Abstract
A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern–Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of π1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern–Simons invariants are illustrated.
Highlights
Each SU(N)-connection, with N ≥ 2, in a closed and oriented 3-manifold M can be represented by a 1-form A = Aμdxμ which takes values in the Lie algebra of SU(N )
It is shown that the associated classical Chern-Simons invariant assumes a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of π1(M ) and by the Heegaard gluing homeomorphism
The term X [A] can be understood as a sort of colored intersection form, because its value is determined by the trace of the representation matrices —belonging to the Lie algebra of the group— which are associated with the boundaries of the meridinal discs of the two handlebodies which intersect each other in the Heegaard diagram
Summary
Simons theory has recently been produced [16, 17] In this case, flat connections dominate the functional integration and the value of the partition function is given by the sum over the gauge orbits of flat connections of the exponential of the classical Chern-Simons invariant. The canonical form of the corresponding classical Chern-Simons invariant is derived, where a two dimensional formula of the Wess-Zumino group volume is produced.
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