nLab self-dual higher gauge theory

Contents

Surveys, textbooks and lecture notes

Differential cohomology

differential cohomology

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

$\nabla \mapsto \int_X F_\nabla \wedge \star_g F_{\nabla} \,,$

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

$dim X = 4 k + 2$

then the Hodge star operator squares to $+1$ (Lorentzian 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

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

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 configurations.)

Since imposing the self-duality constraint on the fields that enter the above action functional makes that functional vanish identically, self-dual higher gauge theory is notorious for being subtle in that either it does not have a Lagrangian field theory description, or else a somewhat intricate indirect one (e.g. Witten 96, p. 7, Belov-Moore 06a). But instead one may regard the self-duality condition rather as part of the quantum theory (Witten 96, p.8, Witten 99, section 3 DMW 00b, page 3), namely as a choice of polarization of the phase space of an unconstrained theory in one dimension higher. By such as “holographic principle” the partition function of the self-dual theory on an $X$ of dimension $4 k +2$ is given by the state (wave function) of an abelian higher dimensional Chern-Simons theory in dimension $4 k + 3$.

The way this works is understood in much mathematical detail for $k = 0$, see at AdS3-CFT2 and CS-WZW correspondence. Motivated by this it has been proposed and studied in a fair bit of mathematical detail for $k = 1$ (Witten 96, Hopkins-Singer 02), see at M5-brane and 6d (2,0)-superconformal QFT and AdS7/CFT6. In both these cases the higher gauge fields are cocycles in ordinary differential cohomology. In (DMW 00, Belov-Moore 06b) it is suggested that similarly taking the self-dual fields to be cocycles in (differential) complex K-theory produces the RR-fields of type II superstring theory in dimension 10.

Definition (outline)

The holographic definition of self-dual abelian higher gauge field theory in dimension $4k+2$ given in (Witten 96, Witten 99) is in outline the following.

Imagine the self-dual $2k$-form field on $\Sigma$ to be coupled to sources given by $(2k+1)$-form gauge fields, hence by cocycles in ordinary differential cohomology $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ of degree $(2k+2)$. One may try to write down a Lagrangian for this coupling (and that is what is discussed in (Belov-Moore 06I)) and the resulting partition function is then locally a function of the source fields, hence a function on the intermediate Jacobian $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ and globally (since the action functional will not be strictly gauge invariant) a holomorphic section of a certain holomorphic line bundle on that space.

However, no choice of Lagrangian will enforce a strictly self-dual gauge field (the anti-self dual component will remain in the space of fields, hence the partition function will depend on it). Therefore the idea is to turn this around, reject the idea that self-dual higher gauge theory is directly itself a Lagrangian quantum field theory, and instead define the partition function of the self-dual higher gauge theory (and thereby, indirectly, the self-dual higher gauge theory itself!) as being a certain section of a certain line bundle on the intermediate Jacobian.

The question then is: which line bundle? Some plausibility arguments show that it must be a Theta characteristic bundle, hence a square root of the canonical bundle on the intermediate Jacobian. These have 1-dimensional spaces of holomorphic sections (theta functions) and hence any choice of that already uniquely fixes the would-be partition function, too, up to a choice of global factor. (For that reason later authors often regard the partition function as a line bundle, somewhat abusing the terminology.)

The claim then is that the correct choice of Theta characteristic to use is a quadratic refinement of the intersection pairing (the Beilinson-Deligne cup-product)

$[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}] \times [\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}] \stackrel{\cup}{\longrightarrow} [\Sigma, \mathbf{B}^{4k+3}\mathbb{G}^\times_{conn}] \stackrel{\int}{\longrightarrow} \mathbf{B}\mathbb{G}^\times_{conn} \,.$

Such quadratic refinement turns out to be given for $k = 0$ by a Spin structure (leading to “Spin Chern-Simons theory”) and for $k =1$ by a Wu structure (and the latter case is the actual example of interest in (Witten 96)). This is what the central theorem of (Hopkins-Singer 02) establishes rigorously.

One notices now that while the self-dual $(4k+2)$-dimensional gauge field theory thus itself is not a Lagrangian quantum field theory, it arises holographically form a field theory in dimension $4k+3$ that is, namely the intersection pairing above is the Lagrangian for higher dimensional Chern-Simons theory and the holomorphic line bundle which it gives rise to on the intermediate Jacobian, is the prequantum line bundle of Chern-Simons theory, whose holomorphic sections are its quantum states (see at quantization of Chern-Simons theory). From this perspective the square root quadratic refinement is the metaplectic correction in the geometric quantization of Chern-Simobs theory.

This kind of relation

self-dual $(4k+2)$d gauge theory$(4k+3)$d Chern-Simons theory
sources$\leftrightarrow$fields
partition function/correlator$\leftrightarrow$wavefunction/quantum state
conformal blocks$\leftrightarrow$space of quantum states

is the hallmark of the holographic principle.

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).

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

$(\delta B_1, \delta B_2) \mapsto \int_\Sigma \delta B_1 \wedge \delta B_2$

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 “canonical coordinates” and their “canonical momenta”).

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

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

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

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

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

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

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

\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 $B_1$ 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

$\star B = i B$

and imaginary anti-self-dual if

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

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

\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 $\Sigma$ 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 96, Hopkins-Singer).

Cohomological description

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

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

Examples

Chiral boson in 2d (on the string)

For $k = 0$ the self-dual theory abelian is that of a scalar field $\phi$ on a real-2-dimensional surface $\Sigma$ such that $\mathbf{d}\phi$ is self-dual. For any complex structure on $\Sigma$, making it a complex torus, this makes $\phi$ a “chiral” boson, in the sense of a chiral half of the $U(1)$-WZW model.

A quick review as a warm-up for the higher dimensional case is in (Witten 96, section 2). A detailed discussion is in (GBMNV) and see the references at AdS3-CFT2 and CS-WZW correspondence.

Chiral 2-form in 6d (on the M5-brane)

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 96).

Relation between 5d Maxwell theory and self-dual 3-forms in 6d

Consider 5d- and 6d-dimensional Minkowski spacetime equipped with global orthonormal coordinate charts $\{x^\kappa\}$, $\{x^\alpha\}$, respectively, adapated to an isometric embedding

$\array{ & \mathbb{R}^{4,1} &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}& \mathbb{R}^{5,1} \\ \kappa = & 0, 1, 2, 3, 4\phantom{,} \\ \alpha = & 0, 1, 2, 3, 4, && 5 }$

With this notation, the pullback of differential forms along this embedding is notationally implicit.

Now any differential 3-form $H_3$ on $\mathbb{R}^{5,1}$ decomposes as

(1)$H_3 \;=\; \widehat{F} \wedge d x^{5} + \widehat{H}$

for unique differential forms of the form

$\widehat F \;=\; \tfrac{1}{2}\hat F_{\kappa_1 \kappa_2}(x^\kappa, x^5) d x^{\kappa_1} \wedge d x^{\kappa_2}$

and

$\widehat{H} \;=\; \tfrac{1}{3!} \widehat{H}_{\kappa_1 \kappa_2 \kappa_3}(x^\kappa, x^5) d x^{\kappa_1} \wedge d x^{\kappa_2} \wedge d x^{\kappa_3} \,.$

In the case that $H_3$ has vanishing Lie derivative along the $x^5$-direction,

(2)$\mathcal{L}_5 H_3 \;=\; 0$

then also these components forms do not depend on $x^5$ are actualls pullbacks of differential forms on $\mathbb{R}^{4,1}$.

In terms of this decomposition, the 6d Hodge dual of $H_3$ is equivalently given by the 5d Hodge duals of these components as (best seen by the relation to Hodge pairing according to this Prop.)

(3)$\star_6 H_3 \;=\; \big( \star_5 \widehat{H}\big) \wedge d x^{5} - \star_5 \widehat{F}$

Since the Hodge star operator squares to unity in the special case that it is applied to differential 3-forms on 6d Minkowski spacetime (by this Prop.)

$\star_6 \star_6 H_3 \;=\; + H_3$

we may ask for $H_3$ to he Hodge self-dual. By (3) this means equivalently that its 5d components are 5d Hodge duals of each other:

$\big( H_3 \;=\; \star_{6} H_3 \big) \;\;\; \overset{ H_3 = \widehat{F} \wedge d x^5 + \widehat{H} }{ \Leftrightarrow } \;\;\; \big( \widehat{H} = \star_5 \widehat{F} \big) \,.$

It follows that if there is no $x^5$-dependence (2) then the condition that $H_3$ be a closed and self-dual 3-form is equivalent to its 5d components $\widehat{F}$ ($\widehat{H}$) being the (dual) field strength/Faraday tensor satisfying the Maxwell equations of D=5 Maxwell theory (without source current):

$\underset{ \color{blue} { {\phantom{A}} \atop {\text{D=6 self-dual 3-form theory}} } }{ \left. \array{ & d H_3 = 0 \\ & \star_6 H_3 = H_3 } \right\} } \;\;\; \overset{ {\mathcal{L}_{5} H_3 = 0} \atop {H_3 = \widehat{F}\wedge d x^5 + \cdots} }{ \Leftrightarrow } \;\;\; \underset{ \color{blue} { {\phantom{A}} \atop \text{D=5 Maxwell theory} } }{ \left\{ \array{ & d \widehat{F} = 0 \\ & d \star_5 \widehat{F} = 0 } \right. }$

This may be summarized as saying that the massless part of the Kaluza-Klein reduction of self-dual 3-form theory from 6d to 5d is D=5 Maxwell theory.

Essentially this relation underlies the formulation of the M5-brane via the Perry-Schwarz Lagrangian.

RR-Fields in 10d (on the “9-brane”)

The RR-field in type II string theory are self-dual. Since the RR-fields are cocycles in (differential) K-theory, the proper discussion of this now involves generalizing from the above story ordinary cohomology and hence generalizing the concept of principally polarized intermediate Jacobians from ordinary cohomology to K-theory.

(Of course one may, as a warmup, “approximate” K-theory classes on a 10-manifold by integral cohomology lifts of the degree-5 component of their Chern character and hence discuss that as abelian self-dual higher gauge theory in dimension 10. This is discussed in (Witten 99, section 4)).

The relevant analog of the intermediate Jacobian for K-theory on a 10-manifold is naturally taken to be

$K(X)\otimes_{\mathbb{Z}} \mathbb{R}/ K(X)$

(where the action is via the Chern character/realification $K \to K \otimes \mathbb{R}$).

This is proposed and discussed in (Witten 99, section 4.3, Moore-Witten 99, section 3, DMW 00, section 7.1, Belov-Moore 06b, section 5, MPS 11):

where for ordinary cohomology the (quadratic refinement of the) intersection product (= Deligne-Beilinson cup product followed by fiber integration in ordinary differential cohomology) provided the symplectic structure/polarization of the intermediate Jacobian, here it is tensoring of Dirac operators followed by fiber integration in differential K-theory, hence the index map in K-theory.

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

References

General

Original articles on the general issue include

Surveys include

• 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)

• Richard Szabo, Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology (arXiv:1209.2530)

An approach based on supergeometry:

For the M5-sigma model

The sigma-model description of the (single) M5-brane of Green-Schwarz action functional-type was found in covariant form in

following

and following the non-covariant form of the self-duality mechanism (Perry-Schwarz action) of

Discussion in the superembedding approach is in

Discussion of the equivalence of these superficially different action functionals is in

For ordinary $U(1)$-higher gauge fields / ordinary differential cohomology

Self-duality for higher abelian gauge fields (in ordinary differential cohomology):

Original reference on self-dual/chiral fields include

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

Discussion of S-duality in 6d self-dual higher gauge theory via non-commutative-deformation:

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)

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

For RR-fields / differential K-theory

The self-dual higher gauge field of the RR-field in terms of a quadratic form on differential K-theory is discussed originally around

and

for type I superstring theory, and for type II superstring theory in

with more refined discussion in twisted differential KR-theory in

See at orientifold for more on this. The relation to 11d Chern-Simons theory is made manifest in

Review is in (Szabo 12, section 3.6 and 4.6).

Last revised on December 21, 2020 at 05:05:51. See the history of this page for a list of all contributions to it.