nLab
self-dual higher gauge theory

Idea

The standard action functional for the higher U(1)-gauge field given by a circle n-bundle with connection $(P,\nabla )$ over a (pseudo) Riemannian manifold $(X,g)$ is

$$\nabla \mapsto {\int }_X{F}_{\nabla }\wedge {\star }_g{F}_{\nabla },$$\nabla \mapsto \int_X F_\nabla \wedge \star_g F_{\nabla} \,,

where $F_{\nabla }$ is the curvature $(n+1)$-form. If the dimension

$$\dim X=4k+2$$dim X = 4 k + 2

then the Hodge star operator squares to $+1$ (Lorentian signature) or $-1$ (Euclidean signature) on ${\Omega }^{k+1}(X)$. Therefore it makes sense in these dimensions to impose the self-duality constraint

$$\pm F_{\nabla }=\star F_{\nabla }.$$\pm F_\nabla = \star F_\nabla \,.

With this duality constraint imposed, one speaks of self-dual higher gauge fields. Their quantum field theory is more subtle than usual: first of all the above standard action functional now vanishes constantly.

But sense can be made of the theory of self-dual gauge fields by other means. Notably – by a version of the holographic principle the – partition function of the self-dual theory on an $X$ of dimension $4k+2$ is given by the state (wave function) of an abelian higher Chern-Simons theory in dimension $4k+3$.

Examples

• 6d (2,0)-superconformal QFT

References

Original reference on self-dual/chiral fields include

• X. Bekaert, Marc Henneaux, Comments on Chiral $p$-Forms (arXiv:hep-th/9806062)

• Mans Henningson, Bengt E.W. Nilsson, Per Salomonson, Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory (arXiv:hep-th/9908107)

• M. Henningson, The quantum Hilbert space of a chiral two-form in $d=5+1$ dimensions (arxiv:hep-th/0111150)

A precise formulation of the phenomenon in terms of differential cohomology is given in

• Dan Freed, Greg Moore, Graeme Segal,

  The Uncertainty of Fluxes Commun.Math.Phys.271:247-274 (2007) (arXiv:hep-th/0605198)

  Heisenberg Groups and Noncommutative Fluxes , AnnalsPhys.322:236-285 (2007) (arXiv:hep-th/0605200)

The idea of describing self-dual higher gauge theory by abelian Chern-Simons theory in one dimension higher originates in

• Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)

• Edward Witten, Duality Relations Among Topological Effects In String Theory (arXiv:hep-th/9912086)

Motivated by this the differential cohomology of self-dual fields had been discussed in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory

More discussion of the general principle is in

• Dmitriy Belov, Greg Moore, Holographic Action for the Self-Dual Field (arXiv:hep-th/0605038)

A quick exposition of the basic idea is in

• Jacques Distler, Actions for self-dual gauge fields (blog)

The application of this to the description of type II string theory in 10-dimensions to 11-dimensional Chern-Simons theory is in the followup

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories (arXiv:hep-th/0611020)

Discussion of the quantum anomaly of self-dual theories is in

• Samuel Monnier, The anomaly line bundle of the self-dual field theory (arXiv:1109.2904)

• Samuel Monnier, The global gravitational anomaly of the self-dual field theory (arXiv:1110.4639)

Discussion of the conformal blocks of self-dual higher gauge theories is in

• Kiyonori Gomi, An analogue of the space of conformal blocks in $(4k+2)$-dimensions (pdf)