nLab higher dimensional Chern-Simons theory

Idea

There are two kinds of higher dimensional generalizations of ordinary 3-dimensional Chern-Simons theory that are often called “higher dimensional Chern-Simons theory” in the literature. Both are special cases of infinity-Chern-Simons theory.

Recall that for $\mathfrak{g}$ a Lie algebra (not necessarily abelian) with non-generate binary invariant polynomial $\langle -,-\rangle$, the corresponding infinity-Chern-Simons theory QFT is ordinary Chern-Simons theory in dimension 3.

But also every other invariant polynomial $\langle-,-,\cdots,-\rangle$ on $\mathfrak{g}$ induces an infinity-Chern-Simons theory, now in higher dimension.

Moreover, every line Lie n-algebra $b^n \mathbb{R}$ carries a canonical invariant polynomial. The infinity-Chern-Simons theory associated with that is often called abelian higher dimensional CS theory .

(…)

More general “higher”-generalization of Chern-Simons theory to infinity-Chern-Simons theory allow $\mathfrak{g}$ to be a (nonabelian) Lie 2-algebra or more generally a (nonabelian) L-infinity algebra or fully generally a L-infinity algebroid.

Higher abelian Chern-Simons theory.

Definition

The definition of higher abelian Chern-Simons theory is simple locally when certain global cohomological effects can be ignored. We first give the simple local definition and then the full global definition.

Let $k \in \mathbb{N}$ be a natural number, let $d = 4 k + 3$ and let $\Sigma$ be a compact smooth manifold of dimension $d$.

Then the simple version of abelian $d$-dimensional Chern-Simons theory is defined as follows.

• the configuration space is the space of differential forms on $\Sigma$ of degree $2k+1$

  $$Conf_{simpl} = \Omega^{2k+1}(\Sigma) \,,$$

• the Lagrangian is

  $$L : B \mapsto B \wedge d_{dR} B \,,$$

• and the action functional

  $$S : \Omega^{2k+1}(\Sigma) \to \mathbb{R}$$

  is therefore

  $$S : B \mapsto \int_\Sigma B \wedge d_{dR}B \,.$$

Notice that generally for an $n$-form $B$ on a closed $(2n+1)$-dimensional manifold $\Sigma$ we have

$$\int_\Sigma B \wedge d_{dR} B = (-1)^{1+n + n(1+n)} \int_\Sigma B \wedge d_{dR} B$$

by first using integration by parts and then switching the order of the wedge factors. Therefore this kind of action vanishes identically when $deg B$ is even. This is the reason for the above assumption that $deg B = 2k+1$ for $k \in \mathbb{N}$ and hence that the Chern-Simons theory is in dimension $4k+3$.

In the full theory instead the configuration space is

$$Conf = \mathbf{H}_{diff}^{2k+2}(\Sigma) \,,$$

the space of circle (2k+1)-bundles with connection (given by cocycles in degree $2k+2$ ordinary differential cohomology). This contains the above simplified configuration space as the subspace of $(2k+1)$-connections whose underlying circle $(2k+1)$-bundle is trivial.

The action functional is given by

$$S : \hat B \mapsto \int_\Sigma \hat B \cup \hat B \,,$$

where now the integral is fiber integration in ordinary differential cohomology and in the integrand we have the cup product in ordinary differential cohomology of differential cocycles.

(See for instance (GT, section 4.1), (FMS, (1.28))).

Formulation in $\infty$-Chern-Simons theory

We discuss how the above definition arises as a special case of the general notion of infinity-Chern-Simons theory.

These theories are defined by

• an L-infinity algebroid $\mathfrak{a}$;

• equipped with an invariant polynomial $\langle \rangle$.

The abelian higher dimensional Chern-Simons theories in dimension $4k+3$ are the special case of this general situation where

• $\mathfrak{a} = b^{2k+1}\mathbb{R}$ is the line Lie (2k+1)-algebra, the $(2k+1)$-fold delooping of the abelian Lie algebra $\mathbb{R}$;

• $\langle - \rangle$ is the canonical quadratic invariant polynomial on this.

(…)

See (FRS, 4.1.4).

Examples

Higher dimensional abelian Chern-Simons theories appear automatically as components of systems of higher supergravity, for instance in 11-dimensional supergravity (they are automatically induced by the requirement of local supersymmetry in these higher dimensional supergravity theories).

• see at M5-brane the section Conformal blocks and 7d Chern-Simons dual.

Properties

Holographic relation to $4k+2$-dimensional theory

Higher Chern-Simons theory in dimension $4k+3$ is related by a holographic principle to self-dual higher gauge theory in dimension $4k+2$ (at least in the abelian case).

• $(k=0)$: ordinary 3-dimensional Chern-Simons theory is related to a string sigma-model on its boundary;

• $(k=1)$: 7-dimensional Chern-Simons theory is related to a fivebrane model on its boundary;

• $(k=2)$: 11-dimensional Chern-Simons theory is related to a parts of a type II string theory on its bounday (or that of the space-filling 9-brane, if one wishes) (BelovMoore)

Background charges and square root action functionals

The supergravity C-field is an example of a general phenomenon of higher abelian Chern-Simons QFTs in the presence of background charge. This phenomenon was originally noticed in (Witten) and then made precise in (HopkinsSinger 05). The holographic dual of this phenomenon is that of self-dual higher gauge theories, which for the supergravity $C$-field is the 2-form theory on the M5-brane – see there for a discussion of this example. Here we discuss this effect generally, for higher abelian Chern-Simons theory in arbitrary dimension $4k+3$.

Fix some natural number $k \in \mathbb{N}$ and an oriented manifold (compact with boundary) $X$ of dimension $4 k + 3$. The gauge equivalence class of a $(2k+1)$-form gauge field $\hat G$ on $X$ is an element in the ordinary differential cohomology group $\hat H^{2k+2}(X)$. The cup product $\hat G \cup \hat G \in \hat H^{4k+4}(X)$ of this with itself has a natural higher holonomy over $X$, denoted

$$\exp(i S (-)) : \hat H^{2k+2}(X) \to U(1)$$

$$\hat G \mapsto \exp(i \int_X \hat G \cup \hat G) \,.$$

This is the exponentiated action functional for bare $(4k+3)$-dimensional abelian Chern-Simons theory, as discussed above.

Observe now that the above action functional may be regarded as a quadratic form on the cohomology group $\hat H^{2k+2}(X)$. The corresponding bilinear form is the (“secondary”, since $X$ is of dimension $4k+3$ instead of $4k+4$) intersection pairing

$$\langle -,-\rangle : \hat H^{2k+2}(X) \times \hat H^{2k+2}(X) \to U(1)$$

$$(\hat a_1 , \hat a_2) \mapsto \exp(i \int_X \hat a_1 \cup \hat a_2 ) \,.$$

But note that from $\exp(i S(-))$ we do not obtain a quadratic refinement
of the pairing. A quadratic refinement is, by definition, a function

$$q : \hat H^{2k+2}(X) \to U(1)$$

(not necessarily homogenous of degree 2 as $\exp(i S(-))$ is), such that the intersection pairing is reobtained from it by the polarization formula

$$\langle \hat a_1, \hat a_2\rangle = q(\hat a_1 + \hat a_2) q(\hat a_1)^{-1} q(\hat a_2)^{-1} q(0) \,.$$

If we took $q := \exp(i S(-))$, then the above formula would yield not $\langle -,-\rangle$, but the square $\langle -,-\rangle^2$, given by (the exponentiation of) twice the integral.

The observation in (Witten96) was that for the correct holographic physics, we need instead an action functional which is indeed a genuine quadratic refinement of the intersection pairing.

But since the differential classes in $\hat H^{2k+2}(X)$ refine integral cohomology, we cannot in general simply divide by 2 and pass from $\exp( i \int_X \hat G \cup \hat G)$ to $\exp( i \int_X \frac{1}{2} \hat G \cup \hat G)$. The integrand in the latter expression does not make sense in general in differential cohomology. If one tried to write it out in the “obvious” local formulas one would find that it is a functional on fields which is not gauge invariant. The analog of this fact is familiar from nonabelian $G$-Chern-Simons theory with simply connected $G$, where also the theory is consistent only at interger levels. The “level” here is nothing but the underlying integral class $G \cup G$.

Therefore the only way to obtain a square root of the quadratic form $\exp(i S(-))$ is to shift its origin. Here we think of the analogy with a quadratic form $q : x \mapsto x^2$ on the real numbers (a parabola in the plane). Replacing this by $q^{\lambda} : x \mapsto x^2 + \lambda x$ for some real number $\lambda$ means keeping the shape of the form, but shifting its minimum from 0 to $-\frac{1}{2}\lambda$. If we think of this as the potential term for a scalar field $x$ previously with rotation-symmetric dynamics about $x = 0$, then the new potential exhibits spontaneous symmetry breaking: its ground state is now at $x = -\frac{1}{2}\lambda$ (and has energy $-\frac{1}{4}\lambda^2$ there). We may say that there is a background field or background charge that pushes the field out of its free equilibrium.

To lift this reasoning to our action quadratic form $\exp(i S(-))$ on differential cocycles, we need a differential class $\hat \lambda \in H^{2k+2}(X)$ such that for every $\hat a \in H^{2k+2}(X)$ the composite class

$$\hat a \cup \hat a + \hat a \cup \hat \lambda \in H^{4k+4}(X)$$

is even, hence is divisible by 2. Because then we could define a shifted action functional

$$\exp(i S^\lambda(-)) : \hat a \mapsto \exp(i \int_X \frac{1}{2}(\hat a \cup \hat a + \hat a \cup \hat \lambda)) \,,$$

where now the fraction $\frac{1}{2}$ in the integrand does make sense. One directly sees that if this exists, then this shifted action is indeed now a quadratic refinement of the intersection pairing

$$\exp(i S^\lambda(\hat a + \hat b)) \exp(i S^\lambda(\hat a))^{-1} \exp(i S^\lambda(\hat b))^{-1} \exp(i S^\lambda(0) = \exp(i \int_X \hat a \cup \hat b)) \,.$$

The condition on the existence of $\hat \lambda$ here means equivalently that the image of the underlying integral class in cohomology with coefficients in $\mathbb{Z}_2$ vanishes:

$$(a)_{\mathbb{Z}_2} \cup (a)_{\mathbb{Z}_2} + (a)_{\mathbb{Z}_2} \cup (\lambda)_{\mathbb{Z}_2} = 0 \in H^{4k+4}(X, \mathbb{Z}_2) \,,$$

Precisely such a class $(\lambda)_{\mathbb{Z}_2}$ does uniquely exist on every oriented manifold. It is called the Wu class $\nu_{2k+2} \in H^{2k+2}(X,\mathbb{Z}_2)$, and may be defined by this condition. Moreover, if $X$ is a Spin-manifold, then every second Wu class, $\nu_{4k}$, has a pre-image in integral cohomology, hence $\lambda$ does exist as required above

$$(\lambda)_{\mathbb{Z}_2} = \nu_{2k+2} \,.$$

It is given by polynomials in the Pontryagin classes of $X$. For instance the degree-4 Wu class (for $k = 1$) is refined by the first fractional Pontryagin class $\frac{1}{2}p_1$

$$(\frac{1}{2}p_1)_{\mathbb{Z}_2} = \nu_4 \,.$$

This was the original observation in Witten96, around (3.3).

Notice that the equations of motion of the shifted action $\exp(i S^\lambda(\hat a))$ are no longer $F_a = 0$, but are now $F_a = - \frac{1}{2}F_\lambda$. Comparing this to the Maxwell equations, we see that $-\frac{1}{2}\hat \lambda$ here plays the role of a background charge (or rather, of the background current that underlies a background charge). We therefore think of $\exp(i S^\lambda(-))$ as the exponentiated action functional for higher dimensional abelian Chern-Simons theory with background charge $-\frac{1}{2}\lambda$.

This of course only makes sense if $X$ is such that $\lambda$ is further divisible by 2, which we will assume now. In (Hopkins-Singer 05) is discussed a way to make sense of the further division in general if one passes to a certain notion of twisted differential cohomology. One can also adopt a different perspective and interpret the condition that $\frac{1}{2}p_1$ is further divisible by 2 precisely as a $\mathrm{String}^{2a}$-structure (SSS3). This is a higher analog of a Spin^c structure.

With respect to the shifted action functional it makes sense to introduce the shifted field

$$\hat G := \hat a + \frac{1}{2}\lambda \,.$$

This is simply a re-parameterization such that the Chern-Simons equations of motion again look homogenous, namely $FieldStrength(\hat G) = 0$. In terms of this shifted field the action $\exp(i S^\lambda(\hat a))$ from above equivalently reads

$$\exp(i S^\lambda(\hat G)) = \exp( i \int_X \frac{1}{2}(\hat G \cup \hat G - (\frac{1}{2}\hat \lambda)^2) ) \,.$$

This is the form of the action functional first given as (Witten96 (3.6)) in for the case $k = 1$.

In the language of twisted differential c-structures, we may summarize this sitation as follows:

in order for the action functional of higher abelian Chern-Simons theory to be correctly divisible, the images of the fields in $\mathbb{Z}_2$-cohomology need to form a twisted Wu structure. Therefore the fields themselves need to constitute a twisted $\lambda$-structure. For $k = 1$ this is a twisted string structure and explains for instance the quantization condition on the supergravity C-field in 11-dimensional supergravity. For that case see also the corresponding discussion at M5-brane.

Higher-dimensional non-abelian CS theory

Chern-Simons actions for Lie algebras $\mathfrak{g}$ but with higher-degree invariant polynomials have in particular received attention for $\mathfrak{g} = \mathfrak{siso}$ the super Poincare Lie algebra. In this case these action functionals can be regarded as defining higher Chern-Simons supergravity.

1d Chern-Simons theory

• 1d Chern-Simons theory

2d Chern-Simons theory

• 2d Chern-Simons theory

  • Poisson sigma-model

3d Chern-Simons theory

• 3d Chern-Simons theory

  • Chern-Simons theory

  • Dijkgraaf-Witten model

  • Courant sigma-model

4d Chern-Simons theory

• 4d Chern-Simons theory

  • Yetter model

5d Chern-Simons theory

• 5d Chern-Simons theory

6d Chern-Simons theory

• 6d Chern-Simons theory

7d Chern-Simons theory

• 7d Chern-Simons theory

11d Chern-Simons theory

• 11d Chern-Simons theory

Infinite-dimensional Chern-Simons theory

• infinite-dimensional Chern-Simons theory

AKSZ $\sigma$-models

• AKSZ sigma-model

String field theory

• string field theory

Related concepts

• Wu classes

• infinity-Chern-Simons theory

  • Chern-Simons theory

  • higher dimensional Chern-Simons theory

    • Chern-Simons gravity

  • AKSZ sigma-model

    • Poisson sigma model

      • A-model, B-model

    • Courant sigma-model

      • Chern-Simons theory

  • 1-dimensional Chern-Simons theory

• higher dimensional WZW models

References

General

For higher dimensional non-abelian Chern-Simons theory see for instance

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

• Máximo Bañados, Luis Garay, Marc Henneaux, Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories Phys. Rev. D 53, R593–R596 (1996) (pdf)

• Máximo Bañados, Luis Garay, Marc Henneaux, The dynamical structure of higher dimensional Chern-Simons theory (arXiv:hep-th/9605159)

• G Giachetta, L Mangiarotti, G Sardanashvily, Noether conservation laws in higher-dimensional Chern-Simons theory Modern Physics Letters A Volume 18(2003) (pdf)

• Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, Higher Chern-Simons based on (2-)crossed modules, JHEP 2023 207 (2023) [arXiv:2212.04667, doi:10.1007/JHEP07(2023)207]

Formulation in differential cohomology

For the formulation of abelian Chern-Simons theory by fiber integration over cup products in ordinary differential cohomology see

• Enore Guadagnini, Frank Thuillier, Deligne-Beilinson cohomology and abelian links invariants SIGMA 4 (2008), 078, 30 (arXiv:0801.1445)

• R. Floreanini, R. Percacci, Higher dimensional abelian Chern-Simons theory Physics Letters B Volume 224, Issue 3, 29 (1989)

• Daniel Freed, Gregory Moore, Graeme Segal, The Uncertainty of Fluxes (arXiv:hep-th/0605198)

Relation to self-dual theories

The idea of describing self-dual higher gauge theory by abelian Chern-Simons theory in one dimension higher originates in

• Edward Witten, Five-Brane Effective Action In M-Theory Journal of Geometry and Physics, Volume 22, Issue 2, May 1997, Pages 103-133 (arXiv:hep-th/9610234)

(there for the 6d (2,0)-susy QFT on the fivebrane) and

• Edward Witten, Duality Relations Among Topological Effects In String Theory (arXiv:hep-th/9912086)

Motivated by this the differential cohomology of self-dual fields had been discussed in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory, 2005

More discussion of the general principle is in

• Dmitriy Belov, Greg Moore, Holographic Action for the Self-Dual Field (arXiv:hep-th/0605038)

The application of this to the description of type II string theory in 10-dimensions to 11-dimensional Chern-Simons theory is in the followup

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories (arXiv:hep-th/0611020)

Higher Chern-Simons (super)gravity

There are various discussions identifying or conjecturing higher dimensional Chern-Simons theories as parts of or related to gravity and supergravity.

An original articles includes

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

An introduction and survey is in

• Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities (arXiv:hep-th/0502193)

For more references see at

• Chern-Simons gravity .

Higher Chern-Simons invariants

• F. Thuillier, Deligne-Beilinson cohomology and abelian link invariants: torsion case J. Math. Phys. 50, 122301 (2009)

• B. Broda, Higher-dimensional Chern-Simons theory and link invariants Physics Letters B Volume 280, Issues 3-4, 30 (1992) Pages 213-218

• L. Gallot, E. Pilon, F. Thuillier, Higher dimensional abelian Chern-Simons theories and their link invariants (arXiv:1207.1270)

Formulation in $\infty$-Chern-Simons theory

Below (5.11) of (Hopkins-Singer 05).

Section 4.1.4 of

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher Chern-Weil Derivation of AKSZ Sigma-Models

Section 4.4 of

• Urs Schreiber, differential cohomology in a cohesive topos

Boundary theories

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

• J. Gegenberg , G. Kunstatter, Boundary Dynamics of Higher Dimensional Chern-Simons Gravity (arXiv:hep-th/0010020)

• J. Gegenberg , G. Kunstatter, Boundary Dynamics of Higher Dimensional AdS Spacetime (arXiv:http://arxiv.org/abs/hep-th/9905228)