nLab
higher dimensional Chern-Simons theory

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

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 𝔤\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 𝔤\mathfrak{g} induces an infinity-Chern-Simons theory, now in higher dimension.

Moreover, every line Lie n-algebra b nb^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 𝔤\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 kk \in \mathbb{N} be a natural number, let d=4k+3d = 4 k + 3 and let Σ\Sigma be a compact smooth manifold of dimension dd.

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

  • the configuration space is the space of differential forms on Σ\Sigma of degree 2k+12k+1

    Conf simpl=Ω 2k+1(Σ), Conf_{simpl} = \Omega^{2k+1}(\Sigma) \,,
  • the Lagrangian is

    L:BBd dRB, L : B \mapsto B \wedge d_{dR} B \,,
  • and the action functional

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

    is therefore

    S:B σBd dRB. S : B \mapsto \int_\sigma B \wedge d_{dR}B \,.

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

ΣBd dRB=(1) 1+n+n(1+n) ΣBd dRB \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 degBdeg B is even. This is the reason for the above assumption that degB=2k+1deg B = 2k+1 for kk \in \mathbb{N} and hence that the Chern-Simons theory is in dimension 4k+34k+3.

In the full theory instead the configuration space is

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

the space of circle (2k+1)-bundles with connection (given by cocycles in degree 2k+22k+2 ordinary differential cohomology). This contains the above simplified configuration space as the subspace of (2k+1)(2k+1)-connections whose underlying circle (2k+1)(2k+1)-bundle is trivial.

The action functional is given by

S:B^ ΣB^B^, 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

The abelian higher dimensional Chern-Simons theories in dimension 4k+34k+3 are the special case of this general situation where

(…)

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).

Properties

Holographic relation to 4k+24k+2-dimensional theory

Higher Chern-Simons theory in dimension 4k+34k+3 is related by a holographic principle to self-dual higher gauge theory in dimension 4k+24k+2 (at least in the abelian case).

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 CC-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+34k+3.

Fix some natural number kk \in \mathbb{N} and an oriented manifold (compact with boundary) XX of dimension 4k+34 k + 3. The gauge equivalence class of a (2k+1)(2k+1)-form gauge field G^\hat G on XX is an element in the ordinary differential cohomology group H^ 2k+2(X)\hat H^{2k+2}(X). The cup product G^G^H^ 4k+4(X)\hat G \cup \hat G \in \hat H^{4k+4}(X) of this with itself has a natural higher holonomy over XX, denoted

exp(iS()):H^ 2k+2(X)U(1) \exp(i S (-)) : \hat H^{2k+2}(X) \to U(1)
G^exp(i XG^G^). \hat G \mapsto \exp(i \int_X \hat G \cup \hat G) \,.

This is the exponentiated action functional for bare (4k+3)(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 H^ 2k+2(X)\hat H^{2k+2}(X). The corresponding bilinear form is the (“secondary”, since XX is of dimension 4k+34k+3 instead of 4k+44k+4) intersection pairing

,:H^ 2k+2(X)×H^ 2k+2(X)U(1) \langle -,-\rangle : \hat H^{2k+2}(X) \times \hat H^{2k+2}(X) \to U(1)
(a^ 1,a^ 2)exp(i Xa^ 1a^ 2). (\hat a_1 , \hat a_2) \mapsto \exp(i \int_X \hat a_1 \cup \hat a_2 ) \,.

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

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

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

a^ 1,a^ 2=q(a^ 1+a^ 2)q(a^ 1) 1q(a^ 2) 1q(0). \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(iS())q := \exp(i S(-)), then the above formula would yield not ,\langle -,-\rangle, but the square , 2\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 H^ 2k+2(X)\hat H^{2k+2}(X) refine integral cohomology, we cannot in general simply divide by 2 and pass from exp(i XG^G^)\exp( i \int_X \hat G \cup \hat G) to exp(i X12G^G^)\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 GG-Chern-Simons theory with simply connected GG, where also the theory is consistent only at interger levels. The “level” here is nothing but the underlying integral class GGG \cup G.

Therefore the only way to obtain a square root of the quadratic form exp(iS())\exp(i S(-)) is to shift its origin. Here we think of the analogy with a quadratic form q:xx 2q : x \mapsto x^2 on the real numbers (a parabola in the plane). Replacing this by q λ:xx 2+λxq^{\lambda} : x \mapsto x^2 + \lambda x for some real number λ\lambda means keeping the shape of the form, but shifting its minimum from 0 to 12λ-\frac{1}{2}\lambda. If we think of this as the potential term for a scalar field xx previously with rotation-symmetric dynamics about x=0x = 0, then the new potential exhibits spontaneous symmetry breaking: its ground state is now at x=12λx = -\frac{1}{2}\lambda (and has energy 14λ 2-\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(iS())\exp(i S(-)) on differential cocycles, we need a differential class λ^H 2k+2(X)\hat \lambda \in H^{2k+2}(X) such that for every a^H 2k+2(X)\hat a \in H^{2k+2}(X) the composite class

a^a^+a^λ^H 4k+4(X) \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(iS λ()):a^exp(i X12(a^a^+a^λ^)), \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 12\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(iS λ(a^+b^))exp(iS λ(a^)) 1exp(iS λ(b^)) 1exp(iS λ(0)=exp(i Xa^b^)). \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 λ^\hat \lambda here means equivalently that the image of the underlying integral class in cohomology with coefficients in 2\mathbb{Z}_2 vanishes:

(a) 2(a) 2+(a) 2(λ) 2=0H 4k+4(X, 2), (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 (λ) 2(\lambda)_{\mathbb{Z}_2} does uniquely exist on every oriented manifold. It is called the Wu class ν 2k+2H 2k+2(X, 2)\nu_{2k+2} \in H^{2k+2}(X,\mathbb{Z}_2), and may be defined by this condition. Moreover, if XX is a Spin-manifold, then every second Wu class, ν 4k\nu_{4k}, has a pre-image in integral cohomology, hence λ\lambda does exist as required above

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

It is given by polynomials in the Pontryagin classes of XX. For instance the degree-4 Wu class (for k=1k = 1) is refined by the first fractional Pontryagin class 12p 1\frac{1}{2}p_1

(12p 1) 2=ν 4. (\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(iS λ(a^))\exp(i S^\lambda(\hat a)) are no longer F a=0F_a = 0, but are now F a=12F λF_a = - \frac{1}{2}F_\lambda. Comparing this to the Maxwell equations, we see that 12λ^-\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(iS λ())\exp(i S^\lambda(-)) as the exponentiated action functional for higher dimensional abelian Chern-Simons theory with background charge 12λ-\frac{1}{2}\lambda.

This of course only makes sense if XX is such that λ\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 12p 1\frac{1}{2}p_1 is further divisible by 2 precisely as a String 2a\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

G^:=a^+12λ. \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(G^)=0FieldStrength(\hat G) = 0. In terms of this shifted field the action exp(iS λ(a^))\exp(i S^\lambda(\hat a)) from above equivalently reads

exp(iS λ(G^))=exp(i X12(G^G^(12λ^) 2)). \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=1k = 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 2\mathbb{Z}_2-cohomology need to form a twisted Wu structure. Therefore the fields themselves need to constitute a twisted λ\lambda-structure. For k=1k = 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 𝔤\mathfrak{g} but with higher-degree invariant polynomials have in particular received attention for 𝔤=𝔰𝔦𝔰𝔬\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

2d Chern-Simons theory

3d Chern-Simons theory

4d Chern-Simons theory

5d Chern-Simons theory

6d Chern-Simons theory

7d Chern-Simons theory

11d Chern-Simons theory

Infinite-dimensional Chern-Simons theory

AKSZ σ\sigma-models

String field theory

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)

Formulation in differential cohomology

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

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

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

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

More discussion of the general principle is in

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

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

For more references see at

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

Section 4.4 of

Boundary theories

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

Revised on June 3, 2014 01:37:54 by Urs Schreiber (82.113.106.149)