Rigid supersymmetry from conformal supergravity in five dimensions

Abstract

We study the rigid limit of 5d conformal supergravity with minimal supersymmetry on Riemannian manifolds. The necessary and sufficient condition for the existence of a solution is the existence of a conformal Killing vector. Whenever a certain SU(2) curvature becomes abelian the backgrounds define a transversally holomorphic foliation. Subsequently we turn to the question under which circumstances these backgrounds admit a kinetic Yang-Mills term in the action of a vector multiplet. Here we find that the conformal Killing vector has to be Killing. We supplement the discussion with various appendices.

A note on instanton operators, instanton particles, and supersymmetry

“…An interesting complementary point of view can be obtained by performing the stateoperator conformal transformation from R 5 to S 4 × R. Regarding the YM coupling as the VEV of a scalar field in a background vector multiplet, it should map as g 0 → g 0 e −τ /2 , where τ is the coordinate along R. This is consistent with the observation made in [16], that in S 4 × R only a position-dependent gauge coupling "constant" g Y M = g 0 e −τ /2 can be turned on in a supersymmetric way. This is a position-space version of the RG flow interpolating between the UV SCFT at early time and the IR supersymmetric gauge theory at late time.…”

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theory on connected sums of S 2 × S 2

“…We consider the asymptotic contributions from the vector and hyper as given above in equations (54), (55). The two terms contribute to the effective action at the point σ = ia on the Coulomb branch.…”

Nekrasov-Shatashvili limit of the 5D superconformal index

“…Having set the stage, we would like to investigate whether there exist limits of the index (2.2) which only receive contributions from certain sectors of the theory, as e.g. is the 10 For a generic SCFT on R × S 4 it is possible to turn on supersymmetrically a position-dependent YM coupling, interpolating between the SCFT and the IR gauge theory [22,23]. 11 For definiteness, we will assume that the hypermultiplet is in the fundamental of the gauge group.…”

“…In that reference, the resultant partition function was identified as the nonperturbative contribution to the twisted superpotential for some associated two-dimensional theory. Subsequently, the authors of [33] also linked the full 2D twisted superpotential-the NS limits of the full perturbative plus nonperturbative partition functions of the 4D theory-with the twisted superpotential for a different, dual 22 This is known as the "homological limit" (see e.g. [30] and references therein) and in notation where one has made explicit the dependence of the fugacities on the Euclideanised time radius, p = e −βǫ1 , q = e −βǫ2 , corresponds to taking β → 0.…”