2012/07/19 by Torsten Aßelmeyer-Maluga, Torsten Asselmeyer-Maluga, Asselmeyer-Maluga, Torsten +2
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #hep-th #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1207.4602
16 pages
arxiv created 2012/07/19 · openalex publication_date 2012/07/19 · arxiv updated 2012/07/20 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We show that superconformal \cal N=4,2 algebras are well-suited to represent some invariant constructions characterizing exotic ℝ4 relative to a given radial family. We examine the case of \cal N=4, c=4 (at r=1 level) superconformal algebra which is realized on flat ℝ4 and curved S3× ℝ. While the first realization corresponds naturally to standard smooth ℝ4 the second describes the algebraic end of some small exotic smooth ℝ4's from the radial family of DeMichelis-Freedman and represents the linear dilaton background SU(2)k× ℝQ of superstring theory. From the modular properties of the characters of the algebras one derives Witten-Reshetikhin-Turaev and Chern-Simons invariants of homology 3-spheres. These invariants are represented rather by false, quasi-modular, Ramanujan mock-type functions. Given the homology 3-spheres one determines exotic smooth structures of Freedman on S3× ℝ. In this way the fake ends are related to the SCA \cal N=4 characters. The case of the ends of small exotic ℝ4's is more complicated. One estimates the complexity of exotic ℝ4 by the minimal complexity of some separating from the infinity 3-dimensional submanifold. These separating manifolds can be chosen, in some exotic ℝ4's, to be homology 3-spheres. The invariants of such homology 3-spheres are, again, obtained from the characters of SCA, \cal N=4. Next we take into account the modification of the algebra of modular forms due to the noncommutativity of the codimension-one foliations of the homology 3-spheres. Then, the modification of modular forms is represented by the Connes-Moscovici construction ...