nLab
twisted Chern character

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Cohomology

cohomology

Special and general types

Special notions

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Extra structure

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Differential cohomology

Contents

Idea

The ordinary Chern character for K-theory sends K-classes to ordinary cohomology with real coefficients. Over a smooth manifold the de Rham theorem makes this equivalently take values in de Rham cohomology.

The twisted Chern character analogously goes from twisted K-theory to twisted de Rham cohomology.

Details

In terms of twisted curvature characteristic forms

For a degree-3 ordinary cohomology-class τ 3H 3(X;,)\tau_3 \in \,H^3(X;, \mathbb{Z})\, (e.g. modeled as a bundle gerbe) on a compact smooth manifold XX, a class in the τ 3\tau_3-twisted K-theory KU τ 3(X)KU^{\tau_3}(X) may be represented as a suitable Grothendieck group-equivalence class [V 1,V 2][V_1,V_2] of a pair V 1,V 2TwVectBund τ 3(X)V_1, V_2 \in TwVectBund^{\tau_3}(X) of complex twisted vector bundles (e.g. modeled as bundle gerbe modules, possibly of infinite rank).

Then for any lift of the twist τ 3\tau_3 to a Deligne cocycle [h 0,A 1,B 2,H 3] U[h_0, A_1, B_2, H_3]_U (a connection on a bundle gerbe) with respect to some surjective submersion p:UXp \colon U \twoheadrightarrow X, hence in particular with a differential 2-form (the local B-field)

(1)B 2Ω 2(U), B_2 \,\in\, \Omega^2(U) \,,

one may make a choice of connections on twisted vector bundles ( 1, 2)(\nabla_1, \nabla_2) on (V 1,V 2)(V_1, V_2), and this induces endomorphism ring-valued curvature 2-forms

(2)F iΩ 2(U,End(V i)) F_i \,\in\, \Omega^2 \big( U, \, End(V_i) \big)

on the cover.

Now the twisted characteristic form (3) obtained as the trace (in the square matrix-coefficients) of the difference of the wedge product-exponential series of these two twisted curvature forms (2), wedged with the wedge-exponential series of the B-field (1)

(3)exp(B 2)tr(exp(F 1)exp(F 2))=p *ch B 2( 1, 2)Ω (U) \exp(B_2) \wedge tr \big( \exp(F_1) - \exp(F_2) \big) \;\; = p^\ast ch^{B_2}(\nabla_1, \nabla_2) \;\; \;\;\; \in \; \Omega^\bullet(U)

turns out to

  1. be well-defined, in that the traces all exist;

  2. be the pullback of an even-degree differential form on XX

    ch B 2( 1, 2)Ω 2(X), ch^{B_2}(\nabla_1,\nabla_2) \;\;\; \in \; \Omega^{2\bullet}(X) \,,
  3. which is closed in the H 3H_3-twisted de Rham complex on XX, in that

    (dH 3)ch B 2( 1, 2)=0, \big( d - H_3 \wedge \big) ch^{B_2}(\nabla_1, \nabla_2) \;=\; 0 \,,
  4. and whose twisted de Rham class

    (4)ch τ 3[V 1,V 2][ch B 2( 1, 2)]H dR 3+H 3(X) ch^{\tau_3}[V_1, V_2] \;\coloneqq\; \big[ ch^{B_2}(\nabla_1, \nabla_2) \big] \;\;\; \in \; H^{3 + H_3}_{dR}(X)

    is independent of the choices (B 2B_2, 1\nabla_1, 2\nabla_2) made.

(BCMMS 2002, Prop. 9.1)

This class (4) is the twisted Chern character of the twisted K-theory class [V 1,V 2][V_1, V_2] (BCMMS 2002, p. 26).

References

In twisted orbifold K-theory:

Last revised on August 18, 2021 at 13:13:59. See the history of this page for a list of all contributions to it.