Schreiber supergravity C-field

Here we describe a natural construction in ∞-Chern-Weil theory? which assigns to every smooth manifold? 3-groupoid? of structures consisting of pairs of a circle 3-bundle? and an E8?-principal bundle? that differentially twist each other via the refined Chern-Weil homomorphism? induced by the Killing form? on $\mathfrak{e}_8$. We show that the 1-truncation? of this 3-groupoid is the 1-groupoid of such structures that is considered in (DFM).

See the discussion at differential string structure? for the moment

We first consider the case where the spin connection? vanishes, $\omega = 0$.

Definition

Let

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

be a morphism (unique up to equivalence) in the cohesive (∞,1)-topos? $\mathbf{H} :=$ Smooth∞Grpd? that presents the Chern-Weil homomorphism? induced by the Killing form? invariant polynomial? on $\mathfrak{e}_8$ (see ∞-Chern-Weil homomorphism? for details).

For given smooth manifold? $X$ (or any other object $X \in Smooth \infty Grpd$), let $H^4(X) \to \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1))$ the inclusion of the set of connected components into the intrinsic de Rham cocycle ∞-groupoid of $X$ in degree 4.

The 3-groupoid of $C$-fields on $X$ for vanishing spin connection is the 3-groupoid? $C Field_{\omega = 0}(X)$ defined by the (∞,1)-pullback?

$$\array{ C Field_{\omega = 0}(X) &\to& H^4(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}E_8) &\stackrel{(\mathbf{c}_s)_{dR}}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,.$$

under construction

Claim

The 1-truncation? of this 3-groupoid is equivalent to the DFM-model.

Proof

We compute the (∞,1)-pullback? by a homotopy pullback? in the presentation? of Smooth∞Grpd? by the projective local model structure on simplicial presheaves? over the site? CartSp?${}_{smooth}$.

This in turn is accomplished by presenting $(\mathbf{c}_2)_{dR}$ by a fibration and then computing th ordinary pullback? of simplicial presheaves? along this fibration.

A suitable fibration presentation for this morphism is discussed in some detail at differential string structure?. In the notation of the discussion there, it is given by

$$\array{ \mathbf{cosk}_3 \exp(b \mathbb{R} \to (\mathfrak{e}_8)_\mu )_{diff} &\to& \exp(b^3 \mathbb{R})_{dR} \\ \downarrow^{\mathrlap{\simeq}} \\ (\mathbf{B}E_8)_c } \,.$$

Then for $\{U_i \to X\}$ a differentiably good open cover?, the 3-groupoid in question is the pullback

$$\array{ C Field_{\omega = 0}(X)_c &\to& H^4(X) \\ \downarrow && \downarrow \\ [CartSp^{op}, sSet](C(U_i), \mathbf{cosk}_3 \exp(b \mathbb{R} \to (\mathfrak{e}_8)_\mu )_{diff}) &\to& [CartSp^{op}, sSet](C(U_i),\exp(b^3 \mathbb{R})_{dR}) } \,.$$

Here we may up to equivalence restrict the bottom left Kan complex to those objects that correspond to genuine connections? on $E_8$-principal bundle?s (as opposed to more general pseudo-connection?s).

Also, if $X$ is a smooth manifold?, we may assume that the right vertical morphism takes values in de Rham hypercohomology?-cocycles that are given by globally defined 4-forms.

The computation of the pullback is then quite analogous to the discussion at circle n-bundle with connection? where a very similar pullback yields ordinary differential cohomology?. The difference to the discussion there is that here the $E_8$-coefficients provide a certain twist to this situation.

(Also as discussed there, once we are in the specific presentation $[CartSp^{op}, sSet]$ we can replace $H^4(X)$ by all of $\Omega^4_{cl}(X)$, if desired.)

By the discussion at differential string structure? we have that the deta in the bottom left is locally given by differential form data of the form

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

First of all we find that vertices of the pullback for given curvature 4-form $\mathcal{G}$ are given by tuples $(P,\nabla, (B_i) ,C)$, where $(P,\nabla)$ is an E8?-principal bundle? with connection?, and $C \in \Omega^3(X)$ is globally defined.

The integrated gauge transformations for $C$ are parameterized over patches $U$ by forms

$$\hat A \in \Omega^1(U \times \Delta^1, \mathfrak{g})$$

and 3-forms

$$\hat C = C_U + C_t d t \in \Omega^3(U \times \Delta^1) \,.$$

The horizontality condition on the 4-form curvature yields for the gauge transformation from $C_1$ to $C_2$

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

where

• $\omega = \int_{\Delta^1} C_t d t$

• $CS(A_1, A_2) = \int_{\Delta^1} \langle F_{\hat A} \wedge F_{\hat A}\rangle$ is the relative Chern-Simons form? for the linear path of connections from $A_1$ to $A_2$.