Perturbative BF theory

Abstract

We consider a superrenomalizable gauge theory of topological type, in which the structure group is equal to the inhomogeneous group ISU (2). The generating functional of the correlation functions of the gauge fields is derived and its connection with the generating functional of the Chern-Simons theory is discussed. The complete renomalization of this model defined in R 3 is presented. The structure of the ISU (2) conjugacy classes is determined. Gauge invariant observables are defined by means of appropriately normalized traces of ISU (2) holonomies associated with oriented, framed and coloured knots. The perturbative evaluation of the Wilson lines expectation values is investigated and the up-to-third-order contributions to the perturbative expansion of the observables, which correspond to knot invariants, are produced. The general dependence of the knot observables on the framing is worked out. and S φπ can be written aswhere the bracket ·· JP denotes the non-degenerate bilinear form [9] on the ISU(2) algebra J a P b JP = δ ab , J a J b JP = 0 = P a P b JP .

Renormalized Schwinger–Dyson functional

“…For this reason, in the abelian Chern-Simons theory the Schwinger-Dyson functional provides the solution [22] for the link observables. Finally, let us consider the I SU(2) BF theory [39][40][41][42][43] in R 3 , with fields A a μ (x) and B a μ (s) and gauge-fixed action in the Landau gauge…”

“…For instance, since the vacuum polarization vanishes, from Eq. (3.22) one derives the following relation for the connected mean value Actually in both the Chern-Simons and the BF theories, the connected components of the Schwinger-Dyson functional containing up to six powers of the external classical sources are exhibited in closed form because of the vanishing of the loop corrections to the two-points and three-points correlation functions of the gauge fields [29,34,43].…”

Renormalized Schwinger-Dyson functional

“…Finally, let us consider the ISU(2) BF theory [39,40,41,42,43] in R 3 , with fields A a µ (x) and B a µ (s) and gauge-fixed action in the Landau gauge…”

“…The non-vanishing propagators for the components of the connection and the auxiliary fields are given by [43] A a µ (x)B b ν (y) = δ ab (3.17)…”

“…Actually in both the Chern-Simons and the BF theories, the connected components of the Schwinger-Dyson functional containing up to six powers of the external classical sources are exhibited in closed form because of the vanishing of the loop corrections to the two-points and three-points correlation functions of the gauge fields [29,34,43].…”