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

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

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).

• $(k=0)$: 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 (of fields which are cocycles in (twisted differential) complex K-theory) is related to a parts of a type II string theory on its boundary (or that of the space-filling D9-brane, if one wishes) (Belov-Moore 06b).

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.

(See also at Serre duality.)

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-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

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.

Original discussion is due to Srivastava 1989, GGR 1992. A quick review as a warm-up for the higher dimensional case is in Witten 96, section 2. More discussion is in GBMNV Miao & Müller-Kirsten 2000, see also 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 (by the idea of K-theory classification of D-brane charge) are meant to be self-dual as cocycles in (differential) topological K-theory (cf. at pre-metric electromagnetism the section on RR-fields).

The proper discussion of these fields hence involves generalizing the above story from ordinary cohomology and principally polarized intermediate Jacobians to an analog in Whitehead-generalized cohomology such as topological K-theory and an appropriate generalized form of intermediate Jacobians (cf. Hodge-filtered differential cohomology).

(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.

See at intermediate Jacobian – For complex K-theory.

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 and Hermitian manifold (e.g.Kähler manifold): 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

• non-Lagrangian field theory

• Perry-Schwarz action

• Kervaire invariant

• bosonization

• higher dimensional Chern-Simons theory

• holographic principle, AdS-CFT

• dual graviton

References

General

Original articles on the general issue include

• Marc Henneaux, Claudio Teitelboim, Dynamics of chiral (self-dual) $p$-forms, Physics Letters B 206 4 (1988) 650-654 [doi:10.1016/0370-2693(88)90712-5]

• Paolo Pasti, Dmitri Sorokin, Mario Tonin, On Lorentz Invariant Actions for Chiral P-Forms, Phys.Rev. D55 (1997) 6292-6298 (arXiv:hep-th/9611100)

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031,2000 (arXiv:hep-th/9912086)

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

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:

• C. A. Cremonini, Pietro Grassi, Self-Dual Forms in Supergeometry I: The Chiral Boson (arXiv:2012.10243)

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

• Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)

• Paolo Pasti, Dmitri Sorokin and M. Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys. Lett. B398 (1997) 41 (arXiv:hep-th/9701037)

following

• Paul Townsend, section 3.3 of D-branes from M-branes, Phys. Lett. B373 (1996) 68-75 (arXiv:hep-th/9512062)

  (which did not yet have the Hopf-Wess-Zumino term)

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

• Malcolm Perry, John Schwarz, Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory, Nucl. Phys. B489 (1997) 47-64 (arXiv:hep-th/9611065)

• John Schwarz, Coupling a Self-Dual Tensor to Gravity in Six Dimensions, Phys.Lett.B395:191-195,1997

• Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, World-Volume Action of the M Theory Five-Brane (arXiv:hep-th/9701166)

Discussion in the superembedding approach;

• Paul S. Howe, Ergin Sezgin, $D=11$, $p=5$, Phys. Lett. B 394 (1997) 62-66 [arXiv:hep-th/9611008, doi:10.1016/S0370-2693(96)01672-3]

• Dmitri Sorokin, Section 5.2 of Superbranes and Superembeddings, Phys. Rept. 329 (2000) 1-101 [arXiv:hep-th/9906142]

Amplification that the resulting B-field on the M5-brane is not naively self-dual:

• Paul S. Howe, Ergin Sezgin, Peter C. West, The six-dimensional self-dual tensor, Physics Letters B 400 3–4 (1997) 255-259 [doi:10.1016/S0370-2693(97)00365-1]

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

• Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, On the equivalence of different formulations of the M Theory five–brane, Phys. Lett. B408 (1997) 135-141 (arXiv:hep-th/9703127)

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

• Xavier Bekaert, Marc Henneaux, Comments on Chiral $p$-Forms, Int. J. Theor. Phys. 38:1161-1172, 1999 (arXiv:hep-th/9806062)

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

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

The chiral boson in 2d:

• Prem P. Srivastava, Quantization of self-dual field revisited, Phys. Rev. Lett. 63 (1989) 2791 [doi:10.1103/PhysRevLett.63.2791]

• H. O. Girotti, M. Gomes, and V. O. Rivelles, Chiral bosons through linear constraints, Phys. Rev. D 45 (1992) R3329(R) [doi:10.1103/PhysRevD.45.R3329]

• Luis Alvarez-Gaumé, Jean-Benoit Bost, Greg Moore, Philip Nelson, Cumrun Vafa, Bosonization On Higher Genus Riemann Surfaces, Commun. Math. Phys. 112 (1987) 503 [doi:10.1007/BF01218489, pdf]

• Yan-Gang Miao and H. J. W. Müller-Kirsten, Self-duality of various chiral boson actions, Phys. Rev. D 62 (2000) 045014 [doi:10.1103/PhysRevD.62.045014]

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 and differential K-theory:

• Daniel Freed, Greg Moore, Graeme Segal, The Uncertainty of Fluxes, Commun. Math. Phys. 271 (2007) 247-274 [arXiv:hep-th/0605198, doi:10.1007/s00220-006-0181-3]

• Daniel Freed, Greg Moore, Graeme Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236-285 [arXiv:hep-th/0605200, doi:10.1016/j.aop.2006.07.014]

• Christian Becker, Marco Benini, Alexander Schenkel, Richard Szabo, Abelian duality on globally hyperbolic spacetimes, Commun. Math. Phys. 349 (2017) 361-392 [arXiv:1511.00316, doi:10.1007/s00220-016-2669-9]

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

• Edward Witten, Five-Brane Effective Action In M-Theory, J. Geom. Phys. 22 (1997) 103-133 [doi:10.1016/S0393-0440(97)80160-X, arXiv:hep-th/9610234]

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031 (2000) [doi:10.1088/1126-6708/2000/05/031, arXiv:hep-th/9912086]

Conceptual aspects of this are also discussed in section 6.2 of

• S. L. Lyakhovich, A. A. Sharapov, Quantizing non-Lagrangian gauge theories: an augmentation method, JHEP 0701 047 (2007) [arXiv:hep-th/0612086]

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

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

• Varghese Mathai, Hisham Sati, Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality, J. Geom. Phys. 92:240-251, 2015 (arXiv:1404.2257)

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.

Another proposal for a Lagrangian for self-dual higher gauge fields, at the cost of an unusual kinds of decoupled auxiliary fields:

• Ashoke Sen, Self-dual forms: Action, Hamiltonian and Compactification, Journal of Physics A: Mathematical and Theoretical, 53 8 (2020) [arXiv:1903.12196, doi:10.1088/1751-8121/ab5423]

Enhancement to superspace (in D'Auria-Fre formulation of supergravity):

• ACDGMNRT22

Further discussion:

• Neil Lambert, $(2,0)$ Lagrangian Structures, Physics Letters B 798 (2019) 134948 [arXiv:1908.10752, doi;10.1016/j.physletb.2019.134948]

• Pichet Vanichchapongjaroen: Covariant M5-brane action with self-dual 3-form, J. High Energ. Phys. 2021 39 (2021) [doi:10.1007/JHEP05(2021)039, arXiv:2011.14384]

  (attention to the non-linearity of the self-duality on M5)

• Chris Hull, Covariant Action for Self-Dual $p$-Form Gauge Fields in General Spacetimes, J. High Energ. Phys. 2024 11 (2024) [arXiv:2307.04748, doi:10.1007/JHEP04(2024)011]

• Sujiphat Janaun, Anajak Phonchantuek, Pichet Vanichchapongjaroen, Nonlinear chiral forms in the Sen formulation [arXiv:2404.05380]

• Dominik Rist, Christian Saemann, Miro van der Worp: Towards an M5-Brane Model III: Self-Duality from Additional Trivial Fields, J. High Energ. Phys. 2021 36 (2021) [doi:10.1007/JHEP06(2021)036, arXiv:2012.09253]

• Laura Andrianopoli, C. A. Cremonini, Riccardo D'Auria, Pietro A. Grassi, Riccardo Matrecano, Ruggero Noris, Lucrezia Ravera, Mario Trigiante, M5-brane in the superspace approach, Phys. Rev. D 106 2 (2022) 026010 [arXiv:2206.06388, doi:10.1103/PhysRevD.106.026010]

  (lifted to superspace in the D'Auria-Fré-Regge formulation of supergravity)

For RR-fields / differential K-theory

Discussion of (Lagrangian densities for) D=10 type II supergravity with “duality-symmetric”/“democratic”/“pregeometric” for of the RR-fields:

• Gianguido Dall'Agata, Kurt Lechner, Mario Tonin, $D=10$, $N=IIB$ Supergravity: Lorentz-invariant actions and duality, JHEP 9807:017 (1998) $[$arXiv:hep-th/9806140, doi:10.1088/1126-6708/1998/07/017$]$

• Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Section 3 of: Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl. Phys. B 535(1998) 242-292 $[$arXiv:hep-th/9806106, doi:10.1016/S0550-3213(98)00552-5$]$

• Eric Bergshoeff, Renata Kallosh, Tomas Ortin, Diederik Roest, Antoine Van Proeyen, New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls, Class. Quant. Grav. 18 (2001) 3359-3382 $[$arXiv:hep-th/0103233, doi:10.1088/0264-9381/18/17/303$]$

• Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, Various Faces of Type IIA Supergravity, Nucl. Phys. B 676 (2004) 189-228 $[$arXiv:hep-th/0307153, doi:10.1016/j.nuclphysb.2003.10.036$]$

• Karapet Mkrtchyan, Fridrich Valach, Democratic actions for type II supergravities, Phys. Rev. D 107 6 (2023) 066027 [doi:10.1103/PhysRevD.107.066027, arXiv:2207.00626]

Enhancement of the self-duality constraint on pregeometric RR-fields from (twisted) de Rham cohomology to (twisted) topological K-theory (under the hypothesized K-theory classification of D-brane charge) in terms of a quadratic form on differential K-theory:

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032 (2000) $[$arXiv:hep-th/9912279$]$

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031 (2000) $[$arXiv:hep-th/9912086, doi:10.1088/1126-6708/2000/05/032$]$

• Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 $[$arXiv:hep-th/0002027$]$

• Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry 7, Int. Press (2000) 129-194 $[$arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392$]$

An indication of a more refined discussion in twisted differential KR-theory:

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) $[$arXiv:0906.0795, slides$]$

See at orientifold for more on this.

Expressing the self-duality of pregeometric RR-fields in terms of 11d Chern-Simons theory:

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

Some review:

• Richard Szabo, section 3.6 and 4.6 of: Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology, ESI 2385 (2012) $[$arXiv:1209.2530, pdf$]$

Discussion in the context of flux quantization (here: D-brane charge quantization in K-theory):

• SS23, §3.3

For the 11d supergravity C-field

Formulation of the equations of motion of D=11 supergravity in superspace on fields including a flux density $G_7$ a priori independent of the flux density $G_4$ of the supergravity C-field:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, ch III.8.3-III.8.5 in vol 2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, epdf, ch III.8: pdf]

  (Using the D'Auria-Fre formulation of supergravity.)

• Antonio Candiello, Kurt Lechner, §6 in: Duality in supergravity theories, Nuclear Physics B 412 3 (1994) 479-501 [doi:10.1016/0550-3213(94)90389-1]

  (These authors seem not to be aware of CDF91, III.8 and, contrary to the result there, conclude that it is not possible without introducing non-local relations.)

Discussion of Lagrangian densities for D=11 supergravity with an a priori independent dual C-field field and introduction of the “duality-symmetric” terminology:

• Igor Bandos, Nathan Berkovits, Dmitri Sorokin, Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes, Nucl. Phys. B 522 (1998) 214-233 [doi:10.1016/S0550-3213(98)00102-3, arXiv:hep-th/9711055]

• Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Section 2 of Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl.Phys. B 535 (1998) 242-292 [doi:10.1016/S0550-3213(98)00552-5, arXiv:hep-th/9806106]

• Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, Section 2 of: Various Faces of Type IIA Supergravity, Nucl. Phys. B 676 (2004) 189-228 [doi:10.1016/j.nuclphysb.2003.10.036, arXiv:hep-th/0307153]

• Alexei J. Nurmagambetov, The Sigma-Model Representation for the Duality-Symmetric $D=11$ Supergravity, eConf C0306234 (2003) 894-901 [arXiv:hep-th/0312157, inspire:635585]

Discussion in the context of shifted C-field flux quantization:

• SS23, §3.4