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Branes wrapped on quadrilaterals

Federico Faedo, Alessio Fontanarossa, Dario Martelli

TL;DR

This work constructs new supersymmetric AdS$_2\times\mathbb{M}_4$ (D=6) and AdS$_3\times\mathbb{M}_4$ (D=7) solutions where $\mathbb{M}_4$ are toric orbifolds with four fixed points (quadrilaterals). The authors present a unified geometric framework that leverages orbifold gravitational blocks and a toric-data language to encode the four-fixed-point structure, enabling an off-shell extremization that reproduces the entropy and gravitational central charge. The solutions admit uplifts to massive type IIA and 11D supergravity, representing near-horizon D4- and M5-brane configurations wrapped on $\mathbb{M}_4$, with global flux quantization governed by Diophantine constraints and divisor data. A detailed treatment of toric data, divisors, complex structure, and fluxes underpins the extremization program, and the resulting off-shell free energies correctly match holographic observables. The work thus strengthens the link between geometric extremization principles and explicit orbifold supergravity solutions, and points to extensions to more general orbifolds and unequal-charge configurations.

Abstract

We construct new families of supersymmetric AdS$_2\times\mathbb{M}_4$ solutions of $D=6$ gauged supergravity and AdS$_3\times\mathbb{M}_4$ solutions of $D=7$ gauged supergravity, where $\mathbb{M}_4$ are four-dimensional toric orbifolds with four fixed points. These are presented in a unified fashion, that highlights their common underlying geometry. The $D=6$ solutions uplift to massive type IIA and describe the near-horizon limit of D4-branes wrapped on $\mathbb{M}_4$, while the $D=7$ solutions uplift to $D=11$ supergravity and describe the near-horizon limit of M5-branes wrapped on $\mathbb{M}_4$. We reproduce the entropy and gravitational central charge of the two families by extremizing a function constructed gluing the orbifold gravitational blocks proposed in arXiv:2210.16128.

Branes wrapped on quadrilaterals

TL;DR

This work constructs new supersymmetric AdS (D=6) and AdS (D=7) solutions where are toric orbifolds with four fixed points (quadrilaterals). The authors present a unified geometric framework that leverages orbifold gravitational blocks and a toric-data language to encode the four-fixed-point structure, enabling an off-shell extremization that reproduces the entropy and gravitational central charge. The solutions admit uplifts to massive type IIA and 11D supergravity, representing near-horizon D4- and M5-brane configurations wrapped on , with global flux quantization governed by Diophantine constraints and divisor data. A detailed treatment of toric data, divisors, complex structure, and fluxes underpins the extremization program, and the resulting off-shell free energies correctly match holographic observables. The work thus strengthens the link between geometric extremization principles and explicit orbifold supergravity solutions, and points to extensions to more general orbifolds and unequal-charge configurations.

Abstract

We construct new families of supersymmetric AdS solutions of gauged supergravity and AdS solutions of gauged supergravity, where are four-dimensional toric orbifolds with four fixed points. These are presented in a unified fashion, that highlights their common underlying geometry. The solutions uplift to massive type IIA and describe the near-horizon limit of D4-branes wrapped on , while the solutions uplift to supergravity and describe the near-horizon limit of M5-branes wrapped on . We reproduce the entropy and gravitational central charge of the two families by extremizing a function constructed gluing the orbifold gravitational blocks proposed in arXiv:2210.16128.
Paper Structure (28 sections, 3 theorems, 260 equations, 3 figures, 1 table)

This paper contains 28 sections, 3 theorems, 260 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Four-dimensional toric orbifolds
  3. Toric geometry ingredients
  4. Quadrilaterals
  5. Supergravity solutions
  6. Solutions in $D$ dimensions
  7. Solutions with equal charges
  8. Conditions of existence
  9. Global structure of the solutions
  10. Warm-up: revisiting the solutions of Faedo:2022rqx
  11. Degenerating Killing vectors
  12. Complex structure and divisors
  13. Toric data
  14. Quantization conditions
  15. Off-shell free energies and extremization
  16. ...and 13 more sections

Key Result

Theorem 1

Let us consider a toric Kähler orbifold $\mathbb{M}$, a ramification divisor $D_a$ and the line bundle $L_a$ associated with it through Poincaré duality. The following adjunction formula holds

Figures (3)

  • Figure 1: Outward-pointing fan and polytope of a generic quadrilateral. The vectors $\vec{v}_a$ are not primitive.
  • Figure 2: Example of graphs of $P_x^\pm(x)$ in the configuration in which $P_x$ admits four positive roots.
  • Figure 3: Polytope of the $\mathbbl{\Sigma}_1\ltimes\mathbbl{\Sigma}_2$ orbifold corresponding to the vectors \ref{['2spin_vec-n-stack']} and \ref{['2spin_vec-n-stack-new']}, respectively.

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark