nLab differential fivebrane structure

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

$$\frac{1}{6} p_2 : B String \to B^8 \mathbb{Z}$$

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

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

– the smooth second fractional Pontryagin class.

The induced morphism on cocycle ∞-groupoids

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

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

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

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

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

The 7-groupoid of differential fivebrane structures is the homotopy fiber of this refinement $\frac{1}{6}\hat \mathbf{p}_2$ 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 $X$ is characterized by a tuple consisting of

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

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

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

$$\lambda : \frac{1}{6}\hat \mathbf{p}_2(\nabla) \stackrel{\simeq}{\to} (0,H_7) \,.$$

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

Definition

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

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

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

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

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

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

$$\frac{1}{6}p_2 : B String \to B^8 \mathbb{Z}$$

in Top. Moreover, the corresponding refined differential characteristic class

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

is in cohomology the corresponding refined Chern-Weil homomorphism

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

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

Definition

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

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

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

where the right vertical morphism is a choice of (any) one point in each connected component (differential cohomology class) of the cocycle ∞-groupoid $\mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ (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 $L_\infty$-Cech cocycles

We use the presentation of the (∞,1)-topos Smooth∞Grpd (as described there) by the local model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ 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 $\frac{1}{6} \hat \mathbf{p}_2$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ by the top morphism of simplicial presheaves in

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

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 $\frac{1}{6}\hat \mathbf{p}_2$ we now find a resolution of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. 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

(…)

Related concepts

Whitehead tower of the orthogonal group

• orientation

• spin structure

• string structure

• fivebrane structure

• twisted differential c-structure

  • differential string structure

  • supergravity C-field

  • differential fivebrane structure

  • differential T-duality

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential string and fivebrane structures (web)

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

A general discussion is at

• Urs Schreiber, differential cohomology in a cohesive topos

in section 4.2.

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