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Betti multiplets, flows across dimensions and c-extremization

Antonio Amariti, Chiara Toldo

TL;DR

The paper develops a general holographic framework for flows from AdS_5 to AdS_3 that realize c-extremization of 2d N=(0,2) SCFTs arising from the partial topological twisting of 4d N=1 theories on Riemann surfaces. It shows that the 2d central charge and the mixing of the R-current with global (including baryonic) symmetries can be captured by a 5d N=2 gauged supergravity description, with the attractor mechanism of the scalar sector encoding c-extremization. The authors apply the formalism to explicit truncations: a Klebanov–Witten/Y^{10}=T^{1,1} model with a Betti vector, the N=4 SYM case via U(1)^3 gauged supergravity, and M5-brane constructions, obtaining central charges that match field-theory computations and revealing when baryonic mixing occurs in 2d but not in 4d. The results provide a unified holographic account of dimension-changing flows and offer a concrete route to explore higher-derivative corrections and more general truncations in the AdS/CFT context.

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.

Betti multiplets, flows across dimensions and c-extremization

TL;DR

The paper develops a general holographic framework for flows from AdS_5 to AdS_3 that realize c-extremization of 2d N=(0,2) SCFTs arising from the partial topological twisting of 4d N=1 theories on Riemann surfaces. It shows that the 2d central charge and the mixing of the R-current with global (including baryonic) symmetries can be captured by a 5d N=2 gauged supergravity description, with the attractor mechanism of the scalar sector encoding c-extremization. The authors apply the formalism to explicit truncations: a Klebanov–Witten/Y^{10}=T^{1,1} model with a Betti vector, the N=4 SYM case via U(1)^3 gauged supergravity, and M5-brane constructions, obtaining central charges that match field-theory computations and revealing when baryonic mixing occurs in 2d but not in 4d. The results provide a unified holographic account of dimension-changing flows and offer a concrete route to explore higher-derivative corrections and more general truncations in the AdS/CFT context.

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.

Paper Structure

This paper contains 18 sections, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Flowing from 4d $\mathcal{N}=1$ to 2d $\mathcal{N}=(0,2)$ SCFTs
  3. Flowing from AdS$_5$ to AdS$_3$ in $\mathcal{N}=2$ gauged supergravity
  4. Inclusion of the hypermultiplets
  5. $\mathcal{N}=4$ SYM and $U(1)^3$ gauged supergravity
  6. $T^{1,1}$ with a Betti multiplet
  7. The 5d truncation
  8. The ansatz and the BPS equations
  9. The AdS$_3$ vacuum
  10. Holographic RG flow
  11. $c$-extremization from gauged supergravity
  12. M5 branes
  13. Conclusions
  14. $\mathcal{N}=2$ 5d gauged supergravity
  15. BPS equations
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Quiver gauge theory for the Klebanov-Witten theory, describing a stack of D3 branes probing the $Y^{10}=T^{11}$ space.
  • Figure 2: Holographic RG flow for the of the scalar fields $\phi_i$ interpolating between the AdS$_5$ vacuum in the UV, $\rho \rightarrow \infty$ and the AdS$_3$ vacuum in the IR, $\rho \rightarrow -\infty$. The colors refers to the values $b=0.08$ (blue), $b=0.1$ (red), $b=0.15$ (yellow) and $b=0.2$ (green).