nLab
differential fivebrane structure

Context

String theory

Differential cohomology

\infty-Chern-Weil theory

Contents

Idea

Where a fivebrane structure is a trivialization of a class in integral cohomology, a differential fivebrane structure is the trivialization of this class refined to ordinary differential cohomology:

the second fractional Pontryagin class

16p 2:BStringB 8 \frac{1}{6} p_2 : B String \to B^8 \mathbb{Z}

in the (∞,1)-topos ∞Grpd \simeq Top has a refinement to H=\mathbf{H} = Smooth∞Grpd of the form

16p 2:BStringB 7U(1) \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)

– the smooth second fractional Pontryagin class.

The induced morphism on cocycle ∞-groupoids

16p 2:H(X,BString)H(X,B 7U(1)) \frac{1}{6}\mathbf{p}_2 : \mathbf{H}(X,\mathbf{B} String) \stackrel{}{\to} \mathbf{H}(X,\mathbf{B}^7 U(1))

sends a string 2-group-principal 2-bundle PP to its corresponding Chern-Simons circle 7-bundle 16p 2(P)\frac{1}{6}\mathbf{p}_2(P).

A choice of trivialization of 16p 2(P)\frac{1}{6}p_2(P) is a fivebrane structure. The 6-groupoid of smooth fivebrane structures is the homotopy fiber of 16p 2\frac{1}{6}\mathbf{p}_2 over the trivial circle 7-bundle.

By Chern-Weil theory in Smooth∞Grpd this morphism may be further refined to a differential characteristic class 16p^ 2\frac{1}{6}\hat \mathbf{p}_2 that lands in the ordinary differential cohomology H diff(X,B 7U(1))\mathbf{H}_{diff}(X, \mathbf{B}^7 U(1)), classifying circle 7-bundles with connection

16p^ 2:H conn(X,BString)H diff(X,B 7U(1)) \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(X,\mathbf{B} String) \stackrel{}{\to} \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))

The 7-groupoid of differential fivebrane structures is the homotopy fiber of this refinement 16p^ 2\frac{1}{6}\hat \mathbf{p}_2 over the trivial circle 7-bundle with trivial connection or more generally over the trivial circle 7-bundles with possibly non-trivial connection.

Such a differential fivebrane structure over a smooth manifold XX is characterized by a tuple consisting of

  1. a 2-connection \nabla on a String-principal 2-bundle on XX;

  2. a choice of trivial circle 7-bundle with connection (0,H 7)(0, H_7), hence a differential 7-form H 7Ω 7(X)H_7 \in \Omega^7(X);

  3. a choice of equivalence λ\lambda of the Chern-Simons circle 7-bundle with connection 16p^ 2()\frac{1}{6}\hat\mathbf{p}_2(\nabla) of \nabla with this chosen 7-bundle

λ:16p^ 2()(0,H 7). \lambda : \frac{1}{6}\hat \mathbf{p}_2(\nabla) \stackrel{\simeq}{\to} (0,H_7) \,.

More generally, one can consider the homotopy fibers of 16p^ 2\frac{1}{6}\hat \mathbf{p}_2 over arbitrary circle 7-bundles with connection 𝒢^ 8H diff 8(X,B 3U(1))\hat \mathcal{G}_8 \in \mathbf{H}_{diff}^8(X, \mathbf{B}^3 U(1)) and hence replace (0,H 7)(0,H_7) in the above with 𝒢^ 8\hat \mathcal{G}_8. According to the general notion of twisted cohomology, these may be thought of as twisted differential string structures, where the class [𝒢 8]H diff 8(X)[\mathcal{G}_8] \in H^8_{diff}(X) is the twist.

Definition

We will assume that the reader is familiar with basics of the discussion at Smooth∞Grpd. We often write H:=SmoothGrpd\mathbf{H} := Smooth \infty Grpd for short.

Let String(n)String(n) \in Smooth∞Grpd be the smooth String 2-group, for some n>6n \gt 6,1 regarded as a Lie 2-group and thus canonically as an ∞-group object in Smooth∞Grpd. We shall notationally suppress the nn in the following. Write BString\mathbf{B}String for its delooping of SpinSpin in Smooth∞Grpd. (See the discussion here). Let moreover B 6U(1)SmoothGrpd\mathbf{B}^6 U(1) \in Smooth \infty Grpd be the circle Lie 7-group and B 7U(1)\mathbf{B}^7 U(1) its delooping.

At Chern-Weil theory in Smooth∞Grpd the following statement is proven (FSS).

Proposition

The image under Lie integration of the canonical Lie algebra 7-cocycle

μ=,[,],[,],[,]:𝔰𝔬b 6 \mu = \langle -,[-,-], [-,-], [-,-]\rangle : \mathfrak{so} \to b^6 \mathbb{R}

on the semisimple Lie algebra 𝔰𝔬\mathfrak{so} of the Spin group – the special orthogonal Lie algebra – is a morphism in Smooth∞Grpd of the form

16p 2:BStringB 7U(1) \frac{1}{6} \mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)

whose image under the the fundamental ∞-groupoid (∞,1)-functor/ geometric realization Π:SmoothGrpd\Pi : Smooth \infty Grpd \to ∞Grpd is the ordinary second fractional Pontryagin class

16p 2:BStringB 8 \frac{1}{6}p_2 : B String \to B^8 \mathbb{Z}

in Top. Moreover, the corresponding refined differential characteristic class

16p^ 2:H conn(,BString)H diff(,B 7U(1)) \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(-,\mathbf{B}String) \to \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1))

is in cohomology the corresponding refined Chern-Weil homomorphism

[16p^ String]:H Smooth 1(X,String)H diff 8(X) [\frac{1}{6}\hat \mathbf{p}_String] : H^1_{Smooth}(X,String) \to H_{diff}^8(X)

with values in ordinary differential cohomology that corresponds to the second Killing form invariant polynomial ,,,\langle - , - ,-,-\rangle on 𝔰𝔬\mathfrak{so}.

Definition

For any XX \in Smooth∞Grpd, the 6-groupoid of differential fivebrane-structures on XXFivebrane diff(X)Fivebrane_{diff}(X) – is the homotopy fiber of 16p^ 2(X)\frac{1}{6}\hat \mathbf{p}_2(X) over the trivial differential cocycle.

More generally (see twisted cohomology) the 6-groupoid of twisted differential fivebrane structures is the (∞,1)-pullback Fivebrane diff,tw(X)Fivebrane_{diff,tw}(X) in

Fivebrane diff,tw(X) H diff 8(X) H conn(X,BString) 16p^ 2 H diff(X,B 7U(1)), \array{ Fivebrane_{diff,tw}(X) &\to& H_{diff}^8(X) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}String) &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) } \,,

where the right vertical morphism is a choice of (any) one point in each connected component (differential cohomology class) of the cocycle ∞-groupoid H diff(X,B 7U(1))\mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) (the homotopy type of the (∞,1)-pullback is independent of this choice).

Remark

In terms of local ∞-Lie algebra valued differential forms data this has been considered in (SSSIII), as we shall discuss below.

Construction in terms of L L_\infty-Cech cocycles

We use the presentation of the (∞,1)-topos Smooth∞Grpd (as described there) by the local model structure on simplicial presheaves [CartSp smooth op,sSet] proj,loc[CartSp_{smooth}^{op}, sSet]_{proj,loc} to give an explicit construction of twisted differential fivebrane structures in terms of Cech-cocycles with coefficients in ∞-Lie algebra valued differential forms.

Recall the following fact from Chern-Weil theory in Smooth∞Grpd (FSS).

Proposition

The differential second fractional Pontryagin class 16p^ 2\frac{1}{6} \hat \mathbf{p}_2 is presented in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj} by the top morphism of simplicial presheaves in

cosk 7exp(𝔰𝔬) ChW,smp exp(μ,cs) B 7/ ChW,smp cosk 7exp(𝔰𝔬) diff,smp exp(μ,cs) B 7/ smp BString c. \array{ \mathbf{cosk}_7\exp(\mathfrak{so})_{ChW,smp} &\stackrel{\exp(\mu, cs)}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{ChW,smp} \\ \downarrow && \downarrow \\ \mathbf{cosk}_7\exp(\mathfrak{so})_{diff,smp} &\stackrel{\exp(\mu, cs)}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{smp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}String_{c} } \,.

Here the middle morphism is the direct Lie integration of the L-∞ algebra cocycle while the top morphisms is its restriction to coefficients for ∞-connections.

In order to compute the homotopy fibers of 16p^ 2\frac{1}{6}\hat \mathbf{p}_2 we now find a resolution of this morphism exp(μ,cs)\exp(\mu,cs) by a fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}. By the fact that this is a simplicial model category then also the hom of any cofibrant object into this morphism, computing the cocycle \infty-groupoids, is a fibration, and therefore, by the general discussion at homotopy pullback, we obtain the homotopy fibers as the ordinary fibers of this fibration.

Presentation of the differential class by a fibration

(…)

The discussion is analogous to that at differential string structure. To be filled in

(…)

Whitehead tower of the orthogonal group

References

The local data for the ∞-Lie algebra valued differential forms for the description of twisted differential fivebrane structures as above was given in

The full Cech-Deligne cocycles induced by this and their homotopy fibers were discussed in

A general discussion is at

in section 4.2.


  1. for n=3,4,5,6n=3,4,5,6 String(n)String(n) is not 6-connected, so one must take extra care with the intervening stages in the Whitehead tower.

Revised on November 27, 2014 06:45:19 by David Roberts (129.127.210.201)