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.

Contents

Idea

Differential twisted cohomology is the pairing of the notions of

with

Recalling that

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 ABCA \to B \to C a fibration sequence and c:XCc : X \to C a given twisting cocycle, a cc-twisted AA-cocycle is nothing but a B^\hat B-cocycle that satisfies a certain constraint. Therefore differential twisted AA-cocycles are just differential BB-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 H\mathbf{H}. Consider two fibration sequences

ABC A \to B \to C

and

ABC A' \to B \to C'

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

H [c,c](X,A× BA):=π 0H [c,c](X,A× BA) H_{[c,c']}(X,A \times_{B} A') := \pi_0 \mathbf{H}_{[c,c']}(X,A \times_{B} A')

of the homotopy pullback in

H [c,c](X,A× BA) * c×c H(X,B) H(X,C×C) \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][c,c']-bitwisted cohomology of XX.

Differential twisted cohomology is an example of this as follows:

Definition (differential twisted cohomology)

Fix some (∞,1)-topos H\mathbf{H}.

Consider a fibration sequences

ABC. A \to B \to C \,.

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

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

(Recall the definition of B flatB_flat from path ∞-groupoid and the definition of B dRB_{dR} from nonabelian deRham cohomology.)

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

Examples (theorems)

Fix a smooth (∞,1)-topos H\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

BString(n)BSpin(n)12p 1B 3U(1) \mathbf{B} String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)

and

BFivebrane(n)BString(n)16p 2B 7U(1). \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

12p 1(TX):XBSpin(n)12p 1B 3U(1) \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

16p 1(TX):XBString(n)16p 2B 7U(1) \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)String(n)- and twisted Fivebrane(n)Fivebrane(n)-cocycles, respectively.

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

  • The local differential form data of a twisted Fivebrane(n)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)String(n)-cocycles. The FivebraneFivebrane-case works entirely analogously.

The crucial point to notice is that the twisting morphism

12p 1:BString(n)B 3(U(1)) \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) locSPSh(C)_{loc} that models our (,1)(\infty,1)-category H\mathbf{H} of Lie ∞-groupoid by the span of crossed complexes

(11Spin(n)*)(U(1)Ω^Spin(n)PSpin(n)1)12p 1(U(1)11*). (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

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

Therefore for g^ tw\hat g_{tw} a cocyle for a smooth 12p 1(TX)\frac{1}{2}p_1(T X)-twisted String(n)String(n)-principal ∞-bundle out of the canonical hypercover YPY \stackrel{\simeq}{\to} P given by the total space PXP \to X is

B(BU(1)String(n)) g^ tw Y g BSpin(n) 12p 1 B 3U(1) P \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

Ω vert (P) A vert CE(b𝔲(1)𝔰𝔱𝔯𝔦𝔫𝔤(n)) Ω (P) (A,F A) W(b𝔲(1)𝔰𝔱𝔯𝔦𝔫𝔤(n)) Ω (X) P(F A) inv(b𝔲(1)𝔰𝔱𝔯𝔦𝔫𝔤(n)) \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.

Revised on October 14, 2009 15:15:49 by Urs Schreiber (195.37.209.182)