Abstract

In this note, we define a holographic dual to four-dimensional superconformal field theories formulated on arbitrary Riemannian manifolds equipped with a Killing vector. Moreover, assuming smoothness of the bulk solution, we study the variation of the holographically renormalized supergravity action in the class of metrics on the boundary four-manifold with a prescribed isometry.

Highlights

  • JHEP10(2019)115 rotations in the two transverse planes [7], and presented in generality by Nekrasov and Okounkov on an arbitrary manifold [8].1 Its applications have been numerous and influential, including a crucial role in the formulation of the AGT correspondence [9]

  • As for the topological twist, this construction can be formulated in terms of coupling to a background N = 2 off-shell supergravity [5], and one can formulate the dual supergravity background, as was outlined in the conclusions of [6]

  • We showed that under certain assumptions of smoothness and existence of the bulk solution, the holographic Ward identity corresponding to the supersymmetric topological twist held

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Summary

Lagrangian and equations of motion

The results obtained here apply to Ω twists of N = 4 SYM and to (some) conformal field theories of class S [20] (the choice of theory being dependent on the uplift).2 In this context, a computation involving a supersymmetric black hole solution to Romans’ N = 4+ theory has already been precisely matched to a supersymmetric Renyi entropy computed in N = 4 SYM [24]. Since we have gauged the subgroup SU(2)R × U(1)R, we naturally split the generators of the Clifford algebra Cliff(5, 0) corresponding to Spin(5) into ΓI ,. Denoting by ± the projection onto the ±i eigenspaces of Γ45, respectively In this way, there is a natural splitting of the equations between the two eigenspaces

Perturbative expansion
Holographic renormalization
Expansion of the supersymmetry equations
Variation of the action
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