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Universal RG Flows Across Dimensions and Holography

Nikolay Bobev, P. Marcos Crichigno

TL;DR

The work establishes universal renormalization-group flows across dimensions for SCFTs triggered by twists that couple only to the stress-energy multiplet, yielding exact anomaly-based relations for even-dimensional UV/IR pairs and universal large-$N$ free-energy relations for odd dimensions. It shows these flows admit holographic descriptions as domain-wall solutions in minimal gauged supergravity, and these solutions can be uplifted to string/M-theory in many ways, reinforcing the universality across UV theories. Field theory and holographic analyses consistently relate UV conformal data $(a_d,\vec{c}_d)$ to IR data $(a_p,\vec{c}_p)$ via topology- and twist-dependent maps, and similar relations connect IR free energies to UV anomalies in odd dimensions. The entropy and central charges of the universal twisted flows align across dimensions, providing a coherent holographic picture of RG flows across $d\rightarrow p$ with detailed checks in 6d, 4d, 2d, and their holographic duals. The results offer a framework to count microstates of universal black branes and motivate further explorations of subleading corrections, higher-curvature effects, and potential monotonicity theorems along these across-dimension flows.

Abstract

We study RG flows between superconformal field theories living in different spacetime dimensions which exhibit universal properties, independent of the details of the UV and IR theories. In particular, when the UV and IR theories are both even-dimensional we establish exact universal relations between their conformal anomaly coefficients. We also provide strong evidence for similar relations between appropriately defined free energies for RG flows between odd-dimensional theories in the large $N$ limit. Holographically, these RG flows across dimensions are described by asymptotically AdS black branes in a gauged supergravity theory, which we exhibit explicitly. We also discuss the uplift of these solutions to string and M-theory and comment on how the entropy of such black branes is captured by the dual field theory.

Universal RG Flows Across Dimensions and Holography

TL;DR

The work establishes universal renormalization-group flows across dimensions for SCFTs triggered by twists that couple only to the stress-energy multiplet, yielding exact anomaly-based relations for even-dimensional UV/IR pairs and universal large- free-energy relations for odd dimensions. It shows these flows admit holographic descriptions as domain-wall solutions in minimal gauged supergravity, and these solutions can be uplifted to string/M-theory in many ways, reinforcing the universality across UV theories. Field theory and holographic analyses consistently relate UV conformal data to IR data via topology- and twist-dependent maps, and similar relations connect IR free energies to UV anomalies in odd dimensions. The entropy and central charges of the universal twisted flows align across dimensions, providing a coherent holographic picture of RG flows across with detailed checks in 6d, 4d, 2d, and their holographic duals. The results offer a framework to count microstates of universal black branes and motivate further explorations of subleading corrections, higher-curvature effects, and potential monotonicity theorems along these across-dimension flows.

Abstract

We study RG flows between superconformal field theories living in different spacetime dimensions which exhibit universal properties, independent of the details of the UV and IR theories. In particular, when the UV and IR theories are both even-dimensional we establish exact universal relations between their conformal anomaly coefficients. We also provide strong evidence for similar relations between appropriately defined free energies for RG flows between odd-dimensional theories in the large limit. Holographically, these RG flows across dimensions are described by asymptotically AdS black branes in a gauged supergravity theory, which we exhibit explicitly. We also discuss the uplift of these solutions to string and M-theory and comment on how the entropy of such black branes is captured by the dual field theory.

Paper Structure

This paper contains 55 sections, 132 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Main results
  3. Generalities
  4. 't Hooft anomalies
  5. Weyl anomaly
  6. Topological twist
  7. Field theory
  8. 6d SCFTs
  9. ${\cal N}=(1,0)$
  10. On Riemann surfaces.
  11. On Kähler four-manifolds.
  12. ${\cal N}=(2,0)$
  13. On Riemann surfaces.
  14. On Kähler four-manifolds.
  15. On product four-manifolds $M_{4}=\Sigma_{1}\times \Sigma_{2}\,$.
  16. ...and 40 more sections

Figures (2)

  • Figure 1: Mapping of various values of $a/c$ under \ref{['ac4dn=120']} from 6d ${\cal N}=(2,0)$ to a 4d ${\cal N}=1$ theory, as well as the flow from 4d to 2d obeying \ref{['universalmat']} and preserving ${\cal N}=(0,2)$ supersymmetry. Positivity of $c_{r}$ in 2d is mapped into the lower bound $3/7$ in 6d.
  • Figure 2: Mapping of various values of $a/c$ under \ref{['matMN2']} from 6d ${\cal N}=(2,0)$ to a 4d ${\cal N}=2$ theory. The map to 2d, either by the $\alpha$-twist \ref{['4d2d2,2']} or the $\beta$-twist \ref{['clcr2004B']}, relates the lower HM bound in 4d to the 2d unitarity bound $c_{r}\geq0$ ($c_{l}\geq0$ is automatically satisfied).