Betti multiplets, flows across dimensions and c-extremization

TL;DR

This paper explores the holographic dual of 4d to 2d SCFT flow via supergravity, confirming field theory predictions of R-current mixing and central charge extremization through explicit AdS solutions and a new formalism.

Contribution

It develops a general supergravity formalism linking 5d N=2 gauged supergravity to 2d SCFT central charge extremization, extending previous understanding of dimensional flows.

Findings

Verified existence of AdS_3 vacua connecting to AdS_5 in supergravity.

Matched supergravity-derived central charge with field theory results.

Demonstrated the attractor mechanism for scalars in the context of c-extremization.

Abstract

We consider 4d N=1 SCFTs, topologically twisted on compact constant curvature Riemann surfaces, giving rise to 2d N=(0,2) SCFTs. The exact R-current of these 2d SCFT extremizes the central charge c_{2d}, similarly to the 4d picture, where the exact R-current maximizes the central charge a_{4d}. There are global currents that do not mix with the R-current in 4d but their mixing becomes non trivial in 2d. In this paper we study the holographic dual of this process by analyzing a 5d N=2 truncation of T^{1,1} with one Betti vector multiplet, dual to the baryonic current on the CFT side. The holographic realization of the flow across dimensions connects AdS_5 to AdS_3 vacua in the supergravity picture. We verify the existence of the flow to AdS_3 solutions and we retrieve the field theory results for the mixing of the Betti vector with the graviphoton. Moreover, we extract the central charge from the Brown-Henneaux formula, matching with the results obtained in field theory. We develop a general formalism to obtain the central charge of a 2d SCFT from 5d N=2 gauged supergravity with a generic number of vector multiplets, showing that its extremization corresponds to an attractor mechanism for the scalars in the supergravity picture.