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Holographic Uniformization

Michael T. Anderson, Christopher Beem, Nikolay Bobev, Leonardo Rastelli

TL;DR

This work introduces and analyzes a novel class of holographic BPS flow equations that act as elliptic geometric flows on a Riemann surface ${\cal C}$ arising from M5/D3 branes wrapped on ${\cal C}$. By reducing the supergravity BPS conditions to a single intrinsic PDE for a conformal factor on ${\cal C}$, the authors show that flows exist for arbitrary UV metrics and flow to a constant-curvature IR metric, thereby realizing a holographic uniformization. They formulate and solve the 1/2 and 1/4 BPS sectors for M5 and D3 branes, derive exact or near-exact solutions (e.g., Maldacena–Nuñez type) where possible, and perform a global existence proof using degree theory, including area monotonicity results. The results provide a rigorous holographic realization of the intuition that UV geometric data on ${\cal C}$ is washed out in the IR and establish a framework for extending uniformization-like flows to broader wrapped-brane settings with potential links to a holographic c-function. These techniques open avenues for studying flow equations on higher-dimensional manifolds and for incorporating punctures or less-supersymmetric cases.

Abstract

We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the UV and the constant-curvature metric in the IR. We confirm this conjecture with a rigorous global existence proof.

Holographic Uniformization

TL;DR

This work introduces and analyzes a novel class of holographic BPS flow equations that act as elliptic geometric flows on a Riemann surface arising from M5/D3 branes wrapped on . By reducing the supergravity BPS conditions to a single intrinsic PDE for a conformal factor on , the authors show that flows exist for arbitrary UV metrics and flow to a constant-curvature IR metric, thereby realizing a holographic uniformization. They formulate and solve the 1/2 and 1/4 BPS sectors for M5 and D3 branes, derive exact or near-exact solutions (e.g., Maldacena–Nuñez type) where possible, and perform a global existence proof using degree theory, including area monotonicity results. The results provide a rigorous holographic realization of the intuition that UV geometric data on is washed out in the IR and establish a framework for extending uniformization-like flows to broader wrapped-brane settings with potential links to a holographic c-function. These techniques open avenues for studying flow equations on higher-dimensional manifolds and for incorporating punctures or less-supersymmetric cases.

Abstract

We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the UV and the constant-curvature metric in the IR. We confirm this conjecture with a rigorous global existence proof.

Paper Structure

This paper contains 35 sections, 201 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Field Theory, Branes, Supergravity
  3. Partially twisted field theories
  4. Twists of the $(2,0)$ SCFT in six dimensions
  5. Twists of ${\cal N}=4$ SYM in four dimensions
  6. Brane realization
  7. Supergravity Ansätze
  8. M5 brane Ansätze
  9. D3 brane Ansätze
  10. Holographic Flows for Twisted M5 Branes
  11. 1/2 BPS flows
  12. Infrared analysis
  13. Ultraviolet analysis
  14. Exact solution and fluctuations
  15. 1/4 BPS flows
  16. ...and 20 more sections

Figures (3)

  • Figure 1: The functions $\gamma_n(\eta)$ (left) and $\ell_n(\eta)$ (right) for $B^{(n)}=1$ and $\mu_n=(1,2,3,4,5,6)$ (increasing as blue goes to red). The IR is at $\eta\to0$ and the UV is at $\eta\to\infty$.
  • Figure 2: A numerical solution for $\varphi(\rho)$ for $1/4$ BPS M5 brane backgrounds. In the IR ($\rho\to-\infty$), the function approaches a constant, while in the UV ($\rho\to +\infty$), it diverges linearly (logarithmically in $r$).
  • Figure 3: A numerical solution for $\varphi(\rho)$ for ${\cal N}=(2,2)$ D3 brane solutions. In the IR ($\rho\to-\infty$), the function approaches a constant, while in the UV ($\rho\to +\infty$), it diverges linearly (logarithmically in $r$).