Witten and Pestun-like theories as examples). The main difficulty is to obtain flux contributions from the localisation procedure. We achieve this by taking a detour via the 5d \( \mathcal{N} \) = 1 vector multiplet on closed simply-connected toric Sasaki-manifolds M which are principal S1-bundles over B. The perturbative partition function can be expressed as a product over slices of the toric cone. By taking finite quotients M/ℤh along the S1, the locus picks up non-trivial flat connections which, in the limit h → ∞, provide the sought-after fluxes on B. We compute the one-loop partition functions around each topological sector on M/ℤh and B explicitly, and then factorise them into contributions from the torus fixed points. This enables us to also write down the conjectured instanton part of the partition function on B.
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Acknowledgments
We are grateful to Guido Festuccia, Jian Qiu, and Maxim Zabzine for stimulating discussions on the subject. We thank them, Matteo Sacchi, Leonardo Santilli and Itamar Yaakov for comments on the manuscript. We also thank Seidon Alsaody and Thomas Kragh for mathematical advice. RM acknowledges support from the Centre for Interdisciplinary Mathematics at Uppsala University. LR acknowledges support from the Shuimu Tsinghua Scholar Program.
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Lundin, J., Mauch, R. & Ruggeri, L. From 5d flat connections to 4d fluxes (the art of slicing the cone). J. High Energ. Phys. 2023, 155 (2023). https://doi.org/10.1007/JHEP10(2023)155
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DOI: https://doi.org/10.1007/JHEP10(2023)155