Abstract
In this article we show that the use of Deligne–Beilinson cohomology in the context of the U(1) BF theory on a closed 3-manifold M yields a discrete ZN BF theory whose partition function is an abelian TV invariant of M. By comparing the expectation values of the U(1) and ZN holonomies in both BF theories we obtain a reciprocity formula.
Highlights
The impact of Deligne-Beilinson cohomology in the context of Quantum Field Theory was carefully investigated in [1]
The use of DB cohomology proves to be very effective in the U(1) BF theory since unlike the non-abelian SU(2) case we find that: 1) the discretisation of the original U(1) BF theory is a consequence of the construction and not an input; 2) no regularisation of the expectation values is required in the discrete abelian case because all sums occurring are finite whereas a Quantum Group has to be introduced by hand in the non-abelian case to get well-defined expressions [14, 15]
In this article we showed how the use of Deligne-Beilinson cohomology allows to prove that the U(1) BF theory can be turned into a discrete ZN BF theory without resorting to the usual guessworks of the non-abelian case
Summary
The impact of Deligne-Beilinson cohomology in the context of Quantum Field Theory was carefully investigated in [1]. In a previous article [2] a study of the U(1) BF theory within the Deligne-Beilinson cohomology [3, 4] framework was initiated, following what was done in the U(1) Chern-Simons (CS) theory case [5, 6, 7, 8, 9] In this first article the partition function of the BF theory was computed and compared with the absolute square of the Chern-Simons partition function highlighting significant differences from the non-abelian case. In the second section of this article we complete the study of the U(1) BF theory on a closed 3-manifold M by computing expectation values of U(1) holonomies, still in the Deligne-Beilinson (DB) cohomology framework. We use =Z to denote equality in R/Z, that is to say modulo integers, as well as Einstein summation convention
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