# nLab Chern-Simons gravity

### Context

#### Gravity

gravity, supergravity

## Quantum theory

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

In the first order formulation of gravity the Einstein-Hilbert action in dimension 3 happens to be equivalent to the action functional of Chern-Simons theory for connections with values in the Poincare Lie algebra (spin connection and vielbein, for a suitable quadratic invariant polynomial. Similarly for 3-dimensional supergravity. See 3d quantum gravity.

Generally, for every higher degree ($n$-ary) invariant polynomial on the Poincaré Lie algebra there is the corresponding higher dimensional Chern-Simons theory in dimension $2n-1$. These Chern-Simons type action functionals all include the Einstein-Hilbert action linear in the curvature as a summand, but generally contain higher degree curvature invariants. A theory of Chern-Simons gravity is a field theory governed by such an action functional.

While there is no experimental evidence for such higher curvature terms in the theory of gravity, there exist parameter regions in which their predicted effects are smaller than could have been observed. Moreover, there are varous suggestions that under Inönü-Wigner contraction? of the AdS Lie algebra? to the Poincare Lie algebra, and $so(n,2)$-Chern-Simons gravity theory could be close to an ordinary einstein-Hilbert theory.

In view of this, the fact that Chern-Simons action functionals are singled out by their nice formal properties has led to various speculations that possibly the fundamental theory of gravity is secretly a theory of Chern-Simons gravity after all (see the References). Whether that is true or not, certainly in the general mathematical context of infinity-Chern-Simons theory the study of Chern-Simons gravity is natural and interesting.

Warning In parts of the literature the term Chern-Simons grvaity is used for the modificaton of the Einstein-Hilbert action in 4-dimensions obtained by adding to the standard action the Pontryagin class term $\int_X \langle R \wedge R\rangle$ for $R$ the Riemann curvature. This is however not a Chern-Simons action functional in the strict sense of the term.

## References

### In 3 dimensions

The original articles that considered 3-dimensional gravity as a Chern-Simons theory are

• A. Achúcarro and P. Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories , Phys. Lett. B180 (1986) 89.

• Edward Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System Nucl. Phys. B311 (1988) 46. (web)

Further developments are in

A review of some aspects is in

• Bastian Wemmenhove, Quantisation of 2+1 dimensional Gravity as a Chern-Simons theory thesis (2002) (pdf)

Boundaries in 3d Chern-Simons gravity and relation to the WZW model are discussed for instance in

### General and higher dimensions

An introduction and survey is in

More along these lines is in

Original articles include

• Ali Chamseddine, Topological gravity and supergravity in various dimensions Nuclear Physics B346 (1990) 213—234 (web)

• Máximo Bañados, Higher dimensional Chern-Simons theories and black holes (pdf)

• Máximo Bañados, Ricardo Troncoso, Jorge Zanelli, Higher dimensional Chern-Simons supergravity Phys. Rev. D 54, 2605–2611 (1996)

• Jorge Zanelli, Chern–Simons forms and transgression actions or the universe as a subsystem Journal of Physics: Conference Series Volume 68 Volume 68

• Hitoshi Nishino and Subhash Rajpoot, Supersymmetric Lorentz Chern-Simons terms coupled to supergravity Phys. Rev. D 81, 085029 (2010)

### In 11d

A speculation that 11-dimensional supergravity is naturally to be understood as a contraction limit of a Chern-Simons supergravity theory was put forward in

with further developments in

Other approaches to 11d Chern-Simons supergravity include

• Fernando Izaurieta, Eduardo Rodríguez, On eleven-dimensional Supergravity and Chern-Simons theory (arXiv:1103.2182)

There is an 6-ary invariant polynomial of degree $12 = 2 \cdot 6$ on the M-theory super Lie algebra. Using its Chern-Simons element as the Lagrangian for an infinity-Chern-Simons theory yields an 11-dimensional supersymmetric field theory different from but maybe related to 11-dimensional supergravity. This is discussed in

A discussion of Chern-Simons elements in higher supergravity and their relation not quite to Chern-Simons forms but to their curvature-first-order-analog – the cosmo-cocycle condition – is at

### Boundary theories

Boundary higher dimensional WZW models for nonabelian higher dimensional Chern-Simons theory are discussed in

Revised on May 22, 2013 16:41:03 by Urs Schreiber (84.153.217.239)