HU-SEFT R 1996-10 AS-ITP-1996-12 Gauss Law Constraints in Chern-Simons Theory From BRST Quantization

The physical state condition in the BRST quantization of Chern-Simons field theory is used to derive Gauss law constraints in the presence of Wilson loops, which play important role in explicitly establishing the connection of Chern-Simons field theory with 2-dimensional conformal field theory. When we discuss knot invariants in terms of Chern-Simons theory [1–6] and the relationship between Chern-Simons field theory and conformal field theory , an important relation is Gauss law constraint in presence of Wilson line, which is given firstly in ref. [1]. This relation plays important role in proving that states of Chern-Simons theory satisfy Knizhnik-Zamolodchikov equation [9–13]. In this short paper we intend to derive the Gauss law constraints from BRST quantization of Chern-Simons field theory. We think this investigation is significant since in some sense BRST quantization formulation is defined better ICSC-World Laboratory, Switzerland 1 than formal manipulation without gauge fixing . We will show that when Wilson lines exist the physical state condition in BRST quantization will lead to Gauss law constraints with source terms just as those have been given in ref. [1]. The procedure we will take is similar to the one in ref. [15], where the equivalence of Dirac first-class constraints and BRST treatment for Yang-Mills theory is formally proved. Let us write down the BRST quantization of Chern-Simons theory first. The action of Chern-Simons field theory takes following form SCS = k 4π ∫