Chern-Simons Gauge Theory in R^3

• Chern-Simons theory is a topological quantum field theory defined on R^3 that links gauge invariance with knot invariants, such as the Jones and HOMFLY polynomials.
• It employs gauge fixing and BRST quantization to handle symmetries, allowing computation of observables like Wilson loop averages with distinct propagator structures.
• The theory emerges from higher-dimensional supersymmetric frameworks and extends to gravity, providing deep insights into low-dimensional topology and gauge-gravity unification.

Chern-Simons gauge theory in $\mathbb{R}^3$ is a three-dimensional, topological quantum field theory defined by an action functional that is characterized by its independence from the background metric and by the deep connection between gauge theory, knot invariants, and topological quantum field theory. It plays a pivotal role in both mathematical physics and low-dimensional topology, being the source of quantum invariants such as the Jones and HOMFLY polynomials, as well as enabling connections with higher-dimensional supersymmetric field theories and factorization algebras.

1. Classical Chern-Simons Action and Gauge Properties

Let $G$ be a compact simple Lie group (e.g., $SU(N)$), with Lie algebra $\mathfrak{g}$ and invariant nondegenerate trace $\operatorname{Tr}$. The classical Chern-Simons action on $\mathbb{R}^3$ for a $\mathfrak{g}$-valued one-form $A = A_\mu dx^\mu$ is

$$S_{CS}[A] = \frac{k}{4\pi} \int_{\mathbb{R}^3} \varepsilon^{\mu\nu\rho} \operatorname{Tr} \left(A_\mu \partial_\nu A_\rho + \frac{2}{3} A_\mu A_\nu A_\rho\right) d^3x,$$

where $k\in\mathbb{Z}$ is the quantized level, and normalization is such that the action shifts by $2\pi k n$ under gauge transformations of winding number $n$—ensuring single-valuedness of the quantum path integral for integer $k$ (Lee et al., 2013). The action is invariant under gauge transformations $A_\mu \mapsto g^{-1}A_\mu g + g^{-1}\partial_\mu g$ for $g:\mathbb{R}^3\to G$.

The gauge symmetry can be handled with a BRST (Becchi–Rouet–Stora–Tyutin) quantization, introducing ghost ($c$), antighost ($\bar c$), and Nakanishi–Lautrup auxiliary fields ($B$), leading to a gauge-fixed action

$$S = S_{CS}[A] + \int d^3x\, \operatorname{Tr} \left( B \, \partial^\mu A_\mu + \bar{c} \, \partial^\mu D_\mu c \right),$$

with the Landau gauge fixing $\partial^\mu A_\mu = 0$ (Lee et al., 2013).

2. Emergence from Higher-Dimensional Theories and Complexification

Chern-Simons theory in $\mathbb{R}^3$ admits a direct derivation as a low-energy effective action from certain compactifications of higher-dimensional supersymmetric gauge theories. Specifically, starting from five-dimensional $\mathcal{N}=2$ maximally supersymmetric Yang–Mills theory with gauge group $G$ on $S^2 \times \mathbb{R}^3$, supersymmetric localization with respect to an appropriate supercharge localizes the 5d path integral to the 3d Chern–Simons theory for the complexified gauge group $G_\mathbb{C}$ on $\mathbb{R}^3$. The relevant saddle-point equations restrict the field content to a complex connection $\mathcal{A}_\mu = A_\mu + i\phi_\mu$ (with three scalars $\phi_\mu$ from the 5d perspective). The effective action then takes the form

$$S_{\text{eff}}[\mathcal{A}] = \frac{i t}{8\pi} \int_{\mathbb{R}^3} \operatorname{Tr}\left(\mathcal{A}\wedge d\mathcal{A} + \frac{2}{3}\mathcal{A}^3\right) + \mathrm{c.c.},$$

with $i t = \frac{8\pi^2 r}{g^2}$, where $r$ is the $S^2$ radius and $g$ the 5d gauge coupling (Lee et al., 2013). Restricting to the real part recovers the standard (real) Chern–Simons theory; for $t\in\mathbb{Z}$, the effective level $k$ is identified with $t$.

3. Gauge Fixing, Quantization, and Propagator Structure

Chern-Simons theory admits a variety of gauge-fixing choices, each with distinctive implications for the structure of observables and the computation of topological invariants. The interpolating gauge (Gallot et al., 2017), defined by

$$\mathcal{F}(A) = \alpha\, \partial^\mu A_\mu + \beta\, (n \cdot \partial)(n^\mu A_\mu),$$

where $n^\mu$ is a fixed direction and parameters $\alpha, \beta$ select between Landau, Coulomb, and axial gauges, connects different geometric realizations of link invariants.

The gauge field propagator in momentum space for gauge-fixed abelian theory takes the general form

$$\widehat G_{\mu\nu}(k) = -\frac{i}{k \cdot q}\varepsilon_{\mu\rho\nu}q^\rho, \qquad q^\rho = \alpha\,k_\perp^\rho + \gamma\,k_\parallel^\rho, \ \gamma = \alpha + \beta,$$

with position-space reconstruction yielding kernels whose shape (sphere, cylinder, or collapsed plane) depends on the gauge choice. This structure yields distinct geometric interpretations for the linking number, as summarized below.

Gauge	$\alpha$	$\gamma$	Interpretation
Covariant	$1$	$1$	Gauss map degree on $S^2$
Coulomb	$1$	$0$	Intersections within a cylinder
Axial-limit	$0$	$1$	Planar projections and crossings

In the non-Abelian case, the Landau (covariant) gauge yields the standard propagator

$$\langle A^a_\mu(p)\,A^b_\nu(-p)\rangle = \frac{2\pi}{k}\frac{\varepsilon_{\mu\nu\rho}p^\rho}{p^2}\delta^{ab},$$

with standard cubic and ghost–gluon vertices (Lee et al., 2013). In the temporal and axial gauges, the action becomes quadratic with ultralocal or step-function propagators, facilitating exact Gaussian computation of Wilson loop averages (Weitsman, 2024, Morozov et al., 2010).

4. Lattice Regularization and Discrete Topology

The Abelian Chern–Simons theory admits a lattice regularization on a three-dimensional cubic lattice which avoids ambiguities associated with forward/backward differences and doubling. The gauge field variables $A_\mu(n)$ reside on links, and the field strength is defined on plaquettes. Introduction of dual-lattice link variables $\bar{A}_p$, constructed as linear combinations of the neighboring $A_\mu$, yields a lattice action

$$S_{\mathrm{lat}} = \frac{k}{4\pi} \sum_{n,\mu<\nu} \bar{A}_{\mu\nu}(n+\tfrac{1}{2}e_\rho) F_{\mu\nu}(n),$$

where $k\in\mathbb{Z}$ again encodes the quantized level. The continuum limit $a\to0$ reproduces the standard continuum Chern–Simons action. Expectation values of lattice Wilson loops match the continuum result,

$$\bigl\langle W(C_1)\,W(C_2)\bigr\rangle = \exp\left(\frac{2\pi i}{k} \,\mathrm{Link}(C_1, C_2)\right),$$

demonstrating the manifestly topological nature and the cubic symmetry of the construction (Zhang, 2021).

5. Observables, Knot Theory, and Quantum Invariants

Wilson loops form the fundamental gauge-invariant observables, with

$$W_R(K) = \operatorname{Tr}_R P \exp\left(\oint_K A \right),$$

for a knot $K \subset \mathbb{R}^3$ and representation $R$. Their expectation values capture link invariants: for SU(N), non-perturbative computations on $S^3$ yield the Jones and HOMFLY polynomials, while on $\mathbb{R}^3$, the perturbative $1/k$ expansion organizes all Vassiliev finite-type invariants (Lee et al., 2013). In the Abelian theory, expectation values are explicitly

$$\langle W(C_1) W(C_2) \rangle = \exp\left(\frac{2\pi i}{k} \,\mathrm{Link}(C_1, C_2)\right).$$

Various gauge-fixing choices yield different approaches to the computation of knot invariants. In axial and temporal gauge, the theory becomes Gaussian and the computations reduce to combinatorics of crossings and turning points in the planar projection of knots, with crossing operators realizing the universal quantum $R$-matrix structure of $U_q(g)$ (Morozov et al., 2010, Weitsman, 2024). The explicit connection to skein relations underpinning the Jones and Kauffman bracket polynomials is realized directly by exponentiating the local crossing operator matrix, yielding invariants for arbitrary complex coupling (i.e., no integrality requirement for couplings in these gauges) (Weitsman, 2024). For higher-rank groups, exact HOMFLY-type skein relations emerge.

6. Quantization, One-Loop Exactness, and Factorization Structures

Quantization of Chern–Simons theory is most fully controlled in the Batalin–Vilkovisky (BV) formalism, especially in holomorphic gauge. The theory is one-loop exact: all higher-loop Feynman diagrams beyond one-loop wheels vanish, and the effective action acquires only a level shift $k \mapsto k + h^\vee$—the dual Coxeter number of the underlying semisimple Lie algebra (Gwilliam et al., 2019). The propagator kernel is determined by the scalar Laplacian, and the renormalized quantum observables organize into a factorization algebra, which restricts to the chiral Kac–Moody algebra (the WZW model) on the boundary. This realizes the celebrated Chern–Simons/WZW bulk–boundary correspondence, where boundary chiral currents are reproduced by bulk holonomies ending on the boundary (Gwilliam et al., 2019).

7. Extensions: Unification with Gravity, Topological Invariants, and Higher Correspondences

Chern–Simons gauge theory in three dimensions admits an extension to include gravity by enlarging the gauge group to contain both internal ($A$) and gravitational ($\omega$, $e$) components. For example, with negative cosmological constant, the enlarged gauge group is $SO(2,2)$, and the connection is $\mathcal{C} = \omega + e/\ell$. The unified action

$$S_{\textrm{uni}}[\mathcal{C}] = \frac{K}{4\pi}\int_{\mathbb{R}^3}\left\langle \mathcal{C}\wedge d\mathcal{C} + \frac{2}{3} \mathcal{C}\wedge \mathcal{C}\wedge \mathcal{C}\right\rangle$$

splits into gauge and gravity Chern–Simons components. The CS partition function then localizes to flat connections labelled by moduli, and on closed manifolds is a sum over contributions $e^{i K I_{CS}(\mathcal{C}_\alpha)}$ weighted by Ray–Singer torsion and $\eta$-invariants. Wilson lines in mixed gauge–gravitational representations yield new two-variable knot invariants reflecting both gauge and geometric data, with the gravitational Chern–Simons contribution encoding framing-dependent phases (Saghir et al., 2017).

• "3d Chern-Simons Theory from M5-branes" (Lee et al., 2013)
• "A one-loop exact quantization of Chern-Simons theory" (Gwilliam et al., 2019)
• "Geometric aspects of interpolating gauge fixing in Chern-Simons Theory" (Gallot et al., 2017)
• "Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix" (Morozov et al., 2010)
• "Abelian Chern-Simons Gauge Theory on The Lattice" (Zhang, 2021)
• "The Chern-Simons Functional Integral, Kauffman's Bracket Polynomial, and other link invariants" (Weitsman, 2024)
• "Unification of Gauge and Gravity Chern-Simons Theories in 3-D Space time" (Saghir et al., 2017)