# 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

### General

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 gravity is used for the modification 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.

### Problems in the non-perturbative regime

The above discussion of the relation between 3-dimensional Einstein-Hilbert gravity and Chern-Simons theory certainly makes sense in perturbation theory. But non-perturbatively the relation between the two theories is at best much more subtle and apparently in fact not existent.

Citing from (Witten 07, pages 4,5,6):

nonperturbatively, the relation between three-dimensional gravity and Chern-Simons gauge theory is unclear. For one thing, in Chern-Simons theory, nonperturbatively the vierbein may cease to be invertible. For example, there is a classical solution with $A = \omega = e = 0$. The viewpoint in (Witten 88) was that such non-geometrical configurations must be included to make sense of three-dimensional quantum gravity nonperturbatively. But it has has been pointed out (notably by N. Seiberg) that when we do know how to make sense of quantum gravity, we take the invertibility of the vierbein seriously. For example, in perturbative string theory, understood as a model of quantum gravity in two spacetime dimensions, the integration over moduli space of Riemann surfaces that leads to a sensible theory is derived assuming that the metric should be non-degenerate. There are other possible problems in the nonperturbative relation between threedimensional gravity and Chern-Simons theory. The equivalence between diffeomorphisms and gauge transformations is limited to diffeomorphisms that are continuously connected to the identity. However, in gravity, we believe that more general diffeomorphisms (such as modular transformations in perturbative string theory) play an important role. These are not naturally incorporated in the Chern-Simons description. One can by hand supplement the gauge theory description by imposing invariance under disconnected diffeomorphisms, but it is not clear how natural this is. Similarly, in quantum gravity, one expects that it is necessary to sum over the different topologies of spacetime. Nothing in the Chern-Simons description requires us to make such a sum. We can supplement the Chern-Simons action with an instruction to sum over threemanifolds, but it is not clear why we should do this. From the point of view of the Chern-Simons description, it seems natural to fix a particular Riemann surface $\Sigma$, say of genus $g$, and construct a quantum Hilbert space by quantizing the Chern-Simons gauge fields on $\Sigma$. (Indeed, there has been remarkable progress in learning how to do this and to relate the results to Liouville theory [8-11].) In quantum gravity, we expect topology-changing processes, such that it might not be possible to associate a Hilbert space with a particular spatial manifold. Regardless of one’s opinion of questions such as these, there is a more serious problem with the idea that gravity and gauge theory are equivalent nonperturbatively in three dimensions. Some years after the gauge/gravity relation was suggested, it was discovered by Bañados, Teitelboim, and Zanelli [12] that in three-dimensional gravity with negative cosmological constant, there are black hole solutions. The existence of these objects, generally called BTZ black holes, is surprising given that the classical theory is “trivial.” Subsequent work [13,14] has made it clear that three-dimensional black holes should be taken seriously, particularly in the context of the AdS/CFT correspondence [15]. The BTZ black hole has a horizon of positive length and a corresponding Bekenstein-Hawking entropy. If, therefore, three dimensional gravity does correspond to a quantum theory, this theory should have a huge degeneracy of black hole states. It seems unlikely that this degeneracy can be understood in Chern-Simons gauge theory, because this essentially topological theory has too few degrees of freedom.

See also at AdS3-CFT2 and CS-WZW correspondence for (pointers to) discussion of how some variant of Chern-Simons theory appears as one “sector” of AdS/CFT in 3 dimensions/2-dimensions (and indeed not as the quantum gravity sector).

## 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 February 16, 2014 12:14:54 by Urs Schreiber (89.204.139.247)