Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomology

Abstract

There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots in [HKS]. As a consequence, we obtain a non-trivial description of the Bott-Cattaneo-Rossi invariant in terms of the Alexander polynomial. The results for higher codimension knots are also given. In those cases similar differential forms to define Bott-Cattaneo-Rossi invariant yields infinitely many cohomology classes of Emb(R^n, R^m) if m,n>= 3 odd and m>n+2. We observe that half of these classes are non-trivial, along a line similar to Cattaneo-CottaRamusino-Longoni [CCL].