twisted de Rham cohomology

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###Context###
#### Differential cohomology
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[[!include differential cohomology - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
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## Idea

Twisted de Rham cohomology is the [[twisted cohomology]]-version of [[de Rham cohomology]], a simple example of [[twisted differential cohomology]].

For degree-3 twists this is the [[codomain]] of the [[twisted Chern character]] on [[twisted K-theory]], and in its [[orbifold cohomology]]-generalization it is the [[codomain]] of the [[twisted equivariant Chern character]] on [[twisted equivariant K-theory]].

## Definition

### Plain

\begin{definition}\label{1TwistedDeRhamCohomology}
  **(1-twisted de Rham cohomology)**\linebreak
  (...)
\end{definition}

\begin{definition}\label{3TwistedDeRhamCohomology}
  **(3-twisted de Rham cohomology)** \linebreak
For $X$ a [[smooth manifold]] and $H \in \Omega^3(X)$ a closed [[differential form|differential 3-form]], the $H$ **twisted de Rham complex** is the $\mathbb{Z}_2$-[[graded vector space]] $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted [[de Rham differential]]

$$
  d + H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X)
  \,,
$$

Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.
\end{definition}

\begin{remark}
There is also the cohomology of the [[chain complex]] whose maps are just multiplication by $H$.
$$
  H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X)
  \,,
$$
This is also called _[[H-cohomology]]_ ([Cavalcanti 03, p. 19](#Cavalcanti03)).
\end{remark}

### Equivariant
 {#Equivariant}

We discuss notions of twisted de Rham cohomology on ([[global quotient orbifold|global quotient]]) [[orbifolds]], as they are used for the [[codomain]] of the [[twisted equivariant Chern character]] on [[twisted equivariant K-theory]].

This combines the above twistings in degrees 1 and 3, the latter induced from the [[curvature]] [[differential 3-form|3-form]] on a twisting 3-class, the former (an "[[inner local system]]") induced from the [[flat connection]] on the [[circle principal bundle]] (over the [[inertia orbifold]]) which is classified by the [[transgression]] of the 3-twist to a 2-class. This requires that this connection be [[flat connection|flat]], hence that the transgressed 2-class is [[torsion subgroup|torsion]], which is guaranteed by a Lemma that we discuss first, in *[Transgression of the 3-twist to a 1-twist on Inertia](#EquivariantPreliminaries)*.

\linebreak

In all of the the following:

* Let $G$ be a [[finite group]].

  * For $g \in G$ any [[element]], write $C_G(g) \,\subset\, G$ for its [[centralizer subgroup]].

  * Write $C_G(g) \times \mathbb{Z}$ for the [[direct product group]] with the additive [[abelian group]] of [[integers]].

* Let

  \[
    \label{AProperSmoothGManifold}
    X \in Proper G Actions(SmthMfds)
  \]

  be a proper smooth [[G-manifold]], hence a [[smooth manifold]] equipped with a [[proper action]] (from the right, say) of $G$ by [[diffeomorphisms]].

  * Notice that for any $g \in G$ the [[fixed locus]]

    \[
      \label{gFixedLocusInGManifold}
      X^g \xhookrightarrow{\;i_g\;} X
    \]

    is a smooth [[submanifold]] (by [this Prop.](equivariant+differential}topology#FixedLociOfSmoothProperActionsAreSubmanifolds)).

All twisting classes, in the following are in ordinary Borel-equivariant cohomology, hence in the [[ordinary cohomology]] of a [[Borel construction]]. The "3-twist" and "torsion 2-twist" are in [[integral cohomology]] (of the Borel construction) and the "1-twist" has [[coefficients]] a finite [[cyclic group]].

#### Transgression of 3-twist to 1-twist on inertia
 {#EquivariantPreliminaries}

We discuss the transgression of a 3-twist on a ([[global quotient orbifold|global quotient]]) [[orbifold]] to a 1-twist on its [[inertia orbifold]], namely to a torsion 2-class:

1. [via the Künneth theorem](#TransgressionOf3TwistTo1TwistVieKuenneth)

   as argued in [Becerra & Uribe 2009, Section 3.2](orbifold+K-theory#BecerraUribe09);

1. [via a Serre spectral sequence](#TransgressionOf3TwistTo1TwistViaSerreSpectralSequence)

   as sketched in [Freed, Hopkins & Teleman 07, (3.5)](#FreedHopkinsTeleman07).

There is also a corresponding argument in terms of [[bundle gerbes]], given in [Tu & Xu 2006, Prop. 2.6 & 3.6](#TuXu06).

##### Via Künneth theorem
 {#TransgressionOf3TwistTo1TwistVieKuenneth}

Throughout, fix an element $g \in G$.

Consider the right [[group action]] of the [[direct product group]] $C_G(g) \times \mathbb{Z}$ (of the [[centralizer subgroup]] with the [[integers]]) on the [[fixed locus]] $X^g$ (eq:gFixedLocusInGManifold) given by

\[
  \label{ActionOfCentralizerTimesIntegersOnFixedLocus}
  \array{
    X^g \times C_G(g) \times \mathbb{Z}
    &\xrightarrow{\;\;\;\;}&
    X^g
    \\
    \big(
      x, \, (h, n)
    \big)
    &\mapsto&
    x \cdot h \cdot g^n
    \mathrlap{\,.}
  }
\]

and notice that the following [[function]] is a [[group homomorphism]] (by the fact that all elements of $C_G(g)$ commute with $g$ in $G$):

\[
  \label{GroupHomomorphismFromCentralizerTimesIntegersIntoG}
  \array{
    C_G(g) \times \mathbb{Z}
    &\xrightarrow{\;\;\phi_g\;\;}&
    G
    \\
    (h,n) &\mapsto& h \cdot g^n
    \mathrlap{\,.}
  }
\]

In view of the action (eq:ActionOfCentralizerTimesIntegersOnFixedLocus), the homomorphism (eq:GroupHomomorphismFromCentralizerTimesIntegersIntoG) induces a map of [[Borel constructions]]

$$
  \big(
    X^g \sslash C_G(g)
    \times
    B \mathbb{Z}
  \big)
  \;\simeq\;
  X^g
    \sslash
  \big(
    C_G(g) \times \mathbb{Z}
  \big)
  \xrightarrow{ \;\; i_g \sslash \phi_g \;\; }
  X
    \sslash
  G
  \,,
$$

(where the [[homotopy equivalence]] shown on the left follows since the [[group action]] of  $\mathbb{Z}$ on $X^g$ is the [[trivial action]], by definition (eq:ActionOfCentralizerTimesIntegersOnFixedLocus), and using that the [[Borel construction]] on a point is the [[classifying space]] $ \ast \sslash \mathbb{Z} \,\simeq\, B \mathbb{Z} \simeq S^1$, hence the [[circle]], in the present case)

and hence the corresponding [[pullback in cohomology|pullback in]] [[integral cohomology]]:

\[
  \label{PullbackInCohomologyAlongMorphismOfBorelConstructions}
  \begin{array}{rrl}
  H^\bullet
  \big(
    X \sslash G
    \;
    \,
    \mathbb{Z}
  \big)
  &
  \xrightarrow{ \; (i_g \sslash \phi_g)^\ast \; }
  &
  H^\bullet
  \big(
    ( X^g \sslash C_G(g) )
    \times
    B \mathbb{Z}
    \;
    \,
    \mathbb{Z}
  \big)
  \\
  &
  \,\simeq\,
  &
  H^\bullet
  \big(
    ( X^g \sslash C_G(g) )
    \;
    \,
    \mathbb{Z}
  \big)
    \otimes_{\mathbb{Z}}
  \underset{
    \mathclap{
      \simeq
      \left\{
      \array{
        \mathbb{Z} &\vert& \bullet \in \{0,1\}
        \\
        0 &\vert& else
      }
      \right.
    }
  }{
    \underbrace{
      H^\bullet
      \big(
        B \mathbb{Z}
        ;\,
        \mathbb{Z}
      \big)
    }
  }
  \\
  &\simeq &
  H^\bullet
  \big(
    ( X^g \sslash C_G(g) )
    \;
    \,
    \mathbb{Z}
  \big)
  \,\oplus\,
  H^{\bullet-1}
  \big(
    ( X^g \sslash C_G(g) )
    \;
    \,
    \mathbb{Z}
  \big)
  \,,
  \end{array}
\]

where the second line uses the [[Künneth theorem]].

(The following definition becomes a proposition if one uses conceptualization of [[transgression]] as laid out in *[[transgression in group cohomology]]*.)

\begin{definition}\label{TransgressionOntoFixedLocus}
  For $g \in G$ write
  $$
    \tau_g
    \;\colon\;
    H^\bullet
    \big(
      X \sslash G
      ;\,
      \mathbb{Z}
    \big)
    \xrightarrow{\;\;}
    H^{\bullet-1}
    \big(
      ( X^g \sslash C_G(g) )
      ;\,
      \mathbb{Z}
    \big)
  $$
  for the [[composition|composite]] of (eq:PullbackInCohomologyAlongMorphismOfBorelConstructions) with [[projection]] onto the second [[direct sum|direct summand]].
\end{definition}

\begin{prop}\label{ImageOfTransgressionMapIsInTorsionSubgroup}
  The [[image]] of the transgression map $\tau_g$ (Def. \ref{TransgressionOntoFixedLocus}) is in the [[torsion subgroup]].
\end{prop}
([Becerra & Uribe 2009, Lemma 3.2](orbifold+K-theory#BecerraUribe09))
\begin{proof}
  The group homomorphism (eq:GroupHomomorphismFromCentralizerTimesIntegersIntoG) factors through the [[cyclic group]] $\mathbb{Z} \to \langle g\rangle \to G$ which is generated by $g \in G$. By the assumption that $G$ is a [[finite group]], so that its [[integral cohomology]] is entirely torsion. Since the transgression map factors through a tensor product with pullback along this factorization, by definition, the claim follows.
\end{proof}

##### Via the Serre spectral sequence
 {#TransgressionOf3TwistTo1TwistViaSerreSpectralSequence}

\begin{lemma}
  For each $g \in G$ and each point in the [[Borel construction]] $X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}$, there is a [[homotopy fiber sequence]] of the form

\[
  \label{HomotopyFibrationOfBorelConstructionsOnFixedLoci}
  \array{
    B \langle{g}\rangle
    &\longrightarrow&
    X^g \!\sslash\! \langle{g}\rangle
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \ast
    &\longrightarrow&
    X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}
  }
\]
\end{lemma}
\begin{proof}

First notice that the [[short exact sequence]]
$$
  1
  \to
  \langle{g}\rangle
    \hookrightarrow
  C_G(g)
    \twoheadrightarrow
  G/\langle{g}\rangle
  \to
  1
$$
[[delooping|deloops]] to a [[homotopy fiber]]-sequence of the form
\[
  \label{HomotopyFiberSequenceOfClassifyingSpacesForResidualGroupActions}
  \array{
    B \langle{g}\rangle
    &\longrightarrow&
    B C_G(g)
    \\
    &&
    \big\downarrow
    \\
    &&
    B \frac{C_G(g)}{\langle{g}\rangle}
  }
\]
(using [this Prop.](simplicial+classifying+space#DeloopedKanFibrationOfHomomorphismOfSimplicialGroups)).

Then notice that we have a [[pasting diagram]] of [[homotopy pullbacks]] as follows:

$$
  \array{
    B \langle{g}\rangle
    &\longrightarrow&
    \ast
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    X^g \!\sslash\! C_G(g)
    &\xrightarrow{\;}&
    X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    B G
    &\xrightarrow{\;}&
    B \frac{C_G(g)}{\langle{g}\rangle}
  }
$$

Here:

* the bottom square being a homotopy pullback expresses that the [[group action]] of $\langle{g}\rangle$ on $X^g$ is trivial, so that the [[group action]] of $C_G(g)$ on $X^g$ is induced by that of $C_G(g)/\langle{g}\rangle$;

* the total rectangle is a homotopy pullback by (eq:HomotopyFiberSequenceOfClassifyingSpacesForResidualGroupActions);

* hence the top square is a homotopy pullback by the [[pasting law]].

\end{proof}

{#ItFollows} It follows -- once we know (do we?) that the action of the [[fundamental group]] of $X^g \sslash \frac{C_G(g)}{\langle{g}\rangle}$ on the [[integral cohomology]] of $B\langle{g}\rangle$ is [[trivial action|trivial]] -- that we have a [[Serre spectral sequence]]

\[
  \label{SerreSpectralSequenceForBorelConstructionOfResidualActions}
  H^p
  \left(
    X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}
    ;\,
    H^q
    \big(
      B\langle{g}\rangle{g}
      ;\,
      \mathbb{Z}
    \big)
  \right)
  \;\;
    \overset{\;\;\;\;}{\Rightarrow}
  \;\;
  H^{p+q}
  \big(
    X^g \!\sslash\! C_G(g)
    ;\,
    \mathbb{Z}
  \big)
  \,.
\]

But since $\langle{g}\rangle \simeq \mathbb{Z}/(ord(g))$ is a [[cyclic group]] (by the assumption that $G$ is a [[finite group]]) it follows (by [this Prop.](finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2) or Lem. 4.51 in [BSST 07](Pontrjagin+dual#BSST07)) that its [[group cohomology]] is concentrated in even degrees:

$$
  H^n
  \big(
    B \langle{g}\rangle
    ;\,
    \mathbb{Z}
  \big)
  \;=\;
  H^n_{grp}
  \big(
    \langle{g}\rangle
    ;\,
    \mathbb{Z}
  \big)
  \;\simeq\;
  \left\{
  \begin{array}{ccl}
    \mathbb{Z} &\vert& n = 0
    \\
    \mathbb{Z}/ord(g) &\vert& n = \,\text{positive multiple of}\, 2
    \\
    0 &\vert& \text{otherwise}
  \end{array}
  \right.
$$

Therefore the spectral sequence (eq:SerreSpectralSequenceForBorelConstructionOfResidualActions) associates with any 3-twist $\alpha \mapsto \alpha_{\vert X^g} \in H^3\big( X^g \sslash C_G(g);\, \mathbb{Z}  \big)$ a "transgressed" degree-1 class

\[
  \label{Transgression1ClassFromSerreSpectralSequence}
  \tau_g(\alpha)
  \;\in\;
  H^1
  \left(
    X^g
    ;\,
    \mathbb{Z}/ord(g)
  \right)
  \xrightarrow{\;\;\beta\;\;}
  H^2
  \left(
    X^g
    ;\,
    \mathbb{Z}
  \right)
  \,,
\]

which the [[Bockstein homomorphism]] identifies with a [[torsion subgroup|torsion]] 2-class in [[integral cohomology]].

Essentially this conclusion is claimed as [FHT 07, (3.5)](#FreedHopkinsTeleman07).

#### Definition
 {#DefinitionEquivariantTwistedDeRhamCohomology}

Now we can indicate the definition of the twisted equivariant de Rham cohomology. In outline:

Write $\Lambda (\prec (X \sslash G))$ for the [[inertia orbifold]] of the [[global quotient orbifold]] of $X$ (eq:AProperSmoothGManifold).

For $\alpha \in H^3\big( X \sslash G; \, \mathbb{Z}  \big)$ a "3-twist" in the degree-3 [[integral cohomology]] of the [[homotopy quotient]] ([[Borel construction]]) of $X$ by $G$ (eq:AProperSmoothGManifold)

* let $H_3 \in \Omega^3\big(  \Lambda (\prec (X \sslash G))  \big)$ be a [[de Rham isomorphism|de Rham image]] of $\alpha$ [[pullback in cohomology|pulled back]] to the [[inertia orbifold]],

* let $\nabla$ be a [[circle bundle with connection|connection]] on the transgression (Def. \ref{TransgressionOntoFixedLocus}) of $\alpha$ to the [[inertia orbifold]], which is [[flat connection|flat]] by Prop. \ref{ImageOfTransgressionMapIsInTorsionSubgroup} or, alternatively, by (eq:Transgression1ClassFromSerreSpectralSequence).

Then equivariant twisted de Rham cohomology of $X$ is the de Rham cohomology of $\Lambda (\prec (X \sslash G))$ which is both

* 1-twisted by $\nabla$ and

* 3-twisted by $H_3$

([Tu & Xu 2006, Def. 3.10](#TuXu06), [Freed, Hopkins & Teleman 2007, (3.19)](#FreedHopkinsTeleman07), [Bunke, Spitzweck & Schick 08, Def. 3.15](#BunkeSpitzweckSchick08)).

\linebreak

## Properties

### Of 1-Twisted de Rham cohomology

#### Equivalence with $\pi_1$-invariant dR cohomology
 {#EquivalenceWithPiOneInvariantdRCohomology}

In the following, let $\mathrm{X}$ be a [[smooth manifold]], which we assume, without real restriction of generality, to be [[connected topological space|connected]]. Therefore we write $\pi_1(\mathrm{X})$ for its [[fundamental group]], for any fixed choice of [[basepoint]] $x_0 \,\in\, \mathrm{X}$.

We write $\widehat{\mathrm{X}} \xrightarrow{\;} \mathrm{X}$ for its [[universal cover]] (also canonically a smooth manifold). This is a $\pi_1(\mathrm{X})$-[[principal bundle]], in particular it has a canonical $\pi_1(\mathrm{X})$-[[group action|action]] by [[deck transformations]].

For $\mathcal{L}$ a [[complex line bundle]] over $\mathrm{X}$ with [[flat connection]] $\nabla$, write
\[
  \label{HolonomyMap}
  hol_\nabla
  \,\colon\,
  \pi_1(\mathrm{X})
  \xrightarrow{\;\;}
  \mathbb{C}^\times
\]
for the [[group homomorphism]] from the [[fundamental group]] to the multiplicative [[group of units]] $\mathbb{C}^\times \,=\, \mathbb{C} \setminus \{0\}$ which is given by sending a [[smooth curve]] $\lambda \colon [0,1] \to \mathrm{X}$ (with $\lambda(0) = \lambda(1)$) to its [[holonomy]] under the [[parallel transport]] with respect to $\nabla$.

\begin{proposition}
\label{OneTwisteddRCohomologyEquivalentToPiOneInvariantOnUniversalCover}
**(1-Twisted dR cohomology equivalent to $\pi_1$-invariant dR cohomology on universal cover)**
\linebreak
  For $\mathcal{L}$ a [[complex line bundle]] over $\mathrm{X}$ with [[flat connection]] $\nabla$, there is a [[natural isomorphism|natural]] [[isomorphism]] between

1. the $\nabla$-twisted de Rham cohomology on $\mathrm{X}$:

   $$
     H^\bullet
     \Big(
       \Omega_{dR}^\bullet
       \big(
         \mathrm{X}
          ;\,
          \mathcal{L}
       \big)
       ,\,
       \nabla
     \Big)
   $$

1. the untwisted but $\pi_1$-[[invariant]] complex-valued de Rham cohomology on the [[universal cover]] $\widehat{\mathrm{X}}$

   $$
     H^\bullet
     \bigg(
       \Big(
         \Omega^\bullet_{dR}
         \big(
           \widehat{\mathrm{X}}
           ;\,
           \mathbb{C}
         \big)
       \Big)^{\pi_1(\mathrm{X})}
       ,\,
       \mathrm{d}
     \bigg)
   $$

   where [[fundamental group|$\pi_1(\mathrm{X})$]] [[group action|acts]] on [[differential forms]] by [[pullback of differential forms|pullback]] along [[deck transformations]] combined with multiplication by the [[holonomy]] (eq:HolonomyMap) of $\nabla$:

   $$
     \array{
       \pi_1(\mathrm{X})
       \times
       \Omega^\bullet_{\mathrm{dR}}
       \big(
         \widehat{\mathrm{X}}
         ;\,
         \mathbb{C}
       \big)
       &\xrightarrow{\phantom{--}}&
       \Omega^\bullet_{\mathrm{dR}}
       \big(
         \widehat{\mathrm{X}}
         ;\,
         \mathbb{C}
       \big)
       \\
       \big(
         [\lambda]
         ,\,
         A
       \big)
       &\mapsto&
       hol_\nabla(\lambda)
       \cdot
       [\lambda]^\ast(A)
     }
   $$

\end{proposition}
A **proof** is spelled out in this [[OneTwistedAsDeckEquivariantDeRham.pdf:file]]. It proceeds by observing that the bundle and its connection trivialize after pullback to the universal cover $\widehat{\mathrm{X}}$, in fact that the local connection form $\widehat{\omega}$ on $\widehat{\mathrm{X}}$ becomes exact (by the [[Poincaré lemma]] for simply connected domains, [here](Poincaré+lemma#ForOneFormsOnSimplyConnectedDomain)):
\[
  \label{MasterFunction}
  \underset{
    \mathclap{\phantom{\vert^{\vert}}}
    { \ell \in }
    \atop
    {
      C^\infty\big(
        \widehat{\mathrm{X}}
        ;\,
        \mathbb{C}^\times
     \big)
    }
  }{\exists}
  \;\;\;
  \widehat{\omega}
  \;=\;
  - \mathrm{d} log \ell
  \,,
  \;\;\;\;
  \in
  \;
  \Omega^1_{dR}
  \big(
    \widehat{\mathrm{X}}
    ,\,
    \mathbb{C}
  \big)
  \,,
\]
and checking that multiplication by this potential $\ell$ (eq:MasterFunction) constitutes a [[isomorphism|isomorphic]] ([[bijection|bijective]]) [[chain map]] between the [[cochain complexes]] in question.

\begin{example}
**(Hypergeometric integral solutions of KZ-equation)**
\linebreak
  For the special case that the [[complex line bundle]] $\mathcal{L}$ is [[trivial bundle|trivial]] (so that the [[flat connection]] $\nabla$ is represented by a globally defined [[differential 1-form]] already on the base manifold $\mathrm{X}$) the statement of Prop. \ref{OneTwisteddRCohomologyEquivalentToPiOneInvariantOnUniversalCover} (or rather its [[holomorphic line bundle|holomorphic]] version) plays a central role in the discussion of the "[hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)" of solutions to the [[Knizhnik-Zamolodchikov equation]], where it is applied to the case that $\mathrm{X}$ is an $n$-punctured [[Riemann sphere]] (e.g. a [[trinion]]). In fact it is so central to this construction that the function $\ell$ (eq:MasterFunction)
which trivializes the connection form on the universal cover and thereby induces the isomorphism in Prop. \ref{OneTwisteddRCohomologyEquivalentToPiOneInvariantOnUniversalCover} came to be called the "master function", in this context ([Slinkin & Varchenko 2019, §2.1](#SlinkinVarchenko19)).

On the other hand, none of the many references listed [there](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) really make the Proposition explicit.
\end{example}

\linebreak

### Of 3-Twisted de Rham cohomology

+-- {: .num_prop #TwistedDeRhamCohomologyCoincidesWithHCohomologyOnInfinitesimllayThickenedPoint}
###### Proposition

If the [[de Rham complex]] $(\Omega^\bullet(X),d_{dr})$ is [[formal dg-algebra|formal]], then for $H \in \Omega_{cl}^3(X)$ a [[closed differential form|closed]] [[differential 3-form]], the $H$-[[twisted de Rham cohomology]] (Def. \ref{3TwistedDeRhamCohomology}) of $X$ coincides with its [[H-cohomology]], for any closed 3-form $H$.

=--

([Cavalcanti 03, theorem 1.6](#Cavalcanti03)).

## Related concepts

* [[twisted Chern character]], [[twisted equivariant Chern character]]

* [[inner local system]]

* [[Hanany-Witten effect]]

## References

### Twist in degree 1
 {#ReferencesForTwistInDegree1}

The classical case of twisted de Rham cohomology with twists in *degree 1*, given by [[flat connections]] on [[flat line bundles]] and more generally on [[flat vector bundles]]), and its equivalence to [[sheaf cohomology]] with [[coefficients]] in [[abelian sheaves]] of [[flat sections]] ([[local systems]]):

* {#Voisin02} [[Claire Voisin]] (translated by [[Leila Schneps]]), Section II 5.1.1 of: _[[Hodge theory and Complex algebraic geometry]] II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3 ([doi:10.1017/CBO9780511615177](https://doi.org/10.1017/CBO9780511615177))

* [[Alexandru Dimca]], Section 2.5 of: *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$

Review:

* [[Anatoly Libgober]], [[Sergey Yuzvinsky]], *Cohomology of local systems*, Advanced Studies in Pure Mathematics **27**, Mathematics Society of Japan (2000) 169-184 [[pdf](http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf), [doi:10.2969/aspm/02710169](https://doi.org/10.2969/aspm/02710169)]

* Cailan Li, *Cohomology of Local Systems on $X_\Gamma$* ([pdf](http://math.columbia.edu/~mundy/L3.pdf), [[Li_CohomologyOfLocalSystems.pdf:file]])

* Youming Chen, Song Yang, Section 2.1 in: *On the blow-up formula of twisted de Rham cohomology*. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) ([arXiv:1810.09653](https://arxiv.org/abs/1810.09653), [doi:10.1007/s10455-019-09667-8](https://doi.org/10.1007/s10455-019-09667-8))

For extensive application, see also the "[hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)" of solutions to the [[Knizhnik-Zamolodchikov equation]].

In this context there is also:

* {#SlinkinVarchenko19} [[Alexey Slinkin]], [[Alexander Varchenko]], *Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules*, SIGMA **15** (2019) 075 [[arXiv:1812.09791](https://arxiv.org/abs/1812.09791), [doi:10.3842/SIGMA.2019.075](https://doi.org/10.3842/SIGMA.2019.075)]

See also at [[supersymmetric quantum mechanics]] and see

* [[Albert Schwarz]], [[Ilya Shapiro]], *Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"*, Nucl. Phys. B **809** (2009) 547-560 [[arXiv:0809.0086](https://arxiv.org/abs/0809.0086), [doi:10.1016/j.nuclphysb.2008.10.005](https://doi.org/10.1016/j.nuclphysb.2008.10.005)]

### Twist in degree 3
 {#ReferencesForTwistInDegree3}

#### Plain

The concept of $H_3$-twisted de Rham cohomology  was introduced (in discussion of the [[B-field]] in [[string theory]]), in:

* [[Ryan Rohm]], [[Edward Witten]], around (23) and appendix of: _The antisymmetric tensor field in superstring theory_, Annals of Physics Volume 170, Issue 2, September 1986, Pages 454-489 (<a href="https://doi.org/10.1016/0003-4916(86)90099-0">doi:10.1016/0003-4916(86)90099-0</a>)

Further discussion (often as the [[codomain]] of the [[twisted Chern character]] on [[twisted K-theory]]):

* {#CBMMS}  [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]], [[Danny Stevenson]], Section 9.3 of: _Twisted K-theory and K-theory of bundle gerbes_ ,  Commun Math Phys, 228 (2002) 17-49 ([arXiv:hep-th/0106194](http://arxiv.org/abs/hep-th/0106194), [doi:10.1007/s002200200646](https://doi.org/10.1007/s002200200646))

* [[Varghese Mathai]], [[Danny Stevenson]], Section 3 of: _Chern character in twisted K-theory: equivariant and holomorphic cases_, Commun. Math. Phys. 236:161-186, 2003 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010))

* {#FreedHopkinsTeleman07} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], Section 2 of: _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1, 2007 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

* {#Teleman04} [[Constantin Teleman]], around Prop. 3.7 in: _K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra_, in: [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_, Cambridge 2004 ([arXiv:math/0306347](https://arxiv.org/abs/math/0306347), [spire:660158](https://inspirehep.net/literature/660158))

* {#Cavalcanti03} [[Gil Cavalcanti]], Section I.4 of: _New aspects of the $d d^c$-lemma_, Oxford  2005 ([arXiv:math/0501406](https://arxiv.org/abs/math/0501406))

#### Equivariant
 {#ReferencesEquivariant3Twisted}

The generalization of 3-twisted de Rham cohomology to [[orbifolds]] (often as the [[codomain]] of the [[twisted equivariant Chern character]] on [[twisted equivariant K-theory]]):

* {#MathaiStevenson03} [[Varghese Mathai]], [[Danny Stevenson]], p. 18 of: *Chern character in twisted K-theory: equivariant and holomorphic cases*, Commun. Math. Phys. **236** (2003) 161-186 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7))

  > (via [[equivariant de Rham cohomology]])

* {#TuXu06} [[Jean-Louis Tu]], [[Ping Xu]], Def. 3.10 in: *Chern character for twisted K-theory of orbifolds*, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 ([arXiv:math/0505267](https://arxiv.org/abs/math/0505267), [doi:10.1016/j.aim.2005.12.001](https://doi.org/10.1016/j.aim.2005.12.001))

* {#BunkeSpitzweckSchick08} [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], Section 3.2 of: _Inertia and delocalized twisted cohomology_, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) ([arXiv:math/0609576](https://arxiv.org/abs/math/0609576), [doi:10.4310/HHA.2008.v10.n1.a6](https://dx.doi.org/10.4310/HHA.2008.v10.n1.a6))

See also

* [Freed, Hopkins & Teleman 07, Paragraph (3.8)](#FreedHopkinsTeleman07)

### Twist in higher degrees
 {#ReferencesTwistInHigherDegrees}

Discussion of higher-degree twisted de Rham cohomology (often as the [[Chern character]]-like image of [higher twisted K-theory](twisted+K-theory#ReferencesHigherTwists)):

* [Teleman 04, Sec. 3](#Teleman04)

* [[Hisham Sati]], *A Higher Twist in String Theory*, J. Geom. Phys. 59:369-373, 2009 ([arXiv:hep-th/0701232](https://arxiv.org/abs/hep-th/0701232))

* [[Varghese Mathai]], [[Siye Wu]], _Analytic torsion for twisted de Rham complexes_, J. Diff. Geom. 88:297-332, 2011 ([arXiv:0810.4204](https://arxiv.org/abs/0810.4204))

  > (relation to [[analytic torsion]])

A [[spectral sequence]] for higher twisted de Rham cohomology (analgous to the [[Atiyah-Hirzebruch spectral sequence]] for [[twisted K-theory]] from [Atiyah & Segal 2005](Atiyah–Hirzebruch+spectral+sequence#AtiyahSegal05)):

* {#LiLiuWang09} Weiping Li, Xiugui Liu, [[He Wang]], _On a spectral sequence for twisted cohomologies_,  Chin. Ann. Math. Ser. B 35, 633–658 (2014) ([arXiv:0911.1417](https://arxiv.org/abs/0911.1417), [doi:10.1007/s11401-014-0842-z](https://doi.org/10.1007/s11401-014-0842-z)).

In the broader context of [[twisted cohomology theory]] and the [[Chern-Dold character map]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3.3 of: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The character map in (twisted differential) non-abelian cohomology]]_ ([arXiv:2009.11909](https://arxiv.org/abs/2009.11909))

[[!include hypergeometric KZ-solutions -- references]]

[[!redirects twisted de Rham complex]]


twisted differential c-structure

+-- {: .rightHandSide}
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###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $c$ any [[characteristic class]], its [[homotopy fibers]] on [[cocycle]] [[∞-groupoid]]s represent $c$-[[twisted cohomology]] (for instance [[twisted bundle]]s, [[twisted spin structures]], etc.).

If $c$ is refined to a characteristic class $\mathbf{c}$ in [[Smooth∞Grpd]] there may exist further refinements $\hat {\mathbf{c}}$ to [[ordinary differential cohomology]]. The twisted cohomology of these _differential characteristic classes_ may be called _twisted differential structures_ . For instance _[[differential string structure]]s_ . See [below](#Examples) for more examples.

These structures have a natural interpretation and play a natural roles as _[[physical fields]]_ (see there for a comprehensive discussion).

## Definition

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]], usually $\mathbf{H} = $ [[Smooth∞Grpd]] or [[SynthDiff∞Grpd]] or the like.

Let $K, G$
be [[∞-group]] objects in $\mathbf{H}$ and let

$$
  \mathbf{c} : \mathbf{B}G \to \mathbf{B}K
$$

be a morphism of their [[delooping]] objects / [[moduli stacks]].

+-- {: .num_defn}
###### Definition

For $X \in \mathbf{H}$ any object and $P \to X$
an $K$-[[principal ∞-bundle]] over $X$, the [[∞-groupoid]]

$$
  \mathbf{c}Struc_{[P]}(X)
  :=
  \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\}
  \,,
$$

hence the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{c}Struc_{[P]}(X)
    &\to&
    *
    \\
    \downarrow^{\mathrlap{P}} && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G)
    &\stackrel{\mathbf{H}(X, \mathbf{c})}{\to}&
    \mathbf{H}(X, \mathbf{B}K)
  }
$$

we may call equivalently

* the $\infty$-groupoid of $K$-structures on $P$ (with respect to the given $\mathbf{c}$);

* the $\infty$-groupoid of $[P]$-twisted $\mathbf{c}$-structures.

=--

+-- {: .num_remark}
###### Remark

As discussed at _[[twisted cohomology]]_, we may think of an object in
$\mathbf{c}Struc_{[P]}(X)$ as a [[section]] (up to [[homotopy]]) $\sigma$

$$
  \array{
    && \mathbf{B}G
    \\
    & {}^{\sigma}\nearrow& \downarrow^{\mathbf{c}}
    \\
    X &\stackrel{g}{\to}& \mathbf{B}K
  }
$$

where we think of $\mathbf{c}$ as being the **universal twisting $\infty$-bundle** and where $g : X \to \mathbf{B}K$ is a morphism presenting $P$.

=--

The following definition looks at a differential refinement of this situation.

+-- {: .num_defn}
###### Definition

For $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ a [[characteristic class|characteristic map]]
in $\mathbf{H}$ and $\hat {\mathbf{c}} : \mathbf{B}G_{\mathrm{conn}}
  \to \mathbf{B}^n U(1)_{\mathrm{conn}}$ its differential refinement,
sending [[connections on ∞-bundles]] to [[circle n-bundles with connection]] (see [[∞-Chern-Weil homomorphism]], we may think of this also as an [[extended Lagrangian]] for a [[higher gauge theory]]).

We write $\hat {\mathbf{c}}\mathrm{Struc}_{\mathrm{tw}}(X)$ for the
  corresponding [[twisted cohomology]],

$$
  \array{
    \hat {\mathbf{c}}Struc_{tw}(X) &\stackrel{tw}{\to}&
    H^{n+1}_{diff}(X)
    \\
    {}^{\mathllap{\chi}}\downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G_{conn})
     &
     \stackrel{\hat \mathbf{c}}{\to}
     &
    \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})
  }
  \,.
$$

=--

## Examples
 {#Examples}

Twisted differential $\mathbf{c}$-structures appear in various guises in the [[background gauge field]]s of  [[string theory]] application.

* [[reduction of structure groups]]

  * [[orthogonal structure]] / [[Riemannian metric]]; see the discussion at _[[vielbein]]_ .

  * [[generalized complex geometry]]

  * [[exceptional generalized geometry]]

* [[lift of structure groups]]

* Higher differential spin structures

  * [[twisted spin structure]], [[differential spin structure]]

  * [[twisted spin^c-structure]]

  * [[twisted differential string structure]]

  * [[supergravity C-field]]

  * [[twisted differential fivebrane structure]]

  * [[twisted Wu structure]]

* [[p1-structure]]

  [[2-framing]]

* [[w4-structure]]

  [[w4-orientation of EO(2)-theory]]

* [[differential T-duality]]

## Related concepts

* [[reduction and lift of structure groups]]

* [[field (physics)]]

## References

The notion was introduced in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_
  {#SSSIII}

and expanded on in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_
 {#FSS}

An exposition is in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

Lecture notes include

* [[Urs Schreiber]], _[[twisted smooth cohomology in string theory]]_, Lectures at the ESI-institute (June 2012)

A general account is in section 5.2 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

In

* [[Daniel Freed]], [[Constantin Teleman]], _Relative quantum field theory_ ([arXiv:1212.1692](http://arxiv.org/abs/1212.1692))
 {#FreedTeleman}

it is proposed to call such twisted structures "relative fields".

[[!redirects c-structure]]

[[!redirects twisted differential c-structure]]
[[!redirects twisted differential c-structures]]
[[!redirects tiwsted differential c-structure]]

[[!redirects twisted differential structure]]
[[!redirects twisted differential structures]]


twisted differential cohomology

+-- {: .rightHandSide}
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###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The flavour of [[generalized cohomology]] which is both [[twisted cohomology|twisted]] as well as [[differential cohomology|differential]] is called _twisted differential cohomology_ (or: _differential twisted cohomology_).

A key motivation for twisted differential cohomology is its role in [[string theory]] as the type of [[generalized cohomology theory]] that classifies [[higher gauge fields]] (following [[Dirac charge quantization and generalized differential cohomology|Freed 00]]), notably in the example of [[twisted differential K-theory]] classifying the [[RR-field]] twisted by the [[B-field]] (see [below](#TwistedDifferentialKTheory)).

## Definition

Observing that

1. [[differential cohomology]] may be understood as being the intrinsic [[cohomology]] inside an [[(∞,1)-topos]] $\mathbf{H}$ which is [[cohesive (∞,1)-topos|cohesive]] (see at _[[differential cohomology diagram]]_);

1. [[twisted cohomology]] may be understood as being the intrinsic [[cohomology]] inside the [[tangent (∞,1)-topos]] $T \mathbf{H}$ of the given ambient [[(∞,1)-topos]] $\mathbf{H}$

and since

* the [[tangent (∞,1)-topos]] $T \mathbf{H}$ of an [[cohesive (∞,1)-topos]] $\mathbf{H}$ is itself again cohesive,

one may define twisted differential cohomology in generality to be the intrinsic [[cohomology]] inside [[cohesive (∞,1)-topos|cohesive]] [[tangent (∞,1)-toposes]] $T \mathbf{H}$ ([Schreiber 13, Sec. 4.1.2](#Schreiber13), [Schreiber 14, slide 16](#Schreiber14)).

The [[objects]] of such a $T \mathbf{H}$ may be understood as [[sheaves of spectra|sheaves of]] [[parametrized spectra]] ([Braunack-Mayer 19, Theorem 3.4](#BraunackMayer19)). For example with $\mathbf{H} =$ [[Smooth∞Grpd]] the objects of $T Smth\infty Grpd$ are [[smooth spectra|smooth]] [[parametrized spectra]]. In this sense the formulation of twisted differential cohomology via [[cohesive (∞,1)-topos|cohesive]] [[tangent (∞,1)-topos]] generalizes the classical [[Brown representability theorem]] (identifying plain [[generalized (Eilenberg-Steenrod) cohomology theories]] with plain [[spectra]]) to twisted and differential cohomology.

By means of a suitable [[model category]] [[presentable (∞,1)-category|presentation]] of [[tangent (∞,1)-toposes]], [Braunack-Mayer 19, Example 3.23](#BraunackMayer19) proves that a more explicit component-based definition of twisted differential cohomology due to [Bunke-Nikolaus 14](#BunkeNikolaus14) is a special case of this general definition from [Schreiber 13, Sec. 4.1.2](#Schreiber13).

## Examples

### Twisted de Rham cohomology

A simple example of twisted cohomology is _[[twisted de Rham cohomology]]_ (see there for more), the [[twisted cohomology|twisted]] generalization of [[de Rham cohomology]].

### Twisted differential K-theory
 {#TwistedDifferentialKTheory}

The archetypical example is [[twisted differential K-theory]], which combines [[twisted K-theory]] with [[differential K-theory]]. To the extent that [[D-brane charge]] is classified by [[K-theory]] (see [there](D-brane#ReferencesKTheoryDescription)), it is [[twisted differential K-theory]] that is relevant: the [[differential K-theory|differential]] aspect captures the [[higher gauge field]] called the [[RR-field]], and the [[twisted K-theory|twisted]] aspects captures the [[higher gauge field]] called the [[B-field]], in [[string theory]].

This example of [[D-brane charge]] used to be one of the main motivations for finding a definition and construction of twisted differential cohomology theories. Earlier account of D-brane charge theory had to assume without a construction that such a theory exists (e.g. [DFM 09](twisted+differential+K-theory#DFM09)).

Other [[higher gauge fields]] in [[string theory]]/[[M-theory]] are expected to be [[cocycles]] in twisted differential cohomology theories for other [[generalized cohomology theories]] apart from K-thery. For instance the hypothesis that the [[M-theory]] [[supergravity C-field|C-field]] is topologically a cocycle in [[twisted cohomotopy]] ([[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS19b]], [[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|FSS19c]]) means that with all differential form data added. it is actually a cocycle in _twisted differential cohomotopy_.

## References

The general abstract definition of [[twisted differential cohomology]] as the intrinsic [[cohomology]] of [[cohesive (∞,1)-topos|cohesive]] [[tangent (∞,1)-toposes]] is due to

* {#Schreiber13} [[Urs Schreiber]], section 4.1.2 of _[[schreiber:Differential cohomology in a cohesive (∞,1)-topos]]_, ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930v1))

* {#Schreiber14} [[Urs Schreiber]], around slide 17 of _[[schreiber:Differential generalized cohomology in Cohesive homotopy type theory]]_ , talk at   IHP trimester on _[Semantics of proofs and certified mathematics](https://ihp2014.pps.univ-paris-diderot.fr/doku.php)_ Workshop 1: _[Formalization of Mathematics](https://ihp2014.pps.univ-paris-diderot.fr/doku.php?id=workshop_1)_, [Institut Henri Poincaré](http://www.ihp.fr/en),  Paris, 5-9 May 2014 ([pdf slides](https://ncatlab.org/schreiber/files/SchreiberParis2014.pdf))

* [[Daniel Grady]], [[Hisham Sati]], *Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence*, Algebr. Geom. Topol. 19 (2019) 2899-2960 ([arXiv:1711.06650](https://arxiv.org/abs/1711.06650), [doi:10.2140/agt.2019.19.2899](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-19/issue-6/Twisted-differential-generalized-cohomology-theories-and-their-AtiyahHirzebruch-spectral-sequence/10.2140/agt.2019.19.2899.short))

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The character map in (twisted differential) non-abelian cohomology]]_ ([arXiv:2009.11909](https://arxiv.org/abs/2009.11909))

Discussion of [[twisted differential K-theory]] along these lines:

* [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

  (specifically: [[KO-theory]])

A more component-based definition was given in:

* {#BunkeNikolaus14} [[Ulrich Bunke]], [[Thomas Nikolaus]], _Twisted differential cohomology_, Algebr. Geom. Topol. **19** 4 (2019) 1631-1710 [[arXiv:1406.3231](http://arxiv.org/abs/1406.3231), [euclid:agt/1566439272](https://projecteuclid.org/euclid.agt/1566439272)]

A proof that the definition of [Bunke-Nikolaus 14](#BunkeNikolaus14) is a special case of that of [Schreiber 13, Sec. 4.2.1](#Schreiber13) is due to

* {#BraunackMayer19} [[Vincent Braunack-Mayer]], Section 3.1 of _Combinatorial parametrised spectra_, based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]] ([arXiv:1907.08496](https://arxiv.org/abs/1907.08496))

For more discussion of twisted differential cohomology as the intrinsic [[cohomology]] of _[[tangent cohesive (∞,1)-topos|Goodwillie-tangent spaces of cohesive (∞,1)-topos]]_ see there.

[[!redirects differential twisted cohomology]]


twisted differential K-theory

+-- {: .rightHandSide}
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+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Twisted differential K-theory_ is the [[twisted differential cohomology]] version [[K-theory]]: the combination of [[differential K-theory]] and [[twisted K-theory]].

Roughly this is an extension of [[twisted de Rham cohomology]] by [[twisted K-theory]] via the [[twisted Chern character]] map.

To the extent that [[D-brane charge]] is classified by [[K-theory]] (see [there](D-brane#ReferencesKTheoryDescription)), it is [[twisted differential K-theory]] that is relevant: the [[differential K-theory|differential]] aspect captures the [[higher gauge field]] called the [[RR-field]], and the [[twisted K-theory|twisted]] aspects captures the [[higher gauge field]] called the [[B-field]], in [[string theory]].

This example of [[D-brane charge]] used to be one of the main motivations for finding a definition and construction of twisted differential cohomology theories. Earlier articles on D-brane charge had to assume that such a theory exists (e.g. [DFM 09](twisted+differential+K-theory#DFM09)). Actual models now exist ([Grady-Sati 19a](#GradySati19a), [Grady-Sati 19b](#GradySati19b))

## Related concepts

* [[K-theory]]

* [[topological K-theory]]

* [[twisted K-theory]]

* [[differential K-theory]]

* [[twisted equivariant K-theory]]

* [[twisted ordinary differential cohomology]]

## References

Preliminary discussion included

* {#Carey} [[Alan Carey]], _Differential twisted K-theory and applications_ , ESI preprint ([pdf](http://www.esi.ac.at/preprints/esi1945.pdf))

* {#KahleValentino} [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_

and specifically the need for twisted differential K-theory for the description of [[type II string theory]] backgrounds with their [[RR-fields]] and [[B-fields]] on [[orientifolds]] was highlighted in

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

Actual constructions appear with

* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))

Review in

* [[Ulrich Bunke]], [[Thomas Schick]], Section 7 in: _Differential K-theory: A survey_, in:  [[Christian Bär]], Joachim Lohkamp, Matthias Schwarz (eds.), _Global Differential Geometry_, Springer 2012 ([doi:10.1007/978-3-642-22842-1](https://doi.org/10.1007/978-3-642-22842-1))

A definition and proof of its pertinent properties of [[twisted differential cohomology]] in general and of twisted differential K-theory in particular is due to

* {#BunkeNikolaus14} [[Ulrich Bunke]], [[Thomas Nikolaus]], _Twisted differential cohomology_ ([arXiv:1406.3231](http://arxiv.org/abs/1406.3231))

See also:

* [[Byungdo Park]], _Geometric models of twisted differential K-theory I_,  J. Homotopy Relat. Struct. 13, 143–167 (2018) ([arXiv:1602.02292](https://arxiv.org/abs/1602.02292), [doi:10.1007/s40062-017-0177-z](https://doi.org/10.1007/s40062-017-0177-z))

More detailed discussion of twisted differential K-theory for [[KU]]/[[KO]] and its application to [[K-theory classification of D-brane charge]]:

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], *Ramond-Ramond fields and twisted differential K-theory*, Advances in Theoretical and Mathematical Physics **26** 5 (2022) [[doi:10.4310/ATMP.2022.v26.n5.a2](https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a2), [arXiv:1903.08843](https://arxiv.org/abs/1903.08843)]

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

* [[Fabio Ferrari Ruffino]], [[Juan Carlos Rocha Barriga]], _Twisted differential K-characters and D-branes_, Nuclear Physics B Volume 960, November 2020, 115169 ([arXiv:2009.04223](https://arxiv.org/abs/2009.04223). [doi:10.1016/j.nuclphysb.2020.115169](https://doi.org/10.1016/j.nuclphysb.2020.115169))

[[!redirects differential twisted K-theory]]


twisted Eilenberg-MacLane space

[[!redirects twisted Eilenberg -  Mac Lane space]]

##Idea

[[Eilenberg - Mac Lane space]]s represent the various classical cohomology theories, but for the classical form of [[twisted cohomology]] the coefficients are in a [[local system]], so one needs to 'twist' the representing object over a base corresponding to the classifying space of some groupoid.

This has sometimes been called a generalized Eilenberg-MacLane space, but is perhaps better termed a _twisted Eilenberg-MacLane space_ to avoid possible confusion.

The construction of these is to be found in the entry on [[twisted cohomology]].

[[!redirects twisted Eilenberg - Mac Lane space]]


twisted equivariant Chern character

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[Chern character]] from [[K-theory]] to [[twisted equivariant K-theory]], hence the joint generalization of the [[twisted Chern character]] and the [[equivariant Chern character]], typically understood in the context of [[topological K-theory]] and then taking values in [[twisted de Rham cohomology|twisted]] [[equivariant de Rham cohomology]] (including a twist from [[inner local systems]]).

## Related concepts

* [[inner local system]]

* [[Chern character]] ([[twisted Chern character|twisted]], [[equivariant Chern character|equivariant]])

* [[topological K-theory]] ([[twisted K-theory|twisted]], [[equivariant K-theory|equivariant]], [[twisted equivariant K-theory|twisted & equivariant]])

* [[Chern-Dold character]]

## References
 {#References}

* {#AdemRuan03} [[Alejandro Adem]], [[Yongbin Ruan]], Section 7 of: *Twisted Orbifold K-Theory*, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))

* {#MathaiStevenson03} [[Varghese Mathai]], [[Danny Stevenson]], *Chern character in twisted K-theory: equivariant and holomorphic cases*, Commun. Math. Phys. **236** (2003) 161-186 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7))

* {#TuXu06} [[Jean-Louis Tu]], [[Ping Xu]], *Chern character for twisted K-theory of orbifolds*, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 ([arXiv:math/0505267](https://arxiv.org/abs/math/0505267), [doi:10.1016/j.aim.2005.12.001](https://doi.org/10.1016/j.aim.2005.12.001))

* [[Daniel S. Freed]], [[Michael J. Hopkins]], [[Constantin Teleman]], Section 1 of: *Twisted equivariant K-theory with complex coefficients* **1** 1 (2008) 16-44 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

[[!redirects twisted equivariant Chern character]]


twisted equivariant cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Twisted equivariant cohomology is the joint generalization of

* [[twisted cohomology]]

and

* [[equivariant cohomology]].

## Examples

* [[twisted equivariant K-theory]]

## References

* {#Lind14} [[John Lind]], *Equivariantly Twisted Cohomology Theories*, Baltimore 2014 (slides  [pdf](https://jlind.yourweb.csuchico.edu/papers/equivtwists_jmm2014.pdf), [[LindEquivariantlyTwistedCohomology.pdf:file]])

[[!redirects twisted equivariant cohomology theory]]
[[!redirects twisted equivariant cohomology theories]]

[[!redirects equivariant twisted cohomology theory]]
[[!redirects equivariant twisted cohomology theories]]


twisted equivariant differential K-theory

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Twisted Equivariant Differential (TED) K-theory_ is meant to be the joint generalization of all three variations of [[topological K-theory]] given by:

1. [[twisted K-theory]]

1. [[equivariant K-theory]]

1. [[differential K-theory]]

hence the joint generalization of:

1. [[twisted equivariant K-theory]]

1. [[twisted differential K-theory]]

1. [[equivariant differential K-theory]].

It is this joint generalization that is, ultimately, required by

1. the [[K-theory classification of D-brane charge]],

   conjectured to apply to [[M-theory|non-perturbative]] [[type II string theory]];

1. the [[K-theory classification of topological phases of matter]],

   conjectured to apply in [[topological phases of matter|topological]] [[condensed matter theory]].

## References

The construction of full _twisted & equivariant & differential_ cohomology theories had long been left open. It does follow as an immediate consequence of the _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|twisted]] & [[schreiber:The Character Map in Equivariant Twistorial Cohomotopy|equivariant]]_ [[Chern-Dold character]] and as such is discussed in:

* [[Hisham Sati]], [[Urs Schreiber]], _

  **[[schreiber:Twisted Equivariant Differential non-abelian generalized cohomology|Twisted Equivariant Differential non-abelian generalized cohomology]]**

  (in preparation)

The twisted + equivariant + *flat* differential sector of K-theory is more readily described:

* [[Hisham Sati]], [[Urs Schreiber]],

  **[[schreiber:Anyonic defect branes in TED-K-theory]]*

  Rev. Math. Phys. (2023)

  [arXiv:2203.11838](https://arxiv.org/abs/2203.11838)

  [doi:10.1142/S0129055X23500095](https://doi.org/10.1142/S0129055X23500095)


twisted equivariant K-theory

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Twisted equivariant K-theory is the special case of [[twisted equivariant cohomology]] for the case that the [[cohomology theory]] is [[K-theory]] (often understood to be [[topological K-theory]], by default), hence the  joint generalization of [[twisted K-theory]] and [[equivariant K-theory]].

This is often considered in the special case of [[twisted ad-equivariant K-theory]].

## Related concepts

* [[twisted equivariant Chern character]]

* [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]

* [[orbifold K-theory]]

* [[twisted ad-equivariant K-theory]]

* [[twisted ad-equivariant Tate K-theory]]

* [[twisted differential K-theory]]

* [[twisted equivariant differential K-theory]]

* [[K-theory classification of D-brane charge]]

* [[K-theory classification of topological phases of matter]], [[topological phases of matter]]

## References

In a context of [[orbifold K-theory]] and for [[torsion subgroup|torsion]] class twists ("[[discrete torsion]]"):

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168), [doi:10.1007/s00220-003-0849-x](https://doi.org/10.1007/s00220-003-0849-x))

With an eye towards [[twisted ad-equivariant K-theory]]:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], *Twisted equivariant K-theory with complex coefficients*, Journal of Topology **1** 1 (2007) [[arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001)]

In terms of [[bundle gerbes]] on [[differentiable stacks]]:

* [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_, Annales Scientifiques de l’École Normale Supérieure Volume 37, Issue 6, November–December 2004, Pages 841-910 ([arXiv:math/0306138](https://arxiv.org/abs/math/0306138), [doi:10.1016/j.ansens.2004.10.002](https://doi.org/10.1016/j.ansens.2004.10.002))

General definition in terms of [[equivariant function|equivariant]] [[sections]] of [[equivariant bundles]] of [equivariant classifying spaces](equivariant+K-theory#ReferencesRepresentingSpectrum) for [[equivariant K-theory]]:

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], Section 6 of: _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

* {#Cantarero09} [[Jose Cantarero]], _Equivariant K-theory, groupoids and proper actions_, Thesis 2009 ([ubctheses:1.0068026](https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0068026), [[CantareroEquivariantKTheory.pdf:file]])

Reviewed in:

* [[Valentin Zakharevich]], Section 2.2 of: _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf), [[ZakharevichKTheoryAndVerlindeRing.pdf:file]])

Alternatively, in terms of [[Grothendieck groups]] of [[equivariant bundle|equivariant]] [[twisted vector bundles]]:

* [[Christopher Dwyer]], _Twisted equivariant K-theory for proper actions of discrete groups_, K-Theory v. 38, n. 2 (January, 2008): 95-111 ([arXiv:0710.2136](https://arxiv.org/abs/0710.2136), [doi:10.1007/s10977-007-9016-z](http://dx.doi.org/10.1007/s10977-007-9016-z))

On the [[cohomology ring]]-structure:

* [[Jean-Louis Tu]], [[Ping Xu]], *The ring structure for equivariant twisted K-theory*, J. Reine Angew. Math. 635 (2009), 97–148 ([arXiv:math/0604160](https://arxiv.org/abs/math/0604160), [doi:10.1515/CRELLE.2009.077](https://doi.org/10.1515/CRELLE.2009.077))

On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

* [[Jesus Espinoza]], [[Bernardo Uribe]], *Topological properties of spaces of projective unitary representations*, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 ([arXiv:1511.06785](https://arxiv.org/abs/1511.06785), [scielo:S0370-39082016000200013](http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013), [doi:10.18257/raccefyn.317](https://doi.org/10.18257/raccefyn.317))

* {#UribeLueck14} [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

The corresponding version of the [[Atiyah-Hirzebruch spectral sequence]]:

* [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Bernardo Uribe]], [[Mario Velásquez]], Section 3.2 of: _Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions_, Proceedings of the Edinburgh Mathematical Society **61** 1 (2018) ([arXiv:1307.1003](https://arxiv.org/abs/1307.1003), [doi:10.1017/S0013091517000281](https://doi.org/10.1017/S0013091517000281))

Relation to [[projective representations]]:

* José Manuel Gómez, Johana Ramírez, _A Decomposition of twisted equivariant K-theory_ ([arXiv:2001.02164](https://arxiv.org/abs/2001.02164))

[[!redirects equivariant twisted K-theory]]

[[!redirects twisted equivariant K-cohomology]]

[[!redirects twisted equivariant topological K-theory]]

[[!redirects TE-K theory]]

[[!redirects TE-K cohomology]]


twisted form

+-- {: .rightHandSide}
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### Context
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Typically, if one has a generalized [[topological space|topology]] $\mathcal{T}$ and one has an object $M$ "over" that topology (e.g. a [[sheaf]], [[bundle]] or [[stack]]) one can ask about other objects over $\mathcal{T}$ which are [[local isomorphism|locally isomorphic]] to $M$. Here, all of these terms (e.g. topology, equivalent, over) depend on the situation at hand. However, we often say than an object $M'$ which is locally isomorphic to $M$ is a _twisted form of $M$_. For some authors, the notion of being _locally isomorphic_ is only for a chosen [[cover]] $U\to X$ of an object $X$. At least for the extent of this wiki, discussion of such twisted forms will always reference the cover. In other words, to say that $M'$ is a _twisted form of $M$ over $X$_ is to say that $M$ and $M'$ are locally isomorphic for all covers. However, to say that $M'$ is a _twisted form of $M$ for $U\to X$_ would indicate that $M$ and $M'$ are only isomorphic when [[base change|pulled back]] along the given cover.

The notion of a twisted form is very general, and has manifestations in [[differential geometry]], [[commutative algebra]], [[category theory]], [[algebraic geometry]], and more recently in [[homotopy theory]]. However, one unifying property in all of these cases is that such forms should be classified by an appropriate [[cohomology]]. Typically, this will be a sort of [[sheaf cohomology]] or [[Čech cohomology]] with coefficients in a sheaf of [[automorphisms]] of the object of interest. Moreover, computations of such cohomologies can often be simplified by identifying them as [[Galois cohomology]] or [[Hopf-Galois cohomology]]. In many cases, twisted forms are in bijection with some relevant notion of [[torsor]] for the automorphism object and when a specific cover is referenced, twisted forms are equivalent to [[descent data]] for the object along that cover.

## History

Probably the most famous twisted form computation is [[Hilbert's Theorem 90]]. That theorem can be reinterpreted as saying that for a [[Galois extension]] of fields $K\to L$, and an $L$-[[vector space]] $W$, there is exactly one $K$-vector space $V$ such that $W\cong V\otimes_K L$. In that case, the relevant [[nonabelian cohomology]] has been reinterpreted to look like [[Galois cohomology]]. [[Serre]] discussed twisted forms at some length (though he just called them "forms") in his book _Corps Locaux_.

## Twisted forms in differential geometry

In [[differential geometry]], one often twists [[differential forms]] by a [[line bundle]]; see [differential form#twisted](differential+form#twisted).

## Twisted forms in commutative algebra

Given a [[homomorphism]] of [[commutative rings]] $\phi:R\to S$, we have an [[extension of scalars]] [[functor]] $-\otimes_R S:Mod_R\to Mod_S$. For a given $R$-[[module]] $M$, a _twisted form_ of $M$ is another $R$-module $M'$ such that $M\otimes_R S$ is [[isomorphism|isomorphic]] to $M'\otimes_R S$.  It turns out that if $\phi$ is of [[descent morphism|effective descent]] for modules, then isomorphism classes of twisted forms of a given $S$-module $N\cong M\otimes_R S$ are in [[bijection]] with the set of [[descent]] data on $N$! What's more, this set can be computed using [[nonabelian cohomology]].

[[!redirects twisted form]]
[[!redirects twisted forms]]
[[!redirects forme tordue]]
[[!redirects formes tordues]]


twisted generalized cohomology in linear homotopy type theory -- table

**[[twisted cohomology|twisted]] [[generalized cohomology theory]] is conjecturally [[categorical semantics|∞-categorical]] [[semantics]] of [[linear homotopy type theory]]**:

| [[linear homotopy type theory]] | [[generalized cohomology theory]] | [[quantum theory]] |
|---------------------------------|-----------------------------------|--------------------|
| [[linear type]] | ([[module spectrum|module]]-)[[spectrum]] |  |
| [[multiplicative conjunction]] | [[smash product of spectra]] |  [[composite system]] |
| [[dependent linear type]]           | [[(infinity,1)-module bundle|module spectrum bundle]] |   |
| [[Frobenius reciprocity]] | [[six operation yoga]] in [[Wirthmüller context]] |  |
| [[dual object|dual type]] (linear negation) | [[Spanier-Whitehead duality]] |  |
| [[invertible type]]  | [[twisted generalized cohomology|twist]]  | [[prequantum line bundle]]  |
| [[dependent sum]] | [[generalized homology]] [[spectrum]] | [[space of quantum states]] ("[[bra-ket|bra]]") |
| [[dual type|dual]] of [[dependent sum]] | [[generalized cohomology]] [[spectrum]] | [[space of quantum states]] ("[[bra-ket|ket]]") |
| [[linear implication]] | [[bivariant cohomology]] | [[quantum operators]] |
| [[exponential modality]] |  |  [[Fock space]] |
| [[dependent sum]] over [[finite homotopy type]] (of twist) | [[suspension spectrum]] ([[Thom spectrum]]) |  |
| [[dualizable object|dualizable]] [[dependent sum]] over [[finite homotopy type]] | [[Atiyah duality]] between [[Thom spectrum]] and [[suspension spectrum]] |  |
| (twisted) [[self-dual object|self-dual type]] | [[Poincaré duality]] | [[inner product]] |
| [[dependent sum]] coinciding with [[dependent product]] | [[ambidextrous adjunction|ambidexterity]], [[semiadditive (∞,1)-category|semiadditivity]] |  |
| [[dependent sum]] coinciding with [[dependent product]] up to [[invertible type]] | [[Wirthmüller isomorphism]] |  |
| $(\sum_f \dashv f^\ast)$-[[counit of an adjunction|counit]] | pushforward in [[generalized homology]] |  |
| (twisted-)[[self-dual object|self-duality]]-[induced dagger](self-dual+object#RelationToDaggerCompactStructure) of this counit | ([[twisted Umkehr map|twisted]]-)[[Umkehr map]]/[[fiber integration]] |  |
|  [[linear polynomial functor]]  |  [[correspondence]]  |  space of [[trajectories]]  |
|  [[linear polynomial functor]] with [[linear implication]] | [[integral kernel]] ([[pure motive]]) | [[prequantized Lagrangian correspondence]]/[[action functional]]  |
| composite of this [[linear implication]] with daggered-counit followed by unit | [[integral transform]] | [[motivic quantization|motivic]]/cohomological [[path integral]] |
| [[trace]]  | [[Euler characteristic]] | [[partition function]] |


twisted infinity-bundle

#Contents#
* table of contents
{:toc}

## Idea

Where a [[principal ∞-bundle]] is the "geometric" incarnation of a [[cocycle]] in [[cohomology]], more generally a _twisted $\infty$-bundle_ is the geometric incarnation of a cocycle in [[twisted cohomology]].

## Definition

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Let $\hat G \to G$ in $Grp(\mathbf{H})$ be a morphism of [[∞-groups]] in $\mathbf{H}$, write $\mathbf{c} : \mathbf{B}\hat G \to \mathbf{B}G$ for the corresponding [[delooping]] and $\mathbf{B}A \to \mathbf{B}\hat G$ for the corresponding [[homotopy fiber]], so that we have a universal coefficient bundle

$$
  \array{
    \mathbf{B}A &\to& \mathbf{B}\hat G
    \\
    && \downarrow^{\mathrlap{\mathbf{c}}}
    \\
    && \mathbf{B}G
  }
  \,.
$$

Then let $\phi : X \to \mathbf{B}G$ be a twisting cocycle with corresponding $G$-[[principal ∞-bundle]] $P \to X$; and let $\sigma : X \to \mathbf{B}\hat G$ be a [[cocycle]] in $\phi$-[[twisted cohomology]]. By the [[pasting law]] this induces a twisted $G$-equivariant $A$-principal $\infty$-bundle $\tilde P$ on the total space of $P$

$$
  \array{
    \tilde P &\to& *
    \\
    \downarrow && \downarrow
    \\
    P &\stackrel{\tilde \sigma}{\to}& \mathbf{B}A &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X
     &\stackrel{\sigma}{\to}&
   \mathbf{B}\hat G
     &\stackrel{\mathbf{c}}{\to}&
   \mathbf{B}G
  }
  \,.
$$

This is the twisted $\infty$-bundle classified by $\sigma$

## Examples

* [[projective representation]]

* [[twisted bundle]]

* [[Jandl gerbe]]

* [[twisted differential string structure|twisted String 2-bundle]]

## Related concepts

* [[principal bundle]] / [[torsor]] / [[associated bundle]] / [[twisted bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]] / [[∞-gerbe]],

* [[vector bundle]]

* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]

## References

Section I 4.3 in

* [[schreiber:Principal ∞-bundles -- theory, presentations and applications]]

[[!redirects twisted ∞-bundle]]

[[!redirects twisted ∞-bundles]]
[[!redirects twisted infinity-bundles]]


twisted K-homology

## Related entries

* [[twisted K-theory]]

## References

* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_ (2007) ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))

* [[Eckhard Meinrenken]], _Twisted K-homology and group-valued moment maps_ (2010) ([arXiv:1008.1261](http://arxiv.org/abs/1008.1261))

[[!redirects twisted geometric cycle]]
[[!redirects twisted geometric cycles]]

* [[Jean-Louis Tu]], _Twisted K-theory and Poincaré duality_ (2006) ([arXiv:math/0609556](http://arxiv.org/abs/math/0609556))


twisted K-theory

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Twisted K-theory_ is a [[twisted cohomology]] version of ([[topological K-theory|topological]]) [[K-theory]].

The most famous twist is by a class in degree 3 [[ordinary cohomology]] (geometrically a $U(1)$-[[bundle gerbe]] or [[circle 2-group]]-[[principal 2-bundle]]), but there are various other twists.

## Definition

### By sections of associated $K U$-bundles

Write [[KU]] for the [[spectrum]] of complex [[topological K-theory]]. Its degree-0 space is, up to [[weak homotopy equivalence]],  the space

$$
  B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z}
$$

or the space $Fred(\mathcal{H})$ of  [[Fredholm operator]]s on some separable [[Hilbert space]] $\mathcal{H}$.

$$
  (K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H})
  \,.
$$

The ordinary [[topological K-theory]] of a suitable [[topological space]] $X$ is given by the set of [[homotopy classes]] of maps from (the [[suspension spectrum]] of) $X$ to $KU$:

$$
  K(X)_\bullet \simeq [X, (K U)_\bullet]
  \,.
$$

The [[projective unitary group]] $P U(\mathcal{H})$ (a [[topological group]]) acts canonically by [[automorphism]]s on $(K U)_0$. (This follows by the identification of $K U_0$ with the space of [[Fredholm operators]], see [below](#ByBundlesOfFredholmOperators)) Therefore for $P \to X$ any $PU(\mathcal{H})$-[[principal bundle]], we can form the [[associated bundle]] $P \times_{P U(\mathcal{H})} (K U)_0$.

Since the [[homotopy type]] of $P U(\mathcal{H})$ is that of an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$, there is precisely one isomorphism class of such bundles representing a class $\alpha \in H^3(X, \mathbb{Z})$.

+-- {: .num_defn #SpectrumBundDefinition}
###### Definition

The **twisted K-theory** with twist $\alpha \in H^3(X, \mathbb{Z})$ is the set of [[homotopy]]-classes of [[section]]s of such a bundle

$$
  K_\alpha(X)^0
  :=
  \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0)
  \,.
$$

Similarily the reduced $\alpha$-twisted K-theory is the subset

$$
  \tilde K_\alpha(X)^0
  :=
  \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U)
  \,.
$$

=--

### By bundles of Fredholm operators
 {#ByBundlesOfFredholmOperators}

The following is due to ([Atiyah-Singer 69](#AtiyahSinger69), [Atiyah-Segal 04](#AtiyahSegal04)).

Write

* $Cl_n \coloneqq Cl^{\mathbb{C}}(\mathbb{R}^n,\langle -,-\rangle)$

  for the [[complexification]] of the [[Clifford algebra]] of the [[Cartesian space]] $\mathbb{R}^n$ with its standard [[inner product]];

* $S_n$ for its $\mathbb{Z}/2\mathbb{Z}$-[[graded module|graded]] [[irreducible module]] (see at _[[spin representation]]_);

* $H_0$ for [[generalized the|the]] $\mathbb{Z}/2\mathbb{Z}$-graded [[separable Hilbert space]] whose even and odd part are both infinite-dimensional.

+-- {: .num_defn #Fredn}
###### Definition

For $n \in \mathbb{N}$, the [[topological space]] $Fred^{(n)}$ of [[Fredholm operators]] on $S_n \otimes H_0$ is the [[set]]

$$
  Fred^{(n)}
  \coloneqq
  \left\{
    F \in \mathcal{B}(S_n \otimes H_0)
    \;|\;
    F \, odd\,,
    F^\ast = F \,,
    F^2 - 1 \in \mathcal{K}(S_n \otimes H_0)\,,
    [F,\gamma] = 0 \, for\, \gamma \in Cl_n
  \right\}
$$

(where $\mathcal{B}$ denotes [[bounded operators]] and $\mathcal{K}$ denotes [[compact operators]] and where $[-,-]$ denotes the [[graded commutator]]) and the [[topological space|topology]] on this set is the [[subspace topology]] induced by the embedding

$$
  Fred^{(n)}
  \hookrightarrow
  \mathcal{B}(S_n \otimes H_0)
  \times
  \mathcal{K}(S_n \otimes H_0)
$$

given by

$$
  F\mapsto (F, F^2 - 1)
  \,,
$$

where $\mathcal{B}$ is equipped with the [[compact-open topology]] and $\mathcal{K}$ with the [[norm topology]].

=--

([Atiyah-Singer 69, p. 7](#AtiyahSinger69), [Atiyah-Segal 04, p. 21](#AtiyahSegal04), [Freed-Hopkins-Teleman 11, def. A.40](#FreedHopkinsTeleman11))

These spaces indeed form a model for the [[KU]] [[spectrum]]:

+-- {: .num_prop }
###### Proposition

For all $n \in \mathbb{N}$ there are [[natural equivalence|natural]] [[weak homotopy equivalences]]

$$
  Fred^{(n+1)} \stackrel{\simeq}{\longrightarrow} \Omega Fred^{(n)}
$$

and

$$
  Fred^{(n+2)} \stackrel{\simeq}{\longrightarrow} Fred^{(n)}
$$

between the spaces of graded [[Fredholm operators]] of def. \ref{Fredn} and their [[loop spaces]].

=--

([Atiyah-Singer 69, theorem B(k)](#AtiyahSinger69), [Atiyah-Segal 04 (4.2)](#AtiyahSegal04), [Freed-Hopkins-Teleman 11, below def. A.40](#FreedHopkinsTeleman11))

Regard the [[stable unitary group]] $U(H_0)$ as equipped with the [[subspace topology]] induced by the inclusion

$$
  U(H_0) \stackrel{(id,(-)^{-1})}{\hookrightarrow} \mathcal{B}(H_0)\times\mathcal{B}(H_0)
$$

from the [[compact-open topology]] on the [[bounded linear operators]].

+-- {: .num_prop }
###### Proposition

The [[conjugation action]] of the [[stable unitary group]] $U(H_0)$ on $Fred^{(n)}$, def. \ref{Fredn}, is [[continuous functions|continuous]].

=--

This follows with ([Atiyah-Segal 04, prop. A1.1](#AtiyahSegal04)).

+-- {: .num_defn }
###### Definition

Given a class $\chi \in H^3(X,\mathbb{Z})$ represented by a $PU(H_0)$-bundle $P \to X$ with [[associated bundle|associated]] Fredholm bundle

$$
  Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)}
  \,,
$$

then the corresponding $\chi$-twisted [[cohomology]] [[spectrum]] is that consisting of the [[spaces of sections]]

$$
  \Gamma(X, Fred^{(n)+ \chi})
  \,.
$$

=--

([Freed-Hopkins-Teleman 11, def. 3.14](#FreedHopkinsTeleman11))

### By twisted vector bundles (gerbe modules)

+-- {: .num_defn #TwBundDefinition}
###### Definition

Let $\alpha \in H^3(X, \mathbb{Z})$ be a class in degree-3 [[integral cohomology]] and let $P \in \mathbf{H}^3(X, \mathbf{B}^2 U(1))$ be any [[cocycle]] representative, which we may think of either as giving a
[[circle n-bundle with connection|circle 2-bundle]] or a [[bundle gerbe]].

Write $TwBund(X, P)$ for the [[groupoid]] of [[twisted bundle]]s on $X$ with twist given by $P$. Then let

$$
  \tilde K_\alpha(X) := TwBund(X,P)
$$

be the set of [[isomorphism]] classes of twisted bundles. Call this the **twisted K-theory** of $X$ with twist $\alpha$.
=--

> (Some technical details need to be added for the non-torsion case.)

+-- {: .num_prop}
###### Proposition

This definition of twisted $K_0$ is equivalent to that of prop. \ref{SpectrumBundDefinition}.

=--

This is ([CBMMS, prop. 6.4, prop. 7.3](#CBMMS)).

### By KK-theory of twisted convolution algebras

A [[circle 2-group]] [[principal 2-bundle]] is also incarnated as a [[centrally extended Lie groupoid]]. The corresponding [[twisted groupoid convolution algebra]] has as its [[operator K-theory]] the twisted K-theory of the base space (or base-[[stack]]). See at _[[KK-theory]]_ for more on this.

## Other constructions

Let $Vectr$ be the [[stack]] of [[vectorial bundles]]. (If we just take [[vector bundle]]s we get a notion of twisted K-theory that only allows twists that are finite order elements in their [[cohomology group]]).

There is a canonical morphism

$$
  \rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr
$$

coming from the standard [[representation]] of the [[group]] $U(1)$.

Let $\mathbf{B}_{\otimes} Vectr$ be the [[delooping]] of $Vectr$ with respect to the [[tensor product]] [[monoidal category|monoidal structure]] (not the additive structure).

Then we have a [[fibration sequence]]

$$
  Vectr \to {*} \to \mathbf{B}_\otimes Vectr
$$

of [[(infinity,1)-category|(infinity,1)-categories]] (instead of [[infinity-groupoid]]s).

The entire morphism above deloops

$$
  \mathbf{B}\rho : \mathbf{B}^2 U(1) \to \mathbf{B}_\otimes Vect \hookrightarrow \mathbf{B}_{\otimes} Vectr
$$

being the standard representation of the [[2-group]] $\mathbf{B}U(1)$.

From the general nonsense of [[twisted cohomology]] this induces canonically now for every $\mathbf{B}^2 U(1)$-[[cohomology|cocycle]] $c$ (for instance given by a [[bundle gerbe]]) a notion of $c$-twisted $Vectr$-cohomology:

$$
 \array{
   \mathbf{H}^c(X, Vectr)
   &\to&
   {*}
   \\
   \downarrow
   &&
   \downarrow^{\mathbf{B}\rho \circ c}
   \\
   {*} &\to& \mathbf{H}(X,\mathbf{B}_\otimes Vectr)
 }
  \,.
$$

After unwrapping what this means, the result of ([Gomi](#Gomi))
shows that [[concordance]] classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory.

## Twists

By the general discussion of [[twisted cohomology]] the [[moduli space]] for the twists of [[periodic complex K-theory]] $KU$ is the [[Picard ∞-group]] in $KU Mod$. The "geometric" twists among these have as moduli space the non-connected delooping $bgl_1^\ast(KU)$ of the [[∞-group of units]] of $KU$.

A model for this in 4-truncation is given by [[super line 2-bundles]]. For the moment see there for further discussion and further references.

## Related concepts

* [[K-theory]]

* [[topological K-theory]], [[KR-theory]]

* **twisted K-theory**

  * [[twisted equivariant K-theory]]

    * [[twisted ad-equivariant K-theory]]

  * [[differential K-theory]]

  * [[twisted differential K-theory]]

  * [[fiber integration in K-theory]]

* [[twisted ordinary cohomology]]

* [[twisted Bredon cohomology]]

* [[twisted Cohomotopy]]

## References

### General

The concept of twisted K-theory originates in

* {#Karoubi68} [[Max Karoubi]], _Algèbres de Clifford et K-théorie._ Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968) ([numdam:ASENS_1968_4_1_2_161_0](http://www.numdam.org/item/ASENS_1968_4_1_2_161_0))

* {#DonovanKaroubi70} [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and $K$-theory with local coefficients_, Publications Mathématiques de l'IHÉS, 38 (1970), p. 5-25  ([numdam:PMIHES_1970__38__5_0](http://www.numdam.org/item?id=PMIHES_1970__38__5_0))

which discusses twists of $KO$ and $KU$ over some $X$ by elements in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$.

Including the twist in degree 1 (see also at *[[PU(ℋ)]]*):

* {#Parker88} [[Ellen Maycock Parker]], around Prop. 2.2 of: *The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras*, Transactions of the American Mathematical Society **308** 1 (1988) [[jstor:2000953](https://www.jstor.org/stable/2000953)]

reviewed in

* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393  ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))

The formulation in terms of sections of Fredholm bundles seems to go back to

* {#Rosenberg89} [[Jonathan Rosenberg]], _Continuous-trace algebras from the bundle theoretic point of view_, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381 ([doi:10.1017/S1446788700033097](https://doi.org/10.1017/S1446788700033097))

and is expanded on in:

* {#FreedHopkinsTeleman02} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], Diagram (2.6) in: _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1, 2007 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory_, Ukrainian Math. Bull. **1** 3 (2004) [[arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf)]

* [[Peter May]], [[Johann Sigurdsson]], Section 22.3 of: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006 ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

* {#AtiyahSegal05} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory and cohomology_ ([arXiv:math/0510674](http://arxiv.org/abs/math/0510674))

* {#AndoBlumbergGepner10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], sections 2.1 and 7 of: _Twists of K-theory and TMF_, in [[Jonathan Rosenberg]] et al. (eds.), _Superstrings, Geometry, Topology, and $C^\ast$-algebras_, volume 81 of _Proceedings of Symposia in Pure Mathematics_, 2009 ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004), [doi:10.1090/pspum/081](https://doi.org/10.1090/pspum/081))

A comprehensive account of twisted K-theory with twists in $H^3(X, \mathbb{Z})$ is in:

* {#AtiyahSinger69} [[Michael Atiyah]], [[Isadore Singer]], _Index theory for skew-adjoint Fredholm operators_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques
January 1969, Volume 37, Issue 1, pp 5-26 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/askew.pdf))

and for more general twists in

* {#FreedHopkinsTeleman11} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _[[Loop Groups and Twisted K-Theory]] I_ , J. Topology, 4 (2011), 737-789  ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906))

See also

* [[Max Karoubi]], _Twisted bundles and twisted K-theory_, in:  Guillermo Cortiñas (ed.) _Topics in Noncommutative Geometry_ ([arXiv:1012.2512](https://arxiv.org/abs/1012.2512), [ISBN:978-0-8218-6864-5](https://bookstore.ams.org/cmip-16))

Textbook accounts:

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Part IV of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

Discussion of [[twisted ad-equivariant K-theory]] and relation to [[loop group]] [[representations]] and the [[Verlinde ring]], now again with twists in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$, is in the series of articles

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Twisted K-theory and loop group representations_ [arXiv:math/0312155](http://arxiv.org/abs/math/0312155);  _[[Loop Groups and Twisted K-Theory]] I_ ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906)) {#FHT07}; _[[Loop Groups and Twisted K-Theory]] II_ ([arXiv:math/0511232](http://arxiv.org/abs/math/0511232))

The result on twisted K-groups has been lifted to an equivalence of categories in

* [[Daniel Freed]], [[Constantin Teleman]], _Dirac families for loop groups as matrix factorizations_, [arxiv/1409.6051](http://arxiv.org/abs/1409.6051)

Discussion in terms of [[Karoubi K-theory]]/[[Clifford module bundles]] is in

* [[Max Karoubi]], _Clifford modules and twisted K-theory_, Proceedings of the International Conference on Clifford algebras (ICCA7) ([arXiv:0801.2794](http://arxiv.org/abs/0801.2794))

See the references at _[[(infinity,1)-vector bundle]]_ for more on this.

Discussion in terms of [[twisted bundles]]/[[bundle gerbe modules]] is in

* {#CBMMS}  [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]], [[Danny Stevenson]], _Twisted K-theory and K-theory of bundle gerbes_,  Commun Math Phys, 228 (2002) 17-49 ([arXiv:hep-th/0106194](http://arxiv.org/abs/hep-th/0106194), [doi:10.1007/s002200200646](https://doi.org/10.1007/s002200200646))

but apparently contains a mistake, as pointed out in

* [[Alan Carey]], [[Bai-Ling Wang]], top of p. 10 in _Thom isomorphism and Push-forward map in twisted K-theory_, Journal of K-Theory **1** 2 (2008) 357-393 ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))

The generalization of this to [[groupoid K-theory]] is in ([FHT 07, around p. 26](#FHT07)) and

* {#TuXuLG03} [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_ ([arXiv:math/0306138](http://arxiv.org/abs/math/0306138))

(which establishes the relation to [[KK-theory]]).

* {#Karoubi} [[Max Karoubi]], _Twisted bundles and twisted K-theory_, [arxiv/1012.2512](http://arxiv.org/abs/1012.2512)

* {#Pennig} [[Ulrich Pennig]], _Twisted K-theory with coefficients in $C^\ast$-algebras_, ([arXiv:1103.4096](http://arxiv.org/abs/1103.4096))

Comparison between [[homotopy theory|homotopy theoretic]] and [[operator algebra|operator algebraic]] constructions:

* {#HebestreitSagave19} [[Fabian Hebestreit]], [[Steffen Sagave]], *Homotopical and operator algebraic twisted K-theory*, Mathematische Annalen **378** (2020) 1021-1059 ([arXiv:1904.01872](https://arxiv.org/abs/1904.01872), [doi:10.1007/s00208-020-02066-6](https://doi.org/10.1007/s00208-020-02066-6))

Discussion in terms of [[vectorial bundles]] is in

* {#Gomi} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))

* {#GomiTerashima} [[Kiyonori Gomi]], [[Yuji Terashima]], _Chern-Weil Construction for Twisted K-Theory_, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 ([doi:10.1007/s00220-010-1080-1](https://doi.org/10.1007/s00220-010-1080-1))

Discussion of combined [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[KR-theory]] on [[orbifold|orbi-]] [[orientifolds]]:

* [[El-kaïoum M. Moutuou]], _Twistings of KR for Real groupoids_ ([arXiv:1110.6836](http://arxiv.org/abs/1110.6836))

* [[El-kaïoum M. Moutuou]], _Graded Brauer groups of a groupoid with involution_, J. Funct. Anal. 266 (2014), no.5 ([arXiv:1202.2057](https://arxiv.org/abs/1202.2057))

* {#Freed12} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]])

* [[Daniel Freed]], [[Gregory Moore]], Section 7 of: _Twisted equivariant matter_,  Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

* {#Gomi17} [[Kiyonori Gomi]], _Freed-Moore K-theory_ ([arXiv:1705.09134](https://arxiv.org/abs/1705.09134), [spire:1601772](http://inspirehep.net/record/1601772))

Review:

* [[Jonathan Rosenberg]], *Twisted cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]

Discussion of twisted [[K-homology]]:

* [[Bai-Ling Wang]], _Gemometric cycles, index theory and twisted K-homology_ ([arXiv:0710.1625](https://arxiv.org/abs/0710.1625))

* [[Eckhard Meinrenken]], _Twisted K-homology and group-valued moment maps_,  International Mathematics Research Notices 2012 (20) (2012), 4563--4618 ([arXiv:1008.1261](https://arxiv.org/abs/1008.1261))

* Bei Liu, _Twisted K-homology,Geometric cycles and T-duality_ ([arXiv:1411.1575](https://arxiv.org/abs/1411.1575))

Discussion of combined [[twisted K-theory|twisted]] and [[equivariant K-theory|equivariant]] and [[real K-theory|real]] K-theory

* {#Gomi17} [[Kiyonori Gomi]], _Freed-Moore K-theory_ ([arXiv:1705.09134](https://arxiv.org/abs/1705.09134))

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]] (see at *[[K-theory classification of D-brane charge]]*):

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], _Ramond-Ramond fields and twisted differential K-theory_ ([arXiv:1903.08843](https://arxiv.org/abs/1903.08843))

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

### Higher twists
 {#ReferencesHigherTwists}

Discussion of higher twists of K-theory (above degree 3):

* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))

* [[Constantin Teleman]], Section 3 of: *K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra* ([arXiv:math/0306347](https://arxiv.org/abs/math/0306347)), in [[Ulrike Tillmann]] (ed.) *Topology, Geometry and Quantum Field Theory*, Cambridge University Press 2004 ([doi:10.1017/CBO9780511526398](https://doi.org/10.1017/CBO9780511526398))

Via [[C-star algebras|$C^\ast$-algebras]]:

* {#Pennig16} [[Ulrich Pennig]], *A noncommutative model for higher twisted K-Theory*, J Topology (2016) 9 (1): 27-50 ([arXiv:1502.02807](https://arxiv.org/abs/1502.02807))

* {#DardalatPennig16} [[Marius Dadarlat]], [[Ulrich Pennig]], *A Dixmier-Douady theory for strongly self-absorbing $C^\ast$-algebras*, J. Reine Angew. Math. 718 (2016) 153-181 ([arXiv:1302.4468](https://arxiv.org/abs/1302.4468), [doi:10.1515/crelle-2014-0044](https://doi.org/10.1515/crelle-2014-0044))

* {#DardalatPennig15} [[Marius Dadarlat]], [[Ulrich Pennig]], *Unit spectra of K-theory from strongly self-absorbing $C^\ast$-algebras*, Algebr. Geom. Topol. 15 (2015) 137-168 [[arXiv:1306.2583](https://arxiv.org/abs/1306.2583), [doi:10.2140/agt.2015.15.137](http://dx.doi.org/10.2140/agt.2015.15.137)]

* [[David Brook]], *Computations in higher twisted K-theory*  ([arXiv:2007.08964A](https://arxiv.org/abs/2007.08964A))

* [[David Brook]], *Higher twisted K-theory* ([dspace:2440/125740](https://digital.library.adelaide.edu.au/dspace/handle/2440/125740))

* [[David E. Evans]], [[Ulrich Pennig]], *Spectral Sequence Computation of Higher Twisted K-Groups of $SU(n)$* [[arXiv:2307.00423](https://arxiv.org/abs/2307.00423)]

see also:

* [[Ulrich Pennig]], *Equivariant Higher Twisted K-Theory of $SU(n)$ via Exponential Functors*, [talk at](CQTS#PennigSep2023) [[CQTS]] (20 Sep 2023) [video:[YT](https://youtu.be/JXO-owcJgTE)]

Discussion of the [[twisted Chern character]] for higher twisted K-theory:

* Lachlan Macdonald, [[Varghese Mathai]], Hemanth Saratchandran, *On the Chern character in Higher Twisted K-theory and spherical T-duality*, Commun. Math. Phys. 385, 331-368 (2021) ([arXiv:2007.02507](https://arxiv.org/abs/2007.02507))

[[!redirects twisted K-theories]]

[[!redirects twisted topological K-theory]]
[[!redirects twisted topological K-theories]]

[[!redirects higher twisted K-theory]]
[[!redirects higher twisted K-theories]]


twisted module of homomorphisms

Let $C$ be a dg-[[coalgebra]], $A$ a [[dg-algebra]], $N$ a left $C$-dg-[[comodule]] with [[coaction]] $\delta_N:N\to C\otimes N$, $P$ a left $A$-dg-module with action $m_P:A\otimes P\to P$ and $\tau:C\to A$ a [[twisting cochain]]. The __twisted module of homomorphisms__ $\mathrm{Hom}_\tau(N,P)$ is a [[chain complex]] which as a [[graded vector space|graded module]] coincides with the ordinary module of homomorphisms of the underlying chain complex $\mathrm{Hom}(N,P)$, and with the differential $d_\tau$ given by
$$
d_\tau(f) = d(f) + m_P\circ(\tau\otimes f)\circ\delta_N,
$$
where $f\in\mathrm{Hom}(N,P)$.


twisted ordinary cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contentens#
* table of contents
{:toc}

## Idea

The [[twisted cohomology]] version of [[ordinary cohomology]].

## Related concepts

* [[twisted de Rham cohomology]]

* [[twisted ordinary differential cohomology]]

* [[twisted K-theory]]

* [[twisted Cohomotopy]]

* [[twisted Bredon cohomology]] (twisted ordinary [[equivariant cohomology]])

## References
 {#References}

Original articles:

* [[Kurt Reidemeister]], _Überdeckungen von Komplexen_, Crelle **173** (1935), 164-173 [[doi:10.1515/crll.1935.173.164](http://dx.doi.org/10.1515/crll.1935.173.164)]

* [[Norman Steenrod]], _Homology With Local Coefficients_, Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627 ([jstor:1969099](https://www.jstor.org/stable/1969099))

Classification of twisted cohomology by [[fibrations]]/[[bundles]] of [[Eilenberg-MacLane spaces]] over the [[classifying space]] of a twisting group (universally: the [[fundamental group]] of the [[domain]] space):

* [[Samuel Gitler]], _Cohomology operations with local coefficients_, American Journal of Mathematics **85** 2 (1963) 156–188  [[doi:10.2307/2373208](http://dx.doi.org/10.2307/2373208)]

* Yasumasa Hirashima, _A note on cohomology with local coefficients_,  Osaka J. Math., **16** 1 (1979) 219-231 &lbrack[euclid:ojm/1200771839](https://projecteuclid.org/euclid.ojm/1200771839)]

* [[Paul Goerss]], [[J. F. Jardine]], p. 332 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4)]

* M. Bullejos, E. Faro, M. A. García-Muñoz, _Homotopy colimits and cohomology with local coefficients_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 [[numdam:CTGDC_2003__44_1_63_0](http://www.numdam.org/item?id=CTGDC_2003__44_1_63_0)]

Review:

* [[Jonathan Rosenberg]], *Twisted cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]

Discussion as a special case of a more general notion of [[twisted cohomology|twisted]] [[non-abelian cohomology]] by [[(infinity,1)-categorical hom-spaces|$\infty$-categorical hom-spaces]] in [[slice (infinity,1)-toposes|slice $\infty$-toposes]]:

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Rem. 4.22 in: *[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal $\infty$-bundles -- _General Theory]]*, Journal of Homotopy and Related Structures, **10** 4 (2015) 749-801 [[doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Ex. 2.32 in: *[[schreiber:The Character Map in Twisted Non-Abelian Cohomology]]* [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909)]

[[!redirects twisted ordinary homology]]


twisted ordinary differential cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Twisted ordinary differential cohomology is [[twisted differential cohomology]] specialized to the case of [[ordinary differential cohomology]].

## Related concepts

* [[twisted ordinary cohomology]]

* [[ordinary differential cohomology]]

* [[equivariant ordinary differential cohomology]]

* [[twisted differential K-theory]]

## References

* [[Daniel Grady]], [[Hisham Sati]], _Twisted smooth Deligne cohomology_, [arXiv:1706.02742](https://arxiv.org/abs/1706.02742).

[[!redirects twisted Deligne cohomology]]


twisted Pontrjagin theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _twisted Pontrjagin theorem_ should generalize the [[Pontrjagin theorem]] from plain [[homotopy theory]]/[[cobordism theory]] to [[parametrized homotopy theory]]/twisted [[cobordism theory]]:

Where the plain [[Pontrjagin theorem]] identifies the [[Cohomotopy]] of a [[differentiable manifold]] with its [[cobordism classes]] of [[normally framed submanifolds]], the twisted Pontrjagin theorem should identify [[twisted Cohomotopy]] with [[cobordism classes]] of [[normally twisted-framed submanifolds]] ([Cruickshank 03, Lemma 5.2](#Cruickshank03))

## Related concepts

* [[Pontrjagin theorem]], [[equivariant Pontrjagin theorem]]

## References

[[!include Pontryagin-Thom construction -- references]]

[[!redirects twisted Pontrjagin theorems]]

[[!redirects twisted Pontryagin theorem]]
[[!redirects twisted Pontryagin theorems]]