group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The twisted cohomology-version of de Rham cohomology (a simple example of twisted differential cohomology):
For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed 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
Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.
There is also the cohomology of the chain complex whose maps are just multiplication by $H$.
This is also called H-cohomology (Cavalcanti 03, p. 19).
If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology of $X$ coincides with its H-cohomology, for any closed 3-form $H$.
The concept of $H_3$-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory) in
Further discussion:
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, doi: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)
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)
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, spire:660158)
Gil Cavalcanti, Section I.4 of: New aspects of the $d d^c$-lemma, Oxford 2005 (arXiv:math/0501406)
Discussion in the context of analytic torsion:
Last revised on September 3, 2020 at 11:56:24. See the history of this page for a list of all contributions to it.