nLab D=5 Chern-Simons theory

Idea

higher dimensional Chern-Simons theory in dimension 5.

Properties

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

A general account of the structure is in

• Olivera Miskovic, Ricardo Troncoso, Jorge Zanelli, Canonical sectors of five-dimensional Chern-Simons theories (arXiv:hep-th/0504055)

Remarks on applications in phenomemenological models are in

• C. Hill, Physics of $D = 5$ Chern-Simons term (pdf)

The relation to 5-dimensional supergravity is discussed in

• A. Ceresole, A. Lerda, and P. van Nieuwenhuizen , Five-dimensional gravitational super Chern-Simons terms, Phys. Rev. D 34, 1744–1748 (1986)

• Yuji Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting (arXiv:hep-th/0401184)

and via KK-compactification of the supergravity C-field of 11-dimensional supergravity in

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

More on supersymmetric 5d CS theory is in

• 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 [arXiv:2405.08083]