3D topological models and Heegaard splitting. I. Partition function

Abstract

The aim of this article is twofold: firstly, we show how to recover the smooth Deligne-Beilinson cohomology groups from a Heegaard splitting of a closed oriented smooth 3manifold by extending the usualČech-de Rham construction; secondly, thanks to the above and still relying on a Heegaard splitting, we explain how to compute the partition functions of the U(1) Chern-Simons and BF theories.

3D topological models and Heegaard splitting. II. Pontryagin duality and observables

“…j 2 ) does not intersect the singular support of d † j 2 (resp. d † j 1 ), then regularizations of j 1 and j 2 allow to define properly (j 1 ∧ j 2 ) [ [1]]. In particular,…”

“…] is well defined when the singular supports of d † j 1 and d † j 2 do not intersect. Another example where (j 1 ∧ j 2 ) [ [1]] is well defined is when j 2 (or j 1 ) is regular since in this case its singular support is empty.…”

“…Indeed, (j c Σ ∧ J α ) [ [1]] as a meaning because the singular support of J α is empty. Moreover, for any c ∈ C p (Σ) we have:…”

“…In a first article [1] we explained how to recover the smooth Deligne-Beilinson (DB) cohomology groups H p D (M) by using a Heegaard splitting X L ∪ f X R of an oriented closed smooth 3-manifold M. Then, we identified a decomposition of DB 1-cocycles of X L ∪ f X R which turned out to be very efficient in the Chern-Simon (CS) and BF U(1) topological theories. Doing so, we were able to check that the partition functions for these two theories coincide with those obtain in the so-called "standard" approach as exposed in [2][3][4][5].…”

“…• In section 3 we present a description of chains, forms and de Rham currents of a Heegaard splitting X L ∪ f X R . We end this section by briefly recalling the construction [1] of H 1 D (X L ∪ f X R ), the first smooth DB cohomology group of X L ∪ f X R . Beside fixing notations, this section tries to detail some useful points about Heegaard splitting and also provide an expression for the linking number of two 1-cycles of X L ∪ f X R .…”