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

Abstract

We consider N = 2 supersymmetric pure gauge theories on toric Kähler manifolds, with particular emphasis on CP 2 . By choosing a vector generating a U(1) action inside the torus of the manifold, we construct equivariantly twisted theories. Then, using localization, we compute their supersymmetric partition functions. As expected, these receive contributions from a classical, a one-loop, and an instanton term. It turns out that the oneloop term is trivial and that the instanton contributions are localized at the fixed points of the U(1). In fact the full partition function can be re-written in a factorized form with contributions from each of the fixed points. The full significance of this is yet to be understood.

Twisting with a Flip (The Art of Pestunization)

“…Hence generically P + ω Ω 2 is not isomorphic to Ω 2+ and the corresponding cohomological theory is not related to standard equivariant Donaldson-Witten theory. It is important to remember that the bundle P + ω Ω 2 is defined up to isomorphism, so that in general we could use different vector fields in the transformations (27) and to define the projector (23). To avoid confusion we will postpone discussing this point to section 4.3.…”

“…The case when we associate instantons to all fixed points corresponds to equivariant Donaldson invariants, (see e.g. [19,20] and earlier works [21][22][23]). We present a general class of supersymmetric field theories that includes on equal footing both equivariant Donaldson-Witten theory and the theories on S 4 ǫ 1 ,ǫ 2 reviewed above.…”

Rigid 4D N = 2 $$ \mathcal{N}=2 $$ supersymmetric backgrounds and actions

“…This excluded such cases as S 2 × S 2 recently discussed in [67][68][69] and reflects the well-known fact that the D(2, 1; α) superalgebra possesses no real form with bosonic part SU(2) × SU(2) × SU(2) [73,74]. 21 Nevertheless, one may still exploit our results to investigate such cases by relaxing the requirement of real supercharges.…”

“…[10,11,[66][67][68][69]). Of particular relevance is the work of Cortes et al which constructs vector multiplets in Euclidean signature via a timelike reduction from 5D [64].…”

“…PIIF-GA-2012-627976. 21 Note that one can obtain four real supercharges on S 2 × S 2 via an equivariant twist [67,69]. 22 Alternatively, one might consider S 2 × S 2 with split signature, η ab = ( − 1, −1, +1, +1).…”