nLab D=5 Chern-Simons theory

Idea

The standard abelian form of higher dimensional Chern-Simons theory in dimension 5 is the Lagrangian field theory with a single abelian ($G = \mathrm{U}(1)$) gauge field $A$ (locally a differential 1-form with flux density $F_A \coloneqq \mathrm{d}A$) and local Lagrangian density proportional to

(1)$$L_{CS}^{5D}(A) \;\propto\; A \wedge F_A \wedge F_A \,.$$

One way this appears is as a “topological mass” term to Maxwell theory in 5D:

For spacetime domain $Y^{1,4}$ a Lorentzian manifold, $\mathrm{dim}\big(Y^{1,4}\big) = 5$, with associated Hodge star operator

(2)$$\star_{{}_Y} \;\colon\; \Omega^\bullet_{dR}\big( Y^{1,4} \big) \longrightarrow \Omega^{5-\bullet}_{dR}\big( Y^{1,4} \big) \,,$$

the Lagrangian density of 5D Maxwell-Chern-Simons theory is proportional to

(3)$$L_{MCS}^{5D}(A) \;\propto\; \tfrac{1}{2} F_A \wedge \star_5 F_A \;-\; \tfrac{c}{3} A \wedge F_A \wedge F_A \,,$$

for some relative prefactor $c$

This appears notably in the bosonic sector of minimal D=5 supergravity, which in total is 5D Einstein-Maxwell-Chern-Simons theory.

The Euler-Lagrange equations of motion of 5D Chern-Simons-Maxwell theory (3) are (setting $c=1$, for definiteness):

(4)$$\begin{aligned} \mathrm{d}\, F_2 & = 0 \mathrlap{\,,} \\ \mathrm{d}\, \star_{{}_Y} F_2 & = \tfrac{1}{2} F_2 \wedge F_2 \mathrlap{\,.} \end{aligned}$$

These are like Maxwell's equations where an electric current is sourced by the gauge field itself.

The structure of the these equations of motion is, up to dimension and form degree, the same as that of the C-field in D=11 supergravity. The two are closely related under double dimensional reduction (cf. references here).

Properties

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

Consider the dimensional reduction of the 5D Maxwell-Chern-Simons equations of motion (4) to 3D, specifically:

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

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

of:

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

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

  which in suitable coordinates $l$ and $v$ have metrics $\begin{aligned} g_L & = \ell_l^2 \mathrm{d}l \otimes \mathrm{d}l \\ g_V & = \ell_v^2 \mathrm{d}v \otimes \mathrm{d}v \end{aligned}$ for positive real numbers $l,v \in \mathbb{R}_+$, respectively.

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

(6)$$\begin{aligned} F_2 = & F_2^{(X X)} \\ & + F_1^{(X l)} \wedge \mathrm{d} l \\ & + F_1^{(X v)} \wedge \mathrm{d} v \\ & + F^{(l v)}_0 \mathrm{d}l \wedge \mathrm{d}v \end{aligned} \;\;\;\;\; \text{for} \;\;\;\;\; \left\{ \begin{aligned} F_2^{(X X)} & \in \Omega^2_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^2_{dR}\big(Y^{1,4}\big) \\ F_1^{(X l)}, F_1^{(X v)} & \in \Omega^1_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^1_{dR}\big(Y^{1,4}\big) \\ F_0^{(l v)}, & \in \Omega^0_{dR}\big(X^{1,2}\big) \xhookrightarrow{ p_X^\ast } \Omega^0_{dR}\big(Y^{1,4}\big) \end{aligned} \right.$$

and such that

(7)$$\underset{x \in X}{\forall} \; F_0^{(l v)}(x) \neq 0$$

(whence we have a flux compactification).

Proposition

In this situation and in the limit $\ell_v \to 0$, the equations of motion (4) are equivalent to the following system of equations:

(8)$$\begin{aligned} & F_2^{(X X)} = - \tfrac{1}{F_0^{(l v)}} F_1^{(X l)} \wedge F_1^{(X v)} \\ & \mathrm{d}_X F_1^{(X v)} = 0, \; \mathrm{d}_X \star_X F_1^{(X v)} = 0, \\ & \mathrm{d}_X F_1^{(X l)} = 0 \mathrlap{\,.} \end{aligned}$$

In particular, when either of the $F_1^{(X -)}$ vanishes, then $F_2^{(X X)}$ satisfies the equations of motion of 3D abelian Chern-Simons theory, in the limit $\ell_v \to 0$.

Proof

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

Due to the product spacetime structure (5), the Hodge dual of $F_2$ (6) with respect to $Y^{1,4}$ is expressed in terms of the Hodge star operator $\star_{X}$ associated with $X^{1,2}$ as follows:

$$\begin{aligned} \star F_2 = & \ell_l \ell_v \big( \star_X F_2^{(X X)} \big) \wedge \mathrm{d}l \wedge \mathrm{d}v \\ & + \tfrac{\ell_v}{\ell_l} \big( \star_X F_1^{(X l)} \big) \wedge \mathrm{d} v \\ & - \tfrac{\ell_l}{\ell_v} \big( \star_X F_1^{(X v)} \big) \wedge \mathrm{d} l \\ & + \tfrac{1}{\ell_l \ell_v} \star_X F_0^{(l v)} \,, \end{aligned}$$

whence the second equation of motion (4) is seen to be equivalent to

(9)$$\begin{aligned} \ell_l \ell_v \mathrm{d}_X \big( \star_X F_2^{(X X)} \big) & = F_2^{(X X)} \wedge F_0^{(l v)} + F_1^{(X l)} \wedge F_1^{(X v)} \\ \tfrac{\ell_v}{\ell_l} \mathrm{d}_X \big( \star_X F_1^{(X l)} \big) & = F_2^{(X X)} \wedge F_1^{(X v)} \\ \tfrac{\ell_l}{\ell_v} \mathrm{d}_X \big( \star_X F_1^{(X v)} \big) & = - F_2^{(X X)} \wedge F_1^{(X l)} \\ \tfrac{1}{\ell_l \ell_v} \underset{ = 0 }{ \underbrace{ \mathrm{d}_X \big( \star_X F_0^{(l v)} \big) }} & = \tfrac{1}{2} \underset{ =0 }{ \underbrace{ F_2^{(X X)} \wedge F_2^{(X X)} }} \mathrlap{\,,} \end{aligned}$$

where the terms over the brace vanish by degree reasons.

In the limit $\ell_v \to 0$ the first equation in (9) goes to

$$\begin{aligned} F_2^{(X X)} \;\to\; - \tfrac{1}{F_0^{(l v)}} F_1^{(X l)} \wedge F_1^{(X v)} \end{aligned}$$

and thus implies the vanishing of the right hand sides of the second and third equations in (9), whence the only remaining condition expressed by (9) is

$$\mathrm{d}_X \big( \star_X F_1^{(X v)} \big) \;\to\; 0 \,.$$

Finally, the first equation of motion (4) says that the component forms (6) are closed. The closure of the 0-form component $F_0^{(l v)}$ means that it is locally constant, and the closure of $F_1^{(X -)}$ implies that of their wedge product $F_2^{(X X)}$. This completes the proof of the claim (8).

Moduli of fields (abelian case)

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

Related concepts

• higher dimensional Chern-Simons theory

  • D=1 Chern-Simons theory

  • D=2 Chern-Simons theory

  • D=3 Chern-Simons theory

  • D=4 Chern-Simons theory

  • D=5 Chern-Simons theory

    • 5d supergravity

  • D=6 Chern-Simons theory

  • D=7 Chern-Simons theory

  • D=11 Chern-Simons theory

  • AKSZ sigma-models

  • string field theory

  • infinite-dimensional Chern-Simons theory

• self-dual Yang-Mills theory

• D=5 gravity

• D=5 supergravity

References

Original discussion:

• Roberto Floreanini, Roberto Percacci: Higher dimensional abelian Chern-Simons theory, Physics Letters B 224 3 (1989) 29 [doi:10.1016/0370-2693(89)91232-X, inSpire:280512]

In the broader context of higher dimensional Chern-Simons theory:

• Máximo Bañados, Luis Garay, Marc Henneaux: Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories, Phys. Rev. D 53 (1996) R593(R) [doi:10.1103/PhysRevD.53.R593, arXiv:hep-th/9506187]

• Máximo Bañados, Luis Garay, Marc Henneaux: The dynamical structure of higher dimensional Chern-Simons theory, Nuclear Physics B 476 3 (1996) 611-635 [doi:10.1016/0550-3213(96)00384-7, arXiv:hep-th/9605159]

• Máximo Bañados: Higher Dimensional Chern-Simons Theories and Topological Black Holes, talk at 6th Conference on Quantum Mechanics of Fundamental Systems: Black Holes and the Structure of the Universe (1997) 1-11 [spire:453985, arXiv:gr-qc/9803002]

In the context of $D=5$ quantum electrodynamics (5d Maxwell-Chern-Simons theory):

• Christopher T. Hill: Anomalies, Chern-Simons Terms and Chiral Delocalization in Extra Dimensions, Phys. Rev. D 73 (2006) 085001 [arXiv:hep-th/0601154, doi:10.1103/PhysRevD.73.085001]

• Christopher T. Hill: Physics of $D = 5$ Chern-Simons term (2006) [pdf, pdf]

Canonical phase space analysis:

• Olivera Miskovic, Ricardo Troncoso, Jorge Zanelli, Canonical sectors of five-dimensional Chern-Simons theories, Phys. Lett. B 615 (2005) 277-284 [arXiv:hep-th/0504055, doi:10.1016/j.physletb.2005.04.043]

In relation to D=5 supergravity:

• Anna Ceresole, A. Lerda, Peter van Nieuwenhuizen: Five-dimensional gravitational super Chern-Simons terms, Phys. Rev. D 34 (1986) 1744-1748 [doi:10.1103/PhysRevD.34.1744]

and obtained from dimensional reduction of D=11 supergravity (“M-theory”):

• Yuji Tachikawa: Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, Journal of High Energy Physics, JHEP02 (2004) [arXiv:hep-th/0401184, doi:10.1088/1126-6708/2004/02/050]

• Federico Bonetti, Thomas Grimm, Stefan Hohenegger: One-loop Chern-Simons terms in five dimensions [arXiv:1302.2918]

On further dimensional reduction of $D=5$ Chern-Simons theory:

• M. Temple‐Raston: The reduction of five dimensional Chern–Simons theories, J. Math. Phys. 35 (1994) 759–768 [doi:10.1063/1.530665, pdf]

On its edge modes in higher dimensional generalization of the relation between 3D abelian Chern-Simons theory and fractional quantum Hall systems:

• K. S. Gupta, Allen B. Stern: 4D edge currents from 5D Chern-Simons theory, Nuclear Physics B 442 1–2 (1995) 157-178 [doi:10.1016/S0550-3213(95)00075-5, arXiv:hep-th/9410216]

More on supersymmetric 5d CS:

• Sergei M. Kuzenko, Joseph Novak, On supersymmetric Chern-Simons-type theories in five dimensions, JHEP 1402 (2014) 096 (arXiv:1309.6803)

See also:

• Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, Higher Chern-Simons based on (2-)crossed modules, JHEP 2023 207 (2023) [arXiv:2212.04667, doi:10.1007/JHEP07(2023)207]

A 5d higher gauge CS-Theory analogous to semi-topological 4d Chern-Simons theory:

• Alexander Schenkel, Benoit Vicedo: 5d 2-Chern-Simons theory and 3d integrable field theories, Commun. Math. Phys. 405 (2024) 293 [arXiv:2405.08083, doi:10.1007/s00220-024-05170-9]