$\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ and Liouville Gravity

Abstract

We consider 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 S 2 × S 2 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.

Partition functions for equivariantly twisted N = 2 $$ \mathcal{N}=2 $$ gauge theories on toric Kähler manifolds

“…We found the one-loop determinant to be trivial as fermionic and bosonic modes cancel exactly. This is in agreement with both [19] and the dimensional reduction of five-dimensional results. In the latter case, the one-loop function counts functions that are holomorphic with respect to the so-called tangential Cauchy-Riemann operator that are charged along the Reeb [15].…”

“…This agrees with [19] in the special case m = n = 0. Furthermore, it is also consistent with the 5d result in [29], which becomes non-trivial only when the S 1 is fibered on top of the CP 2 so as to make an S 5 .…”

“…Since our results follow directly from supergravity via rigid supersymmetry, it is important to point out that our result for the classical action appears different from that of [19] whose authors did not take the supergravity approach. Note that the evaluation of the classical action is also intimately related to factorization, hence providing yet another motivation for a further study of this point.…”

“…The last point is especially interesting as it raises the question of how to generalize our results to non-vanishing first Chern class in order to make a better connection with [9,19]. Alternatively one could wonder whether it is appropriate to add an operator e F ∧ω to the path integral, where ω is a suitably chosen two-form.…”

“…In the closing stages of this project we received [19,20], which consider M 4 = P 1 × P 1 . Especially [19] obtains, where applicable, similar results to ours.…”