Gauge theories on compact toric manifolds

• Published in: Letters in mathematical physics Vol. 111; no. 3
• Main Authors: Bonelli, Giulio, Fucito, Francesco, Morales, Jose Francisco, Ronzani, Massimiliano, Sysoeva, Ekaterina, Tanzini, Alessandro
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2021; Springer Nature B.V
• Subjects: Complex Systems; Gauge theory; Geometry; Group Theory and Generalizations; Instantons; Mathematical and Computational Physics; Partitions (mathematics); Physics; Physics and Astronomy; Residues; Theoretical; 14; Extended Supersymmetry; Differential and Algebraic geometry; Topological field theories; 81; Supersymmeetric gauge theory; 53
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-021-01419-9

We compute the N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the C 2 partition function. As particular cases, our formulae compute the SU (2) and SU (3) equivariant Donaldson invariants of P 2 and F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU (2) case. Finally, we show that the U (1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a N = 2 analog of the N = 4 holomorphic anomaly equations.