By M. Aizenman (Chief Editor)

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A bivariant Chern character designed in an equivariant setting may shed some light on the interplay between BRS cohomology and the recently discovered cyclic cohomology of Hopf algebras [10, 11]. • Twisted K-theory and K-homology recently appeared in the physics literature through the classification of D-branes [23, 35]. This also falls into the scope of a bivariant Chern character. Bivariant Chern Character for Families of Spectral Triples 47 First we have to consider the right category of algebras.

The multiplication map and d are bounded. It is the universal complete bornological DG algebra generated by A. On A are defined various operators. First of all, the Hochschild boundary b : n+1 A → n A is b(ωda) = (−)n [ω, a] for ω ∈ n A, and b = 0 on 0 A = A. One easily shows that b is bounded and b2 = 0. Then the Karoubi operator κ : n A → n A is constructed out of b and d: 1 − κ = db + bd. (3) Therefore κ is bounded and commutes with b and d. The last operator is Connes’ B : n A → n+1 A, B = (1 + κ + .

In this case, the ψ-component of δ(c) is the coadjoint action of the parameter space on the space of maps from into V , while the h-component is simply the “covariant derivative” of c relative to the connection defined by the gauge field h. Thus, by introducing more general polynomials WAB , Ikeda is introducing a generalization of ordinary gauge theory by requiring that the gauge symmetries be defined via the polynomials WAB . For this generalization to work, Ikeda imposes restrictions on the polynomials WAB which amount to making P a Lie algebra, hence his terminology of “non-linear Lie algebra”.

### Communications in Mathematical Physics - Volume 231 by M. Aizenman (Chief Editor)

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