nLab
twisted bundle

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A twisted principal-bundle is the object classified by a cocycle in twisted cohomology the way an ordinary principal bundle is the object classified by a cocycle in plain cohomology (generally in nonabelian cohomology).

For G^\hat G a group, a G^\hat G-principal bundle is classified in degree 1 nonabelian cohomology with coefficients in the delooped groupoid BG^\mathbf{B} \hat G.

Given a realization of G^\hat G as an abelian extension

AG^G A \to \hat G \to G

of groups, i.e. given a fibration sequence

BABG^BG \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G

of groupoids such that BA\mathbf{B}A is once deloopable so that the fibration sequence continues to the right at least one step as

BG^BGB 2A \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A

the general mechanism of twisted cohomology induces a notion of twisted G^\hat G-cohomology. The fibrations classified by this are the twisted G^\hat G-bundles.

Definition

We give a discussion of twisted bundles as a realization of twisted cohomology in any cohesive (∞,1)-topos H\mathbf{H} as described in the section cohesive (∞,1)-topos – twisted cohomology. For the cases that H=\mathbf{H} = ETop∞Grpd or H=\mathbf{H} = Smooth∞Grpd this reproduces the traditional notion of topological and smooth twisted bundles, respectively, whose twists are correspondingly topological or smooth bundle gerbes/circle n-bundles.

Setup

Let B n1U(1)H\mathbf{B}^{n-1}U(1) \in \mathbf{H} be the circle n-group. We shall concentrate here for definiteness on twists in B 2U(1)\mathbf{B}^2 U(1)-cohomology, since that reproduces the usual notions of twisted bundles found in the literature. But every other choice would work, too, and yield a corresponding notion of twisted bundles.

Fix once and for all an ∞-group GHG \in \mathbf{H} and a cocycle

c:BGB 2U(1) \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 U(1)

representing a characteristic class

[c]H Smooth 2(BG,U(1)) [\mathbf{c}] \in H_{Smooth}^2(\mathbf{B}G,U(1))

Notice that if GG is a compact Lie group, as usual for the discussion of twisted bundles where G=PU(n)G = P U(n) is the projective unitary group in some dimension nn, then by this theorem we have that

H Smooth 2(BG,U(1))H 3(BG,), H_{Smooth}^2(\mathbf{B}G, U(1)) \simeq H^3(B G, \mathbb{Z}) \,,

where on the right we have the ordinary integral cohomology of the classifying space BGB G \in Top of GG.

The abstract definition

Let GG and c\mathbf{c} be as above.

Definition

Write

BG^BGcB 2U(1) \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 U(1)

for the homotopy fiber of c\mathbf{c}.

This identifies G^\hat G as the group extension of GG by the 2-cocycle c\mathbf{c}.

Note

Equivalently this means that

BU(1)BG^BG \mathbf{B}U(1) \to \mathbf{B}\hat G \to \mathbf{B}G

is the smooth circle 2-bundle/bundle gerbe classified by c\mathbf{c}; and its loop space object

U(1)G^G U(1) \to \hat G \to G

the corresponding circle group principal bundle on GG.

Let XHX \in \mathbf{H} be any object. From twisted cohomology we have the following notion.

Definition

The degree-1 total twisted cohomology H tw 1(X,G^)H_{tw}^1(X, \hat G) of XX with coefficients in G^\hat G, def. , relative to the characteristic class [c][\mathbf{c}] is the set

H tw 1(X,G^):=π 0H tw(X,GH^) H^1_{tw}(X, \hat G) := \pi_0 \mathbf{H}_{tw}(X, \mathbf{G}\hat H)

of connected components of the (∞,1)-pullback

H tw(X,BG^) tw H Smooth 2(X,U(1)) H(X,BG) c * H(X,B 2U(1)), \array{ \mathbf{H}_{tw}(X, \mathbf{B}\hat G) &\stackrel{tw}{\to}& H_{Smooth}^2(X,U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{c}_*}{\to}& \mathbf{H}(X, \mathbf{B}^2 U(1)) } \,,

where the right verticsl morphism is any section of the truncation projection from cocycles to cohomology classes.

Given a twisting class [α]H Smooth 2(U(1))[\alpha] \in H^2_{Smooth}(U(1)) we say that

H [α] 1(X,G^):=H tw 1(X,G^)× [α]* H_{[\alpha]}^1(X,\hat G) := H^1_{tw}(X, \hat G) \times_{[\alpha]} *

is the [α][\alpha]-twisted cohomology of XX with coefficients in G^\hat G relative to c\mathbf{c}.

Note

For [α]=0[\alpha] = 0 the trivial twist, [α][\alpha]-twisted cohomology coincides with ordinary cohomology:

H [α]=0 1(X,G^)H Smooth 1(X,G^). H^1_{[\alpha] = 0}(X, \hat G) \simeq H^1_{Smooth}(X, \hat G) \,.

By the discussion at principal ∞-bundle we may identify the elements of H Smooth 1(X,G^)H^1_{Smooth}(X, \hat G) with G^\hat G-principal ∞-bundles PXP \to X. In particular if G^\hat G is an ordinary Lie group and XX is an ordinary smooth manifold, then these are ordinary G^\hat G-principal bundles over XX. This justifies equivalently calling the elements of H tw 1(X,G^)H^1_{tw}(X,\hat G) twisted principal \infty-bundles; and we shall write

G^TwBund(X):=H tw 1(X,G^), \hat G TwBund(X) := H^1_{tw}(X, \hat G) \,,

where throughout we leave the characteristic class [c][\mathbf{c}] with respect to which the twisting is defined implcitly understood.

Explicit cocycles

We unwind the abstract definition, def. , to obtain the explicit definition of twisted bundles by Cech cocycles the way they appear in the traditional literature (see the General References below).

Proposition

Let U(1)G^GU(1) \to \hat G \to G be a group extension of topological groups.

Let XX \in Mfd \hookrightarrow ETop∞Grpd =:H=: \mathbf{H} be a paracompact topological manifold with good open cover {U iX}\{U_i \to X\}.

  1. Relative to this every twisting cocycle [α]H ETop 2(X,U(1))[\alpha] \in H^2_{ETop}(X, U(1)) is a Cech cohomology representative given by a collection of functions

    {α ijk:U iU jU kU(1)} \{ \alpha_{i j k} : U_i \cap U_j \cap U_k \to U(1) \}

    satisfying on every quadruple intersection the equation

    α ijkα ikl=α jklα ijl. \alpha_{i j k} \alpha_{i k l} = \alpha_{j k l} \alpha_{i j l} \,.
  2. I terms of this cocycle data the twisted cohomology H [α] 1(X,G^)H^1_{[\alpha]}(X, \hat G) is given by equivalence classes of cocycles consisting of

    1. collections of functions

      {g ij:U iU jG^} \{g_{i j} : U_i \cap U_j \to \hat G \}

      subject to the condition that on each triple overlap the equation

      g ijg˙ jk=g ikα ijk g_{i j} \dot g_{j k} = g_{i k} \cdot \alpha_{i j k}

      holds, where on the right we are injecting α ijk\alpha_{i j k} via U(1)G^U(1) \to \hat G into G^\hat G
      and then form the product there;

    2. subject to the equivalence relation that identifies two such collections of cocycle data {g ij}\{g_{i j}\} and {g ij}\{g'_{i j}\} if there exists functions

      {h i:U iG^} \{h_i : U_i \to \hat G\}

      and

      {β ij:U iU jU^(1)} \{\beta_{i j} : U_i \cap U_j \to \hat U(1)\}

      such that

      β ijβ jk=β ik \beta_{i j} \beta_{j k} = \beta_{i k}

      and

      g ij=h i 1g ijh jβ ij. g'_{i j} = h_i^{-1} \cdot g_{i j} \cdot h_j \cdot \beta_{i j} \,.
Proof

We pass to the standard presentation of ETop∞Grpd by the projective local model structure on simplicial presheaves over the site CartSp. We then compute the defining (∞,1)-pullback by a homotopy pullback there.

Write BG^ c,B 2U(1) c[CartSp op,sSet]\mathbf{B}\hat G_{c}, \mathbf{B}^2 U(1)_c \in [CartSp^{op}, sSet] etc. for the standard models of the abstract objects of these names by simplicial presheaves. Write accordingly B(U(1)G^) c\mathbf{B}(U(1) \to \hat G)_c for the delooping of the crossed module associated to the central extension G^G\hat G \to G.

In terms of this the characteristic class c\mathbf{c} is represented by the ∞-anafunctor

B(U(1)G^) c c B(U(1)1) c=B 2U(1) c BG c, \array{ \mathbf{B}(U(1) \to \hat G)_c &\stackrel{\mathbf{c}}{\to}& \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G_c } \,,

where the top horizontal morphism is the evident projection onto the U(1)U(1)-labels. Moreover, the Cech nerve of the good open cover {U iX}\{U_i \to X\} forms a cofibrant resolution

C({U i})X \emptyset \hookrightarrow C(\{U_i\}) \stackrel{\simeq}{\to} X

and so α\alpha is presented by an ∞-anafunctor

C({U i}) α B 2U(1) c X. \array{ C(\{U_i\}) &\stackrel{\alpha}{\to}& \mathbf{B}^2 U(1)_c \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

Using that [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} is a simplicial model category this means in conclusion that the homotopy pullback in question is given by the ordinary pullback of simplicial sets

H [α] 1(X,G^) * α [CartSp op,sSet](C({U i}),B(U(1)G^) c) c * [CartSp op,sSet](C({U i}),B 2U(1) c). \array{ \mathbf{H}^1_{[\alpha]}(X,\hat G) &\to& * \\ \downarrow && \downarrow^{\mathrlap{\alpha}} \\ [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}(U(1) \to \hat G)_c) &\stackrel{\mathbf{c}_*}{\to}& [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}^2 U(1)_c) } \,.

An object of the resulting simplicial set is then seen to be a simplicial map g:C({U i})B(U(1)G^) cg : C(\{U_i\}) \to \mathbf{B}(U(1) \to \hat G)_c that assigns

g: (x,j) (x,i) (x,k) g ij(x) α ijk(x) g jk(x) g ik(x) g \;\; : \;\; \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\;\; \mapsto \;\;\;\; \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow &\Downarrow^{\alpha_{i j k}(x)}& \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\underset{g_{i k}(x)}{\to}&& \bullet }

such that projection out along B(U(1)G^) cB(U(1)1) c=B 2U(1) c\mathbf{B}(U(1) \to \hat G)_c \to \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c produces α\alpha.

Similarily for the morphisms. Writing out what these diagrams in B(U(1)G^) c\mathbf{B}(U(1) \to \hat G)_c mean in equations, one finds the formulas claimed above.

Properties

General

(…)

Consider the extension U(1)U(n)PU(n)U(1) \to U(n) \to P U(n) of the projective unitary group to the unitary group for all nn. Then direct sum of matrices gives a sum operation

H [α] 1(X,PU(n 1))×H [α] 1(X,PU(n 2))H [α] 1(X,PU(n 1+n 2)) H^1_{[\alpha]}(X, P U(n_1)) \times H^1_{[\alpha]}(X, P U(n_2)) \to H^1_{[\alpha]}(X, P U(n_1 + n_2))

and a tensor product operation

H [α 1] 1(X,PU(n))×H [α 2] 1(X,PU(n))H [α 1]+[α 2] 1(X,PU(n 1n 2)) H^1_{[\alpha_1]}(X, P U(n)) \times H^1_{[\alpha_2]}(X, P U(n)) \to H^1_{[\alpha_1]+ [\alpha_2]}(X, P U(n_1 \cdot n_2))

(…)

Twisted K-theory

Equivalence classes of twisted U(n)U(n)-bundles for fixed BU(1)\mathbf{B}U(1)-twist α\alpha form a model for topological α\alpha-twisted K-theory. See there for details.

References

General

The concept of twisted vector bundles was introduced, as a model for twisted K-theory (in the generality of equivariant bundles and orbifold K-theory) in:

The Cech cocycle-incarnation of twisted vector bundles (effectively due to Lupercio & Uribe 2001, Def. 7.2.1) was considered in:

An equivalent characterization of twisted vector bundles (identified in Lupercio & Uribe 2001, (v2-) Prop. 7.2.2) appeared under the name bundle gerbe module in:

Discussion of a splitting principle for twisted vector bundles (phrased in terms of gerbe modules) is in

  • Atsushi Tomoda, On the splitting principle of bundle gerbe modules, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (Euclid, talk slides pdf)

In twisted K-theory

Just as vector bundles model cocycles in K-theory, twisted vector bundles model cocycles in twisted K-theory.

The generalization of this construction to non-torsion twists requires using vectorial bundles instead of plain vector bundles. Full twisted K-theory in terms of twisted vectorial bundles was realized in

Then there is

As 2-sections of 2-bundles

The observation that twisted vector bundles may be understood as higher-order sections of 2-vector bundles associated with circle 2-bundles/bundle gerbes appears in

  • Urs Schreiber, Quantum 2-States: Sections of 2-vector bundles Talk at Higher categories and their applications, Fields institute (2007) (pdf).

A discussion of this with 2-connections taken into account is in section 4.4.3 of

A discussion in the context of principal infinity-bundles (as opposed to higher vector bundles), is in section “2.3.5 Twisted cohomology and sections” and then in section “3.3.7.2 Twisted 1-bundles – twisted K-theory”

The observation then re-appears independently in

Last revised on July 30, 2021 at 10:29:35. See the history of this page for a list of all contributions to it.