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 𝔒 8\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, Ο‰=0\omega = 0.



(c 2) dR:BE 8β†’β™­ dRB 4U(1) (\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? H:=\mathbf{H} := Smooth∞Grpd? that presents the Chern-Weil homomorphism? induced by the Killing form? invariant polynomial? on 𝔒 8\mathfrak{e}_8 (see ∞-Chern-Weil homomorphism? for details).

For given smooth manifold? XX (or any other object X∈Smooth∞GrpdX \in Smooth \infty Grpd), let H 4(X)β†’H(X,β™­ dRB 4U(1))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 XX in degree 4.

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

CField Ο‰=0(X) β†’ H 4(X) ↓ ↓ H(X,BE 8) β†’(c s) dR H(X,β™­ dRB 4U(1)). \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


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


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{}_{smooth}.

This in turn is accomplished by presenting (c 2) dR(\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

cosk 3exp(bℝ→(𝔒 8) ΞΌ) diff β†’ exp(b 3ℝ) dR ↓ ≃ (BE 8) c. \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→X}\{U_i \to X\} a differentiably good open cover?, the 3-groupoid in question is the pullback

CField Ο‰=0(X) c β†’ H 4(X) ↓ ↓ [CartSp op,sSet](C(U i),cosk 3exp(bℝ→(𝔒 8) ΞΌ) diff) β†’ [CartSp op,sSet](C(U i),exp(b 3ℝ) dR). \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 8E_8-principal bundle?s (as opposed to more general pseudo-connection?s).

Also, if XX 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 8E_8-coefficients provide a certain twist to this situation.

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

(F A= dA+[A∧A] H 3= βˆ‡B:=dB+CS(A)+C 3 dF A= βˆ’[A∧F A] dH 3= 𝒒 4βˆ’βŸ¨F A∧F A⟩ d𝒒 4= 0) i←t a ↦A a r a ↦F A a b ↦B c ↦C 3 h ↦H 3 g ↦𝒒 4|(dt a= C a bct b∧t c+r a h= db+cs+c dr a= βˆ’C a bct b∧r a dc= gβˆ’βŸ¨βˆ’,βˆ’βŸ© dh= g). \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 𝒒\mathcal{G} are given by tuples (P,βˆ‡,(B i),C)(P,\nabla, (B_i) ,C), where (P,βˆ‡)(P,\nabla) is an E8?-principal bundle? with connection?, and C∈Ω 3(X)C \in \Omega^3(X) is globally defined.

The integrated gauge transformations for CC are parameterized over patches UU by forms

A^∈Ω 1(UΓ—Ξ” 1,𝔀) \hat A \in \Omega^1(U \times \Delta^1, \mathfrak{g})

and 3-forms

C^=C U+C tdt∈Ω 3(UΓ—Ξ” 1). \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 1C_1 to C 2C_2

C 2=C 1+dω+CS(A 1,A 2), C_2 = C_1 + d \omega + CS(A_1, A_2) \,,


  • Ο‰=∫ Ξ” 1C tdt\omega = \int_{\Delta^1} C_t d t

  • CS(A 1,A 2)=∫ Ξ” 1⟨F A^∧F A^⟩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 1A_1 to A 2A_2.

Last revised on April 4, 2011 at 21:32:47. See the history of this page for a list of all contributions to it.