self-dual higher gauge theory



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Differential cohomology



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

XF gF , \nabla \mapsto \int_X F_\nabla \wedge \star_g F_{\nabla} \,,

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

dimX=4k+2 dim X = 4 k + 2

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

±F =F . \pm F_\nabla = \star F_\nabla \,.

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 XX of dimension 4k+24 k +2 is given by the state (wave function) of an abelian higher Chern-Simons theory in dimension 4k+34 k + 3.


Holographic relation to higher Chern-Simons theory

Idea and examples

Generally, higher dimensional Chern-Simons theory in dimension 4k+34k+3 (for kk \in \mathbb{N}) is holographically related to self-dual higher gauge theory in dimension 4k+24k+2 (at least in the abelian case).

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+34k+3 on Σ×\Sigma \times \mathbb{R} can be identified with the space of flat 2k+12k+1-forms on Σ\Sigma. The presymplectic form on this space is given by the pairing

(δB 1,δB 2) ΣδB 1δB 2 (\delta B_1, \delta B_2) \mapsto \int_\Sigma \delta B_1 \wedge \delta B_2

obtained as the integration of differential forms over Σ\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 Σ\Sigma. The corresponding Hodge star operator

:Ω 2k+1(Σ)Ω 2k+1(Σ) \star : \Omega^{2k+1}(\Sigma) \to \Omega^{2k+1}(\Sigma)

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

B=(1) (2k+1)(4k+3)B=B. \star \star B = (-1)^{(2k+1)(4k+3)} B = - B \,.

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

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

XB 1B 2 = XB 1B 2. \begin{aligned} \int_X B_1 \wedge B_2 & = - \int_X B_1 \wedge \star \star B_2 \end{aligned} \,.

Here on the right now the Hodge inner product of B 1B_1 with B 2\star B_2 appears, which is invariant under applying the Hodge star to both arguments.

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

B=iB \star B = i B

and imaginary anti-self-dual if

B=iB. \star B = - i B \,.

Then for imaginary self-dual B 1B_1 and B 2B_2 we find that the symplectic pairing is

(B 1,B 2) =i XB 1B 2 =i X(B 1)(B 2) =+i XB 1B 2. \begin{aligned} (B_1, B_2) &= -i \int_X B_1 \wedge \star B_2 \\ & = -i \int_X (\star B_1) \wedge \star (\star B_2) \\ & = +i \int_X B_1 \wedge \star B_2 \end{aligned} \,.

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


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).

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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure


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 pp-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+1d = 5 + 1 dimensions (arxiv:hep-th/0111150)

The chiral boson in 2d is discussed in detail in

A quick exposition of the basic idea is in

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

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

Conceptual aspects of this are also discussed in section 6.2 of

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

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

More discussion of the general principle is in

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

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

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)(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.

Revised on September 12, 2013 23:42:01 by Urs Schreiber (