nLab Maxwell-Chern-Simons theory

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Contents

Idea

Under Maxwell-Chern-Simons theories one understands Lagrangian field theories whose field content is a single U ( 1 ) \mathrm{U}(1) gauge field (locally on spacetime given by a differential 1-form AA with flux density F=dAF = \mathrm{d}A) and whose Lagrangian density is a sum of that of (vacuum) Maxwell theory with that of an abelian Chern-Simons theory (hence 3D Yang-Mills-Chern-Simons theory for abelian gauge group).

This means, over a spacetime of dimension D=3D = 3, that the Lagrangian density is proportional to

L MCS 3D(A)∝12F∧⋆F−c2A∧F. L_{MCS}^{3D}(A) \;\propto\; \tfrac{1}{2} F \wedge \star F \;-\; \tfrac{c}{2} A \wedge F \,.

The corresponding Euler-Lagrange equations of motion are (where we include the Bianchi identity in the first line in order to highlight the comparison to Maxwell's equations):

dF =0 d⋆F =cF. \begin{aligned} \mathrm{d} F & = 0 \\ \mathrm{d} \star F & = c \,F \,. \end{aligned}

Similarly, there are higher dimensional versions. For instance over a spacetime of dimension D=5D =5 and using the Lagrangian density of abelian D=5 Chern-Simons theory, one has

L MCS 5D(A)∝12F∧⋆F−c3A∧F∧F L_{MCS}^{5D}(A) \;\propto\; \tfrac{1}{2} F \wedge \star F \;-\; \tfrac{c}{3} A \wedge F \wedge F

with Euler-Lagrange equations of motion

(1)dF =0 d⋆F =cF∧F. \begin{aligned} \mathrm{d} F & = 0 \\ \mathrm{d} \star F & = c \,F \wedge F \,. \end{aligned}

This appears notably in the bosonic sector of D=5 supergravity (which is an Einstein-Maxwell-D=5 Chern-Simons theory).

Properties

Dimensional reduction of 5D Maxwell-Chern-Simons to 3D

Consider the dimensional reduction of the 5D Maxwell-Chern-Simons equations of motion (1) to 3D, specifically:

Consider the case that the spacetime Y 1,4Y^{1,4} is a product manifold (with product metric)

(2)Y 1,4=X 1,2×L 1×V 1, Y^{1,4} = X^{1,2} \times L^1 \times V^1 \,,

of:

  • a Lorentzian manifold X 1,2X^{1,2}, dim(X 1,2)=3dim\big(X^{1,2}\big) = 3,

  • 1-dimensional Riemannian manifolds L 1L^1 and V 1V^1 (not necessarily connected or closed)

    which in suitable coordinates ll and vv have metrics g L =ℓ l 2dl⊗dl g V =ℓ v 2dv⊗dv \begin{aligned} g_L & = \ell_l^2 \mathrm{d}l \otimes \mathrm{d}l \\ g_V & = \ell_v^2 \mathrm{d}v \otimes \mathrm{d}v \end{aligned} for positive real numbers l,v∈ℝ +l,v \in \mathbb{R}_+, respectively.

and that the flux density is constant along the fibers, in that

(3)F 2= F 2 (XX) +F 1 (Xl)∧dl +F 1 (Xv)∧dv +F 0 (lv)dl∧dvfor{F 2 (XX) ∈Ω dR 2(X 1,2)↪p X *Ω dR 2(Y 1,4) F 1 (Xl),F 1 (Xv) ∈Ω dR 1(X 1,2)↪p X *Ω dR 1(Y 1,4) F 0 (lv), ∈Ω dR 0(X 1,2)↪p X *Ω dR 0(Y 1,4) \begin{aligned} F_2 = & F_2^{(X X)} \\ & + F_1^{(X l)} \wedge \mathrm{d} l \\ & + F_1^{(X v)} \wedge \mathrm{d} v \\ & + F^{(l v)}_0 \mathrm{d}l \wedge \mathrm{d}v \end{aligned} \;\;\;\;\; \text{for} \;\;\;\;\; \left\{ \begin{aligned} F_2^{(X X)} & \in \Omega^2_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^2_{dR}\big(Y^{1,4}\big) \\ F_1^{(X l)}, F_1^{(X v)} & \in \Omega^1_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^1_{dR}\big(Y^{1,4}\big) \\ F_0^{(l v)}, & \in \Omega^0_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^0_{dR}\big(Y^{1,4}\big) \end{aligned} \right.

and such that

(4)∀x∈XF 0 (lv)(x)≠0 \underset{x \in X}{\forall} \; F_0^{(l v)}(x) \neq 0

(whence we have a flux compactification).

Proposition

In this situation and in the limit ℓ v→0\ell_v \to 0, the equations of motion (1) are equivalent to the following system of equations:

(5) F 2 (XX)=−1F 0 (lv)F 1 (Xl)∧F 1 (Xv) d XF 1 (Xv)=0,d X⋆ XF 1 (Xv)=0, d XF 1 (Xl)=0. \begin{aligned} & F_2^{(X X)} = - \tfrac{1}{F_0^{(l v)}} F_1^{(X l)} \wedge F_1^{(X v)} \\ & \mathrm{d}_X F_1^{(X v)} = 0, \; \mathrm{d}_X \star_X F_1^{(X v)} = 0, \\ & \mathrm{d}_X F_1^{(X l)} = 0 \mathrlap{\,.} \end{aligned}

In particular, when either of the F 1 (X−)F_1^{(X -)} vanishes, then F 2 (XX)F_2^{(X X)} satisfies the equations of motion of 3D abelian Chern-Simons theory, in the limit ℓ v→0\ell_v \to 0.

Proof

This is a standard kind of argument, but seems not to be citable from the literature:

Due to the product spacetime structure (2), the Hodge dual of F 2F_2 (3) with respect to Y 1,4Y^{1,4} is expressed in terms of the Hodge star operator ⋆ X\star_{X} associated with X 1,2X^{1,2} as follows:

⋆F 2= ℓ lℓ v(⋆ XF 2 (XX))∧dl∧dv +ℓ vℓ l(⋆ XF 1 (Xl))∧dv −ℓ lℓ v(⋆ XF 1 (Xv))∧dl +1ℓ lℓ v⋆ XF 0 (lv), \begin{aligned} \star F_2 = & \ell_l \ell_v \big( \star_X F_2^{(X X)} \big) \wedge \mathrm{d}l \wedge \mathrm{d}v \\ & + \tfrac{\ell_v}{\ell_l} \big( \star_X F_1^{(X l)} \big) \wedge \mathrm{d} v \\ & - \tfrac{\ell_l}{\ell_v} \big( \star_X F_1^{(X v)} \big) \wedge \mathrm{d} l \\ & + \tfrac{1}{\ell_l \ell_v} \star_X F_0^{(l v)} \,, \end{aligned}

whence the second equation of motion (1) is seen to be equivalent to

(6)ℓ lℓ vd X(⋆ XF 2 (XX)) =F 2 (XX)∧F 0 (lv)+F 1 (Xl)∧F 1 (Xv) ℓ vℓ ld X(⋆ XF 1 (Xl)) =F 2 (XX)∧F 1 (Xv) ℓ lℓ vd X(⋆ XF 1 (Xv)) =−F 2 (XX)∧F 1 (Xl) 1ℓ lℓ vd X(⋆ XF 0 (lv))⏟=0 =12F 2 (XX)∧F 2 (XX)⏟=0, \begin{aligned} \ell_l \ell_v \mathrm{d}_X \big( \star_X F_2^{(X X)} \big) & = F_2^{(X X)} \wedge F_0^{(l v)} + F_1^{(X l)} \wedge F_1^{(X v)} \\ \tfrac{\ell_v}{\ell_l} \mathrm{d}_X \big( \star_X F_1^{(X l)} \big) & = F_2^{(X X)} \wedge F_1^{(X v)} \\ \tfrac{\ell_l}{\ell_v} \mathrm{d}_X \big( \star_X F_1^{(X v)} \big) & = - F_2^{(X X)} \wedge F_1^{(X l)} \\ \tfrac{1}{\ell_l \ell_v} \underset{ = 0 }{ \underbrace{ \mathrm{d}_X \big( \star_X F_0^{(l v)} \big) }} & = \tfrac{1}{2} \underset{ =0 }{ \underbrace{ F_2^{(X X)} \wedge F_2^{(X X)} }} \mathrlap{\,,} \end{aligned}

where the terms over the brace vanish by degree reasons.

In the limit ℓ v→0\ell_v \to 0 the first equation in (6) goes to

F 2 (XX)→−1F 0 (lv)F 1 (Xl)∧F 1 (Xv) \begin{aligned} F_2^{(X X)} \;\to\; - \tfrac{1}{F_0^{(l v)}} F_1^{(X l)} \wedge F_1^{(X v)} \end{aligned}

and thus implies the vanishing of the right hand sides of the second and third equations in (6), whence (and this conclusion is generally obtained by fixing temporal gauge) the only remaining condition expressed by (6) is

d X(⋆ XF 1 (Xv))→0. \mathrm{d}_X \big( \star_X F_1^{(X v)} \big) \;\to\; 0 \,.

Finally, the first equation of motion (1) says that the component forms (3) are closed. The closure of the 0-form component F 0 (lv)F_0^{(l v)} means that it is locally constant, and the closure of F 1 (X−)F_1^{(X -)} implies the closure of their wedge product F 2 (XX)F_2^{(X X)}. This completes the proof of the claim (5).

References

In 3D

Plain

Early discussion of 3D Maxwell-Chern-Simons theory:

See also early discussions of abelian Chern-Simons theory as an effective field theory for fractional quantum Hall systems, here, many of which start with Maxwell-Chern-Simons theory; but beware of the claim in Haller & Lim 1992, 1996 that “anyon statistics” does not actually arise in MCS as often claimed.

Review:

In the context of effective description of superconductivity:

In the context of effective field theory for FQH systems (cf. abelian Chern-Simons theory for FQH systems):

  • Eduardo Fradkin; around (11.41) in: Field Theories of Condensed Matter Physics, Cambridge University Press (2013) [doi:10.1017/CBO9781139015509, ISBN:9781139015509]

  • Josef Willsher; §2.3 of : The Chern–Simons Action & Quantum Hall Effect: Effective Theory, Anomalies, and Dualities of a Topological Quantum Fluid, PhD thesis, Imperial College London (2020) [pdf, pdf]

  • Eduardo Fradkin; section 10 of: Field Theoretic Aspects of Condensed Matter Physics: An Overview, Encyclopedia of Condensed Matter Physics (2nd ed.) 1 (2024) 27-131 [doi:10.1016/B978-0-323-90800-9.00269-9, arXiv:2301.13234]

On Maxwell-Chern-Simons theory as the result of “bosonization” of interacting fermions:

On area-preserving diffeomorphisms symmetry Maxwell-Chern-Simons theory:

  • Ian I. Kogan: Area Preserving Diffeomorphisms and Symmetry in a Chern-Simons Theory [arXiv:hep-th/9208028]

On the quantization of the theory:

Review:

On the quantum states of (tacitly) renormalized 3D MCS theory representing area-preserving diffeomorphisms, via methods of constructive field theory:

and making this explicit:

With boundaries and edge modes:

In view of electric-magnetic duality (“S-duality”):

The lattice formulation:

See also:

  • M. Henningson, Canonical coordinates for Yang-Mills-Chern-Simons theory, J. High Energ. Phys. 2024 138 (2024) [doi:10.1007/JHEP10(2024)138]

With fermions

Discussion of coupling to fermion fields hence of Maxwell-Chern-Simons-Dirac theory (MCSD) or topologically massive QCD:

As an effective description of superconductivity:

In 5d

In the context of minimal D=5 supergravity:

Concerning the coupling constant:

In the context of quantum electrodynamics in 5d:

With defects:

Last revised on April 14, 2026 at 13:31:40. See the history of this page for a list of all contributions to it.

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