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Session 26. Physics and Differential Topology
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Exotic smoothness on open 4-manifolds in string theory and generalized geometries |
Jerzy Król, Silesian University, Katowice, Poland
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Superstring theory is a rich collection of mathematical procedures leading to the quantization of gravity. It is consistent in 10 dimensions (\(9+1\)) and a usual way toward 4 physical dimensions is via compactifications and dualities. The result, however, is highly degenerate - there are \(\sim 10^{500}\) 4 dim. possible backgrounds. Radial family of small exotic \(\mathbb{R}^4\) is the continuum many distinct smooth exotic \(\mathbb{R}^4\) which all embed in the standard \(\mathbb{R}^4\) and are parameterized by the real radius \(\rho\). Such radial family \(\mathbb{R}^4_{\rho}\) determines the family of codimension-1 foliations of \(S^3\), with Godbillon-Vey number equal to \(\rho^2\) (see the Torsten Asselmeyer-Maluga talk at this session).
I show how to translate certain exact 10 dim. superstring solutions into 4 dim. radial family.
The solutions are, in particular, the Callan-Harvey-Strominger linear dilaton (1991) and the Kounas-Kiritsis solution (1994).
The connection with the generalized geometry of Hitchin-Gualtieri is also discussed.
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