2011/07/18 by Torsten Aßelmeyer-Maluga, Asselmeyer-Maluga, Torsten, Jerzy Król +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1107.3458
openalex publication_date 2011/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper shows deep connections between exotic smoothings of a small R4 (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic R4 to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterized by the real-valued Godbillon-Vey invariant. Special care is taken for the integer case which is related to flat PSL(2,R)-bundles. Then we discuss the leaf space of the foliation using noncommutative geometry. This leaf space contains the hyperfinite III1 factor of Araki and Woods important for quantum field theory (QFT) and the I∞ factor. Using Tomitas modular theory, one obtains a relation to a factor II∞ algebra given by the horocycle foliation of the unit tangent bundle of a surface S of genus g>1. The relation to the exotic R4 is used to construct the (classical) observable algebra as Poisson algebra of functions over the character variety of representations of the fundamental group π1(S) into the SL(2,C). The Turaev-Drinfeld quantization (as deformation quantization) of this Poisson algebra is a (complex) skein algebra which is isomorphic to the hyperfinite factor II1 algebra determining the factor II∞=II1⊗ I∞ algebra of the horocycle foliation. Therefore our geometrically motivated hyperfinite III1 factor algebra comes from the quantization of a Poisson algebra. Finally we discuss the states and operators to be knots and knot concordances, respectively.