Schreiber differential twisted cohomology

This entry discusses the definition of differential twisted cohomology classifying connections on twisted principal $\infty$-bundles.

We list examples identifying gauge fields in string theory subject to a Green-Schwarz mechanism with cocycles in differential twisted cohomology.

Idea

Differential twisted cohomology is the pairing of the notions of

• twisted cohomology

with

• differential cohomology? .

Recalling that

• a cocycle in differential cohomology? is to the underlying bare cocycle as a connection on an ∞-bundle is to a principal ∞-bundle;

• a cocycle in twisted cohomology is to an ordinary cocycle as a twisted ∞-bundle is to a principal ∞-bundle

we expect cocycles in differential twisted cohomology to classify twisted principal $\infty$-bundles with connection.

The definition of differential twisted cohomology is obvious when we remember that for $A \to B \to C$ a fibration sequence and $c : X \to C$ a given twisting cocycle, a $c$-twisted $A$-cocycle is nothing but a $\hat B$-cocycle that satisfies a certain constraint. Therefore differential twisted $A$-cocycles are just differential $B$-cocycles satisfying a constraint.

More formally, we can restate this using the fact that differential cohomology? itself is defined as a kind of twisted cohomology, where the twist is given by curvature characteristic classes.

Therefore differential twisted cohomology may be simply understood as an example of bi-twisted cohomology.

Definition

Definition (bi-twisted cohomology)

Fix some (∞,1)-topos $\mathbf{H}$. Consider two fibration sequences

$$A \to B \to C$$

and

$$A' \to B \to C'$$

with the same object $B$ in the middle. Then given a $C$- and a $C'$-cocycle $c : X \to C$ and $c' : X \to C'$, respectively, we say that the set of connected components

$$H_{[c,c']}(X,A \times_{B} A') := \pi_0 \mathbf{H}_{[c,c']}(X,A \times_{B} A')$$

of the homotopy pullback in

$$\array{ \mathbf{H}_{[c,c']}(X,A \times_{B} A') &\to& {*} \\ \downarrow && \downarrow^{c \times c'} \\ \mathbf{H}(X,B) &\to& \mathbf{H}(X,C \times C') }$$

is the $[c,c']$-bitwisted cohomology of $X$.

Differential twisted cohomology is an example of this as follows:

Definition (differential twisted cohomology)

Fix some (∞,1)-topos $\mathbf{H}$.

Consider a fibration sequences

$$A \to B \to C \,.$$

As at differential cohomology? assume first for simplicity that $B$ is once deloopable (semi-abelian). Then according to the discussion at differential cohomology? it sits also in the fibration sequence

$$B_{flat} \to B \to \mathbf{B} B_{dR} \,.$$

(Recall the definition of $B_flat$ from path ∞-groupoid and the definition of $B_{dR}$ from nonabelian deRham cohomology.)

Then given a $C$-cocycle $c : X \to C$ and a curvature characteristic class $P : X \to \mathbf{B}B_{dR}$, we say that the differential $[c]$-twisted cohomology with curvature class $[P]$ is the $[c,P]$-bitwisted cohomology $H_{[c,P]}(X, A \times_{B} B_{flat})$ of $X$.

Examples (theorems)

Fix a smooth (∞,1)-topos $\mathbf{H}$ of Lie infinity-groupoid and consider the ∞-Lie theory that it provides.

In this context we have smooth versions of the String- and the Fivebrane fibration sequence

$$\mathbf{B} String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$$

and

$$\mathbf{B} Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1) \,.$$

For a given twisting class

$$\frac{1}{2}p_1(T X) : X \to \mathbf{B}Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$$

and

$$\frac{1}{6}p_1(T X) : X \to \mathbf{B}String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)$$

we determine the local differential form data and twisted Bianchi identity of differential refinements of the corresponding twisted $String(n)$- and twisted $Fivebrane(n)$-cocycles, respectively.

Theorem

• The local differential form data of a twisted $String(n)$-bundle with connection is (see section 5.2 of 5twist).

• The local differential form data of a twisted $Fivebrane(n)$-bundle with connection is (see second part of section 5.4 of 5twist).

Remark

This is the differential form data known from the Green-Schwarz mechanism and dual Green-Schwarz mechanism.

Proof sketch

We consider the case of differential twisted $String(n)$-cocycles. The $Fivebrane$-case works entirely analogously.

The crucial point to notice is that the twisting morphism

$$\frac{1}{2}p_1 : \mathbf{B} String(n) \to \mathbf{B}^3(U(1))$$

may be modeled in the model structure on simplicial presheaves $SPSh(C)_{loc}$ that models our $(\infty,1)$-category $\mathbf{H}$ of Lie ∞-groupoid by the span of crossed complexes

$$(1 \to 1 \to Spin(n) \stackrel{\to}{\to} {*}) \stackrel{\simeq}{\leftarrow} (U(1) \to \hat \Omega Spin(n) \to P Spin(n) \stackrel{\to}{\to} 1) \stackrel{\frac{1}{2}p_1}{\to} (U(1) \to 1 \to 1 \stackrel{\to}{\to} {*}) \,.$$

The corresponding morphism of L ∞-algebras is

$$\mathfrak{so}(n) \stackrel{\simeq}{\leftarrow} (b \mathfrak{u}(1) \to string(n)) \to b^2 u(1) \,.$$

Therefore for $\hat g_{tw}$ a cocyle for a smooth $\frac{1}{2}p_1(T X)$-twisted $String(n)$-principal ∞-bundle out of the canonical hypercover $Y \stackrel{\simeq}{\to} P$ given by the total space $P \to X$ is

$$\array{ && \mathbf{B}(\mathbf{B}U(1) \to String(n)) \\ & {}^{\hat g_{tw}}\nearrow && \searrow^{} \\ Y &\stackrel{g}{\to}& \mathbf{B}Spin(n) & \stackrel{\frac{1}{2}p_1}{\to} & \mathbf{B}^3 U(1) \\ \downarrow^{\simeq} \\ P }$$

the corresponding Cartan-Ehresmann ∞-connection is given by the diagram

$$\array{ \Omega^\bullet_{vert}(P) &\stackrel{A_{vert}}{\leftarrow}& CE(b \mathfrak{u}(1) \to \mathfrak{string}(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& W(b \mathfrak{u}(1) \to \mathfrak{string}(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{P(F_A)}{\leftarrow}& inv(b \mathfrak{u}(1) \to \mathfrak{string}(n)) }$$

of Chevalley-Eilenberg algebras such that the corresponding twist is the prescribed one. This correct projection of the twist is ensured by inserting this diagram into the big diagram on page 73 of 5twist.

It remains to read off the corresppnding local differential form data. This is done in the diagram on page 75 of 5twist.