Abstract
We compute the mathcal{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. 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 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of mathbb {P}^2 and mathbb {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 analog of the mathcal {N}=4 holomorphic anomaly equations.
Highlights
The study of N = 2 supersymmetric Yang-Mills gauge theories in four dimensions (SYM) led to many interesting and deep results which opened a new perspective in the understanding of non-perturbative effects in gauge theories [1,2]
Our work follows a gauge theory approach with the use of recursion formulae to establish the analytic properties of the partition function, alternative to wall crossing computations, in which the conditions for bundle stability come from the consistency of the computation
The computation of the gauge theory partition function on compact manifolds presents a main additional difficulty with respect to the non compact case, namely one has to perform an integration over the Coulomb branch parameters, which are in this case integrable zero modes of the dynamical fields
Summary
Keywords Extended Supersymmetry · Supersymmeetric gauge theory · Differential and Algebraic geometry · Topological field theories. In the topologically twisted N = 2 theory the coupling appears in the gauge fixing term which is metric dependent This in turn implies an induced anomalous dependence of the partition function on the metric of the manifold and the related wall crossing behaviour. The anomalous dependence on the gauge coupling discussed above is expected to be the UV ancestor of the holomorphic anomaly equations in the IR, closely connected to analogous results first found in string theory models [26] in the case of the computation of the partition function of the N = 4 SYM [27] and in the Donaldson invariants generating functions [28,29,30,31]. To compute such residues we use a surprising “duality” relation between the residues computed at the poles of the one-loop and instanton part of the partition
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