nLab
supergravity C-field

Idea

The field content of 11-dimensional supergravity contains a higher U(1)-gauge field called the supergravity C-field or M-theory 3-form , which is locally a 3-form and globally some variant of a circle 3-bundle with connection. There have been several suggestions for what precisely its correct global description must be.

The DFM-model

Construction via $E_8$ gauge fields

In (DFM, section 3) the following definition is considered and argued to be a good model of the supergravity $C$-field.

Note

The homotopy groups of the classifying space $B{E}_8$ of the Lie group E8 satisfy

$${\pi }_{i}B{E}_8=\{\begin{array}{c}\mathbb{Z}\mid i=4\\ 0\mid i\ne 4,i\le 15\end{array}.$$\pi_i B E_8 = \left\{ \array{ \mathbb{Z} | i = 4 \\ 0 | i \neq 4, i \leq 15 } \right. \,.

Therefore for $X$ a manifold of dimension $\dim X\le 15$ there is a canonical morphism

$$H^1(X,E_8)\simeq H^4(X,\mathbb{Z}).$$H^1(X, E_8) \simeq H^4(X, \mathbb{Z}) \,.

Definition

Let $X$ be a smooth manifold of dimension $\dim X<15$. For each $a\in H^4(X,\mathbb{Z})$. choose an E8-principal bundle $P\to X$ which represents $a$ under the above isomorphism.

Write then

$$E(X)\in \mathrm{Grpd}$$\mathbf{E}(X) \in Grpd

for the groupoid whose

• objects are triples $(P,\nabla ,c)$ where

  • $P$ is one of the chosen $E_8$-bundles,

  • $\nabla$ is a connection on $P$;

  • $c\in {\Omega }^3(X)$ is a degree-3 differential form on $X$.

• morphisms $\omega :(P,{\nabla }_1,c_1)\to (P,{\nabla }_2,c_2)$ are parameterized by their source and target triples together with a closed 3-form $\omega \in {\Omega }_{\mathbb{Z}}^3(X)$ with integral periods, subject to the condition that

  $$c_2-c_1=\mathrm{CS}({\nabla }_1,{\nabla }_2)+\omega ,$$c_2 -c_1 = CS(\nabla_1,\nabla_2) + \omega \,,

  where $\mathrm{CS}({\nabla }_1,{\nabla }_2)$ is the relative Chern-Simons form corresponding to the linear path of connections from ${\nabla }_1$ to ${\nabla }_2$

• the composition of morphisms

  $$({\omega }_2\circ {\omega }_1):(P,{\nabla }_1,C_1)\stackrel{{\omega }_1}{\to }(P,{\nabla }_2,C_2)\stackrel{{\omega }_2}{\to }(P,{\nabla }_3,C_3)$$(\omega_2 \circ \omega_1 ) : (P,\nabla_1, C_1) \stackrel{\omega_1}{\to} (P, \nabla_2, C_2) \stackrel{\omega_2}{\to} (P, \nabla_3, C_3)

  is given by

  $${\omega }_1+{\omega }_2+⟨({\nabla }_2-{\nabla }_1)\wedge ({\nabla }_3-\nabla 2)⟩.$$\omega_1 + \omega_2 + \langle (\nabla_2-\nabla_1)\wedge(\nabla_3-\nabla2) \rangle \,.

See (DFM, (3.22), (3.23)).

Here we think of $X$ as equipped with a pseudo Riemannian structure and spin connection $\omega$ and think of each object $(P,\nabla ,C)$ of $E(X)$ as inducing an degree-4 cocycle in ordinary differential cohomology with curvature 4-form

$${𝒢}_{\nabla ,c}=\mathrm{tr}{F}_{\nabla }\wedge F_{\nabla }-\frac{1}{2}\mathrm{tr}{R}_{\omega }\wedge R_{\omega }+dc.$$\mathcal{G}_{\nabla,c} = tr F_\nabla \wedge F_\nabla - \frac{1}{2} tr R_\omega \wedge R_\omega + d c \,.

Notice that with the normalization implicit here the second terms is one half of the image of something in integral cohomology. So this is not itself a differential character, but can be regarded as “shifted differential character”: a trivialization of the trivial 5-character with global connection 4-form given by $\frac{1}{2}\mathrm{tr}{R}_{\omega }\wedge R_{\omega }$. See below for more on this.

Claim

The above groupoid has homotopy groups

• ${\pi }_0\simeq H_{\mathrm{diff}}^4(Y)$

• ${\pi }_1(-,(\nabla ,c))\simeq H^2(Y,U(1))$ .

The first, the set of connected components (gauge equivalence classes of $C$-fields) is isomorphic to the set of ordinary differential cohomology in degree 4 of $X$. In fact ${\pi }_0$ is naturally a torsor over this abelian group: the torsor of $\frac{1}{2}\mathrm{tr}{R}^2$-shifted differential characters.

The second, the fundamental group, is that of flat circle bundles.

Orientation and fractional classes

Ordinarily, given a $\mathrm{Spin}\times E_8$-bundle $P\to Y$ with first fractional Pontryagin class

$$\lambda :=\frac{1}{2}{p}_1(P)$$\lambda := \frac{1}{2}p_1(P)

and second Chern class

$$a:=c_2(P)$$a := c_2(P)

the $C$-field is supposed to have a curvature class in de Rham cohomology given by

$$a_{\mathrm{dR}}+\frac{1}{2}{\lambda }_{\mathrm{dR}}\in H_{\mathrm{dR}}^4(Y).$$a_{dR} + \frac{1}{2} \lambda_{dR} \in H_{dR}^4(Y) \,.

Since in general $\lambda =\frac{1}{2}{p}_1(P)$ is not further divisible in integral cohomology, this means that this cannot be the curvature of any differential character/bundle 2-gerbe/circle 3-bundle with connection, since these are necessarily the images in de Rham cohomology of their integral classes.

See also (DFM, section 12.1) where it is argued that this is related to boundaries and orientation double covers.

Restriction to the boundary

By DFM, section 12 on a manifold $Y$ with boundary $X=\partial Y$ we are to impose $C{\mid }_{\partial Y}=0$.

See the discussion below for how this reproduces the Green-Schwarz mechanism for heterotic supergravity on the boundary.

Description in $\mathrm{\infty }$-Chern-Weil theory

Some remarks on ways to regard the $C$-field from the point of view of ∞-Chern-Weil theory.

Abstract definition

We shall consider the sum of two $C$ fields, whose curvature is the image in de Rham cohomology of the proper integral class $2a-\lambda$

Recall from the discussion at circle n-bundle with connection that in the cohesive (∞,1)-topos $H:=$ Smooth∞Grpd the circle 3-bundles with local 3-form connection over an object $Y\in H$ (for instance a smooth manifold, or an orbifold) are objects in the 3-groupoid $H_{\mathrm{diff}}(X,B^{3}U(1))$ that is the (∞,1)-pullback

$$\begin{array}{ccc}{H}_{\mathrm{diff}}(Y,B^{3}U(1))& \to & H_{\mathrm{dR}}^4(Y)\\ \downarrow & & \downarrow \\ H(X,B^{3}U(1))& \stackrel{\mathrm{curv}}{\to }& H(X,{♭}_{\mathrm{dR}}{B}^{4}U(1))\end{array}$$\array{ \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\to& H^4_{dR}(Y) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) }

in ∞Grpd.

(Recall from the discussion there that if desired one may pass to the canonical presentation of this by the model structure on simplicial presheaves over CartSp and that in this explicit presentation we may replace $H_{\mathrm{dR}}^4(Y)$ with the more familiar ${\Omega }_{\mathrm{cl}}^4(Y)$. )

We consider now the analog of this definition for the universal curvature form on $B^{3}U(1)$ replaced by the difference of the differentially refined second Chern class of E8 and the first fractional Pontryagin class of the spin group. The resulting $(\mathrm{\infty },1)$-pullback we tentatively call $C\mathrm{Field}(Y)$, though we shall have to discuss to which extend this faithfully models the $C$-field, and which aspects of it.

Definition

For $Y\in$ Smooth∞Grpd, let $C\mathrm{Field}(Y)\in$ ∞Grpd be the (∞,1)-pullback

$$\begin{array}{ccc}C\mathrm{Field}(Y)& \to & H_{\mathrm{dR}}^4(Y)\\ \downarrow & & \downarrow \\ H(Y,B(\mathrm{Spin}\times E_8))& \stackrel{(2c_2{)}_{\mathrm{dR}}-(\frac{1}{2}{p}_1{)}_{\mathrm{dR}}}{\to }& H(Y,{♭}_{\mathrm{dR}}{B}^{4}U(1))\end{array}.$$\array{ C Field(Y) &\to& H^4_{dR}(Y) \\ \downarrow && \downarrow \\ \mathbf{H}(Y, \mathbf{B} (Spin \times E_8)) &\stackrel{(2\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_1)_{dR}}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,.

Note

By its intrinsic definition we have that the differential characteristic class $(c_2{)}_{\mathrm{dR}}$ is the composite

$$(c_2{)}_{\mathrm{dR}}:B{E}_8\stackrel{c_2}{\to }{B}^{3}U(1)\stackrel{\mathrm{curv}}{\to }{♭}_{\mathrm{dR}}{B}^{4}U(1)$$(\mathbf{c}_2)_{dR} : \mathbf{B}E_8 \stackrel{\mathbf{c}_2}{\to} \mathbf{B}^3 U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)

of the smooth refinement of the second Chern class with the universal curvature form on $B^{3}U(1)$. Similarly for $(\frac{1}{2}{p}_2{)}_{\mathrm{dR}}$.

Therefore we may either compute the (∞,1)-pullback in def. 2 directly, or in two consecutive steps. Both methods lead to their insights.

In

• General properties

we consider general abstract consequences of the above definition, mainly making use of the factorization. In

• Presentation by differential form data

we find a presentation by simplicial presheaves of the direct homotopy pullback.

In the first approach connections on the E8-principal bundles never appear explicitly. In the second approach they appear as pseudo-connections, or as genuine connections whose morphisms are however allowed to shift them arbitrarily. This means that these connections are purely auxiliary data that serve to present the required homotopies. They do not survive in cohomology. This is as in the DFM model above.

Finally in

• Restriction to the boundary

we comment how genuine $E_8$-connections may appear inside the second presentation of the $C$-model.

General properties

This implies by the pasting law for (∞,1)-pullbacks that the $(\mathrm{\infty },1)$-pullback from def. 2 may be decomposed into two consecutive pullbacks of the form

$$\begin{array}{ccccc}C\mathrm{Field}(X)& \stackrel{\hat{\chi }}{\to }& H_{\mathrm{diff}}(X,B^{3}U(1))& \stackrel{\omega }{\to }& H_{\mathrm{dR}}^4(X)\\ \downarrow & & \downarrow & & \downarrow \\ H(X,B(E_8\times \mathrm{Spin}(10,1)))& \stackrel{2c_2-\frac{1}{2}{p}_2}{\to }& H(X,B^{3}U(1))& \stackrel{\mathrm{curv}}{\to }& H(X,{♭}_{\mathrm{dR}}{B}^{4}U(1))\end{array},$$\array{ C Field(X) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) &\stackrel{\omega}{\to}& H^4_{dR}(X) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}(E_8 \times Spin(10,1))) &\stackrel{2\mathbf{c}_2- \frac{1}{2}\mathbf{p}_2}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,,

where on the right we find the defining pullback for (the cocycle 3-groupoid of) ordinary differential cohomology.

This implies the following structure and properties.

Note/Definition

By the above there exists canonically a morphism

$$\hat{\chi }:C\mathrm{Field}(X)\to H(X,B^{3}U(1))\stackrel{{\tau }_{\le 0}}{\to }{H}_{\mathrm{diff}}^4(X)$$\hat \chi : C Field(X) \to \mathbf{H}(X,\mathbf{B}^3 U(1)) \stackrel{\tau_{\leq 0}}{\to} H^4_{diff}(X)

that maps $C$-field configurations to ordinary differential cohomology in degree 4, whose curvature $\omega (\hat{\chi })$ is the image $(c_2{)}_{\mathrm{dR}}-(\frac{1}{2}{p}_2{)}_{\mathrm{dR}}:=\mathrm{curv}(2c_2-\frac{1}{2}{p}_2)$ in de Rham cohomology of the second Chern-class of some $E_8$-bundle.

The differential cocycle $\hat{\chi }(C)$ has all the general properties that make its higher parallel transport over membrane worldvolumes be well-defined. (Apart from the coefficient of $\lambda$, this is the only requirement from which DFM deduce their model.)

The following proposition describes the first two homotopy groups of the 3-groupoid $C\mathrm{Field}(Y)$.

Proposition

Over a fixed $\mathrm{Spin}$-principal bundle $P_{\mathrm{Spin}}$ we have a short exact sequence (of pointed sets)

$$*\to H^3(Y,U(1))\to {\pi }_{0}C{\mathrm{Field}}_{P_{\mathrm{Spin}}}(Y)\to H_{\mathrm{dR}}^4(Y{)}_{2\mathbb{Z}}\to *$$* \to H^3(Y, U(1)) \to \pi_0 C Field_{P_{Spin}}(Y) \to H^4_{dR}(Y)_{2 \mathbb{Z}} \to *

and

${\pi }_{0}C{\mathrm{Field}}_{P_{\mathrm{Spin}}}(Y)$ is the group of pairs $([c],f)\in H^2(X,U(1))\times C^{\mathrm{\infty }}(X,E_8)$ where $f$ is a smooth refinement under $E_8{\simeq }_{14}{B}^{2}U(1)\simeq K(\mathbb{Z},3)$ of the integral image of $[c]$.

Proof

Notice that we have the pasting diagram of (∞,1)-pullbacks

$$\begin{array}{ccccc}C{\mathrm{Field}}_{\omega (\hat{\chi }(-))=0}(Y)& \to & H(Y,♭B^{3}U(1))& \to & *\\ \downarrow & & \downarrow & & {\downarrow }^0\\ C\mathrm{Field}(Y)& \stackrel{\hat{\chi }}{\to }& H_{\mathrm{diff}}(Y,B^{3}U(1))& \stackrel{\omega }{\to }& H_{\mathrm{dR}}^4(Y)\\ \downarrow & & \downarrow & & \downarrow \\ H(Y,B{E}_8)& \stackrel{2c_2}{\to }& H(Y,B^{3}U(1))& \stackrel{\mathrm{curv}}{\to }& H(Y,{♭}_{\mathrm{dR}}{B}^{4}U(1))\end{array},$$\array{ C Field_{\omega(\hat \chi(-)) = 0}(Y) &\to& \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) &\to& {*} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{0}} \\ C Field(Y) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\stackrel{\omega}{\to}& H^4_{dR}(Y) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(Y, \mathbf{B}E_8) &\stackrel{2\mathbf{c}_2}{\to}& \mathbf{H}(Y, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,,

where the top right square is discussed at cohesive (∞,1)-topos -- Differential cohomology. By the discussion at smooth ∞-groupoid -- Flat cohomology we have that ${\pi }_{0}H(Y,♭B^{3}U(1))\simeq H^3(Y,U(1))$, where on the right we have ordinary cohomology (for instance realized as singular cohomology). Finally observe that ${\pi }_{0}H(Y,E_8)\simeq {\pi }_{0}H(Y.B^{3}U(1))$, by the above remark. Therefore after passing to connected components by applying ${\pi }_0(-)$ we get on cohomology

$$\begin{array}{ccccc}{H}^3(Y,U(1))& \stackrel{\cdot 2}{\to }& H^3(Y,U(1))& \to & *\\ \downarrow & & \downarrow & & {\downarrow }^0\\ {\pi }_{0}C\mathrm{Field}(Y)& \stackrel{\hat{\chi }}{\to }& H_{\mathrm{diff}}^4(Y)& \stackrel{\omega }{\to }& H_{\mathrm{dR}}^4(Y)\\ \downarrow & & \downarrow & & \downarrow \\ H^1(Y,E_8)& \stackrel{\cdot 2}{\to }& H^4(Y,\mathbb{Z})& \stackrel{\mathrm{curv}}{\to }& H_{\mathrm{dR}}^4(Y)\end{array}$$\array{ H^3(Y, U(1)) & \stackrel{\cdot 2}{\to}& H^3(Y, U(1)) &\to& {*} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{0}} \\ \pi_0 C Field(Y) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}^4(Y) &\stackrel{\omega}{\to}& H^4_{dR}(Y) \\ \downarrow && \downarrow && \downarrow \\ H^1(Y, E_8) & \stackrel{\cdot 2}{\to}& H^4(Y, \mathbb{Z}) &\stackrel{curv}{\to}& H^4_{dR}(Y) }

by reasoning as discussed at fiber sequence. In parallel to the familiar short exact sequence for ordinary differential cohomology

$$*\to H^3(Y,U(1))\to H_{\mathrm{diff}}^4(Y)\to H_{\mathrm{dR}}^4(Y{)}_{\mathbb{Z}}\to *.$$* \to H^3(Y, U(1)) \to H^4_{diff}(Y) \to H^4_{dR}(Y)_{\mathbb{Z}} \to * \,.

this therefore implies also the short exact sequence

$$*\to H^3(Y,U(1))\to {\pi }_{0}C\mathrm{Field}\to H_{\mathrm{dR}}^4(Y{)}_{2\mathbb{Z}}\to *.$$* \to H^3(Y, U(1)) \to \pi_0 C Field \to H^4_{dR}(Y)_{2 \mathbb{Z}} \to * \,.

Next we redo the entire discussion after applying the loop space object-construction to everything. Using that

$$\Omega H(Y,BQ)\simeq H(Y,\Omega BQ)\simeq H(Y,Q)$$\Omega \mathbf{H}(Y, \mathbf{B}Q) \simeq \mathbf{H}(Y, \Omega \mathbf{B}Q) \simeq \mathbf{H}(Y, Q)

on general grounds (see fiber sequence for details) and that also

$$\Omega (♭B^{n}U(1))\simeq ♭B^{n-1}U(1)$$\Omega (\mathbf{\flat}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}\mathbf{B}^{n-1} U(1)

and

$$\Omega ({♭}_{\mathrm{dR}}{B}^{n}U(1))\simeq {♭}_{\mathrm{dR}}{B}^{n-1}U(1)$$\Omega (\mathbf{\flat}_{dR}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}_{dR}\mathbf{B}^{n-1} U(1)

(since $♭$ and ${♭}_{\mathrm{dR}}$ are right adjoint (∞,1)-functors – by the discussion at cohesive (∞,1)-topos – and hence commute with the (∞,1)-pullback that defines $\Omega$), we have then the looped pasting diagram of (∞,1)-pullbacks

$$\begin{array}{ccccc}\Omega C\mathrm{Field}(Y{)}_{P_{\mathrm{Spin}}}& \stackrel{\Omega \hat{\chi }}{\to }& H_{\mathrm{flat}}(Y,B^{2}U(1))& \stackrel{\omega }{\to }& *\\ \downarrow & & \downarrow & & \downarrow \\ H(Y,E_8)& \stackrel{2\Omega c_2}{\to }& H(Y,B^{2}U(1))& \stackrel{\mathrm{curv}}{\to }& H(Y,{♭}_{\mathrm{dR}}{B}^{3}U(1))\end{array}.$$\array{ \Omega C Field(Y)_{P_{Spin}} &\stackrel{\Omega \hat \chi}{\to}& \mathbf{H}_{flat}(Y, \mathbf{B}^2 U(1)) &\stackrel{\omega}{\to}& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(Y, E_8) &\stackrel{2\Omega \mathbf{c}_2}{\to}& \mathbf{H}(Y, \mathbf{B}^2 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)) } \,.

Observe that $E_8$ here is a smooth but 0-truncated object: so that

$$H(Y,E_8)\simeq H^0(Y,E_8)=C^{\mathrm{\infty }}(Y,E_8)$$\mathbf{H}(Y, E_8) \simeq H^0(Y, E_8) = C^\infty(Y, E_8)

is the set of smooth functions $Y\to E_8$ (to be thought of as the the set of gauge transformations from the trivial $E_8$-principal bundle on $Y$ to itself).

Presentation by differential form data

In order to compute the $(\mathrm{\infty },1)$-pullback $C\mathrm{Field}(X)$ more explicitly, we follow the discussion at differential string structure, where presentations of this pullback in terms of simplicial presheaves arising from Lie integration is given.

Write now

$$𝔤:={𝔢}_8\times {𝔢}_8\times \mathrm{𝔰𝔬}(10,1)$$\mathfrak{g} := \mathfrak{e}_8 \times \mathfrak{e}_8 \times \mathfrak{so}(10,1)

for the Lie algebra of $G:=E_8\times E_8\times \mathrm{Spin}(10,1)$ and write

$$\mu :={\mu }_{{𝔢}_8}+{\mu }_{{𝔢}_8}-{\mu }_{\mathrm{𝔰𝔬}(10,1)}$$\mu := \mu_{\mathfrak{e}_8} + \mu_{\mathfrak{e}_8} - \mu_{\mathfrak{so}(10,1)}

for the sum of the canonical Lie algebra cocycles in transgression with the respective Killing form invariant polynomials.

Write

$${𝔢}_8\times \mathrm{𝔰𝔬}(10,1)\to 𝔤$$\mathfrak{e}_8 \times \mathfrak{so}(10,1) \to \mathfrak{g}

for the canonical diagonal embedding Write

$$\begin{array}{cccccc}c:=& {\mathrm{cosk}}_3(\exp ({𝔢}_8\times \mathrm{𝔰𝔬}(10,1)))& \stackrel{\Delta }{\to }& {\mathrm{cosk}}_3(\exp (𝔤))& \stackrel{\exp (\mu )}{\to }& B^{3}U(1{)}_c\\ {\downarrow }^{\simeq }\\ BG\end{array}$$\array{ \mathbf{c} := & \mathbf{cosk}_3( \exp(\mathfrak{e}_8 \times \mathfrak{so}(10,1)) ) &\stackrel{\Delta}{\to}& \mathbf{cosk}_3( \exp(\mathfrak{g}) ) & \stackrel{\exp(\mu)}{\to} & \mathbf{B}^3 U(1)_c \\ \downarrow^{\simeq} \\ \mathbf{B}G }

for the corresponding smooth characteristic class. See ∞-Chern-Weil homomorphism for details. By the discussion there we present $\hat{c}$ by

$$\begin{array}{ccc}{\mathrm{cosk}}_3\exp (bℝ\to {𝔤}_{\mu }{)}_{\mathrm{diff}}& \to & B^4{ℝ}_{\mathrm{dR}}\\ {\downarrow }^{\simeq }\\ B{E}_8\end{array}.$$\array{ \mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff} &\to& \mathbf{B}^4 \mathbb{R}_{dR} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}E_8 } \,.

By the discussion at differential string structure we have that the top morphism is a fibration in the global projective model structure on simplicial presheaves $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ (there it is shown that the analogous morphism out of ${\mathrm{cosk}}_3\exp (bℝ\to {𝔢}_8{)}_{\mathrm{ChW}}$ is a fibration, but then so is this one, because the components on the left are the same but with fewer conditions on them, so that the lifts that existed before still exist here).

Over some $U\in$ CartSp and $[k]\in \Delta$ we have that $\exp (b^{ℝ}\to {𝔤}_{\mu }{)}_{\mathrm{diff}}$ is given by differential form data

$${\left(\begin{array}{cc}{F}_A=& dA+\frac{1}{2}[A\wedge A]\\ C_3=& \nabla B:=dB+\mathrm{CS}(A)-H_3\\ {𝒢}_4=& d{H}_3\\ d{F}_A=& -[A\wedge F_A]\\ d{C}_3=& ⟨F_A\wedge F_A⟩-{𝒢}_4\\ d{𝒢}_4=& 0\end{array}\right)}_i\stackrel{\begin{array}{cc}{t}^a& \mapsto A^a\\ r^a& \mapsto F_A^a\\ b& \mapsto B\\ c& \mapsto C_3\\ h& \mapsto H_3\\ g& \mapsto {𝒢}_4\end{array}}{\leftarrow }\mid \left(\begin{array}{cc}{r}^a=& d{t}^a+\frac{1}{2}{C}^a{}_{bc}{t}^b\wedge t^c+\\ c=& db+\mathrm{cs}-h\\ g=& dh\\ d{r}^a=& -C^a{}_{bc}{t}^b\wedge r^a\\ dc=& ⟨-,-⟩-g\\ dg=& 0\end{array}\right)$$\left( \array{ F_A =& d A + \frac{1}{2}[A \wedge A] \\ C_3 =& \nabla B := d B + CS(A) - H_3 \\ \mathcal{G}_4 =& d H_3 \\ d F_A =& - [A \wedge F_A] \\ d C_3 =& \langle F_A \wedge F_A\rangle - \mathcal{G}_4 \\ d \mathcal{G}_4 =& 0 } \right)_i \;\;\;\; \stackrel{ \array{ t^a & \mapsto A^a \\ r^a & \mapsto F^a_A \\ b & \mapsto B \\ c & \mapsto C_3 \\ h & \mapsto H_3 \\ g & \mapsto \mathcal{G}_4 } }{\leftarrow}| \;\;\;\; \left( \array{ r^a =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + \\ c = & d b + cs - h \\ g =& d h \\ d r^a =& - C^a{}_{b c} t^b \wedge r^a \\ d c =& \langle -,-\rangle - g \\ d g =& 0 } \right)

on $U\times {\Delta }^k$. Here, recall, $A$ takes values in $𝔤={𝔢}_8\times 𝔢\times \mathrm{𝔰𝔬}(10,1)$, so that for instance the ${𝒢}_4$-curvature is in detail given by

(1)$${𝒢}_4=d{H}_3=⟨F_{A_{{𝔢}_8}^L}\wedge F_{A_{{𝔢}_8}^L}⟩+⟨F_{A_{{𝔢}_8}^R}\wedge F_{A_{{𝔢}_8}^R}⟩-⟨F_{\omega }\wedge F_{\omega }⟩-d{C}_3,$$\mathcal{G}_4 = d H_3 = \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle + \langle F_{A^R_{\mathfrak{e}_8}} \wedge F_{A^R_{\mathfrak{e}_8}} \rangle - \langle F_{\omega} \wedge F_{\omega} \rangle - d C_3 \,,

where $\omega$ denotes the spin connection.

Let $\{U_i\to X\}$ be a differentiably good open cover. We hit all connected components of $H(X,BG)$ by considering in

$$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](C(U_i),\exp (bℝ\to {𝔤}_{\mu }){)}_{\mathrm{diff}}$$[CartSp^{op}, sSet](C(U_i), \exp(b \mathbb{R} \to \mathfrak{g}_\mu))_{diff}

those cocycles that

• involve genuine $G$-connections (as opposed to the more general pseudo-connections that are also contained);

• have a globally defined $C_3$-form.

Write therefore $(P,\nabla ,C_3)$ for such a cocycle.

For gauge transformations between two such pairs, parameterized by the above form data patchwise on $U\times {\Delta }^1$, the fact that ${𝒢}_4$ vanishes on ${\Delta }^1$ implies the infinitesmal gauge transformation law

$$\frac{d}{dt}C=d_U{\omega }_t+{\iota }_t⟨F_{\hat{A}}\wedge F_{\hat{A}}⟩,$$\frac{d}{d t} C = d_U \omega_t + \iota_t \langle F_{\hat A} \wedge F_{\hat A}\rangle \,,

where $\hat{A}\in {\Omega }^1(U\times {\Delta }^1,{𝔢}_8)$ is the shift of the 1-forms. This integrates to

(2)$$C_2=C_1+d\omega +\mathrm{CS}({\nabla }_1,{\nabla }_2),$$C_2 = C_1 + d \omega + CS(\nabla_1,\nabla_2) \,,

where

• $\omega :={\int }_{{\Delta }^1}{\omega }_t$

• $\mathrm{CS}({\nabla }_1,{\nabla }_2)={\int }_{{\Delta }^1}⟨F_{\hat{\nabla }}\wedge F_{\hat{\nabla }}⟩$ is the relative Chern-Simons form corresponding to the shift of $G$-connection.

Restriction to the boundary

We have seen that $C\mathrm{Field}(Y)$ is the 3-goupoid of those Cech cocycles on $Y$ with coefficients in $\exp (bℝ\to {𝔤}_{\mu }{)}_{\mathrm{diff}}$ such that the curvature 4-form ${𝒢}_4$ has a fixed globally defined value.

Consider the subobject

$$\exp (bℝ-{𝔤}_{\mu }{)}_{\mathrm{diff}}^{C=0}\hookrightarrow \exp (bℝ-{𝔤}_{\mu }{)}_{\mathrm{diff}}$$\exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff}^{C = 0} \hookrightarrow \exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff}

of the simplicial presheaf $\exp (bℝ\to {𝔤}_{\mu })$ on those objects and k-morphisms for which $C=0$.

By the gauge transformation law (2)

$$C_2=C_1+d\omega +\mathrm{CS}(A_1,A_2)$$C_2 = C_1 + d \omega + CS(A_1, A_2)

this means that this picks those morphisms for which the Chern-Simons form vanishes

$$\mathrm{CS}(A_1,A_2)={\int }_{{\Delta }^1}⟨F_A\wedge F_A⟩=0,$$CS(A_1,A_2) = \int_{\Delta^1} \langle F_{A} \wedge F_{A}\rangle = 0 \,,

where $A=A_U+\lambda dt\in {\Omega }^1(U\times {\Delta }^1,𝔤)$ is the 1-form datum (with $t$ the canonical coordinate on the 1-simplex ${\Delta }^1=[0,1]$).

Note

In the literature often the relative Chern-Simons form is considered for “ungauged” paths of connections: for $\lambda =0$ in the above formula, hence for a $𝔤$-valued 1-form on $U\times {\Delta }^1$ with no leg along the simplex (only depending on the simplex coordinate). Here, however, it is crucially important that we consider the general “gauged” paths.

Notice that on the semisimple Lie algebra and compact Lie algebra ${𝔢}_8$ the Killing form $⟨-,-⟩$ is non-degenerate and positive definite (or negative definite, depending on convention). The latter condition means that this integral vanishes precisely if

$${\iota }_{{\partial }_t}⟨F_A\wedge F_A⟩=0.$$\iota_{\partial_t} \langle F_A \wedge F_A \rangle = 0 \,.

This is the case on paths for which ${\iota }_t{F}_A=0$, but this are exactly the paths that induce genuine gauge transformations between $A_1$ and $A_2$, where

$$\frac{d}{dt}A=d_U\lambda +[\lambda ,A].$$\frac{d}{d t} A = d_U \lambda + [\lambda , A] \,.

This means that cocycles with coefficients in this subobject for $C=0$ are cocycles as described at differential string structure, exhibiting the Green-Schwarz mechanism on the heterotic boundary, witnessed by the restriction of the curvature equation (1) to vanishing $C$-field

(3)$$d{H}_3^L=⟨F_{A_{{𝔢}_8}^L}\wedge F_{A_{{𝔢}_8}^L}⟩-\frac{1}{2}⟨F_{\omega }\wedge F_{\omega }⟩$$d H_3^L = \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle - \frac{1}{2}\langle F_{\omega} \wedge F_{\omega} \rangle

Related concepts

• electromagnetic field (”$A$-field”)

• Kalb-Ramond field (”$B$-field”)

• supergravity $C$-field

• twisted differential c-structure

  • differential string structure

  • differential fivebrane structure

  • differential T-duality

Table of branes appearing in supergravity/string theory

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D SuGra/M-theory	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

References

The state-of-the-art in the literature concerning attempts to find the correct mathematical model for the supergravity C-field seems to be

• E. Diaconescu, Dan Freed, Greg Moore, The $M$-theory 3-form and $E_8$-gauge theory (arXiv:hep-th/0312069)

A summary and rview of this is in

• Greg Moore, Anomalies, Gauss laws, and Page charges in M-theory (arXiv:hep-th/0409158)

The discussion in twisted nonabelian differential cohomology is given in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber,

  The moduli 3-stack of the C-field,

  M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory (arXiv:1201.5277)

See also section 4.3.4 of

• Urs Schreiber, differential cohomology in a cohesive topos

A detailed discussion of the quantum anomaly of the supergravity C-field – and its cancellation – is in

• Dan Freed, Greg Moore, Setting the quantum integrand of M-theory , Communications in mathematical physics, Volume 263, Number 1, 89-132, DOI: 10.1007/s00220-005-1482-7 (arXiv:hep-th/0409135)