# nLab twisted Chern character

Contents

cohomology

### Theorems

#### Differential cohomology

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

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

(1)$B_2 \,\in\, \Omega^2(U) \,,$

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

(2)$F_i \,\in\, \Omega^2 \big( U, \, End(V_i) \big)$

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)$\exp(B_2) \wedge tr \big( \exp(F_1) - \exp(F_2) \big) \;\; = p^\ast ch^{B_2}(\nabla_1, \nabla_2) \;\; \;\;\; \in \; \Omega^\bullet(U)$

turns out to

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

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

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

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

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

is independent of the choices ($B_2$, $\nabla_1$, $\nabla_2$) made.

This class (4) is the twisted Chern character of the twisted K-theory class $[V_1, V_2]$ (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.