Configuration space integrals and invariants for 3-manifolds and knots

Abstract

In this paper we give a brief description of the way proposed in [4] of associating invariants of both 3-dimensional rational homology spheres (r.h.s.) and knots in r.h.s.’s to certain combinations of trivalent diagrams. In addition, we discuss the relation between this construction and Kontsevich’s proposal [8]. The same diagrams appear in the LMO invariant [9] for 3-manifolds, and it would be very interesting to know if there exists any relationship between the two approaches in the case of r.h.s.’s. The reason for restricting here to r.h.s.’s is quite technical, as will be clear in Sect. 4. Until that point, without any loss of generality we can assume M to be any connected, compact, closed, oriented 3-manifold. Our construction yields the invariants in terms of integrals over a suitable compactification of the configuration space of points on M. More precisely, the number of points corresponds to the number of vertices in the trivalent diagram, and the integrand is obtained by associating to each edge in the diagram a certain 2-form that represents the integral kernel of an “inverse of the exterior derivative d.” One reason for constructing invariants in terms of “d −1 ” comes from perturbative Witten–Chern–Simons theory [11]. (More precisely, one should invert the covariant derivative with respect to a flat connection; so the present construction is related to the trivial-connection contribution.) Another reason, which is perhaps more transparent to topologists, relies on the definition of the linking number of two curves in R 3 as