nLab Connes fusion

Idea

The operation called Connes fusion (after Alain Connes) is a tensor product operation of Hilbert space bimodules over von Neumann algebras that adjusts the naive algebraic tensor product.

Applications

Connes fusion is used to define fusion of positive energy representations of the loop group $\mathcal{L}S U(N)$ in

• Antony Wassermann: Operator algebras and conformal field theory III. Fusion of positive energy representations of $L S U(N)$ using bounded operators, Invent math 133 (1998) 467–538 [arXiv:math.OA/9806031, doi:10.1007/s002220050253]

It is used in a proposal for a construction of string group-principal bundles in the context of finding geometric models for elliptic cohomology in

• Stephan Stolz and Peter Teichner, What is an elliptic object? (link)

References

The fusion operation on Hilbert modules of von Neumann algebras has been described in

• Alain Connes, On the spatial theory of von Neumann algebras , J. of functional analysis 35 (1980) 153–164;

and in 1983 studied further by V. Jones in context of subfactors.

A useful review of Connes fusion observing some simplifications is

• Andreas Thom, A remark about the Connes fusion tensor product (arXiv:math.OA/0601045)

An attempt to summarize that in a blog entry is here

R. M Brower and N. P. Landsman described a bicategory of von Neumann algebras, Connes’ correspondences as morphisms and (bounded) intertwiners as 2-cells. They proved that the equivalence in that bicategory is the same as usual Morita equivalence of von Neumann algebras as studied by Mark Rieffel and others.

Construction of Connes fusion of fibers of stringor bundles:

• Peter Kristel, Konrad Waldorf: Connes fusion of spinors on loop space, Compositio Mathematica 160 7 (2024) 1596-1650 [arXiv:2012.08142, doi:10.1112/S0010437X24007188]