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Abstract: The paper shows deep connections between exotic smoothings of small R^4,noncommutative algebras of foliations and quantization. At first, based on theclose relation of foliations and noncommutative C*-algebras we show that cycliccohomology invariants characterize some small exotic R^4. Certain exotic smoothR^4-s define a generalized embedding into a space which is K-theoreticequivalent to a noncommutative Banach algebra. Furthermore, we show that afactor III von Neumann algebra is naturally related with nonstandard smoothingof a small R^4 and conjecture that this factor is the unique hyperfinite factorIII 1. We also show how an exotic smoothing of a small R^4 is related to theDrinfeld-Turaev deformation quantization of the Poisson algebraXS,SL2,C,{,} of complex functions on the space of flat connectionsXS,SL2,C over a surface S, and that the result of this quantization is theskein algebra K tS,, for the deformation parameter t=exph-4. This skeinalgebra is retrieved as a II 1 factor of horocycle flows which is Moritaequivalent to the II infty factor von Neumann algebra which in turn determinesthe unique factor III 1 as crossed product. Moreover, the structure of Cassonhandles determine the factor II 1 algebra too. Thus, the quantization of thePoisson algebra of closed circles in a leaf of the codimension 1 foliation ofS^3 gives rise to the factor III 1 associated with exotic smoothness of R^4.Finally, the approach to quantization via exotic 4-smoothness is considered asa fundamental question in dimension 4 and compared with the topos approach toquantum theories.

Autor: Torsten Asselmeyer-Maluga, Jerzy Krol

Fuente: https://arxiv.org/

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