Exotic R^4 and quantum field theory

TL;DR

This paper explores the connection between exotic smooth structures on R^4, operator algebras, and quantum field theory, revealing how exotic smoothness influences observable algebras and relates to deformation quantization and quantum models.

Contribution

It establishes a novel link between exotic R^4, foliation-induced von Neumann algebras, and quantum field theory through deformation quantization and operator algebra frameworks.

Findings

Exotic smooth R^4 relates to von Neumann algebras of type III_1.

The algebra W(S^3) can be viewed as deformation quantization of flat SL(2,C) connections.

Connections to fermionic and gauge field actions in quantum field theory models.

Abstract

Recent work on exotic smooth R^4's, i.e. topological R^4 with exotic differential structure, shows the connection of 4-exotics with the codimension-1 foliations of $S^{3}$, SU(2) WZW models and twisted K-theory $K_{H}(S^{3})$, $H\in H^{3}(S^{3},\mathbb{Z})$. These results made it possible to explicate some physical effects of exotic 4-smoothness. Here we present a relation between exotic smooth R^4 and operator algebras. The correspondence uses the leaf space of the codimension-1 foliation of S^3 inducing a von Neumann algebra $W(S^{3})$ as description. This algebra is a type III_1 factor lying at the heart of any observable algebra of QFT. By using the relation to factor II, we showed that the algebra $W(S^{3})$ can be interpreted as Drinfeld-Turaev deformation quantization of the space of flat SL(2,\mathbb{C}) connections (or holonomies). Thus, we obtain a natural relation to quantum field theory. Finally we discuss the appearance of concrete action functionals for fermions or gauge fields and its connection to quantum-field-theoretical models like the Tree QFT of Rivasseau.