Abstract
We consider $$ \mathcal{N}=2 $$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.
Highlights
Understanding non-perturbative corrections in quantum field theories is an important problem in modern theoretical physics
Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry
For manifolds admitting an isometry, we show that these equations are solved by an equivariant version of the topological twist and we explicitly compute the gauge theory path-integral, which turns out to be given by an appropriate gluing of Nekrasov partition functions
Summary
Understanding non-perturbative corrections in quantum field theories is an important problem in modern theoretical physics. If the space-time manifold admits some isometry, one might be able to further localize the path integral over an invariant locus of the moduli space, improving upon the method of equivariant localization This allowed the exact computation of the instanton partition function of N = 2 supersymmetric theories on R4 [8, 9]. For manifolds admitting an isometry, we show that these equations are solved by an equivariant version of the topological twist and we explicitly compute the gauge theory path-integral, which turns out to be given by an appropriate gluing of Nekrasov partition functions. Appendix B describes the solutions to Killing spinor equations in the general case of a four-manifold admitting a U(1) isometry.
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