Supersymmetry, homology with twisted coefficients and n-dimensional knots

Details Supersymmetry, homology with twisted coefficients and n-dimensional knots Supersymmetry, homology with twisted coefficients and n-dimensional knots

2003-11-16

arxiv.org/abs/hep-th/0311136v1

Int.J.Mod.Phys. A21 (2006) 4185-4196

Physics

High Energy Physics

High Energy Physics - Theory

10pages, no figures

Scientific paper

Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.

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