nLab twisted Chern character

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.

(BCMMS 2002, Prop. 9.1)

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

Related concepts

• Chern character

• equivariant Chern character

• twisted equivariant Chern character

References

• Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, Danny Stevenson, Section 6.3 in: Twisted K-theory and K-theory of bundle gerbes, Commun Math Phys 228 (2002) 17-49 [arXiv:hep-th/0106194, doi:10.1007/s002200200646]

• 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, doi:10.1112/jtopol/jtm001)

• 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, doi:10.1007/s00220-003-0807-7]

• Michael Atiyah, Graeme Segal, Section 7 of: Twisted K-theory, Ukrainian Math. Bull. 1 (2004) (arXiv:math/0407054, journal page, published pdf)

• Varghese Mathai, Danny Stevenson, Section 6 of: On a generalized Connes-Hochschild-Kostant-Rosenberg theorem, Advances in Mathematics, vol. 200 no. 2 (2006) 303-335 (arXiv:math/0404329)

• Paul Bressler, A. Gorokhovsky, R. Nest, Boris Tsygan, Chern Character for Twisted Complexes. In: Kapranov M., Manin Y.I., Moree P., Kolyada S., Potyagailo L. (eds.) Geometry and Dynamics of Groups and Spaces. Progress in Mathematics, vol 265. Birkhäuser Basel (2007) (doi:10.1007/978-3-7643-8608-5_5)

• Kiyonori Gomi, Yuji Terashima, Section 4 of: Chern-Weil Construction for Twisted K-Theory, Commun. Math. Phys. 299, 225–254 (2010) (doi:10.1007/s00220-010-1080-1)

  (in terms of vectorial bundles)

• Max Karoubi, Section 8.3 of: Twisted bundles and twisted K-theory, in: Guillermo Cortiñas (ed.) Topics in Noncommutative Geometry (arXiv:1012.2512, ISBN:978-0-8218-6864-5)

In twisted orbifold K-theory:

• 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, doi:10.1016/j.aim.2005.12.001)

On bundle gerbe modules with higher G-structures (including String structure, etc.) and the corresponding twisted Chern character:

• Fei Han, Ruizhi Huang, Varghese Mathai, Fractional structures on bundle gerbe modules and fractional classifying spaces [arXiv:2203.14439]