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Superstrings, knots, and noncommutative geometry in \({\mathcal E}^{(\infty)}\) space. (English) Zbl 0935.58005
Summary: Within a general theory, a probabilistic justification for a compactification which reduces an infinite-dimensional spacetime \({\mathcal E}^{(\infty)} (n=\infty)\) to a four-dimensional one \((D_T=n=4)\) is proposed. The effective Hausdorff dimension of this space is given by \(\langle\dim_H {\mathcal E}^{(\infty)} \rangle=d_H= 4+\phi^3\), where \(\phi^3=1/[4+ \phi^3]\) is a PV number and \(\phi=(\sqrt 5-1)/2\) is the golden mean. The derivation makes use of various results from knot theory, four-manifolds, noncommutative geometry, quasiperiodic tiling, and Fredholm operators. In addition some relevant analogies between \({\mathcal E}^{(\infty)}\), statistical mechanics, and Jones polynomials are drawn. This allows a better insight into the nature of the proposed compactification, the associated \({\mathcal E}^{(\infty)}\) space, and the Pisot-Vijayvaraghavan number \(1/\phi^3=4.236067977\) representing its dimension. This dimension is in turn shown to be capable of a natural interpretation in terms of the Jones knot invariant and the signature of four-manifolds. This brings the work near to the context of Witten and Donaldson topological quantum field theory.

MSC:
81T75 Noncommutative geometry methods in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
57M99 General low-dimensional topology
58B34 Noncommutative geometry (à la Connes)
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