One of the most important group cocycles is the two-cocycle giving the central extension of the restricted general linear group of a polarised Hilbert space (H,H₊). It has a wide range of applications, ranging from the conformal field theory to invariants of the algebraic K-theory. It can be seen as a two-cocycle associated to the action of the group GL_res(H,H+) on the category of idempotents P∈ß (H) such that [P_H+, P] ∈L²(H).More generally, given an action of a group G on an n-category satisfying certain conditions, one can construct a (n+1)-cocycle on G. A well known example is the n-Tate space, essentially an algebra of the form K = k((s₁))((s₂)) … ((s_n)), where the group is the group of invertibles in K and the n-category structure comes from the natural filtration of K. 

The corresponding cocycles, when evaluated on K^alg_n+1(K), reproduce the Tate tame symbol. However, the constructions are purely algebraic and do not seem to extend to the analytic context, as in the case of n = 1. 

In this talk we will sketch a construction of a family of two-categories associated to a pair of commuting idempotents P and Q on a Hilbert space and construct the associated three cocycle on the associated groups. For example, in the case of a two-Tate space, this produces an extension of the Tate symbol and the corresponding invariant of K^alg₃ from the 2-Tate space to C^∞(T ²). As another example we get a corresponding invariant of K^alg ₃ of the non-commutative torus C^∞(T²θ). The construction is based on the properties of the determinant of Fredholm operators, in particular on the existence of the canonical perturbation isomorphism Det(T)Det(S) associated to a pair of Fredholm operators T and S satisfying T-S∈L¹(H).This is a joint work with Jens Kaad and Jesse Wolfson.                                                                                                                                                            海报