# nLab Chern-Simons theory

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

The words “Chern–Simons theory” (after Shiing-shen Chern and James Simons who have their names attached to the Chern-Simons elements and Chern-Simons forms and Chern-Simons circle 3-bundle involved) can mean various things to various people, but it generally refers to the three-dimensional topological quantum field theory whose configuration space is the space of $G$-principal bundles with connection on a bundle and whose Lagrangian is given by the Chern-Simons form of such a connection (for simply connected $G$), or rather, more generally, whose action functional is given by the higher holonomy of the Chern-Simons circle 3-bundle.

In other words, for $G$ a Lie group, Chern-Simons theory is a sigma-model TQFT whose target space is the smooth moduli stack $\mathbf{B}G_{conn}$ of $G$-principal connections, and whose background gauge field is a circle 3-bundle with connection on $\mathbf{B}G_{conn}$. The higher Chern class/Dixmier-Douady class of this three bundle is the level of the Chern-Simons theory. For $G$ semisimple this is the ∞-Chern-Simons theory induced from the canonical Chern-Simons element on a semisimple Lie algebra $\mathfrak{g}$.

For the special case that $G$ is a discrete group the theory reduces to the (much simpler) Dijkgraaf-Witten theory.

The Chern-Simons TQFT was introduced in (Witten 1989).

## Classical Chern-Simons theory

The properties of the field configuration space of Chern-Simons theory depends on the properties of its gauge group $G$. If $G$ is a simply connected Lie group, then then configuration space is isomorphic simply to the space of Lie algebra valued 1-forms on the given base manifold. Generally, though, it is given by $G$-principal bundles with connection. We discuss the first case separately

### For simply connected gauge groups

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

#### The Lagrangian and action functional

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

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

$\mu_3 = \langle -,[-,-]\rangle \in CE(\mathfrak{g})$

in the Chevalley-Eilenberg algebra of $\mathfrak{g}$.

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

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

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

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

$cs(A) \in \Omega^3(\Sigma)$

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

$\exp(i S(-)) : Conf_\Sigma \to U(1)$

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

$A \mapsto \exp(i \int_\Sigma cs(A)) \,.$

#### 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

$F_A = 0 \,,$

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

$\omega(\delta A_1, \delta A_2) \propto \int_{\Sigma_0} \langle \delta A_1 \wedge \delta A_2 \rangle \,.$

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 symplectic structure on the moduli space of flat connections is discussed in more detail also in (Atiyah-Bott) and in (Weinstein).

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 $GL(2,\mathbb{R})$-Chern-Simons theory the presymplectic form becomes the Weil-Petersson symplectic form?.

#### With Wilson line observables

More generally, configurations of Chern-Simons theory are defined on 3-dimensional manifolds $\Sigma$ with a closed 1-dimensional submanifold $\Sigma^{def} \hookrightarrow \Sigma$ where each connected component (diffeomorphic to a circle) is labeled by an irreducible unitary representation $R_i$ of the gauge group.

In the path integral formulation of Chern-Simons theory this means that to the integrand is added the Wilson loop $W(\Sigma^{def},R, A)$ of the principal connection around $\Sigma^def$

$Z(\Sigma^{def}, \Sigma) \propto \int D A \;\; W(\Sigma^{def}, R, A) \exp(i S_{CS}(A)) \,.$

But more fundamentally, the whole Wilson loop $W(\cdots)$ itself here is to be regarded as the result of a path integral for a 1-dimensional Chern-Simons theory with moduli space a coadjoint orbit of $G$ and with the representation $R$ arising by the orbit method (see there for more details).

This means that the configuration space of Chern-Simons theory over $(\Sigma^{def} \hookrightarrow \Sigma)$ is the space of $G$-principal connections on $\Sigma$ and of maps to the coadjoint orbit on $\Sigma^{def}$, with the action functional now being the sum of 3-dimensional Chern-Simons theory over $\Sigma$ as above and of a 1-dimensional Chern-Simons theory along $\Sigma^{def}$ for which the $G$-principal connection serves as a background gauge field.

This orbit method-formulation of Wilson loops in Chern-Simons theory was vaguely indicated in (Witten89, p. 22). More details were discussed in (EMSS 89), but in the context of other gauge theories (Yang-Mills theory) the same formulation appears much earlier in (Balachandran, Borchardt, Stern 78). A detailed review and further refinements are discussed in section 4 of (Beasley 09). Aspects of the formulation in the context of BV-BRST formalism are discussed in (Alekseev-Barmaz-Mnev 12). The formalizations via extended Lagrangians and extended prequantum field theory is in (Fiorenza-Sati-Schreiber 12).

### For general gauge groups

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 $\mathfrak{g}$ and let $\langle-,-\rangle$ be a binary invariant polynomial. Then the refined Chern-Weil homomorphism produces a map

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

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

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

#### Abelian case

Let $G = U(1)$ the circle group, and $\langle-,-\rangle$ the canonical invariant polynomial.

Then the configuration space is $\mathbf{H}^2(\Sigma)_{diff}$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

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

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

## Quantization

We discuss now aspects of the quantization of Chern-Simons theory. There are two main formalizations for making sense of this, geometric quantization and algebraic deformation quantization. We discuss these separartely:

### Geometric quantization

Of the existing formalizations of quantization, it is geometric quantization that is naturally suited for obtaining the spaces of states of Chern-Simons theory and their identification with the conformal blocks of the holographic dual WZW model. We indicate some aspects.

We discuss the space of states (in geometric quantization) of quantized $G$-Chern-Simons theory for $G$ a simple, simply connected Lie group, as above.

#### Geometric Prequantization

1. consider Chern-Simons theory on a 3-dimensional smooth manifold which is a cylinder $\Sigma_3 \coloneqq \Sigma_2 \times [0,1]$ over a 2-dimensional manifold;

2. compute the covariant phase space over $\Sigma_3$ to be that of flat connections on $\Sigma$, equipped with a certain presymplectic form (as discussed above);

3. after gauge reduction this becomes a symplectic form, for which there is a prequantum circle bundle on the phase space;

#### Geometric quantization

1. in order to complete this prequantization to geometric quantization one needs to choose a polarization of phase space; it turns out that one naturally obtains such from any choice of conformal structure $[g]$ on $\Sigma$ (see for instance Witten-Jeffrey, p. 81, see also self-dual higher gauge theory – Relation to Chern-Simons – Conformal structure from polarization):

this is provided by the Narasimhan–Seshadri theorem which establishes that the moduli space of flat connections on a Riemann surface is naturally a complex manifold. This yields a Kähler polarization (as well as a spin^c-structure for geometric quantization by push-forward (Freed-Hopkins-Teleman)).

2. one finds that the resulting space of states (in geometric quantization) $H_{[g]}$ is naturally isomorphic to the space of conformal blocks of the 2-dimensional WZW model on $\Sigma$, regarded with that conformal structure.

So for a 2-dimensional manifold $\Sigma$, a choice of polarization of the phase space of 3d Chern-Simons theory on $\Sigma$ is naturally induced by a choice $J$ of conformal structure on $\Sigma$. Once such a choice is made, the resulting space of quantum states $\mathcal{H}_\Sigma^{(J)}$ of the Chern-Simons theory over $\Sigma$ is naturally identified with the space of conformal blocks of the WZW model 2d CFT on the Riemann surface $(\Sigma, J)$.

But since from the point of view of the 3d Chern-Simons theory the polarization $J$ is an arbitrary choice, the space of quantum states $\mathcal{H}_\Sigma^{(J)}$ should not depend on this choice, up to specified equivalence. Formally this means that as $J$ varies (over the moduli space of conformal structures on $\Sigma$) the $\mathcal{H}_{\Sigma}^{(J)}$ should form a vector bundle on this moduli space of conformal structures which is equipped with a flat connection whose parallel transport hence provides equivalences between between the fibers $\mathcal{H}_{\Sigma}^{(J)}$ of this vector bundle.

This flat connection is the Knizhnik-Zamolodchikov connection / Hitchin connection. This was maybe first realized and explained in (Witten 89, p. 20) and first actually constructed in (Axelrod-Pietra-Witten 91).

For more see the references below.

#### Extended / in higher codimension

We discuss aspects of Chern-Simons theory in extended prequantum field theory. For more on this see at Higher Chern-Simons local prequantum field theory.

Let $G$ be a simply connected Lie group, such as the spin group.

Write $\mathbf{B}G \in \mathbf{H}$ for the smooth moduli stack of $G$-principal bundles (as discussed here) and write $\mathbf{B}G_{conn} \in \mathbf{H}$ for that of $G$-principal bundles with connection. (Here $\mathbf{H} =$ Smooth∞Grpd is the (∞,1)-topos for smooth higher geometry.) Similarly, for $n \in \mathbb{N}$ write $\mathbf{B}^n U(1)_{conn}$ for the smooth moduli n-stack of circle n-bundles with connection.

In contrast, write $B G \in$ Top for the ordinary classifying space of $G$-principal bundles, the geometric realization of $\mathbf{B}G$.

By the discussion at Lie group cohomology we have

$H^4(B G, \mathbb{Z}) \simeq \mathbb{Z} \simeq \pi_0 \mathbf{H}^3(\mathbf{B}G, U(1)) \,.$

Write

$\mathbf{c} : \mathbf{B}G \to \mathbf{B}^3 U(1)$

for a morphism in $\mathbf{H}$ representing this class. This modulates the cicle 3-group=$\mathbf{B}^2 U(1)$-principal ∞-bundle over $\mathbf{B}G$

$\array{ P \\ \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1) }$

sometimes called the Chern-Simons circle 3-bundle.

By the discussion at differential String structure (FSS) this map has a differential refinement to a morphism of smooth moduli stacks of the form:

$\mathbf{c}_{conn} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \,.$

Let then $\Sigma_k$ be a compact closed oriented smooth manifold of dimension $0 \leq k \leq 3$. We can form the internal hom (mapping $\infty$-stack) $[\Sigma_k,-]$ and produce the morphism

$[\Sigma_k, \mathbf{c}_{conn}] : [\Sigma_k, \mathbf{B}G_{conn}] \to [\Sigma_k, \mathbf{B}^3 U(1)_{conn}] \,.$

This in turn we may postcompose with the operation of fiber integration in ordinary differential cohomology, refined to a morphism of smooth ∞-groupoids

$\exp(2 \pi i \int_{\Sigma_k}(-)) : [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k}U(1)_{conn} \,.$

The result is a morphism

$\exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, (-)]) : [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \mathbf{c}_{conn}]}{\to} [\Sigma_k, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_k}(-))}{\to} \mathbf{B}^{3-k} U(1)_{conn} \,.$

The result is a circle (3-k)-bundle with connection over the smooth moduli stack of Chern-Simons fields on $\Sigma_k$. We explain now how, as $k$-varies, there transgressions of the differential characteristic map $\mathbf{c}_{conn}$ constitute prequantum circle (3-k)-bundles for an higher geometric quantization of Chern-Simons theory, as indicated in the following table.

codimension $k=$prequantum circle (3-k)-bundle
3differentially refined first fractional Pontryagin class (level (Chern-Simons theory))$\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$
2Wess-Zumino-Witten model background B-field$G \stackrel{\bar \nabla_{can}}{\to} [S^1, \mathbf{B}G_{conn}] \stackrel{[S^1, \mathbf{c}_{conn}]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^2}(-))}{\to} \mathbf{B}^2 U(1)_{conn}$
1ordinary prequantum circle bundle of Chern-Simons theory$[\Sigma_2, \flat\mathbf{B}G] \to [\Sigma_2, \mathbf{B}G_{conn}] \stackrel{[\Sigma_2, \mathbf{c}_{conn}]}{\to} [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_2}(-)}{\to} \mathbf{B} U(1)_{conn}$
0action functional of Chern-Simons theory$[\Sigma_3, \mathbf{B}G_{conn}] \stackrel{[\Sigma_3, \mathbf{c}_{conn}]}{\to} [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_3}(-))}{\to} U(1)$

The first line we have already discussed (FSS). The second line is implicit in (CJMSW, def. 3.3, prop. 3.4). There it is shown that there is a natural $G$-principal connection on $S^1 \times$, such that the fiber integration in ordinary differential cohomology of its Chern-Simons circle 3-bundle with connection $\nabla_{con}$ over $S^1$ is the Wess-Zumino-Witten model circle n-bundle with connection on $G$. But in terms of moduli stacks this means that there is a canonical morphism

$\nabla_{can} : G \times S^1 \to \mathbf{B}G_{conn}$

$\bar \nabla_{conn} : G \to [S^1 , \mathbf{B}G_{conn}]$

has the property that postcomposition with $\exp(2 \pi i \int_{S^1}[S^1, \mathbf{c}_{conn}])$ modulates the WZW 2-bundle. This is precisely the content of the second line in the table above.

### Perturbative quantization

#### Path integral quantization

Witten (1989), section 2 indicates the perturbative path integral quantization of Chern-Simons theory and finds that the result after gauge fixing by a choice of Riemannian metric $g$ is the sum over equivalence classes of classical solutions, i.e. of flat connections/local systems $\nabla_{fl}$, of the product of

1. the exponentiated Chern-Simons action functional $\exp(i k \, S_{CS}(\nabla))$ of the classical solution;

2. the analytic torsion $T(g,\nabla_{fl})$;

3. the exponentiated eta invariant $\exp(\pi i\, \eta(\nabla_{fl}))$ (hence the Selberg zeta function twisted by the given local system $\nabla_{fl}$).

$\underset{}{\int} \exp(i k \, S_{CS}(\nabla)) (D\nabla) \;\stackrel{k \to \infty}{\longrightarrow}\; \underset{[\nabla_{fl}]}{\sum} \exp\left(i k S_{CS}\left(\nabla_{fl}\right)\right) \exp\left(i \pi \eta\left(\nabla_{fl}\right) \right) T\left(g,\nabla_{fl}\right) \,.$

For more on this see at eta invariant – Boundaries, determinant line bundles and perturbative Chern-Simons.

Now this expression is not independent of the chosen metric $g$. But it becomes indepdendent of it after adding the $SO$-Chern-Simons term of the Levi-Civita connection of the metric (Witten 89 (2.23)) (and is then well-defined after choosing an Atiyah 2-framing).

Notice that here $i k$ is $\tfrac{i}{\hbar}$,so that the limit $k \to \infty$ is the semiclassical limit, i.e. this is perturbation theory in Planck's constant $\hbar$, as also considered in perturbative deformation quantization.

#### BV deformation quantization

We discuss the perturbative deformation quantization of Chern-Simons theory to a factorization algebra of local observables along the lines of renormalization – Of theories in BV-CS forms (Costello).

We first discuss the classical and quantum BV-BRST complex of the underlying free field theory

and then perturbatively introduce the interactions by renormalizing and solving the quantum master equation:

##### The BV-BRST complex of the underlying free field theory

Fix $(\mathfrak{g}, \langle -,-\rangle_{\mathfrak{g}})$ a Lie algebra equipped with a binary and non-degenerate invariant polynomial (for instance a semisimple Lie algebra with Killing form). Let $\Sigma$ be a smooth closed manifold.

In perturbation theory we consider only infinitesimal gauge transformations between the fields,

In perturbation theory we regard the interaction term

$A \mapsto I(A) \propto \int_\Sigma \langle A \wedge [A \wedge A]\rangle$

as a perturbation of the free field theory with kinetic action functional

$A \mapsto S_{kin}(A) \propto \int_\Sigma \langle A \wedge d A\rangle \,.$

The BRST complex of the full theory is the Chevalley-Eilenberg algebra $CE(Lie(G\mathbf{Conn}(X)//C^\infty(X,G)))$ of the Lie algebroid $Lie(\Omega^1_\Sigma(-,\mathfrak{g})//[X,G]$ which is the Lie differentiation of the action groupoid of the smooth group of gauge transformations acting on the fields, which are the Lie algebra valued 1-forms.

So the BRST complex of Chern-Simons theory has

• fields the 1-forms $\Omega^1(\Sigma, \mathfrak{g})$

• and as ghosts the $\Omega^0(\Sigma, \mathfrak{g})$.

The corresponding BV-BRST complex hence has in addition

• antifields the local duals of the fields, which under the evident wedge product-and-integration pairing is $\Omega^2(\Sigma, \mathfrak{g})$;

• antighosts similarly $\Omega^3(\Sigma, \mathfrak{g})$.

In summary the fields, ghosts, antifields and antighists form the de Rham complex of $\Sigma$ tensored with $\mathfrak{g}$: As a free field theory (with the notation as dicussed there) Chern-Simons theory on $\Sigma$ has the sheaf of sections of the field bundle given by

$\mathcal{E} = \Omega^\bullet_\Sigma(-, \mathfrak{g}) \,.$

Here we should regard $\mathfrak{g}$ as being graded and homogeneously of degree $(-1)$ (this is the natural grading on $\mathfrak{g}$ regarded as an L-infinity algebra. In fact essentially all of the discussion here goes through for general $L_\infty$-algebras equipped with a binary invariant polynomial). With this the evident total grading on $\Omega^\bullet_\Sigma(-,\mathfrak{g})$ is already the correct BV-BRST grading

field:ghost fieldsgenuine fieldsantifieldsantighost fields
$\phi \in$$\Omega_\Sigma^0(-,\mathfrak{g})$$\Omega_\Sigma^1(-,\mathfrak{g})$$\Omega_\Sigma^2(-,\mathfrak{g})$$\Omega_\Sigma^3(-,\mathfrak{g})$
degree-1012

The antibracket is given by the canonical local pairing obtained by taking the wedge product of differential forms, then evaluating the coefficients in $\mathfrak{g} \otimes \mathfrak{g}$ in the invariant polynomial $\langle-,-\rangle_{\mathfrak{g}}$, then projecting onto the 3-form summand and finally forming the integration of differential forms over $\Sigma$: for $\phi_1, \phi_2 \in \Omega^\bullet(-,\mathfrak{g})$ we have

$\{\phi_1, \phi_2\} = \int_\Sigma \langle \phi_1 \wedge \phi_2\rangle_{\mathfrak{g}} \,.$

In perturbation theory we take the observables to be polynomial linear functions on these fields, hence the (graded-)symmetric algebra $Sym \overline{\mathcal{E}}$ of the distributions $\overline{\mathcal{E}}$. By the Atiyah-Bott lemma we may in the present situation take the observables equivalently to be the compactly supported sections of the dual field bundle, with duality induced by the local pairing $\mathcal{E}_c \otimes \mathcal{E}_c \to Dens_\Sigma$. For instance a monomial local linear function on the genuine fields

$f \colon \Omega^0(\Sigma,\mathfrak{g}) \to \mathbb{R}$

is then presented by an element $\bar f \in \Omega^3(\Sigma,\mathfrak{g})$ via the evaluation map:

$f \;\colon\; \phi \mapsto \int_\Sigma \bar f \wedge \phi \,.$

With this identification we have in summary the following situation:

linear local observables:ghost field observablesgenuine field observablesantifield observablesantighost field observables
$f \in$$\Omega_{cp}^3(\Sigma,\mathfrak{g})$$\Omega^2_{cp}(\Sigma,\mathfrak{g})$$\Omega_{cp}^1(\Sigma,\mathfrak{g})$$\Omega_{cp}^0(\Sigma,\mathfrak{g})$
degree10-1-2

The de Rham differential on the sections $\mathcal{E}$ of the field bundle induces a differential on the observables by dualization.

Hence the classical BV-complex of the free field theory is

$Obs^{cl}_{free} = \left( Sym (\Omega^\bullet_{cp}(\Sigma, \mathfrak{g})), Q = (d_{dR})^\ast, \langle -,-\rangle^\ast = \int_\Sigma (-) \wedge (-) \right) \,.$

Equipped with the standard BV-Laplacian $\Delta$ discussed at free field theory - The quantum observables this yields the corrsponding quantum BV-complex of the free field theory

$Obs^{q}_{free} = \left( Sym (\Omega_{cp}^\bullet(\Sigma, \mathfrak{g})[ [\hbar] ], Q = (d_{dR})^\ast + \hbar \Delta, \langle -,-\rangle^\ast = \int_\Sigma (-) \wedge (-) \right) \,.$
##### The renormalized quantum master equation

Next we want to add to the above free field theory the interaction term $I$. This amounts to changing the differential $Q + \hbar \Delta$ of $Obs^q_{free}$ to $Q + \{I,-\} + \hbar \Delta$. For this indeed to still be a differential it must still square to 0, which is the condition expressed by the quantum master equation, This needs renormalization in order to be well defined.

This is discussed for instance in (Costello, section 15).

(…)

## Quantum Chern-Simons theory

### The Reshetikhin-Turaev construction and its equivalence to geometric quantization

The Reshetikhin-Turaev model of 3d TQFT had traditionally been expected to be quantized Chern-Simons theory. A proof if this requires showing that the RT-constuction is equivalent to the result of the geometric quantization from above.

The proof of this is not entirely spelled out in the literature, but all the ingredients seem to be known, involving results such as (Andersen 12).

For more see at quantization of Chern-Simons theory.

See also the MathOverflow-discussion Why hasn’t anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?.

### 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 $SU(2)$-connections on $M$ of the exponential of the Chern–Simons action functional $S[A]$:

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

where

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

is the integral of the Chern–Simons Lagrangian,

(3)$Tr Hol_A (K)$

is the trace of the holonomy of the connection around the knot $K$ in the fundamental representation of $SU(2)$, and

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

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 $SU(2)$ 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:

Proposals by other authors include (Henriques 15).

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 representations 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–Simons 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-representations 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 $S_{CS} : \Omega^1(\Sigma,\mathfrak{g}) \to \mathbb{R}$ with respect to the respective Hodge inner product metric on $\Omega^1(\Sigma,\mathfrak{g})$ characterizes Yang-Mills instanton solutions of the Yang-Mills theory on $\Sigma \times \mathbb{R}$ 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 $Iso(2,1)$ (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 $Iso(2,2)$ and $Iso(3,1)$ 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 $Iso(2,1)$-connection then the configuration space naturally contains degenerate vielbein fields $E$ (notably the 0 vielbein) and hence the induced rank-2 tensor $g = \langle E \otimes E\rangle$ 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 $Iso(2,1)$-principal connections which are in fact $(O(2,1) \hookrightarrow Iso(2,1))$-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.

### 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 $\mathcal{C}_p$ 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 $\mathcal{C}_p$ 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)$I(\lambda) = \int_{\mathbb{R}^n} d^n exp(i\lambda f(x_1, \ldots, x_n)).$

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 $H_n (X, X_{\ll0})$. (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 $\mathbb{R}^n$ appearing in the integral over $\mathbb{R}^n$) as a linear combination of these Morse theory cycles:

(6)$C = \sum_p n_p C_p$

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

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

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,

### Traces as a path integral?

In (Witten 89) is the 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. The quote is reproduced at Geometric quantization and the path integral in Chern-Simons theory.

For technical details on this see at orbit method.

### 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.

### Realization in physical systems

Chern-Simons theory has mostly been studied as a test case example for (pre-)quantum field theories in theoretical physics and mathematics. Also in string theory it appears in various incarnations and governs the hypothetical physics of string, notably through its holographic relation to the WZW model and the higher dimensional generalizations of this.

But there are also physical systems that one can set up in a laboratory experiment which are described by at least some aspects of Chern-Simons theory at least in some limit. The most prominent such example is the quantum Hall effect system in solid state physics, which consists of electrons constrained to an effectively 2-dimensional surface and subject to perpendicular magnetic field. This system and its variant are also being proposed as supporting topological quantum computing.

## References

### General

The local Chern-Simons term as an action functional for quantum field theory appears perhaps first in section III of

The geometric quantization and path integral quantization of Chern-Simons theory and the relation of its Wilson line observables to the Jones polynomial was introduced in

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

A comprehensive and clean account of the classical aspects is in

• 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)

Discussion with emphasis on the symplectic structure on phase space and the expression of the Wilson lines by the orbit method is in

• Anton Alekseev, A. Z. Malkin, Symplectic Geometry of the Chern-Simons theory ESI preprint 80 (1994) (web)

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) .

A detailed discussion of the symplectic structure on the moduli space of flat connections is in

• Alan Weinstein, The symplectic structure on moduli space, The Floer memorial volume, 627{635, Progr. Math., 133, Birkhauser, Basel, (1995) (pdf)

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 and its higher geometric quantization is in

### With Wilson loops, defects and boundaries

The orbit method-formulation of Wilson loops in 3d Chern-Simons theory was vaguely indicated in (Witten89, p. 22). More details were discussed in

• S. Elitzur, Greg Moore, A. Schwimmer, and Nathan Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134.

In the context of other gauge theories (Yang-Mills theory) the same formulation appears much earlier in

• A. P. Balachandran, S. Borchardt, and A. Stern, Lagrangian And Hamiltonian Descriptions of Yang-Mills Particles, Phys. Rev. D 17 (1978) 3247–3256

A detailed review and further refinements are discussed in (Alekseev-Malkin) and in section 4 of

• Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

Aspects of the formulation in the context of BV-BRST formalism are discussed in (Alekseev-Barmaz-Mnev).

Discussion of topological boundaries (branes) and suface defects for Chern-Simons theory (as opposed to the non-topoligical WZW model-boundaries) is in

### Quantization

We list discussions of quantization 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

• Edward Witten (lecture notes by Lisa Jeffrey), New results in Chern-Simons theory, pages 81 onwards in: Simon Donaldson, C. Thomas (eds.) Geometry of low dimensional manifolds 2: Symplectic manifolds and Jones-Witten theory (1989) (pdf)

• 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)

and for the generalization to complex Lie groups in

• Edward Witten, Quantization of Chern-Simons gauge theory with complex gauge group, Comm. Math. Phys. Volume 137, Number 1 (1991), 29-66. (Euclid)

The role of the metaplectic correction is studied in some detail in

• Peter Scheinost, Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)

Also, from p. 154 (11 of 76) on, this article carefully discusses what is really the non-abelian version of the Griffiths intermediate Jacobian structure (see there) for $k = 0$.

Detailed discussion with emphasis on theta functions and the relation to Weyl quantization and skein relations is in

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

Discussion that relates the geometric quantization of $G$-Chern-Simons theory to the Reshetikhin-Turaev construction of a 3d-TQFT from the modular tensor category induced by $G$ is in

and references cited there.

Discussion in terms of geometric quantization by push-forward is in

The “prequantum circle 3-bundle” in codimension 3, was constructed in

and in codimension 2 (Wess-Zumino-Witten model) implicitly in

#### Perturbative quantization

Discussion of perturbative quantization of Chern-Simons theory is in

• Maxim Kontsevich, Feynman diagrams and low-dimensional topology, in First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121, Birkhäuser, Basel, 1994.

Perturbative quantization along the lines of Renormalization - Of theories in BV-CS form is in

building on results summarized in section 1.11 and 1.12 and discussed in more detail in section 3 of

In comparison it says in (Costello, p. 14)

In the case when $H^\bullet(\mathcal{E})$, Kontsevich 94 and Axelrod-Singer 92, 94 (when $dim M = 3$) have already constructed the perturbative Chern-Simons invariants. In some sense, their construction is orthogonal to the construction in this paper. Because we work modulo constants, the construction in this paper doesn’t give anything in the case when $H^\bullet(\mathcal{E}) = 0$. On the other hand, their constructions don’t apply in the situations where our construction gives something non-trivial. There seems to be no fundamental reason why a generalisation of the construction to this paper, including the constant term, should not exist. Such a generalisation would also generalise the results of Kontsevich and Axelrod-Singer. However, the problem of constructing such a generalisation does not seem to be amenable to the techniques used in this paper.

The BV-formalism for Chern-Simons theory on manifolds with boundary is discussed in

based on

On p. 3 there it says:

There is a consensus that perturbative quantization of the classical Chern-Simons theory gives the same asymptotical expansions as the combinatorial topological field theory based on quantized universal enveloping algebras at roots of unity (45), or, equivalently, on the modular category corresponding to the Wess-Zumino-Witten conformal field theory (56, 42) with the first semiclassical computations involving torsion made in (56). However this conjecture is still open despite a number of important results in this direction, see for example (47, 3).

One of the reasons why the conjecture is still open is that for manifolds with boundary the perturbative quantization of Chern-Simons theory has not been developed yet. On the other hand, for closed manifolds the perturbation theory involving Feynman diagrams was developed in (32, 27, 7) and in (5, 35, 13). For the latest development see (19). Closing this gap and developing the perturbative quantization of Chern-Simons theory for manifolds with boundary is one of the main motivations for the project started in this paper.

#### As second quantization of the A- or B-model

In

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.

Discussion of an alternative derivation of this statement is in

A textbook account of this relation is in

• Marcos Mariño, Chern-Simons Theory, Matrix Models, and Topological Strings, Oxford University Press (2005)

### As a relative extended TQFT

Discussion of quantum Chern-Simons theory as a 3-2-1 extended TQFT is for instance in

• R. Gelca, Topological quantum field theory with corners based on the Kauffman bracket (pdf)

Discussion as a (relative) 3-2-1-0 extended TQFT is in

A gentle introduction leading up to one proposal for what Chern-Simons theory assigns to a point (a category of positive energy representations of the based loop group) is in

### Computations

• Auclky, Topological methods to compute Chern-Simons invariants (pdf)

Computations of flat Chern-Simons/Dijkgraaf-Witten theory action functionals for the complex special linear group are discused (and discussed to be related to volumes of hyperbolic 3-manifolds) in

### Extension to supergroups

• Victor Mikhaylov, Aspects of Supergroup Chern-Simons Theories, (thesis)

• Victor Mikhaylov, Analytic Torsion, 3d Mirror Symmetry, And Supergroup Chern-Simons Theories (arXiv:1505.03130)

### Workshops and conferences

Revised on November 21, 2016 05:29:53 by David Corfield (129.12.18.176)