Contents

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

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

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 or chirality constraint

With this duality constraint imposed, one speaks of self-dual higher gauge fields or chiral higher gauge fields or higher gauge fields with self-dual curvature. (These are a higher degree/dimensional generalization of what in Yang-Mills theory are called Yang-Mills instanton field configruations.)

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 $4k+3$.

Properties

Holographic relation to higher Chern-Simons theory

Idea and examples

Generally, higher dimensional Chern-Simons theory in dimension $4k+3$ (for $k\in \mathbb{N}$) is holographically related to self-dual higher gauge theory in dimension $4k+2$ (at least in the abelian case).

• $(k=0)$: ordinary 3-dimensional Chern-Simons theory is related to a string sigma-model on its boundary;

• $(k=1)$: 7-dimensional Chern-Simons theory is related to a fivebrane model on its boundary;

• $(k=2)$: 11-dimensional Chern-Simons theory is related to a parts of a type II string theory on its bounday (or that of the space-filling 9-brane, if one wishes) (BelovMoore).

Conformal structure from polarization

We indicate why higher dimensional Chern-Simons theory is – if holographically related to anything – holographically related to self-dual higher gauge theory.

The phase space of higher dimensional Chern-Simons theory in dimension $4k+3$ on $\Sigma \times ℝ$ can be identified with the space of flat $2k+1$-forms on $\Sigma$. The presymplectic form on this space is given by the pairing

obtained as the integration of differential forms over $\Sigma$ of the wedge product of the two forms.

The geometric quantization of the theory requires that we choose a polarization of the complexification of this space (split the space of forms into “coordinates” and their “canonical momenta”).

One way to achieve this is to choose a conformal structure on $\Sigma$. The corresponding Hodge star operator

provides the polarization by splitting into self-dual and anti-self-dual forms:

notice that (by the formulas at Hodge star operator) we have on mid-dimensional forms

Therefore it provides a complex structure on ${\Omega }^{2k+1}(\Sigma )\otimes ℂ$.

We see that the symplectic structure on the space of forms can equivalently be rewritten as

Here on the right now the Hodge inner product of $B_1$ with $\star B_2$ appears, which is invariant under applying the Hodge star to both arguments.

We then decompose ${\Omega }^{2k+1}(\Sigma )$ into the $\pm i$-eigenspaces of $\star$: say $B\in {\Omega }^{2k+1}(\Sigma )$ is imaginary self-dual if

and imaginary anti-self-dual if

Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is

Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into Lagrangian subspaces.

Therefore a state of higher Chern-Simons theory on $\Sigma$ may locally be thought of as a function of the self-dual forms on $\Sigma$. Under holography this is (therefore) identified with the correlator of a self-dual higher gauge theory on $\Sigma$.

Partition function

By the above discussion (…) the partition function of self-dual higher gauge theory is given by (a multiple of) the unique holomorphic section of a square root of the line bundle classified by the secondary intersection pairing. (Witten I, Hopkins-Singer).

Examples

Chiral boson in 2d

(…)

The simplest case. A review is for instance in (Witten I, section 2). A detailed discussion is in (GBMNV).

(…)

Fivebrane in 11-dimensional supergravity

The worldvolume theory of the M5-brane, the 6d (2,0)-superconformal QFT, contains a self-dual 2-form field. Its AdS7-CFT6 holographic description by 7-dimensional Chern-Simons theory is due to (Witten I).

RR-Fields in type II supergravity

The RR-field in type II string theory are self-dual (as a formal sum of fields). A holographic description is discussed in (Belov-Moore II).

Related concepts

The following table lists classes of examples of square roots of line bundles

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface: spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

• higher dimensional Chern-Simons theory

• holographic principle, AdS-CFT

References

An survey and introduction is in

• Greg Moore, A Minicourse on Generalized Abelian Gauge Theory, Self-Dual Theories, and Differential Cohomology, Lectures at Simons Center for Geometry and Physics (2011) (pdf)

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)

The chiral boson in 2d is discussed in detail in

• Luis Alvarez-Gaumé, J.-B. Bost, Greg Moore, P. Nelson, Cumrun Vafa, Bosonization On Higher Genus Riemann Surfaces, Commun. Math. Phys. 112 (1987) 503. (#GBMNV)

A quick exposition of the basic idea is in

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

A precise formulation of the phenomenon in terms of ordinary 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 ordinary differential cohomology of self-dual fields had been discussed in

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

The generalization of this to generalized differential cohomology is discussed from p. 26 on in

• Dan Freed, Dirac charge quantization and generalized differential cohomology.

More discussion of the general principle is in

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

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, pdf slides)

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

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

• Samuel Monnier, Geometric quantization and the metric dependence of the self-dual field theory (arXiv:1011.5890)

For the case of nonabelian self-dual 1-form gauge fields see the references at Yang-Mills instanton.