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, see at Models below

Consistency conditions

Several subtle consistency conditions (quantum anomaly cancellation-conditions) have been argued for the charge quantization of the supergravity C-field:

1. Shifted flux quantization condition

2. C-Field tadpole cancellation condition

Shifted flux quantization condition

The shifted C-field flux quantization condition is a charge quantization-condition on the supergravity C-field expected in M-theory. It says that the real cohomology class of the flux density (field strength) differential 4-form $G_4 \in \Omega^4(X)$ on spacetime $X$ becomes integral after shifted by one quarter of the first Pontryagin class, hence the condition that with the shifted 4-flux density defined as

(1)$$\widetilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X}) \;\in\; \Omega^4(X)$$

(for $\nabla_{T X}$ any affine connection on spacetime, in particular the Levi-Civita connection) we have (using the de Rham theorem to translate from de Rham cohomology to real cohomology) that $\widetilde G_4$ represents an integral cohomology-class:

$$[\widetilde G_4] \;\in\; H^4(X, \mathbb{Z}) \overset{H^4(X, \mathbb{Z}\hookrightarrow \mathbb{R})}{\longrightarrow} H^4(X, \mathbb{R}) \,.$$

This condition was originally argued for in (Witten 96a, Witten 96b) as a sufficient condition for ensuring that the prequantum line bundle for the 7d Chern-Simons theory on an M5-brane worldvolume is divisible by 2.

Proposals for encoding this condition by a Wu class-shifted variant of stable ordinary differential cohomology were considered in Hopkins-Singer 02, Diaconescu-Freed-Moore 03, FSS 12.

It turns out that the shifted flux quantization condition on the C-field is naturally implied (FSS1 19b, Prop. 4.12) by the requirement that $G_4$ is the differential form datum underlying, via Sullivan's theorem, a cocycle in unstable J- twisted Cohomotopy in degree 4 (Hypothesis H).

C-Field tadpole cancellation condition

In M-theory compactified on 8-dimensional compact fibers $X^{(8)}$ (see M-theory on 8-manifolds) tadpole cancellation condition for the supergravity C-field has been argued (Sethi-Vafa-Witten 96, Becker-Becker 96, Dasgupta-Mukhi 97) to be the condition

$$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - (\tfrac{1}{2}p_1)^2 \big)[X^{8}] } } \;\;\;\; \overset{!}{\in} \mathbb{Z} \,,$$

where

1. $N_{M2}$ is the net number of M2-branes in the spacetime (whose worldvolume appears as points in $X^{(8)}$);

2. $G_4$ is the field strength/flux of the supergravity C-field

3. $p_1$ is the first Pontryagin class and $p_2$ the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.

If $X^{8}$ has

• Spin(7)-structure (hence in particular if it is a Calabi-Yau manifold, which has $SU(4) =$ Spin(6)-structure)

or

• Sp(2).Sp(1)-structure

then

$$\tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi$$

is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently

(2)$$N_{M2} \;=\; - \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;+\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,,$$

where $\chi[X]$ is the Euler characteristic of $X$.

Models

One proposal for a mathematical model of the C-field is as a cocycle in a Wu class-shifted variant of ordinary differential cohomology in degree 4:

• The DFM model

Another proposal is that the C-field is simply a cocycle in J- twisted Cohomotopy (Hypothesis H):

• C-field charge quantization in J-twisted Cohomotopy

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

$$\pi_i B E_8 = \left\{ \array{ \mathbb{Z} | i = 4 \\ 0 | i \neq 4, i \leq 15 } \right. \,.$$

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

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

Definition

Let $X$ be a smooth manifold of dimension $dim X \lt 15$. 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

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

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 $\omega \in \Omega^3_{\mathbb{Z}}(X)$ with integral periods, subject to the condition that

  $$c_2 -c_1 = CS(\nabla_1,\nabla_2) + \omega \,,$$

  where $CS(\nabla_1, \nabla_2)$ is the relative Chern-Simons form corresponding to the linear path of connections from $\nabla_1$ to $\nabla_2$

• the composition of morphisms

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

  is given by

  $$\omega_1 + \omega_2 + \langle (\nabla_2-\nabla_1)\wedge(\nabla_3-\nabla2) \rangle \,.$$

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 $\mathbf{E}(X)$ as inducing an degree-4 cocycle in ordinary differential cohomology with curvature 4-form

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

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 $\frac{1}{2} tr R_\omega \wedge R_\omega$. See below for more on this.

Claim

The above groupoid has homotopy groups

• $\pi_0 \simeq H^4_{diff}(Y)$

• $\pi_1(-,(\nabla,c)) \simeq H^2(Y, U(1))$ .

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 $\frac{1}{2}tr R^2$-shifted differential characters.

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

Orientation and fractional classes

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

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

and second Chern class

$$a := c_2(P)$$

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

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

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 (DFM 03, section 12.1, FSS 12)

Restriction to the boundary

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

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

Description in $\infty$-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 $2 a - \lambda$

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

$$\array{ \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\to& H^4_{dR}(Y) \\ \downarrow && \downarrow \\ \mathbf{H}(Y, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) }$$

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 $H^4_{dR}(Y)$ with the more familiar $\Omega^4_{cl}(Y)$. )

We consider now the analog of this definition for the universal curvature form on $\mathbf{B}^3 U(1)$ 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 $(\infty,1)$-pullback we tentatively call $C Field(Y)$, 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 $C Field(Y) \in$ ∞Grpd be the (∞,1)-pullback

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

Note

By its intrinsic definition we have that the differential characteristic class $(\mathbf{c}_2)_{dR}$ is the composite

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

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

Therefore we may either compute the (∞,1)-pullback in def. 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 $(\infty,1)$-pullback from def. may be decomposed into two consecutive pullbacks of the form

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

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

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

that maps $C$-field configurations to ordinary differential cohomology in degree 4, whose curvature $\omega(\hat \chi)$ is the image $(\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_2)_{dR} := curv(2\mathbf{c}_2 - \frac{1}{2}\mathbf{p}_2)$ 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 $C Field(Y)$.

Proposition

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

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

and

$\pi_0 C Field_{P_{Spin}}(Y)$ is the group of pairs $([c], f) \in H^2(X, U(1)) \times C^\infty(X, E_8)$ 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

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

where the top right square is discussed at cohesive (∞,1)-topos – Differential cohomology. By the discussion at smooth ∞-groupoid – Flat cohomology we have that $\pi_0 \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) \simeq H^3(Y,U(1))$, where on the right we have ordinary cohomology (for instance realized as singular cohomology). Finally observe that $\pi_0 \mathbf{H}(Y, \mathbf{E}_8) \simeq \pi_0 \mathbf{H}(Y.\mathbf{B}^3 U(1))$, by the above remark. Therefore after passing to connected components by applying $\pi_0(-)$ we get on cohomology

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

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

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

this therefore implies also the short exact sequence

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

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

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

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

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

and

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

(since $\mathbf{\flat}$ and $\mathbf{\flat}_{dR}$ 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

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

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

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

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 $(\infty,1)$-pullback $C Field(X)$ 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

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

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

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

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

Write

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

for the canonical diagonal embedding Write

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

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

$$\array{ \mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff} &\to& \mathbf{B}^4 \mathbb{R}_{dR} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}E_8 } \,.$$

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 $[CartSp^{op}, sSet]_{proj}$ (there it is shown that the analogous morphism out of $\mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{e}_8)_{ChW}$ 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 $\exp(b^\mathbb{R} \to \mathfrak{g}_\mu)_{diff}$ is given by differential form data

$$\left( \array{ F_A =& d A + \frac{1}{2}[A \wedge A] \\ C_3 =& \nabla B := d B + CS(A) - H_3 \\ \mathcal{G}_4 =& d H_3 \\ d F_A =& - [A \wedge F_A] \\ d C_3 =& \langle F_A \wedge F_A\rangle - \mathcal{G}_4 \\ d \mathcal{G}_4 =& 0 } \right)_i \;\;\;\; \stackrel{ \array{ t^a & \mapsto A^a \\ r^a & \mapsto F^a_A \\ b & \mapsto B \\ c & \mapsto C_3 \\ h & \mapsto H_3 \\ g & \mapsto \mathcal{G}_4 } }{\leftarrow}| \;\;\;\; \left( \array{ r^a =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + \\ c = & d b + cs - h \\ g =& d h \\ d r^a =& - C^a{}_{b c} t^b \wedge r^a \\ d c =& \langle -,-\rangle - g \\ d g =& 0 } \right)$$

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

(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 $\mathbf{H}(X, \mathbf{B}G)$ by considering in

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

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 $\mathcal{G}_4$ vanishes on $\Delta^1$ implies the infinitesmal gauge transformation law

$$\frac{d}{d t} C = d_U \omega_t + \iota_t \langle F_{\hat A} \wedge F_{\hat A}\rangle \,,$$

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

(4)$$C_2 = C_1 + d \omega + CS(\nabla_1,\nabla_2) \,,$$

where

• $\omega := \int_{\Delta^1} \omega_t$

• $CS(\nabla_1, \nabla_2) = \int_{\Delta^1} \langle F_{\hat \nabla} \wedge F_{\hat \nabla}\rangle$ is the relative Chern-Simons form corresponding to the shift of $G$-connection.

Restriction to the boundary

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

Consider the subobject

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

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

By the gauge transformation law (4)

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

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

$$CS(A_1,A_2) = \int_{\Delta^1} \langle F_{A} \wedge F_{A}\rangle = 0 \,,$$

where $A = A_U + \lambda d t \in \Omega^1(U \times \Delta^1, \mathfrak{g})$ 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 $\mathfrak{g}$-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 $\mathfrak{e}_8$ the Killing form $\langle -,-\rangle$ is non-degenerate and positive definite (or negative definite, depending on convention). The latter condition means that this integral vanishes precisely if

$$\iota_{\partial_t} \langle F_A \wedge F_A \rangle = 0 \,.$$

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

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

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 (3) to vanishing $C$-field

$$d H_3^L = \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle - \frac{1}{2}\langle F_{\omega} \wedge F_{\omega} \rangle$$

Model by twisted Cohomotopy

It turns out that the shifted C-field flux quantization condition is naturally implied (FSS1 19b, Prop. 4.12) by the requirement that $G_4$ is the differential form datum underlying, via Sullivan's theorem, a cocycle in unstable J- twisted Cohomotopy in degree 4 (Hypothesis H).

Related concepts

• electromagnetic field (“$A$-field”)

• Kalb-Ramond field (“$B$-field”)

• supergravity $C$-field

  • C-field tadpole cancellation

• twisted differential c-structure

  • differential string structure

  • differential fivebrane structure

  • differential T-duality

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

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	$\,$	$\,$
D(-2)-brane	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
D8-brane	$\,$	$\,$
$(D = 2n+1)$	type IIB	$\,$	$\,$
D(-1)-brane	$\,$	$\,$	$\,$
D1-brane	$\,$	$\,$	2d CFT with BH entropy
D3-brane	$\,$	$\,$	N=4 D=4 super Yang-Mills theory
D5-brane	$\,$	$\,$	$\,$
D7-brane	$\,$	$\,$	$\,$
D9-brane	$\,$	$\,$	$\,$
(p,q)-string	$\,$	$\,$	$\,$
(D25-brane)	(bosonic string theory)
NS-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
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
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G₂-manifold
topological M5-brane	$\,$	C6-field on G₂-manifold
S-brane
SM2-brane, membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

References

General

The C-field in D=11 supergravity originates as the $A_{\mu\nu\rho}$-field in

• Eugene Cremmer, Bernard Julia, Joël Scherk, Supergravity in theory in 11 dimensions, Phys. Lett. 76B (1978) 409 (doi:10.1016/0370-2693(78)90894-8)

Re-derivation in the D'Auria-Fré formulation of supergravity:

• Riccardo D'Auria, Pietro Fré, (5.11b) of: Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (doi:10.1016/0550-3213(82)90376-5, errata)

Review in:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, III.8.53 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• André Miemiec, Igor Schnakenburg, Section 3.1.3 of: Basics of M-Theory, Fortsch. Phys. 54 (2006) 5-72 (arXiv:hep-th/0509137, doi:10.1002/prop.200510256)

The shifted C-field flux quantization condition was originally proposed in

• Edward Witten, On Flux Quantization In M-Theory And The Effective Action, J. Geom. Phys. 22:1-13, 1997 (arXiv:hep-th/9609122)

• Edward Witten, Five-Brane Effective Action In M-Theory, J. Geom. Phys. 22:103-133, 1997 (arXiv:hep-th/9610234)

Proposals to model the condition by a Wu class-shifted variant of ordinary differential cohomology:

• Duiliu-Emanuel Diaconescu, Dan Freed, Greg Moore, The $M$-theory 3-form and $E_8$-gauge theory, chapter in Haynes Miller, Douglas Ravenel (eds.) Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 [arXiv:hep-th/0312069, doi:10.1017/CBO9780511721489]

picked up e.g. in

• Ron Donagi, Martin Wijnholt: The M-Theory Three-Form and Singular Geometries [arXiv:2310.05838]

A related model of the C-field in terms of nonabelian bundle 2-gerbes:

• Paolo Aschieri, Branislav Jurčo, Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory, JHEP 0410:068 (2004) [arXiv:hep-th/0409200, doi:10.1088/1126-6708/2004/10/068]

Further discussion ofthe quantum anomaly of the supergravity C-field, and its cancellation:

• Dan Freed, Greg Moore, Setting the quantum integrand of M-theory, Communications in Mathematical Physics 263 1 (2006) 89-132, [arXiv:hep-th/0409135, doi:10.1007/s00220-005-1482-7]

A summary and review of this:

• 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, Communications in Mathematical Physics 333 1 (2015) 117-151, [arXiv:1202.2455, doi:10.1007/s00220-014-2228-1]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321 (arXiv:1201.5277, doi:10.4310/ATMP.2014.v18.n2.a1)

Discussion with Dirac charge quantization of the C-field in twisted Cohomotopy (Hypothesis H):

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M-theory anomaly cancellation, Communications in Mathematical Physics 377 (2020) 1961-2025 [arXiv:1904.10207, doi:10.1007/s00220-020-03707-2]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization, Comm. Math. Phys. 384 (2021) 403–432 [arXiv:1906.07417, doi:10.1007/s00220-021-03951-0]

surveyed in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §12 of: The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalized, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]

and in the generality of orbifold spacetimes, in equivariant Cohomotopy:

• Hisham Sati, Urs Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation, J. Geometry and Physics, 156 (2020) 103775 [arXiv:1909.12277, doi:10.1016/j.geomphys.2020.103775]

• Hisham Sati, Urs Schreiber, pp. 99 of: Proper Orbifold Cohomology [arXiv:2008.01101]

Discussion of the dual 6-form field to the 3-form C-field (required notably in the context of exceptional generalized geometry):

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

• Eric Bergshoeff, Mees de Roo, Olaf Hohm, Can dual gravity be reconciled with E11?, Phys. Lett. B 675 (2009) 371-376 [arXiv:0903.4384]

Discussion of discrete torsion (orbifold equivariance) for circle 3-bundles describing the supergravity C-field is discussed in

• Eric Sharpe, Analogues of Discrete Torsion for the M-Theory Three-Form, Phys.Rev. D68 (2003) 126004 (arXiv:hep-th/0008170)

• Shigenori Seki, Discrete Torsion and Branes in M-theory from Mathematical Viewpoint, Nucl.Phys. B606 (2001) 689-698 (arXiv:hep-th/0103117)

and applied to discussion of black M2-brane worldvolume field theory (ABJM model) in

• Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, Fractional M2-branes, JHEP 0811:043,2008 (arXiv:0807.4924)

• Mauricio Romo, Aspects of ABJM orbifolds with discrete torsion, J. High Energ. Phys. (2011) 2011 (arXiv:1011.4733)

Duality-symmetric formulation

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

Supergravity C-Field gauge algebra

Identifying the super-graded gauge algebra of the C-field in D=11 supergravity (with non-trivial super Lie bracket $[v_3, v_3] = -v_6$):

• Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Equation (2.6) 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]

• I. V. Lavrinenko, H. Lu, Christopher N. Pope, Kellogg S. Stelle, (3.4) in: Superdualities, Brane Tensions and Massive IIA/IIB Duality, Nucl. Phys. B 555 (1999) 201-227 [doi:10.1016/S0550-3213(99)00307-7, arXiv:hep-th/9903057]

• Jussi Kalkkinen, Kellogg S. Stelle, (75) of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 [doi:10.1016/S0393-0440(03)00027-5, arXiv:hep-th/0212081]

• Igor A. Bandos, Alexei J. Nurmagambetov, Dmitri P. Sorokin, (86) in: 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]

Identification as an $L_\infty$-algebra (a dg-Lie algebra, in this case):

• Hisham Sati, (4.9) in: Geometric and topological structures related to M-branes, in Superstrings, Geometry, Topology, and $C^\ast$-algebras, Proc. Symp. Pure Math. 81 (2010) 181-236 [ams:pspum/081, arXiv:1001.5020]

and identificatoin with the rational Whitehead $L_\infty$-algebra (the rational Quillen model) of the 4-sphere (cf. Hypothesis H):

• Hisham Sati, Alexander Voronov, (13) in: Mysterious Triality and M-Theory [arXiv:2212.13968]

• Hisham Sati, Urs Schreiber, (22) in: Flux Quantization on Phase Space [arXiv:2312.12517]