11-dimensional supergravity

11-dimensional supergravity
Name	11-dimensional supergravity
Introduced	1978
Contributors	Eugene Cremmer, Bernard Julia, Joel Scherk
Field	Theoretical physics

11-dimensional supergravity is the unique maximal supergravity theory in eleven spacetime dimensions, formulated as a low-energy effective field theory that unifies gravity and supersymmetry. Developed in the late 1970s by Eugene Cremmer, Bernard Julia, and Joel Scherk, it provides a cornerstone for higher-dimensional approaches to unification involving Edward Witten, John Schwarz, and Michael Green. The theory plays a central role in proposals linking M-theory, Type IIA string theory, and various dualities explored by Paul Townsend, Chris Hull, and Andrew Strominger.

Overview

11-dimensional supergravity was first written down in 1978 by Cremmer, Julia and Scherk and stands as the maximal supergravity theory consistent with a single graviton multiplet in eleven dimensions. It is the unique local theory with 32 real supercharges and connects to later developments by Peter Freund, Michael Duff, and Edward Witten in the study of compactification on manifolds such as K3 surface, Calabi–Yau manifold, and G2 manifold. Its spectrum and symmetry structure influenced work on the AdS/CFT correspondence by Juan Maldacena and on brane dynamics studied by Joseph Polchinski, Eric Bergshoeff, and Paul Townsend.

Field content and action

The fundamental fields are the elfbein (vielbein) e_M^A associated with the eleven-dimensional Lorentz group, a Majorana gravitino ψ_M forming the fermionic partner, and a three-form gauge potential A_{MNP} whose four-form field strength is denoted F_{MNPQ}. The action constructed by Cremmer, Julia, and Scherk includes the Einstein–Hilbert term for the metric, the Rarita–Schwinger term for the gravitino as developed by William Rarita and Julian Schwinger, and a Chern–Simons-like term for the three-form paralleling constructions in Chern–Simons theory and work by Edward Witten on topological terms. The precise eleven-dimensional Lagrangian controls interactions that were later related to D-brane charge quantization described by Joseph Polchinski and Clifford Johnson.

Equations of motion and supersymmetry

Supersymmetry variations close on-shell to the eleven-dimensional superalgebra first classified in studies by Haag–Łopuszański–Sohnius and explored in the context of supergravity by Steven Weinberg and Sergio Ferrara. The bosonic equations include Einstein equations with energy–momentum sourced by the four-form F and the gravitino energy, while the fermionic equation is the Rarita–Schwinger equation generalized to eleven dimensions. Preserved supersymmetries for backgrounds are characterized by Killing spinors studied in the mathematical physics literature by Shing-Tung Yau, Simon Donaldson, and Nigel Hitchin, and are central in the classification programs by Gary Gibbons and Chris Pope.

Compactifications and solutions

Classic solutions include the flat eleven-dimensional vacuum, Freund–Rubin compactifications on product spacetimes such as Anti-de Sitter space × sphere studied by Peter Freund and Mark Rubin, and brane solutions: the M2-brane and M5-brane discovered through work by Paul Townsend, Eric Bergshoeff, Roberto Emparan, and Michael Duff. Compactification on S^7 relates to the four-dimensional N=8 supergravity of Bernard Julia and Hugh Osborn, while compactification on G2 manifolds yields lower-dimensional models investigated by Ed Witten and Bobby Acharya. Solutions preserving varying fractions of supersymmetry underpin holographic dualities developed by Juan Maldacena, S. S. Gubser, and Igor Klebanov.

Relation to M-theory and string theory

11-dimensional supergravity emerges as the low-energy limit of M-theory as conjectured by Edward Witten and further articulated by Paul Townsend and Michael Duff. Dimensional reduction on a circle connects to Type IIA string theory via the work of Chris Hull and Paul Townsend, mapping the M2-brane and M5-brane to D-brane and NS5-brane configurations studied by Joseph Polchinski and Ashoke Sen. Dualities including T-duality, S-duality, and U-duality were shaped by research by Hull and Townsend and Clifford V. Johnson, establishing the role of eleven-dimensional supergravity in the panorama of string duality and nonperturbative effects probed by Nathan Seiberg.

Mathematical structures and anomalies

The theory involves geometric structures such as spin bundles, differential forms, and exceptional holonomy groups investigated by Dominic Joyce and Nigel Hitchin. Anomaly cancellation considerations for M5-brane worldvolume theories connect to the work of Alvarez-Gaumé and Witten and to index theorem technology by Atiyah–Singer, while the quantization of fluxes and Chern–Simons terms has been analyzed by Edward Witten and Greg Moore. Cohomological frameworks including K-theory and generalized cohomology appear in studies by Graeme Segal and Daniel S. Freed, informing consistent definitions of partition functions and global anomalies as in research by Justin Harvey and Greg Moore.

Physical implications and phenomenology

Though formulated in eleven dimensions, the theory informs model building and phenomenological scenarios through compactification schemes by Michael Green, John Schwarz, Cumrun Vafa, and Erik Verlinde. Its brane solutions underpin cosmological model constructions examined by Andrei Linde and Alan Guth and proposals for particle spectra inspired by Lisa Randall and Raman Sundrum. Observational tests remain indirect and are tied to searches for signatures of extra dimensions in experiments at CERN, model-dependent predictions from supersymmetry searches led by Howard Baer and collider phenomenology groups, and cosmological probes pursued by Planck (spacecraft) and WMAP teams.

Category:Supergravity