Chern-Simons-like Term in QFT and Gravity

Updated 24 October 2025

Chern-Simons-like term is a Lorentz- and often CPT-violating local modification that generalizes the traditional Chern-Simons action to higher dimensions and non-topological contexts.
Its radiative induction in four-dimensional gauge and gravitational theories is linked to axial anomalies and ambiguous regularization, highlighting symmetry-breaking effects.
In condensed matter and holographic models, CS-like terms drive anomalous electromagnetic transport and modify response characteristics by encoding nontrivial topological phenomena.

A Chern-Simons-like (CS-like) term is a Lorentz-violating, and often CPT-violating, local term in quantum field theory or gravity whose form generalizes the celebrated Chern-Simons (CS) action, but which may appear in higher dimensions, in non-topological or anomalous contexts, or as the result of explicit symmetry-breaking backgrounds or boundary effects. Unlike “pure” CS terms—which are topological invariants built from gauge connections in odd-dimensional manifolds—a CS-like term may arise as an induced effective action in even dimensions (notably in four-dimensional gauge or gravitational theories, as well as at boundaries or via dimensional reduction), typically violating Lorentz or CPT symmetry by coupling to a fixed background tensor or vector. The interest in CS-like terms lies in their roles as markers of symmetry breaking, sources of anomalous transport or topological phenomena, and windows into new physics both in high-energy theory and in condensed matter systems.

1. Algebraic Structure and Induction in (3+1)D Gauge Theories

In standard four-dimensional quantum electrodynamics (QED), the quantum effective action is Lorentz and CPT invariant. However, if the original Lagrangian is extended by introducing a constant background four-vector $b_\mu$ coupled axially to the fermions,

$\mathcal{L}_{\text{mod}} = \bar\psi \left(i\gamma^\mu \partial_\mu - m - e \gamma^\mu A_\mu - \gamma^\mu b_\mu \gamma_5 \right) \psi,$

the quantization procedure (integrating out fermionic fluctuations) radiatively induces a new term in the gauge sector of the form

$S_{\text{CS-like}}^{(3+1)\text{D}} = \frac{e^2}{12\pi^2} \int d^4x\, \epsilon^{\mu\nu\rho\sigma} b_\mu A_\nu \partial_\rho A_\sigma,$

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita tensor (Oliveira, 2010). This term is a direct generalization of the Carroll-Field-Jackiw term and is odd under Lorentz and CPT transformations. The proportionality constant is not uniquely determined: it depends on the regularization prescription (e.g., dimensional regularization, Pauli-Villars, or cut-off). In dimensional regularization, the coefficient takes the form $e^2/(12\pi^2)$ , but other schemes yield different prefactors, reflecting an inherent ambiguity tied to the treatment of divergent loop integrals. Such ambiguity persists even under gauge invariance constraints, as shown in gravitational analogues (Felipe et al., 2014).

The mechanism of induction is traceable to the axial coupling; diagrammatic expansion of the fermion propagator in powers of $b_\mu$ allows for perturbative treatment, with leading-order contributions generating the CS-like structure upon taking traces over Dirac matrices containing $\gamma_5$ . This construction is intimately related to the axial anomaly, with the induced term reflecting the underlying violation of Lorentz and CPT invariance upon quantization.

2. CS-like Terms in Gravity and Topologically Nontrivial Backgrounds

CS-like terms also appear in linearized quantum gravity extended with Lorentz- and CPT-violating axial-vector couplings in the fermionic sector. The gravitational analogue is

$\mathcal{L}_{\text{CS,grav}} = \left[\frac{1}{96\pi^2} - 16i\sigma_0 \right] \kappa^2 b^\lambda h^{\mu\nu} \epsilon_{\alpha\mu\lambda\rho} \partial^\rho \left( \partial^2 h^\alpha_\nu - \partial_\nu \partial_\gamma h^{\gamma\alpha} \right),$

with $\kappa$ the gravitational coupling and $h^{\mu\nu}$ the metric fluctuation (Felipe et al., 2014). As in the gauge sector, the coefficient is ambiguous up to regularization-dependent parameters not fixed by gauge invariance or momentum-routing invariance. These ambiguities manifest as surface terms—the difference of divergent integrals in loop calculations. The physical interpretation is that one-loop radiative corrections alone do not uniquely fix the normalization of the induced Lorentz-violating term, necessitating experimental input or further theoretical constraints.

Moreover, topologically nontrivial backgrounds induce CS-like terms in the effective gauge action. For example, in four-dimensional chiral gauge theory compactified on $\mathbb{R}^3 \times S^1$ (with specific boundary conditions and vanishing $A_4$ component), integrating out chiral fermions generates

$\Gamma_{\text{anom}}[A] = -2\pi F e^2 \left[ \frac{1}{L} \int_0^L dx^4 \int_{\mathbb{R}^3} d^3x\, \frac{1}{16\pi^2} \epsilon^{ijk} A_i \partial_j A_k \right]$

(Ghosh et al., 2017), which violates four-dimensional Lorentz and CPT invariance. The role of nontrivial topology is essential: the CS-like term arises due to the interplay of global geometry, anomaly cancellation, and the structure of the fermion measure (manifest in both regularized continuum and lattice approaches).

3. CS-like Terms in Condensed Matter and Holographic Applications

CS-like terms have extensive relevance in condensed matter physics, notably in Weyl semimetals and topological phases:

In three-dimensional Weyl semimetals, the electromagnetic response in the presence of a nonzero chiral chemical potential $b_\mu$ is governed by a CS-like term,

$\mathcal{L}_{\text{CS}} = \frac{1}{2} k_\mu \epsilon^{\mu\nu\rho\sigma} F_{\nu\rho} A_\sigma, \quad k_\mu = -\frac{1}{8\pi^2} b_\mu,$

where $k_\mu$ (time- or space-like) generates anomalous electromagnetic transport, including the chiral magnetic effect under non-equilibrium conditions (Omid, 2014). This term exists only in nonequilibrium states; in equilibrium, the system relaxes so that $b_\mu$ is absorbed and the effective theory lacks the CS contribution, replaced instead by the anomalous Hall effect.

Within holographic superconductor models, an axionic coupling of the form $\theta F \wedge F$ in the four-dimensional AdS bulk induces a CS-like term on the $(2+1)$ -dimensional boundary theory (Tallarita et al., 2010). The effective coupling results from the near-boundary expansion of the axion field and gives rise to distinct vortex structures (e.g., magnetic field rings, field inversions) characteristic of Maxwell-Chern-Simons hybrid systems.
The inclusion of spatial Chern-Simons interactions in Ginzburg-Landau models for superconductors leads to parity violation and modifies the magnetic penetration depth, producing oscillatory (rather than screening) magnetic responses (“skin effect”) and destroying the conventional Meissner effect in systems with pre-formed pairs (Tao, 2016).

4. Dimensional Reduction and Emergence from Higher-Dimensional Theories

Dimensional reduction provides another route to CS-like terms:

In Kaluza-Klein theories, dimensional reduction of the five-dimensional Gibbons-Hawking surface term yields the Abelian Chern-Simons action in three dimensions after integrating over the compact coordinate (Kim et al., 22 Mar 2024). The process involves writing the higher-dimensional metric in terms of lower-dimensional fields (metric, gauge, scalar/dilaton), reducing both the bulk and surface terms, and utilizing the field equations to identify the emergent CS term as a real physical effect, not a parametrization artifact. The presence of this boundary term may break local gauge invariance in the reduced theory, rendering the photon massive on the plane.
At finite temperature, integrating out fermions in a $(3+1)$ -dimensional theory with vector and axial-vector gauge couplings, keeping only the lowest Matsubara mode and setting the axial field in its vacuum configuration, induces a Chern-Simons term in the resulting $(2+1)$ -dimensional effective action. The coefficient of the induced term depends sensitively on the vacuum value of the axial field (Fosco et al., 2022).

5. CS-like Terms in Lifshitz and Non-Lorentz-Invariant Theories

CS-like terms are not restricted to Lorentz-invariant settings:

Adding CS terms to Lifshitz field theories (with higher spatial derivatives) or to anisotropic quantum electrodynamics in three dimensions leads to generalized CS structures, including “extended” CS terms involving higher derivatives in both space and time (Lozano et al., 2012, Alves et al., 2013). These extended terms remain finite and are necessary to reconcile scale invariance and topological structure in the presence of anisotropic scaling. The isotropic (Lorentz-invariant) limit is subtle and can result in cancellations or vanishing of the “canonical” CS term, leaving only higher-derivative corrections.
In models such as the gauged baby Skyrme model with a Maxwell-CS term, the inclusion of the CS-like action crucially modifies the structure and electromagnetic interaction of solitons. The Chern-Simons coupling leads to electric-magnetic mixing, nontrivial angular momentum, and, in the strong coupling regime, effective quantization of magnetic flux and electric charge, even though these are not topological numbers in the model (Samoilenka et al., 2016).

6. Gauge, BRST, and Anomalous Structure; Absence of Radiative Generation in Certain Sectors

The generation—and non-generation—of CS-like terms by radiative corrections is governed by both symmetry constraints and cohomological properties:

In Lorentz-violating Yang-Mills theory, as well as in Lorentz-violating QED with a CPT-odd axial coupling, algebraic renormalization and BRST cohomology show that CS-like terms are not generated radiatively. Potential CS sources reside in the trivial sector of BRST cohomology and can be eliminated (if present) by suitable choice of counterterms; furthermore, gauge invariance and IR power counting via the Lowenstein-Zimmermann subtraction scheme exclude such terms in the quantum effective action at any loop order (Santos et al., 2016, Cima et al., 2017).
In contrast, the ambiguous normalization of radiatively induced CS-like terms in situations where they do arise (as in the bμ-extended QED or gravity) remains a central unresolved issue, reflecting regularization dependence and the non-uniqueness of anomaly descent outside the strictly cohomological (i.e., gauge anomaly) context (Felipe et al., 2014).

7. Geometric and Discrete Approaches; Gauge-Invariant Discretization

CS-like theories can be formulated and analyzed using geometric and discrete tools:

In three-dimensional gravity, a broad class of massive and higher-derivative gravity models can be recast in a “Chern-Simons-like” form: actions are written as integrals over 3-forms constructed from wedge products of Lorentz-valued 1-forms (dreibein, spin connection, auxiliary fields), organized by a “flavor” metric and totally symmetric coupling constants (Bergshoeff et al., 2014, Merbis, 2014). The Hamiltonian analysis in this context clarifies constraints, degrees of freedom, and conditions for absence of ghosts and consistency with AdS/CFT features.
CS and CS-like couplings can be discretized while maintaining gauge invariance by defining link variables for p-forms on simplicial complexes, where field strengths are associated with (p+1)-simplices and wedge products correspond to suitable compositions of link variables living on simplices with common vertices. The discretized expressions reduce to their continuum counterparts in the fine-mesh limit, and can faithfully represent topological invariants and gauge properties on the lattice (Hatakeyama et al., 30 Apr 2024).
In the geometric theory of defects in elasticity, the CS-functional for the SO(3) connection provides the unique nontrivial action for describing disclinations (rotational line defects in the spin structure) while maintaining a flat Euclidean metric. Solutions with $\delta$ -function sources in the connection correspond to quantized angular rotation fields (Frank vector quantization), capturing the topological nature of these defects (Katanaev, 2017).

In summary, CS-like terms represent a unifying structure at the interface of anomaly, symmetry breaking, topology, and the dynamics of gauge and gravitational fields. They are induced by quantization in the presence of symmetry-breaking backgrounds or boundaries, encode subtle topological or anomalous information, and are central in both high-energy and condensed matter settings as sources of exotic phases, anomalous transport, and novel geometric effects. Their ambiguous normalization in certain settings underscores the importance of regularization and symmetry principles, while their extensions to gravitational, Lifshitz, and discrete frameworks demonstrate their broad mathematical and physical reach.