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Branes, Calibrations and Supergravity

Jerome P. Gauntlett

TL;DR

The notes develop a geometrical program to build supergravity solutions for branes wrapping calibrated cycles in special-holonomy manifolds, focusing on D=11 theory. They show how to use worldvolume calibrations and a chain of consistent Kaluza–Klein reductions, notably to D=7 SO(5) gauged supergravity, to construct near-horizon AdS geometries and their uplifts to 11D. Concrete results include wrapped fivebranes on SLAG 3-cycles yielding AdS4×H^3/Γ and RG-flow solutions, as well as wrapped membranes and D3-branes via higher-dimensional gauged supergravities; these provide new AdS/CFT examples and insights into dual field theories. The work emphasizes a unifying, geometrical approach via calibrations and G-structures to analyze supersymmetry, fluxes, and the structure of normal bundles across multiple brane configurations.

Abstract

These notes are based on lectures given at the Clay School on Geometry and String Theory, Isaac Newton Institute, Cambridge, 25 March - 19 April 2002. They attempt to provide an elementary and somewhat self contained discussion of the construction of supergravity solutions describing branes wrapping calibrated cycles, emphasising the geometrical aspects and focusing on D=11 supergravity. Following a discussion of the role of special holonomy backgrounds in D=11 supergravity, the basic membrane and fivebrane solutions are reviewed and the connection with the AdS/CFT correspondence is made. The world-volume description of branes is introduced and used to argue that branes wrapping calibrated cycles in special holonomy manifolds preserve supersymmetry. The corresponding supergravity solutions are constructed first in an auxiliary gauged supergravity theory which is obtained via Kaluza-Klein reduction.

Branes, Calibrations and Supergravity

TL;DR

The notes develop a geometrical program to build supergravity solutions for branes wrapping calibrated cycles in special-holonomy manifolds, focusing on D=11 theory. They show how to use worldvolume calibrations and a chain of consistent Kaluza–Klein reductions, notably to D=7 SO(5) gauged supergravity, to construct near-horizon AdS geometries and their uplifts to 11D. Concrete results include wrapped fivebranes on SLAG 3-cycles yielding AdS4×H^3/Γ and RG-flow solutions, as well as wrapped membranes and D3-branes via higher-dimensional gauged supergravities; these provide new AdS/CFT examples and insights into dual field theories. The work emphasizes a unifying, geometrical approach via calibrations and G-structures to analyze supersymmetry, fluxes, and the structure of normal bundles across multiple brane configurations.

Abstract

These notes are based on lectures given at the Clay School on Geometry and String Theory, Isaac Newton Institute, Cambridge, 25 March - 19 April 2002. They attempt to provide an elementary and somewhat self contained discussion of the construction of supergravity solutions describing branes wrapping calibrated cycles, emphasising the geometrical aspects and focusing on D=11 supergravity. Following a discussion of the role of special holonomy backgrounds in D=11 supergravity, the basic membrane and fivebrane solutions are reviewed and the connection with the AdS/CFT correspondence is made. The world-volume description of branes is introduced and used to argue that branes wrapping calibrated cycles in special holonomy manifolds preserve supersymmetry. The corresponding supergravity solutions are constructed first in an auxiliary gauged supergravity theory which is obtained via Kaluza-Klein reduction.

Paper Structure

This paper contains 19 sections, 113 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. D=11 supergravity
  3. Special Holonomy
  4. Membranes and Fivebranes
  5. AdS/CFT Correspondence
  6. Brane worldvolumes and calibrations
  7. Calibrations
  8. Membrane world-volume theory
  9. D3-brane and fivebrane world-volume theories
  10. Generalised Calibrations
  11. Supergravity solutions for fivebranes wrapping calibrated cycles
  12. Consistency of Kaluza-Klein reduction
  13. Reduction on $S^1$ to type IIA supergravity
  14. Reduction on $S^4$ to D=7 $SO(5)$ gauged supergravity
  15. Fivebranes wrapping SLAG 3-cycles
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Behaviour of the orbits for five-branes wrapping SLAG 3-cycles with $l=-1$. Note the flow from the $AdS_7$-type region when $F,x$ are large to the IR fixed point and the flows to the good and bad singularities in the IR, IR(GS) and IR(BS), respectively.
  • Figure 2: Behaviour of the orbits for fivebranes wrapping SLAG 3-cycles with $l=1$.