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twisted de Rham cohomology

Contents

Idea

For XX a smooth manifold and HΩ 3(X)H \in \Omega^3(X) a closed differential 3-form, the HH twisted de Rham complex is the 2\mathbb{Z}_2-graded vector space Ω even(X)Ω odd(X)\Omega^{even}(X) \oplus \Omega^{odd}(X) equipped with the HH-twisted de Rham differential

d+H():Ω even/odd(X)Ω odd/even(X), d + H \wedge(-) : \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,

Notice that this is nilpotent, due to the odd degree of HH, such that HH=0H \wedge H = 0, and the closure of HH, dH=0d H = 0.

Properties

Twisted de Rham cohomology is the recipient of the twisted Chern character in twisted differential K-theory.

Created on May 20, 2011 06:49:35 by Urs Schreiber (131.211.238.38)