Exotic R⁴ and quantum field theory
Asselmeyer-Maluga, T. · Mader, R.
Original · EN
Recent work on exotic smooth R⁴'s, i.e. topological R⁴ with exotic differential structure, shows the connection of 4-exotics with the codimension-1 foliations of S³, SU(2) WZW models and twisted K-theory Kₕ(S³), H∈ H³(S³,Z). These results made it possible to explicate some physical effects of exotic 4-smoothness. Here we present a relation between exotic smooth R⁴ and operator algebras. The correspondence uses the leaf space of the codimension-1 foliation of S³ inducing a von Neumann algebra W(S³) as description. This algebra is a type III₁ factor lying at the heart of any observable algebra of QFT. By using the relation to factor II, we showed that the algebra W(S³) can be interpreted as Drinfeld-Turaev deformation quantization of the space of flat SL(2,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.
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