Chern–Simons theory on R3 in axial gauge: a rigorous approach

Abstract

We study pure Chern-Simons models on M ¼ R 3 using a functional integral quantization approach which is based on axial gauge fixing. It is well-known (see, e.g., Comm. Math. Phys. 126 (1989) 167; Comm. Math. Phys. 186 (1997) 563) that in axial gauge the Chern-Simons action function is quadratic and that the Faddeev-Popov determinant of this gauge fixing procedure is a constant function. This means that the Wilson loop observables (WLOs) of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of ''Gaussian type''. We demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from White noise analysis with certain regularization techniques like ''loop smearing'' and ''framing''. For the special case G ¼ Uð1Þ; G being the group of the model, we carry out the details of this approach, which is also applicable to the case of non-Abelian G; and find well-known linking number expressions for the WLOs. r 2004 Elsevier Inc. All rights reserved.

An Analytic Approach to Turaev's Shadow Invariant

“…where exp := exp C ∞ (M,G) : C ∞ (M, g) → C ∞ (M, G) is the "pointwise" exponential map. From the chain rule and the relation (d exp G ) |0 = id g which, informally, implies d exp |0 = id C ∞ (M,g) we obtain for every A 0 ∈ A gf △ F P [A 0 ] = △ F P (A 0 , 1) = det(Ψ • dq |(A 0 ,1) ) = det(dH |(A 0 ,0) ) (C. 21) where dH is now the "usual" total differential of a smooth map between (infinite-dimensional) vector spaces. 36 clearly, △F P (A0, Ω0) depends on Ψ via a multiplicative constant but this is irrelevant for our purposes 37 we emphasize that the translation part Ω −1 0 dΩ0 in q(A0, Ω0) = A0 · Ω0 = Ω −1 0 A0Ω0 + Ω −1 0 dΩ0 does not make an appearance in dq |(A 0 ,Ω 0 ) if the identifications (C.17) above are used 38 since we are arguing on a heuristic level we will not bother to specify a topology here…”

“…One word of caution is appropriate here, though: it is not not totally impossible that something similar will happen as in the axial gauge approach to Chern-Simons models on R 3 , cf. [19,4,21,22]. In [22] it turned out that, in the Non-Abelian case, the expressions for the WLOs obtained for links with double points depended on the precise way in which the loop smearing regularization procedure was implemented.…”

“…In other words: for the approach in [22] it was necessary to "combine" results that hold only for spaces with compact support with results that hold only for the bigger spaces A cax , G and A. It should therefore not be too surprising that the axial gauge approach for Chern-Simons models on R 3 runs into difficulties (at least when using the implementation of [4,21,22]). In fact, the loop smearing dependence was not the only complication/problem in [22].…”

“…for loops without double points 31 . Secondly, in the approach in [4,21,22] it was unclear right from the beginning how quantum groups (resp. the corresponding R-matrices) could enter the computations.…”

“…In fact, due to the remarkable property of Eq. (3.31) that all the heuristic measures that appear there are of "Gaussian type" we can apply similar techniques as in the axial gauge approach to Chern-Simons models on R 3 developed in [19,4,21,22]. In particular, we can make use of white noise analysis and of the two regularization techniques "loop smearing" and "framing".Finally, we study the question if and how the right-hand side of Eq.…”