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.
In (DFM, section 3) the following definition is considered and argued to be a good model of the supergravity $C$-field.
The homotopy groups of the classifying space $B E_8$ of the Lie group E8 satisfy
Therefore for $X$ a manifold of dimension $dim X \leq 15$ there is a canonical morphism
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
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
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
is given by
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
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.
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.
Ordinarily, given a $Spin \times E_8$-bundle $P \to Y$ with first fractional Pontryagin class
and second Chern class
the $C$-field is supposed to have a curvature class in de Rham cohomology given by
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.
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.
Some remarks on ways to regard the $C$-field from the point of view of ∞-Chern-Weil theory.
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 $\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))$ that is the (∞,1)-pullback
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.
For $Y \in$ Smooth∞Grpd, let $C Field(Y) \in$ ∞Grpd be the (∞,1)-pullback
By its intrinsic definition we have that the differential characteristic class $(\mathbf{c}_2)_{dR}$ is the composite
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. 2 directly, or in two consecutive steps. Both methods lead to their insights.
In
we consider general abstract consequences of the above definition, mainly making use of the factorization. In
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
we comment how genuine $E_8$-connections may appear inside the second presentation of the $C$-model.
This implies by the pasting law for (∞,1)-pullbacks that the $(\infty,1)$-pullback from def. 2 may be decomposed into two consecutive pullbacks of the form
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.
By the above there exists canonically a morphism
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 $C Field(Y)$.
Over a fixed $Spin$-principal bundle $P_{Spin}$ we have a short exact sequence (of pointed sets)
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 $[c]$.
Notice that we have the pasting diagram of (∞,1)-pullbacks
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
by reasoning as discussed at fiber sequence. In parallel to the familiar short exact sequence for ordinary differential cohomology
this therefore implies also the short exact sequence
Next we redo the entire discussion after applying the loop space object-construction to everything. Using that
on general grounds (see fiber sequence for details) and that also
and
(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
Observe that $E_8$ here is a smooth but 0-truncated object: so that
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).
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
for the Lie algebra of $G := E_8 \times E_8 \times Spin(10,1)$ and write
for the sum of the canonical Lie algebra cocycles in transgression with the respective Killing form invariant polynomials.
Write
for the canonical diagonal embedding Write
for the corresponding smooth characteristic class. See ∞-Chern-Weil homomorphism for details. By the discussion there we present $\hat \mathbf{c}$ by
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
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
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
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
where $\hat A\in \Omega^1(U \times \Delta^1, \mathfrak{e}_8)$ is the shift of the 1-forms. This integrates to
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.
We have seen that $C Field(Y)$ is the 3-goupoid 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
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 (2)
this means that this picks those morphisms for which the Chern-Simons form vanishes
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 $\Delta^1 = [0,1]$).
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
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
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
electromagnetic field (”$A$-field”)
Kalb-Ramond field (”$B$-field”)
supergravity $C$-field
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The state-of-the-art in the literature concerning attempts to find the correct mathematical model for the supergravity C-field seems to be
A summary and rview of this is in
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
A detailed discussion of the quantum anomaly of the supergravity C-field – and its cancellation – is in