Gauge theories on compact toric manifolds

Abstract

We compute the $$\mathcal{N}=2$$ 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 $$\mathbb {C}^2$$ 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 $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ 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 $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

Scaling black holes and modularity

“…Such non-holomorphic contributions have appeared in similar contexts. In specific cases, the non-holomorphic contributions are derived from different physical points of view, for example the continuum of multi-particle states in R 4 [58], or the quantum field theory on the world volume of the D-branes [44,47,48], or the perspective of D3-instantons in the hypermultiplet moduli space [18].…”

“…This implies potentially interesting arithmetic of the BPS indices, while the non-holomorphic contribution is also interesting independently, and a generalization of similar non-holomorphic terms in partition functions of N = 4 Yang-Mills on four-manifolds [40,41]. The origins and explicit expressions of these terms have been understood better recently [30,[42][43][44][45][46][47][48].…”

Scaling Black Holes and Modularity

“…Such non-holomorphic contributions have appeared in similar contexts. In specific cases, the non-holomorphic contributions are derived from different physical points of view, for example the continuum of multi-particle states in R 4 [57], or the quantum field theory on the world volume of the D-branes [44,47,48], or the perspective of D3-instantons in the hypermultiplet moduli space [18].…”

“…This implies potentially interesting arithmetic of the BPS indices, while the non-holomorphic contribution is also interesting independently, and a generalization of similar non-holomorphic terms in partition functions of N = 4 Yang-Mills on four-manifolds [40,41]. The origins and explicit expressions of these terms have been understood better recently [30,42,43,44,45,46,47,48].…”

Topological correlators of $SU(2)$, $\mathcal{N}=2^*$ SYM on four-manifolds

“…Within the setting of the twisted N = 4 theory and the dual two-dimensional field theory, Dabholkar, Putrov and Witten [30] have given a derivation of the non-holomorphic anomaly. Moreover, Bonelli et al [40] have arrived at the holomorphic anomaly using toric localization in supersymmetric field theory.…”