nLab
Chern-Simons theory

Idea

The words “Chern–Simons theory” (after Shiing-shen Chern and James Simons who have their names attached to the Chern-Simons elements involved) can mean various things to various people, but here it generally refers to the three-dimensional topological quantum field theory introduced by Edward Witten in his seminal paper (Witten 1989), the paper which went a large way to him obtaining the Fields medal.

Witten

Abstractly, Chern-Simons theory is the sigma-model TQFT whose target space is the smooth delooping $BG$ of a simple Lie group $G$ and whose background gauge field is the Chern-Simons 2-gerbe on $BG$.

This is the ∞-Chern-Simons theory induced from the canonical Chern-Simons element on a semisimple Lie algebra $𝔤$.

For $G$ a discrete group the corresponding (much simpler) theory is Dijkgraaf-Witten theory.

Classical Chern-Simons theory (for simply connected groups)

For $G$ a simply connected Lie group we describe the basic setup of $G$-Chern-Simons theory

• Lagrangian and action functional

• Covariant phase space

The Lagrangian and action functional

Let $𝔤$ be a semisimple Lie algebra and write $⟨-,-⟩\in W(𝔤)$ for (some multiple of) its Killing form invariant polynomial (in the Weil algebra of $𝔤$).

Notice that this is in transgression via a Chern-Simons element $\mathrm{cs}\in W(𝔤)$ to (a multiple of) the canonical Lie algebra 3-cocycle

$${\mu }_3=⟨-,[-,-]⟩\in \mathrm{CE}(𝔤)$$\mu_3 = \langle -,[-,-]\rangle \in CE(\mathfrak{g})

in the Chevalley-Eilenberg algebra of $𝔤$.

For $\Sigma$ a compact smooth manifold of dimension 3, write

$${\mathrm{Conf}}_{\Sigma }:={\Omega }^1(\Sigma ,𝔤)=\{{\Omega }^{•}(\Sigma )\leftarrow W(𝔤):A\}$$Conf_\Sigma := \Omega^1(\Sigma, \mathfrak{g}) = \{ \Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{g}) : A \}

for the groupoid of Lie algebra valued 1-forms on $\Sigma$, – we call this the field configuration space of $𝔤$-Chern-Simons theory over $\Sigma$. Notice that this is canonically a smooth groupoid, as discussed there.

By means of the above Chern-Simons element $W(𝔤)\leftarrow W(b^2ℝ):\mathrm{cs}$ there is naturally associated to every field configuration $A$ a 3-form

$$\mathrm{cs}(A)\in {\Omega }^3(\Sigma )$$cs(A) \in \Omega^3(\Sigma)

called the Chern-Simons form of $A$. This 3-form is the Lagrangian of Chern-Simons theory over $\Sigma$.

Definition

The exponentiated action functional of Chern-Simons theory is the morphism

$$\exp (iS(-)):{\mathrm{Conf}}_{\Sigma }\to U(1)$$\exp(i S(-)) : Conf_\Sigma \to U(1)

to the circle group, which sends a field configuration $A$ to the integral over $\Sigma$ of its Chern-Simons form $\mathrm{CS}(A)$.

$$A\mapsto \exp (i{\int }_{\Sigma }\mathrm{cs}(A)).$$A \mapsto \exp(i \int_\Sigma cs(A)) \,.

The covariant phase space

Since the above action functional is a local action functional, its covariant phase space – which is the space of solutions of the corresponding Euler-Lagrange equations – naturally carries a presymplectic structure.

Proposition

The Euler-Lagrange equations for the action functional from def. 1 are

$$F_A=0,$$F_A = 0 \,,

where $F_A$ is the curvature 2-form of the Lie algebra valued form $A$.

The presymplectic structure on the space of solutions relative to any 2-dimensional submanifold ${\Sigma }_0\hookrightarrow \Sigma$ is

$$\omega (\delta A_1,\delta A_2)\propto {\int }_{{\Sigma }_0}⟨\delta A_1\wedge \delta A_2⟩.$$\omega(\delta A_1, \delta A_2) \propto \int_{\Sigma_0} \langle \delta A_1 \wedge \delta A_2 \rangle \,.

The proof can be found spelled out at ∞-Chern-Simons theory.

The statements for equations of motion and gauge fixed Poisson structure appears for instance as (Witten89, (2.3), (3,2)) or (FreedI, prop. 3.1, prop. 3.17). The presymplectic structure on the covariant phase space has apparently first been discussed in (Witten86, section 5) and in (Zuckerman, section 3, example 2).

Zuckerman states that on the reduced phase space of $\mathrm{GL}(2,ℝ)$-Chern-Simons theory the presymplectic form becomes the Weil-Petersson symplectic form?.

Geometric quantization

Of the existing formalizations of quantization, it is geometric quantization that is naturally suited for the treatment of Chern-Simons theory.

For the moment, see the references below.

Classical Chern-Simons theory (general case)

If the Lie group $G$ is not simply-connected a $G$-principal bundle on 3-dimensional $\Sigma$ is not necessarily trivializable. (For instance for $G=U(1)$ the circle group the circle bundles on $\Sigma$ are classified by their Chern class, which can be any element in the integral cohomology $H^2(\Sigma ,\mathbb{Z})$.)

Therefore in these cases the configuration space of Chern-Simons theory is no longer in general just a groupoid of Lie algebra valued forms – which is a groupoid of connections on trivial principal bundles, but a groupoid of more general connections on non-trivial principal bundles.

The general formulation of Chern-Simons theory is then this:

let $G$ be a compact Lie group with Lie algebra $𝔤$ and let $⟨-,-⟩$ be a binary invariant polynomial. Then the refined Chern-Weil homomorphism produces a map

$$\hat{c}:H^1(\Sigma ,G{)}_{\mathrm{conn}}\to H^4(\Sigma {)}_{\mathrm{diff}}$$\hat \mathbf{c} : H^1(\Sigma,G)_{conn} \to H^4(\Sigma)_{diff}

from $G$-bundles with connection to degree-4 ordinary differential cohomology of $\Sigma$, classifying circle 3-bundles with connection on $\Sigma$: the Chern-Simons circle 3-bundles.

The action functional of Chern-Simons theory is the higher holonomy of this circle 3-bundle

$$S_{\mathrm{CS}}:\nabla \mapsto {\int }_{\Sigma }\hat{c}(\nabla )\in U(1).$$S_CS : \nabla \mapsto \int_\Sigma \hat \mathbf{c}(\nabla) \in U(1) \,.

Abelian case

Let $G=U(1)$ the circle Lie group, and $⟨-,-⟩$ the canonical invariant polynomial.

Then the configuration space is $H^2(\Sigma {)}_{\mathrm{diff}}$ – ordinary differential cohomology in degree 2 – and the action functional is given by the fiber integration in ordinary differential cohomology over the Beilinson-Deligne cup product

$$S_{\mathrm{CS}}:\hat{A}\mapsto {\int }_{\Sigma }\hat{A}\cup \hat{A}.$$S_{CS} : \hat A \mapsto \int_\Sigma \hat A \cup \hat A \,.

(For the moment see higher dimensional Chern-Simons theory for references on this case.)

Quantum Chern-Simons theory

Observables: knot polynomials

The Wilson line?-observables in quantum Chern-Simons theory are given by knot invariants.

In Witten (1989) it was shown that the new polynomial invariant of knots invented by Vaughan Jones in the context of von Neumann algebras – the Jones polynomial – can be given a heuristic geometric interpretation: the Jones polynomial V(q) of a knot $K$ in a 3-manifold $M$ can be viewed as the path integral over all $\mathrm{SU}(2)$-connections on $M$ of the exponential of the Chern–Simons action functional $S[A]$:

(1)$$V_K(q)={\int }_{\mathrm{all}\mathrm{connections}A\mathrm{on}M}{\mathrm{hol}}_A(K)\exp (\mathrm{iS}[A])$$V_K(q) = \int_{all\,connections\,A\,on\,M} hol_A(K) \,\, exp(iS[A])

where

(2)$$S[A]=\frac{k}{4\pi }{\int }_M\mathrm{Tr}(A\wedge \mathrm{dA}+\frac{2}{3}A\wedge A\wedge A)$$S[A] = \frac{k}{4\pi} \int_M Tr (A \wedge dA + \frac{2}{3} A \wedge A \wedge A)

is the integral of the Chern–Simons Lagrangian,

(3)$$\mathrm{Tr}{\mathrm{Hol}}_A(K)$$Tr Hol_A (K)

is the trace of the holonomy of the connection around the knot $K$ in the fundamental representation? of $\mathrm{SU}(2)$, and

(4)$$q=e^{\frac{2\pi i}{k+2}}.$$q = e^{\frac{2 \pi i}{k+2}}.

Said heuristically: the Jones polynomial of the knot $K$ can be understood as the “average value” over all connections of the trace of the holonomy of the connection around the knot $K$. Note that this idea can be generalized by varying the gauge group $G$ from $\mathrm{SU}(2)$ to some other Lie group; the representation in which the trace is evaluated can also be altered. Each of these modifications gives rise to a knot invariant.

As an extended TQFT

The beautiful thing about Chern–Simons theory is that Witten was able use the locality property of the path integral to give a nonperturbative way to actually compute it. In this way Chern–Simons theory has become the ‘poster-child’ of extended topological quantum field theory since it exemplifies the main idea: take advantage of the higher gluing laws in order to compute geometric quantities.

One of the major mathematical projects around Chern–Simons theory has therefore been to try and understand it rigorously as a 3-2-1-0 extended topological quantum field theory. For the abelian case the major paper in this regard is Topological quantum field theories from Compact Lie Groups by Freed, Hopkins, Lurie and Teleman. No-one has yet made rigorous sense of the nonabelian theory as an extended TQFT. However, the invariants that the theory assigns to closed manifolds of dimension 0,1,2 and 3 are heuristically expected to be:

A closed 3-manifold $M$ $\mapsto$ the path integral given above (a number).

A closed 2-manifold $\Sigma$ $\mapsto$ the space of sections of the line bundle over the moduli space of flat connections on $\Sigma$ (a finite-dimensional vector space). (Reshetikhin and Turaev give an alternate quantum-groupy description of this space).

A circle $S^1$ $\mapsto$ the category of positive-energy representation?s of the loop group ${\Omega }_k(G)$ at level $k$ (a linear category).

The R-T construction sticks on the circle the modular tensor category of representations of a quantum group at a root of unity, modulo “unphysical representations.” Are these supposed to be the same? Is this just the Kazhdan-Lusztig equivalence?

Urs Schreiber: the Reshetikhin-Turaev construction works with any modular tensor category, I’d say. Using one coming from reps of loops group is expected to produce the Chern–Simonss QFT as a cobordism rep. But I think a full proof of that, i.e. a formalization of the CS path integral that would after turning the crank yield the RT construction, is not available to date. There is just lots of “circumstancial evidence”.

The 2-category assigned to the point is the most interesting piece of data since in principle all the other invariants can be derived from it using the gluing law. In the paper Topological Quantum Field Theories from Compact Lie Groups, it is proposed that

A point $\mapsto$ the category of skyscraper sheaves on —, thought of as a 2-category via —.

Bruce: I’ve run out of time here and I can’t precisely fill in those blanks above. Any help?

Ben Webster: My understanding is that nobody is quite sure how to fill in those blanks. One line of thinking is that it should be an object in a 3-category with is not 2Cat.

Other groups have conceptualized this differently (but most likely equivalently at the end of the day as)

A point $\mapsto$ the 2-category of unitary 2-representation?s of the group $G$.

Still others think of the 2-category assigned to the point in different terms.

Properties

Tunneling between vacua – instanton Floer homology

For $g$ a fixed Riemannian metric on the 3-dimensional base space $\Sigma$, the gradient flow of the Chern-Simons theory action functional $S_{\mathrm{CS}}:{\Omega }^1(\Sigma ,𝔤)\to ℝ$ with respect to the respective Hodge inner product metric on ${\Omega }^1(\Sigma ,𝔤)$ characterizes Yang-Mills instanton solutions of the Yang-Mills theory on $\Sigma \times ℝ$ with metric $g\otimes I$.

This phenomenon is captured by instanton Floer homology.

As an effective background for topological string theory

In (Witten94) an argument was given that Chern-Simons theory can be understood as the effective target space string theory of the A-model or B-model TCFT. This argument has later been made more precise in the language of TCFT. See TCFT – Effective background theories for more on this.

As 3-dimensional gravity

The Chern-Simons action functional for the case that the gauge group is the Poincare group $\mathrm{Iso}(2,1)$ (and the invariant polynomial is taken to be the one that pairs a translation generator with a rotation generator) happens to be equivalent to the Einstein-Hilbert action in the first order formulation of gravity in 3-dimensions.

Moreover, Chern-Simons theory for the groups $\mathrm{Iso}(2,2)$ and $\mathrm{Iso}(3,1)$ is similarly equivalent to gravity with cosmological constant in 3-dimensions.

Since the quantization of Chern-Simons theory is fairly well understood, these identifications imply indeed a definition of quantum gravity in 3-dimensions.

More on this is at Chern-Simons gravity.

Beware that there is a subtlety in the definition of the configuration space: when the field of gravity is identified with an $\mathrm{Iso}(2,1)$-connection then the configuration space naturally contains degenerate vielbein fields $E$ (notably the 0 vielbein) and hence the induced rank-2 tensor $g=⟨E\otimes E⟩$ may also be degenerate. Such degenerate tensors are not technically pseudo-Riemannian metric tensors, since these are required to be non-degenerate. The genuine non-degenerate metric tensors correspond to precisely those $\mathrm{Iso}(2,1)$-principal connections which are in fact $(O(2,1)\hookrightarrow \mathrm{Iso}(2,1))$-Cartan connections.

However, the quantum theory exists nicely if one allows the larger configuration space of possibly degenerate metrics exists nicely, while the constrained one does not. This may be interpreted as saying that at least for purposes of quantum gravity it is wrong require non-degenerate metric tensors.

Other features of Chern–Simons theory

There are plenty of topics not yet touched upon here.

• The idea of transgression, and the significance of the Chern–Simons form

• Large $N$ duality

• The Batalin-Vilkovisky viewpoint

• Much more…

Chern–Simons theory and modular forms

Trying to interest your number theory friends with Chern–Simons theory? How about this: the Chern–Simons path integral $Z(k)$ above is (in a certain precise sense) a modular form. This correspondence between the Chern–Simons quantum invariants and modular forms sheds light in both directions, and is a fascinating idea to me. The key words here (which I don’t understand) are “Eichler integral” and “mock theta function”. See:

• Lawrence and Zagier, Modular forms and quantum invariants of 3-manifolds, Asian Journal of Mathematics vol 3 no 1 (1999).

• Hikami, Quantum invariant, modular forms, and lattice points, arXiv. See also the follow ups to this paper.

The Morse theory of Chern–Simons theory

In a recent talk, Witten outlined a new approach to Chern–Simons theory which perhaps gives an alternative nonperturbative definition of the path integral. Quoting from Not Even Wrong:

The main new idea that Witten was using was that the contributions of different critical points p (including complex ones), could be calculated by choosing appropriate contours ${𝒞}_p$ using Morse theory for the Chern–Simons functional. This sort of Morse theory involving holomorphic Morse functions gets used in mathematics in Picard-Lefshetz theory?. The contour is given by the downward flow from the critical point, and the flow equation turns out to be a variant of the self-duality equation that Witten had previously encountered in his work with Kapustin on geometric Langlands. One tricky aspect of all this is that the contours one needs to integrate over are sums of the ${𝒞}_p$ with integral coefficients and these coefficients jump at “Stokes curves” as one varies the parameter in one’s integral (in this case, $x=k/n$, $k$ and $n$ are large). In his talk, Witten showed the answer that he gets for the case of the figure-eight knot.

For slides of Witten’s talk, click here and for video, click here. Pilfering material from the slides, the basic idea is as follows. Consider a general oscillatory integral in $n$ dimensions:

(5)$$I(\lambda )={\int }_{{ℝ}^n}{d}^n\exp (i\lambda f(x_1,\dots ,x_n)).$$I(\lambda) = \int_{\mathbb{R}^n} d^n exp(i\lambda f(x_1, \ldots, x_n)).

We want to make sense of the integral when the function $f$ is allowed to take on imaginary values (naively, the integral diverges). To do this, we use Morse theory: we choose as our Morse function the real part of the exponent, that is $h=\Re (i\lambda f)$. For every critical point $p$ of $h$, the descending manifold $C_p$ is an $n$-cycle in the relative homology group $H_n(X,X_{\ll 0})$. (Basically this means that it’s a new “contour” for the integral). Moreover Morse theory tells us that the cycles we obtain in this way form a basis for the homology, so we can express our original cycle $C$ (the ${ℝ}^n$ appearing in the integral over ${ℝ}^n$) as a linear combination of these Morse theory cycles:

(6)$$C=\sum _p{n}_p{C}_p$$C = \sum_p n_p C_p

In this way we can make sense of the integral $I$ by {\em defining} it as the integral over these new cycles (“contours”):

(7)$$I(\lambda )=\sum _{\text{crit. points}p}{n}_p{I}_p(\lambda )$$I(\lambda) = \sum_{\text{crit. points} p} n_p I_p(\lambda)

This new definition actually converges, and makes sense. Apparantly the same technique can be used to interpret the Chern–Simons path integral in the case of complex $k$. Witten argues that this viewpoint is useful if we try to interpret Chern–Simons theory as a theory of three-dimensional gravity,

Chern–Simons theory: 20 years on

The year 2009 marked the 20th anniversary of Witten’s seminal paper Quantum field theory and the Jones polynomial and there were a number of conferences marking this event. If anyone has notes for these conferences please say so, I (Bruce) would be very grateful!

• Chern-Simons Gauge Theory: 20 years after was a workshop at the Max Planck Institute in Bonn from 3–7 August 2009.
• (More here…)

Geometric quantization and Chern–Simons theory?

In Quantum field theory and the Jones polynomial, Witten makes the tantalizing observation that the “trace” occuring in the “trace of the holonomy around the knot” term in the path integral should itself be seen as a path integral. In this way one hopes to obtain a much more unified formalism. See Geometric quantization and the path integral in Chern-Simons theory (password required).

Higher dimensional generalizations

One question that’s been bugging me (Ben Webster) recently is what fills in the analogy “Jones polynomial is to Chern–Simons theory as Khovanov homology is to ??”

(Urs: Answer now at Khovanov homology.)

Which is to say What 3/4-dimensional structure is Khovanov homology hinting at? I’m inclined to think there must be one, as it seems that all of the knot homologies associated by Chern–Simons theory to representations have categorifications (I have a mostly finished paper on this). Presumably these all glue together into something, possibly by a similar trick to the Reshetikhin-Turaev construction of 3-manifold invariants, but it’s not so easy for me to see how.

Related concepts

• ∞-Chern-Simons theory

  • higher dimensional Chern-Simons theory

• sigma-model

  • AKSZ sigma-model

    • Poisson sigma-model

      • A-model, B-model

    • Courant sigma-model

      • Chern-Simons theory

        • ABJM theory

References

General

Chern-Simons theory was introduced in

• Edward Witten Quantum Field Theory and the Jones Polynomial Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 (project EUCLID)

It derives its name from the Chern-Simons forms that were originally introduced in

• Shiing-shen Chern, James Simons, Characteristic forms and geometric invariants The Annals of Mathematics, Second Series, Vol. 99, No. 1 (1974) (jstor).

A comprehensive and clean account of the classical aspects is in

• Dan Freed,

  • Classical Chern-Simons theory Part I , Adv. Math., 113 (1995), 237–303 (arXiv:hep-th/9206021)

  • Classical Chern-Simons theory, part II Houston J. Math., 28 (2002), pp. 293–310 (pdf)

  • Remarks on Chern-Simons theory Bulletin (New Series) of the AMS, Volume 46, Number 2, April 2009, Pages 221–254S 0273-0979(09)01243-9 (arXiv)

A decent survey of the constructions within Chern-Simons theory is in

• R. K. Kaul, T. R. Govindarajan, P. Ramadevi, Schwarz Type Topological Quantum Field Theories Encyclopedia of Mathematical Physics (2005) (arXiv:hep-th/0504100)

The covariant phase space of Chern-Simons theory with its presymplectic structure is originally discussed in section 5 of

• Edward Witten Interacting field theory of open superstrings Nuclear Physics B Volume 276, Issue 2, 13 October 1986, Pages 291-324 (1986) (web)

(there in the context of string field theory, but the manipulations of formulas is the same) and in section 3, example 2 of

• G. J. Zuckerman, Action Principles and Global Geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf) .

Quantization

We list discussions of quantizationn? of Chern-Simons theory.

Geometric quantization

The original method of quantization of Chern-Simons theory used already in (Witten 89) is geometric quantization.

More discussion of this is in

• Scott Axelrod, S. Della Pietra, Edward Witten, Geometric quantization of Chern-Simons gauge theory, Jour. Diff. Geom. 33 (1991), 787-902. (EUCLID)

• Scott Axelrod, Geometric Quantization of Chern-Simons Gauge Theory PhD thesis (1991)

Decent expositions are for instance in

• M. B. Young, Chern-Simons theory, knots and moduli spaces of connections 2010 (pdf)

and in section 3.3 of

• Tudor Dimofte, Sergei Gukov, Quantum Field Theory and the Volume Conjecture, in Andersen, Boden, Hahn, Himpel (eds.)_Chern-Simons gauge theory: 20 years after_, AMS (2011) (arXiv:1003.4808)

Hamiltonian quantization

• Anton Alexeev?, Harald Grosse?, Volker Schomerus?, Combinatorial quantization of the Hamiltonian Chern-Simons theory (pdf)

Perturbative quantization

Discussion of quantization using perturbation theory is in

• Scott Axelrod, Isadore Singer,

  Chern-Simons Perturbation Theory (arXiv:hep-th/9110056)

  Chern–Simons Perturbation Theory II (arXiv)

As second quantization of the A- or B-model

In

• Edward Witten. Chern–Simons Theory as a String Theory Prog. Math. 133: 637–678. (arXiv:hep-th/9207094).

an argument was given that Chern-Simons theory can be understood as the effective target space string theory of the A-model or B-model TCFT. This argument has later been made more precise in the language of TCFT. See TCFT – Effective background theories for more on this.

• Anton Kapustin, Natalia Saulina, Surface operators in 3d TFT and 2d Rational CFT in Branislav Jurco, Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory AMS, 2011

A talk about the historical origins of the standard Chern-Simons forms see

• Jim Simons, Origin of Chern-Simons talk at Simons Center for Geometry and Physics (2011) (video)

A discussion of Chern-Simons theory as a canonical object in infinity-Chern-Weil theory is in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory