Non-invertible symmetries in Maxwell-Chern-Simons theories

Statement

Maxwell-Chern-Simons (MCS) theory in 2+1 dimensions occupies a special position in the landscape of gauge theories: it is simultaneously massive (the Chern-Simons term gives the photon a gauge-invariant mass), topological at long distances, dual to the self-dual-model, and admits both electric and magnetic excitations (anyons). Recent developments in the theory of non-invertible symmetries — symmetries whose fusion rules are not those of a group — suggest that such theories may carry non-invertible symmetry structures that generalize the electromagnetic duality. What are the non-invertible symmetries of Maxwell-Chern-Simons theory, and what do they imply for its phase structure, spectrum, and connection to the julia-toulouse-mechanism?

Why It Matters

Non-invertible symmetries have emerged as a major organizing principle in modern QFT, unifying diverse phenomena: duality defects in Maxwell theory (Choi-Cordova-Hsin-Lam-Shao, 2022), Tambara-Yamagami symmetries in 2d, fusion categories in topological phases, and non-invertible higher-form symmetries in gauge theories. In 3+1d Maxwell theory, the electromagnetic duality is not a symmetry in the standard sense (it maps electric to magnetic charges and thus permutes sectors); in the modern language, it is implemented by a topological-defect-line (or surface) that is non-invertible.

Maxwell-Chern-Simons in 2+1d is particularly natural for this analysis because:

1. It is exactly solvable and has a known dual (self-dual model) — a well-established example of a duality that acts on the Hilbert space.
2. The Chern-Simons level $k$ controls anyon statistics: at level $k$, the theory has anyonic excitations with statistical phase $e^{i\pi /k}$. The fusion rules of these anyons are generally non-group-like, suggesting non-invertible symmetries.
3. The earlier work on Maxwell-Chern-Simons duality in noncommutative spaces (Phys. Lett. B 605, 2005) and the connection between Chern-Simons terms and julia-toulouse-mechanism (Phys. Rev. D 88, 2013) provide technical tools that can be deployed.
4. This connects the earliest research line (duality, 2003–2013) to one of the most active current frontiers, providing a natural revival and modernization of that program (Project 18).

What We Know

Maxwell-Chern-Simons — established:

• MCS action in 2+1d: $S=\int d^{3}x\left[-\frac{1}{4e^2}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{k}{4\pi }{ϵ}^{\mu \nu \rho }{A}_{\mu }{\partial }_{\nu }{A}_{\rho }\right]$. The parameter $k$ is the Chern-Simons level; for $k\in \mathbb{Z}$, the theory is gauge-invariant under large gauge transformations.
• The photon acquires a gauge-invariant mass $m=k{e}^2/2\pi$; at energies below $m$, the theory flows to pure Chern-Simons theory, which is a topological field theory with anyon excitations. (Corrected 2026-09-21 from $k{e}^2/4\pi$. With $A\to eA$ the action becomes $-\frac{1}{4}{F}^2+\frac{k{e}^2}{4\pi }ϵA\partial A$, which is the Deser–Jackiw–Templeton form $\frac{m}{2}ϵA\partial A$ with $m=k{e}^2/2\pi$.)
• Global data (see chern-simons-theory).
  • $k\in \mathbb{Z}$ because the Chern–Simons function is defined only mod $\mathbb{Z}$ and $e^{iI}$ must be single-valued (Witten 2026, §3).
  • For odd $k$ the theory needs a spin structure: it is a spin TQFT, since $\int F\wedge F/8{\pi }^2$ is an integer only on spin 4-manifolds (Witten 2026, §3).
  • On a spatial torus the holonomy zero modes form a Landau problem with cyclotron frequency exactly $m$ and a $k$-fold degenerate lowest level (Dunne–Jackiw–Trugenberger 1989). That is the $U(1{)}_k$ TQFT sector, and the self-dual model has no counterpart. The MCS/self-dual duality is therefore a local statement, expected to hold globally only up to the TQFT factor.
  • The spin ($k$ odd) and bosonic ($k$ even) cases must be classified separately. The anomaly of the ${\mathbb{Z}}_k$ 1-form symmetry is read off from the line spins $h_n=n^2/2k$, and those differ between the two cases (Hsin–Lam–Seiberg 2019).
• Duality between MCS and the self-dual model (Deser-Jackiw, 1984; Fradkin-Schaposnik, 1994): the two theories have identical S-matrices and partition functions locally (for the global statement, see the caveat above). In Marcelo’s thesis (2005), this duality was extended to noncommutative space.
• In the presence of magnetic monopoles (Julia-Toulouse context): monopoles in 2+1d are instantons (Polyakov monopoles), and the Chern-Simons term gives them an electric charge, leading to dyonic excitations. This was studied explicitly (Phys. Rev. D 88, 2013; Phys. Lett. B 674, 2009).

Non-invertible symmetries — modern framework:

• In 3+1d Maxwell theory, the duality symmetry $F\to \star F$ is not a conventional symmetry (it mixes electric and magnetic sectors, which are not gauge-invariant individually). It is implemented by a topological surface operator — a “duality defect” — that obeys non-group fusion rules: $\mathcal{D}\times \mathcal{D}={⨁}_q{\mathcal{L}}_q$ where ${\mathcal{L}}_q$ are line operators (Choi-Cordova-Hsin-Lam-Shao, 2022).
• In 2+1d, 0-form non-invertible symmetries are implemented by topological line operators (defect lines). The fusion of two defect lines can produce a sum of defect lines — a fusion category structure.
• For Chern-Simons theory at level $k$, the anyonic excitations define a modular tensor category (MTC). The symmetry of the theory (acting on the Hilbert space) is described by the Drinfeld center of the MTC, which is in general non-invertible.
• The interplay between the Maxwell (non-topological) term and the Chern-Simons term in MCS breaks the full anyon symmetry of pure CS theory — but some non-invertible symmetries may survive.

Gap: The non-invertible symmetry structure of MCS theory (as opposed to pure CS theory) has not been analyzed. The role of the Maxwell term in breaking or preserving non-invertible symmetries is unknown.

Possible Approaches

1. Topological defect line construction for MCS: Following the method of Choi et al. for 3+1d Maxwell theory, construct the topological defect line implementing the MCS duality in 2+1d. This requires: (i) identifying the duality transformation as a map on the field space, (ii) constructing the corresponding topological operator, and (iii) computing its fusion rules with local and line operators of the theory.

2. Level-rank duality connection: MCS at level $k$ with gauge group $U(1)$ is related by level-rank duality to $U(1{)}_k$ CS theory with different matter content. This duality is itself a non-invertible symmetry (it is not a bijection on operators). Using the level-rank duality, map the non-invertible symmetry question in MCS to a known result in pure CS theory.

3. Anyon condensation analysis: The Julia-Toulouse mechanism in 2+1d corresponds to anyon condensation: when anyons with specific statistics condense, the theory transitions to a new topological phase. Anyon condensation is described by the theory of Lagrangian subgroups of the MTC — a well-developed mathematical framework. Applying this to MCS would identify which non-invertible symmetries are preserved and which are broken by Julia-Toulouse condensation.

4. Symmetry TFT (SymTFT) approach: The modern approach to generalized symmetries uses a “symmetry topological field theory” (SymTFT) in one higher dimension to classify all possible symmetry structures of a given theory. For MCS in 2+1d, the SymTFT lives in 3+1d and is related to the 4d ${\mathbb{Z}}_k$ gauge theory. Analyze the SymTFT to classify the non-invertible symmetries of MCS.

5. Noncommutative MCS: The earlier work on MCS duality in noncommutative space (Phys. Lett. B 605, 2005) may provide a deformation parameter that controls the non-invertible symmetry structure. In noncommutative geometry, the symmetry algebra is deformed, and non-invertible symmetries may appear or disappear as the noncommutativity parameter $\theta$ is varied.

Related Questions

• julia-toulouse-higher-form-symmetries — The Julia-Toulouse mechanism in MCS is the starting point; this question asks about the finer symmetry structure
• entanglement-as-confinement-probe — Non-invertible symmetries and their breaking control topological phases, which are characterized by long-range entanglement
• higher-spin-bell-inequalities — The anyonic excitations of MCS are higher-spin objects in the sense of braid statistics; Bell inequalities for anyons are unexplored