Perturbative BF theory

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 R3 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.


Introduction
Among the quantum field theory models of topological type [1], the so-called BF theory [2,3,4,5] has been proposed in order to describe several different phenomena.
Various BF quantization procedures have been examined [52,53,54,55,56,57,58,59,60,61] and the renormalizability of the theory has been proved by power counting and general arguments [62,63,64,65,66,67]. The BF model is actually a superrenormalizable theory; nevertheless, the explicit renormalization -with specified normalization conditions-has never been produced. One of the purposes of the present article is precisely to provide the complete renormalization of the nonabelian BF theory in R 3 .
The definition and computation of topological invariants [68,69,70,71,72,73,74,75,76,77] are central issues in the BF model. The observables that we propose have not been considered in literature. We shall demonstrate that the appropriately normalized traces of the expectation values of the holonomies -for the inhomogeneous group ISU(2)associated with oriented framed knots in R 3 are well defined. The first three orders of the perturbative computation of these observables are presented.
Let us recall that the solution of the abelian BF theory in generic closed oriented 3-manifolds has been produced by Mathieu and Thuillier [78,79,80]. In the present paper we shall concentrate on the perturbative approach to the nonabelian BF theory in R 3 with structure group ISU (2).
The Lie algebra of the inhomogeneous group ISU(2) can be interpreted as a particular extension of the SU(2) algebra which, in the quantum mechanics description of one particle moving in R 3 , is obtained by the introduction of the three components P a of the momentum in addition to the three components J a of the angular momentum. The corresponding ISU (2) connection has then six components A µ = A a µ (x)J a + B a µ (x)P a . The most general action in R 3 which is ISU(2) gauge invariant and metric-independent contains two different terms: the first term B a ∧ F a (A) -where F a (A) are the angular momentum components of the curvaturegives the name to the model and the second term Tr(A ∧ dA + i 2 3 A ∧ A ∧ A) coincides with the Chern-Simons action for the SU(2) subgroup.
Section 2 contains the fundamentals of the perturbative approach for the computation of the BF correlation functions of the connection in the Landau gauge. The general structure of the connected Feynman diagrams is worked out. The computation of the generating functional of the connected correlation functions to all orders of perturbation theory is presented and its Chern-Simons relationship is discussed in Section 3. The complete renormalization of the BF theory is given in Section 4. It is shown that the theory is superrenormalizable, and only six one-loop diagrams need to be examined. These one-particle-irreducible diagrams concern the two-point function and the three-point proper vertex of the connection. It is shown that, as in the case of the Chern-Simons theory, the two-point function of the connection does not receive loop corrections and therefore the bare propagator coincides with the dressed propagator.
In order to introduce Wilson line observables in the BF model, certain unitary representations of ISU(2) are described in Section 5. Since the group ISU(2) is noncompact, these nontrivial representations are infinite dimensional. Wilson line operators are defined by means of normalized traces of the ISU(2) holonomies associated with oriented knots. For completeness, the classical traces of the ISU(2) conjugacy classes are described in Section 6. The proof that the BF expectation values of the Wilson line operators are well defined is contained in Section 7. It is shown that, since the correlation functions of the connection are invariant under global ISU(2) transformations, the expectation value of a knot holonomy is a function of the Casimir operators of ISU (2). This implies that the BF mean values of the Wilson line operators are well defined and describe topological invariants for oriented and framed knots in R 3 .
The perturbative computation of the knot observables up to the third order in powers of is described in Section 8. The knot invariants that are found at first and second order correspond to the knot invariants that also appear in the Chern-Simons theory. While, at the third order of perturbation theory, the BF and Chern-Simons knot invariants differ. A proof that the entire framing dependence of the knot observables is completely determined by an overall multiplicative factor is given. This factor is the exponential of the linking number between the knot and its framing multiplied by the combination of the quadratic Casimir operators which is determined by the two point function of the connection. Section 9 contains the conclusions.

Fields, lagrangian and diagrams
The fundamental fields of the so-called BF theory [1,2,3,4,5,9] are given by the components of the ISU(2) connection where the generators (J a , P a ) (with a = 1, 2, 3) of the algebra of ISU(2) satisfy the commutation relations Let us consider the BF model defined in R 3 . Gauge transformations act as where Ω : R 3 → ISU(2). When Ω ≃ 1 + iβ a J a + iη a P a , the infinitesimal gauge transformations take the form The components of the curvature are given by and The action of the BF theory in R 3 is the sum of the two metric-independent terms which are separately invariant under ISU(2) transformations (2.4) Without loss of generality, the overall normalization of the first term in expression (2.8) can be taken to be (1/2), because the ISU(2) generators P a can be rescaled without any modification of the Lie algebra commutation relations (and consequently B a µ also can be rescaled). The real parameter g is a dimensionless coupling constant which multiplies the Chern-Simons lagrangian term When g = (k/4π) with integer k, one also recovers invariance under large gauge transformations, which anyway play no role in the perturbative approach to the theory. Note that, in order to discuss the renormalization of any gauge theory model, all the possible lagrangian terms which are gauge invariant must be taken into account. This is why the renormalization of the BF model requires that both lagrangian terms -shown in expression (2.8)-must be included in the action.

Gauge fixing
The gauge fixing procedure is implemented according to the BRST method [81,82]. The BRST transformations [83] are given by where {ξ a , ξ a , c a , c a } is the set of anticommuting ghosts and antighosts fields, whereas M a , N a represent the commuting auxiliary fields. In the Landau gauge, the gauge-fixing and ghosts action terms are given by where the flat euclidean metric g µν = δ µν of R 3 has been introduced in order to contract the vector indices. The total action S T OT = S + S φπ is invariant under BRST transformations. In order to recognize the structure constants of the ISU(2) Lie algebra in the gauge-fixing procedure, it is convenient to introduce the ghost field C = c a J a + ξ a P a , the antighost field C = ξ a J a + c a P a and the auxiliary field N = N a J a + M a P a . The BRST transformations take the form

Propagators
The Green functions of the differential operators acting on the fields -and entering the quadratic parts of S T OT in powers of the fields-determine the form of the fields propagators. As far as the bosonic fields are concerned, the nonvanishing components of the propagators are given by and For the anticommuting fields one gets (2.14)

Structure of the diagrams
The Feynman diagrams of the BF theory, which is defined by the action S T OT = S + S φπ in R 3 , have quite peculiar properties that we shall now discuss. Let us consider the generating functional W [J, K] of the connected correlation functions of the components of the connection where J a µ (x) and K a µ (x) are classical sources. We are interested in W [J, K] because in the following sections we shall consider the expectation values of observables which are functions of the fields A a µ and B a µ exclusively. In this section we consider the combinatorial structure -which is determined by the Wick contractions-of the Feynman diagrams contributing to W [J, K]. The renormalization will be treated in Section 4. The first issue to be discussed is an extension of the Oda and Yahikozawa observation presented in [84]. (2.16) The contribution W 0 [J, K] of the tree-level Feynman diagrams can be decomposed into the sum of two terms, AA = 0 and the nonvanishing components of the propagators are shown in equation (2.12). So let us now concentrate on diagrams which contain interaction vertices, which are of the type BAA or of the type AAA; it is convenient to recover all these diagrams in three steps.
1. The diagrams constructed with BAA interaction vertices and AB propagators exclusively are called the basic diagrams; these are the diagrams that remain in the g → 0 limit. They contain one power of the field J a µ and may contain an arbitrary number bigger than unit of K a µ fields. One example is shown in Figure 1(a). Indeed, each tree diagram is obtained by combining subdiagrams called "branches". One branch is a one-dimensional ordered sequence of vertices connected by propagators, as shown in Figure 1(b). Note that the external lines of Figure 1(b) correspond to field components and do not represent propagators; in particular, one branch diagram necessarily has external legs corresponding to one B a µ field and several A a µ fields.  3. Finally, by incorporating the additional possibility of including also vertices of the AAA type, the new diagrams only contain one extra AAA vertex with respect to the basic diagrams, they are linear in g and do not depend on J a µ , as shown in the example of Figure 2 As a result, the set of all the connected tree-level diagrams contains diagrams which are linear in g and do not depend on J a µ and diagrams which linearly depend on J a µ and do not depend on g. This concludes the proof of equation (2.17).
Let us now consider the one-loop connected diagrams entering W [J, K]. As shown in the example of Figure 3(a), connected diagrams with one loop of ghosts only depend on the source field K a µ because of the particular structure of the ghosts couplings (2.11). As far as the diagrams without a ghost loop are concerned, by cutting one internal propagator of each one-loop diagram one can open the loop and obtain a connected zero-loop diagram. In view of the result (2.17), the broken propagator was necessary of the AB type. Consequently, also each one-loop diagram with no ghost propagators does not depend on the J a µ field and does not depend on g, see the example of Figure 3(b). This concludes the proof of equation (2.18).

Generating functionals and Chern-Simons relationship
In order to complete the description of the BF diagrams, in this section we derive the BF generating functional of the connected correlation functions and discuss its relationship with the generating functional of the Chern-Simons theory.

Connected diagrams
In the computation of the path integral which appears in the numerator of expression (2.15), it is convenient to make the linear change of variables together with the gauge-fixing constraints Because of equations (3.2), the classical components A a µ and B a µ are functions of J a µ and K a µ , (and, for localised J a µ and K a µ , both components A a µ and B a µ vanish in the |x| → ∞ limit as ∼ 1/|x| 2 ). One then finds Note that S[A, B, ...] represents the resulting action for the quantum components A a µ and B a µ of the fields in which • the linear terms in the quantum fields are missing. Indeed, as a consequence of equations (3.2) and (3.3), A a µ and B a µ satisfy the classical gauge-fixing constraint and represent a stationary point of the action in the presence of the source terms; • the lagrangian vertices for the quantum fields -which are contained in S[A, B, ...]depend on the J a µ and K a µ through the classical components A a µ and B a µ .
Therefore the generating functional W [J, K] satisfies This expression shows that W [J, K] can be written as the sum of two parts, W = W 0 + W 1 , in which • the connected tree-level Feynman diagrams entering W 0 are described by a Legendre transformation of the classical action, • the connected diagrams containing loops -described by W 1 -are obtained by computing the vacuum-to-vacuum diagrams of the quantum field components. These diagrams are determined by the lagrangian terms contained in the resulting action S, with the normalization given by the vacuum-to-vacuum diagrams computed in the absence of sources, i.e., when A a µ and B a µ vanish.
where the Chern-Simons action S CS [A] is shown in equation (2.9); A a µ is a classical field which only depends on K a µ , it satisfies ∂ µ A a µ (x) = 0 and Proof. Since the BF action (2.8) can be written as the first of equations (3.2) coincides with equation (3.9). This means that A a µ (x) only depends on K a µ and does not depend on J a µ and g. Finally, the action S[A, B] is a linear function of B a µ . Therefore, in the Legendre transform (3.7), the two terms which are linear in B a µ cancel, and one obtains precisely expression (3.8).

Equation (3.8) is in agreement with expression (2.17), and shows that when
where W 0,CS [J] denotes the generating functional of the tree-level connected diagrams of the Chern-Simons theory, which is defined by the action gS CS [A], Let us now consider diagrams with loops.
Proposition 3. The whole set of the vacuum-to-vacuum connected diagrams for the quantum field components is equal to the set iW 1 [K] of the one-loop connected diagrams which only depend on K a µ , Proof. The field propagators that are derived from the S T OT are shown in equations (2.12) and (2.13); in particular, it turns out that A a µ (x)A b ν (y) = 0 and ξ(x)c(y) = 0 = c(x)ξ(y). Consequently, the only connected source-dependent diagrams containing loops are the one-loop connected diagrams entering equation (3.13).
The result (3.13) is in agreement with the statements of Proposition 1 and shows that, when where the factor 2 is due to the combinatorics and the presence of two ghost fields, and W 1,CS [J] denotes the generating functional of the one-loop connected diagrams in the Chern-Simons theory,

Connected one-loop diagrams
As a consequence of equation (3.13), the functional W 1 [K] can be written as corresponds to the sum of the connected diagrams with one loop of the vector fields, whereas W (g) 1 [K] denotes the sum of the connected diagrams with one loop of the ghost fields. In Schwinger notations [85], the AB propagator (2.12) reads and then and Tr denotes the trace in the color indices, vector indices and orbital indices The connected diagrams with one loop of the ghost fields give the contribution iW (g) and have values which are not well defined because of possible ultraviolet divergences; these diagrams will be renormalized in the Section 4.

Renormalization
Since the observables that we shall consider only depend on A a µ and B a µ , and since the corresponding BF connected diagrams have zero loops or one loop only, in order to discuss the relevant aspects of the renormalization we need to consider the functional where iΓ 1 [A, B] denotes to the sum of the one-particle-irreducible diagrams with one loop, in which A a µ and B a µ represent the external legs [86,87,88]. In other words, Γ 1 [A, B] is the sum of the one-loop proper vertices for the field components A a µ and B a µ . Indeed, as it has been shown in Section 2 and in Section 3, in the BF theory the contributions to the proper vertices which are described by diagrams with two or more loops are absent. The zero-loop component of the proper vertices coincides with the lagrangian and the one-loop component only contains primitive divergences. Therefore, in the renormalization procedure, diagrams with external ghost fields can be ignored.
Equations ( Each term of the expansion of Γ 1 [A] in powers of the fields A a µ is well defined apart from the terms with two and three fields. The corresponding six diagrams are not well defined a priori; they possibly have ultraviolet divergences. Since only a finite number of diagrams need to be renormalized, the BF model is a superrenormalizable field theory.

Normalization conditions
As there are no gauge anomalies in three dimensions, it is possible to define a renormalized Γ which is BRST invariant. Let us define In addition to the BRST invariance of Γ, the normalization conditions are taken to be and lim Equations (4.5) and (4.6) establish the normalization of the fields and specify the value of the coupling constant g. Since the one-loop contributions contained in Γ 1 [A] do not depend on the field B a µ , equation (4.6) -which is valid at the tree-level-remains valid to all orders of perturbations theory. Consequently, only equation (4.5) needs to be considered; in renormalized perturbation theory [87], equation (4.5) controls the one-loop counterterms. Let us consider the renormalization procedure [86,89,90,91,92,93] in the space of the coordinates x µ . Of course, the final result coincides with the result obtained by means of the renormalization procedure in momentum space.

One-loop two points function
can be expanded in powers of the fields A a µ ; the quadratic term is given by the sum of the contribution Γ One has and iΓ (g) Precisely like in the Chern-Simons theory [94,95], the sum of the two contributions Γ where The amplitude which appears in equation (4.10), is well defined for x = y. Consequently "the nonlocal component" of Γ is well defined and vanishes because When x = y expression (4.11) is not well defined, so one has to specify the value of H µν (x, y) in the case x = y. In facts, since "the nonlocal component" of Γ consists [86] precisely in specifying the value of "the local component" of Γ , which is defined by H µν (x, y) for x = y. This value is uniquely determined by the normalization condition (4.5), which requires It should be noted that the renormalized value (4.13) of Γ is also in agreement with the point-splitting procedure, that we shall use in the definition of the composite Wilson line operators. Indeed, the point-splitting definition of H µν (x, y) for x = y gives

One-loop three points function
The term of Γ 1 [A] which contains three powers of the field A a µ is the sum of Γ (v) 1 [A], which is described by the Feynman diagram of Figure 5(a), and Γ (g) 1 [A] which is specified by the one-loop contributions of Figure 5 One finds where and (4.20) When x = y, x = z and y = z, the amplitude is well defined and, when it is multiplied by the completely antisymmetric tensor ǫ µνλ τ ρσ , it vanishes, V µνλ (x, y, z) x =y =z = 0 . Therefore, as in the case of the two points functions, "the nonlocal component" of Γ is vanishing. In order to specify the renormalized value of Γ we need to define [89,90,91,92,93] the value of the "diagonal local component" of V µνλ (x, y, z), corresponding to the case in which the external fields are defined at coincident points x = y = z. This is in agreement with the general fact that, in one-loop diagrams, the possibly divergent (not well defined) contribution is local or, to be more precise, the introduction of appropriate local counterterms makes the diagrams well defined.
The renormalized value of Γ (v) is determined by the normalization conditions and by symmetry arguments. Indeed the BRS invariance of Γ requires that the value of the local component of the one-loop contribution to the 3-point proper vertex must be (1/6) the value of the one-loop contribution to the dressed propagator, which vanishes. Therefore relation (4.13) and BRST invariance imply The point-splitting procedure also shows that each "partially local component", say x = y = z, is vanishing because In renormalizable field theories, the "partially local components" of the diagrams are possibly related with the (overlapping) sub-divergences. In the connected diagrams of the BF theory, there are no subdivergences to deal with because the connected diagrams have at most one loop.
Since all the remaining diagrams contributing to Γ are finite, this concludes the renormalization of the BF theory in R 3 . This means that, by taking into account equations (4.13) and (4.23), the expectation values when the fields are defined at not coincident points, are well defined. In the computation of the BF observables, we shall need to remove certain ambiguities of the expectation values which appear in a specific limit in which two fields are defined in the same point. This issue, which is related to the introduction of a framing for the knots, will be discussed in Section 7.

Wilson line observables
Similarly to the case of the Chern-Simons gauge field theory, the gauge invariant observables that we shall consider correspond to appropriately normalized traces of the expectation values of the gauge holonomies which are associated with oriented framed knots in R 3 in a given representation of ISU(2).

Representations of ISU (2)
We shall consider linear unitary representations of ISU(2) in which {P a } are nontrivially represented and which are specified by the values of the two quadratic Casimir operators P a P a and J a P a . More precisely, if |ϕ denotes a vector transforming according to the irreducible (Λ, r) representation, it must satisfy P a P a |ϕ = Λ 2 |ϕ , and J a P a |ϕ = rΛ|ϕ , with fixed positive Λ and fixed semi-integer r (i.e., 2r ∈ Z). In this article we shall concentrate on the "scalar" (Λ, 0) representation and the "fundamental" (Λ, 1/2) representation.
In order to describe these representations, let us first consider the quantum mechanics states space of a spinless particle moving in three dimensional euclidean space. Let P a represent the cartesian components of the momentum operator and let L a denote the components of the orbital angular momentum of the particle, in which [Q a , P b ] = iδ ab . The operators {J a = L a , P a } satisfy the commutation relations (2.2).

(5.5)
When the value of the first Casimir operator P a P a of ISU(2) is chosen to be Λ 2 , one needs to consider the linear space H (Λ,0) which is generated by all the vectors In this case, the possible eigenvalues k of the momentum belong to a spherical surface in R 3 of radius equal to Λ. The set of all the plane waves {ψ k (r)} with kk = Λ 2 is left invariant by the action of the SU(2) group which is generated by the orbital angular momentum components (5.3). Therefore the linear space H (Λ,0) is invariant under the transformations generated by {J a = L a , P a }. Since L a P a = 0, the ISU(2) action on H (Λ,0) which is implemented by the transformations exp {iβ a J a + iη a P a } defines the scalar (Λ, 0) representation of ISU (2).
The commutation relations of the operators {J a = L a , Q a } also coincide with the commutation relations of the ISU(2) algebra. Thus an alternative interpretation of this ISU(2) representation can be obtained by considering the quantum mechanics states of one particle moving on the surface of a 2-sphere in R 3 . For the purposes of the present article, we don't need to discuss the rigged Hilbert space structure [96] associated with H (Λ,0) .

Fundamental representation
Let us now examine the fundamental (Λ, 1/2) representation of ISU (2). Let H spin denote the two dimensional space of the spin states of a spin (1/2) nonrelativistic particle, and let S a represent the components of the spin operator, where σ a denote the Pauli sigma matrices. The operators S a act on the vectors which belong to H spin . In the tensor product H (Λ,0) ⊗ H spin , one can put In addition to the constraint kk = Λ 2 , the specification of the value (1/2)Λ of the second Casimir operator J a P a selects the states in H (Λ,0) ⊗ H spin of positive helicity. Let π + denote the projector on the positive helicity states, Let H (Λ,1/2) be the linear space which is generated by the vectors π + |χ in which |χ ∈ H (Λ,0) ⊗ H spin . (5.10) The ISU(2) action on H (Λ,1/2) , which is carried out by the transformations generated by {J a = L a + S a , P a }, defines the (Λ, 1/2) representation. One can easily verify that the projector π + commutes with the generators of ISU(2). A generic (Λ, r) representation could be constructed by means of a procedure which is similar to the procedure that has been illustrated in the case of the (Λ, 1/2) representation. Each representation (Λ, r), with r = 0 or r = 1/2, is irreducible and infinite dimensional.

Holonomies
Let us consider a classical gauge configuration which is described by the components A a µ (x) and B a µ (x). Given an oriented path γ in R 3 , which connects the starting point x 1 to the final point x 2 , the corresponding ISU(2) holonomy h γ ∈ ISU(2) is defined by where the symbol P denotes the path-ordering of the {J a , P b } operators along the direction specified by the orientation of γ. Under a gauge transformation (2.3), h γ transforms as Thus for each non intersecting closed path C -that is, for each oriented knot C ⊂ R 3 -with a given starting and final point x 0 , the associated holonomy h C transforms covariantly under gauge transformations, Therefore any function, which is defined on the ISU(2) conjugacy classes, determines a classical gauge invariant observable. We shall describe the conjugacy classes of the group ISU(2) in Section 6. For the moment, let us recall the normal construction of classical gauge invariant observables for finite dimensional representations of the structure group. Let [h C ] ρ be the representative of the element h C ∈ ISU(2) in the representation ρ of the gauge group. If the representation ρ is finite dimensional, the cyclic property of the trace implies that Tr[h C ] ρ is gauge invariant. Really, in the BF theory we are interested in the ISU(2) representations (Λ, r), with r = 0 or r = 1/2, which are not finite dimensional. In this case, the ordinary traces of the holonomies in the representation spaces H (Λ,0) and H (Λ,1/2) need to be improved in order to specify a well defined observable.

Trace of holonomies
Let us consider the standard method which is used in physics -for instance in particle physics and in statistical mechanics-to describe the sum over the one-particle quantum states. One can introduce appropriately normalized plane waves where V = L 3 is the volume of a cubic box in which the particle can propagate; then one must consider the V → ∞ limit. From the definition (5.14) it follows With periodic boundary conditions, for instance, the possible values of the momenta are given by k = (2π/L) n, with n j ∈ Z. Therefore, in the large L limit, the sum over the eigenstates of the momentum is given by the integral [L 3 /(2π) 3 ] d 3 k, which also coincides with the counting of the number of quantum states in the semiclassical limit by means of the integral d 3 p d 3 q/(2π) 3 in classical phase space. With this notation, the trace of a given operator O p in the linear space of the one-particle orbital states takes the form which can easily be controlled in the V → ∞ limit because of the presence of the overall multiplicative V factor. The states of the (Λ, 0) representation are characterized by values of the momentum which belong to the 2-dimensional surface k 2 = Λ 2 in momentum space. In order to make contact with the d 3 p d 3 q/(2π) 3 expression for the counting of states in H (Λ,0) , one can introduce a small thickness ∆ P to the k 2 = Λ 2 surface. If, for instance, the relation L∆ P /(2π) = 1 is satisfied, then the ∆ P → 0 limit is recovered in the L → ∞ limit. According to this prescription, the trace of a given operator O p in the space H (Λ,0) of the (Λ, 0) representation of ISU (2) reads where dω = sin θ dθ dφ refers to the solid angle which is defined by the direction of the vector k, k = Λ(sin θ cos φ, sin θ sin φ, cos θ) . (5.19) Note that the presence of the product L 2 Λ 2 in equation (5.18) is required by dimensional reasons. Whereas different prescriptions for the ∆ P → 0 limit may lead to the presence of different adimensional multiplicative factors. These factors play no role because the Wilson line operators will correspond to appropriately normalized traces.
In the definition of the normalized trace of the holonomy h C , the multiplicative factor L 2 Λ 2 /π in front of expression (5.18) can be removed. So (in the L → ∞ limit) we define the Wilson line operator W C in the (Λ, 0) representation by means of the normalized trace

ISU (2) conjugacy classes
The set of the conjugacy classes of the inhomogeneous group ISU(2) has rather peculiar properties that show up also in the values of the corresponding classical characters.

Classes of conjugated elements
A generic element G ∈ ISU(2) can be written as where Θ ′ and X ′ denote the rotated vectors which are obtained according to the adjoint representation of SU(2), i.e. R ab (β) ∈ SO(3). Therefore, the conjugacy class of G is possibly labelled by the rotation invariants |Θ|, |X| and ΘX = Θ a X a . On the other hand, under conjugation with a translation element of ISU(2) Equation (6.5) shows that the parameter Θ is not modified and • when Θ = 0, X is not modified; • when Θ = 0, the component of X which is orthogonal to Θ can be arbitrarily modified. While the component of X along the direction of Θ is not modified.
The set of variables {(r 1 , r 2 )} does not parametrize a two dimensional manifold because of the singularity at r 1 = 0.
The observed discontinuity of the classical trace of G at Θ = 0 matches the structure of the set of ISU(2) conjugacy classes discussed in Section 6.1.

Expectation values
Let us concentrate on the BF topological invariants which are associated with oriented framed coloured knots in R 3 . A knot C in R 3 , with a specified irreducible ISU(2) representation, is called a coloured knot. The invariant W C which is associated with the knot C is defined by the BF expectation value of the Wilson line operator where W C corresponds to the normalized trace of the holonomy h C shown in equations (5.20) and (5.21). In perturbation theory, the determination of W C is obtained by means of the following steps: (1) expansion of the holonomy h C in powers of the gauge fields, (2) computation of the vacuum expectation values of the products of the gauge fields, and (3) evaluation of the normalized trace of the ISU(2) generators.
In the quantum BF field theory, the holonomy h C is a composite operator and its expansion in powers of the connection A contains product of fields at coincident points. As in the case of the quantum Chern-Simons field theory, the ambiguities of the mean value (7.1), which are due to the presence of fields at coincident points, are removed by means of the point-splitting limit procedure [98,99] which is based on the introduction of a framing of the knot C. So, the invariant (7.1) is really defined for framed knots.
The perturbative computation of W C is based on the expansion of h C in powers of the fields where A µ (x) = A a µ (x)J a + B a µ (x)P a and x 0 denotes a given base point on the oriented knot C. In expression (7.2), it is understood that the generators {J a , P b } are multiplied according to the order shown in the formula. More precisely, if {J a , P b } are collectively denoted by {T α }, one has A µ (x) = A α µ (x)T α and in equation (7.2) the products of connections mean, for instance, When the ISU(2) generators are not multiplied, they can be understood as elements of a tensor product in colour space; so, it is convenient to introduce the notation According to equation (7.2), for each ISU(2) representation (Λ, r) with r = 0 or r = 1/2, the normalized trace of h C in the colour space takes the form of a sum of normalized traces of product of generators J a and P b . It should be noted that, since the representations (Λ, r) are infinite dimensional, the cyclic property of the trace is no more valid; consequently, the classical gauge invariance of the trace of h C is not guaranteed. What saves the day is that the field theory expectation values of connection's products are invariant under global ISU(2) transformations.
Proof. The proof is made of two parts. First it shown that equation (7.4) is satisfied in the case in which G = e iβ a J a , and then it is demonstrated that equality (7.4) is satisfied for G = e iη a P a . When G = e iβ a J a , one has the total action S T OT = S + S φπ is invariant. Therefore equation (7.4) is fulfilled when G = e iβ a J a .
In the case G = e iη a P a , one gets the total action S T OT = S + S φπ is invariant as a consequence of the Jacobi identity. Thus equation (7.4) is satisfied for G = e iη a P a . To sum up, equation (7.4) is satisfied when G = e iβ a J a with arbitrary β a and also when G = e iη a P a with arbitrary η a . Therefore equality (7.4) holds for any G ∈ ISU(2).
A first consequence of equation (7.4) is that the two-points function A a µ (x)A b ν (y) must vanish because there is not an ISU(2) invariant which is quadratic in J a .
In the expansion (7.2) of h C in powers of the fields, the generators of ISU(2) are multiplied; hence equation (7.4) implies Thus, as in the case of the Chern-Simons theory, the expectation value of the holonomy associated with a knot C -with colour given by an irreducible representation of the gauge groupis proportional to the identity in colour space or, more precisely, it is a function of the Casimir operators of the structure group. This means that W C , which is the normalized trace of h C in the ISU(2) representations (Λ, 0) and (Λ, 1/2), is well defined, it is gauge invariant and it does not depend on the choice of the base point on C.
Finally, since the holonomy h C does not depend on the metric of R 3 and the only dependence of the total action on the metric is contained in the gauge fixing terms, the expectation value (7.1) corresponds to a topological invariant of oriented framed coloured knots in R 3 .

Perturbative expansion of the observables
The value of the observable W C , which is given by the normalized trace of the expectation value of the holonomy associated with the knot C ⊂ R 3 , can be obtained by computing the expectation value h C by means of an expansion of h C in powers of the fields. It is important to note that, in the evaluation of h C , the presence of a base point x 0 in the knot C must be taken into account. Thus, W C takes the form of a sum of an infinite number of perturbative contributions.
The invariant W C can be approximated by considering only a finite number of terms, but the truncation of the perturbative series cannot be introduced arbitrarily. In order to obtain topological invariants, one needs to sum all the diagrams which are necessary to ensure the gauge invariance of the result. This can be achieved by summing all the diagrams which are of the same order in powers of the parameters which multiply the gauge invariant lagrangian terms. The BF action is the sum of two terms which are separately invariant under gauge transformations; so, two independent parameters are required. One parameter can be chosen to be , and the second parameter can be taken to be the coupling constant g.
In the previous sections, the convention = 1 has been used. In this section, the dependence of the Feynman diagrams on is made explicit. Let us recall that a given Feynman diagram with P propagators and V interaction vertices is of order P−V . The dependence of a diagram on the coupling constant g can easily be determined because g multiplies the BB component of the propagator, equation (2.12), and the AAA interaction lagrangian term.
Note that the possible values of the group generators J a and P b represent "colour quantum numbers" that have vanishing field theory dimensions. If one wishes to give a physical interpretation to the vectors of the ISU(2) representations as particle state vectors, one can imagine that the eigenvalues of "momentum" P a refer to a given momentum scale, so that Λ is dimensionless.
In what follows, the perturbative contributions to W C of order n with n = 0, 1, 2, 3 are in order. The contribution of order n is indicated by W C (n) and contains all the nonvanishing components which are labelled by powers of g. The colour of the knot is specified by the (Λ, r) representation of ISU(2) with r = 0, 1/2.

Lowest order
With the chosen normalization of the traces shown in equations (8.1) and (8.2), the component of W C of order 0 is just the unit W C (0) = 1 .

First order
The contributions of order are given by the integration of the two components of the field propagator along the knot C, as sketched in Figure 6. The double line of Figure 6 generically indicates a framed knot C with its base point x 0 pointed out. The embedding of C in R 3 is not shown. A simple line represents a gauge field propagator (2.12).
x 0 C Figure 6. First order contribution to W C .
In this case, the point-splitting procedure, which is defined by means of the framing C f of the knot C, is used. Since the AB component of the propagator is of order and the BB component of the propagator is of order g, one finds where ℓk(C, C f ) denotes the linking number of C and its framing C f . Indeed, the linking number of two oriented knots C 1 and C 2 can be expressed [100] by means of the Gauss integral

Second order
The nonvanishing contributions of order 2 to W C are related with diagrams with two field propagators, shown in Figure 7, and diagrams with one vertex and three field propagators shown in Figure 8. As shown in Section 4, diagrams with one loop give vanishing results of order 2 .
In the computation of h C , diagrams with two field propagators give contributions which are proportional to the combinations of Casimir operators: (JP ) 2 = (J a P a ) 2 , (JP )(P 2 ) = (J a P a )(P b P b ) and (P 2 ) 2 = (P a P a ) 2 . Moreover, from the diagrams of the type shown in the second picture of Figure 7, one gets an additional contribution which is proportional to the Casimir operator P a P a . This is a consequence of the identity which follows from the structure of the ISU(2) algebra.
x 0 x 0 x 0 Figure 7. Second order contribution to W C with two field propagators.
The contributions to h C coming from the diagrams of Figure 7 are x 0 Figure 8. Second order contribution to W C with one vertex.
The nonvanishing contribution to h C coming from the diagram of Figure 8 is proportional to the Casimir operator P 2 , as a consequence of the identity ǫ abc P b J a P c = −2iP a P a , (8.8) and takes the form The sum of all the terms of order 2 is given by where ρ(C) is the knot invariant that has been found [98] in the study of the knot polynomials which are derived from the Chern-Simons field theory, where The ρ(C) knot invariant [98] gives the analytic expression of the second coefficient of the Alexander-Conway polynomial [100,101,102].

Third order
The value of W C (3) is given by the sum of the amplitudes which are associated with diagrams containing 3, 4 and 5 field propagators (2.12). In the computation of h C at order 3 , diagrams with one loop produce vanishing results. The contributions corresponding to the diagrams with 5 propagators and two lagrangian vertices, shown in Figure 9, are vanishing as a consequence of the algebra structure (2.2) of the ISU(2) generators. Diagrams with 4 propagators contain one vertex and are of the type shown in Figure 10. The corresponding amplitudes contain the combinations (JP )P 2 and (P 2 ) 2 of the Casimir operators. The sum of these contributions to h C is given by Diagrams with 3 propagators are sketched in Figure 11. The combinations of Casimir operators that one finds in this case are (JP ) 3 , (JP ) 2 P 2 , (JP )(P 2 ) 2 , (P 2 ) 3 , (JP )P 2 and (P 2 ) 2 . The resulting h C amplitude which is associated with the diagrams of Figure 11 is given by + permutations x 0 Figure 11. Third order diagrams with three propagators.
Finally, the sum of all the contributions of order 3 takes the form

Chern-Simons comparison
The knot invariants contained in W C (1) and W C (2) are precisely the invariants that one also finds in the Chern-Simons field theory (multiplying different Casimir operators, of course). At the third order, the knot invariants of the BF and of the Chern-Simos theory differ significantly. Indeed, the third order term W C (3) in the Chern-Simons theory -which has been computed correctly by Hirshfeld and Sassenberg [103]-contains a new knot invariant ρ III that does not appear in the BF theory. This seems to be caused by the special structure of the commutation algebra of the ISU(2) generators.

Framing dependence
Up to terms of order 3 , the normalized trace of the expectation value of the knot holonomy in the BF theory is given by the sum 3 n=0 W C (n) and can be written as Let us recall that, in the Chern-Simons theory, the framing factor [98,99] of the knot invariants is given by CS framing factor = e −i 2g ℓk(C,C f )C 2 (R) , (8.18) where C 2 (R) denotes the value of the quadratic Casimir operator in the R representation -of the structure group-which is associated with the knot, and g = (k/4π) is the CS coupling constant [95] which multiplies the Chern-Simons action. The framing dependence of the knot observables has a common origin in both the BF and the CS theories.
Proposition 5. The BF knot invariant W C of the framed knot C has the form where Q C does not depend on the framing C f of the knot C.
Proof. Let us recall that the framing of the knot C can be defined by means of a knot C f which belongs to the boundary of a tubular neighbourhood of C. If C is oriented, the orientation of C f is chosen to agree with the orientation of C.
It should be noted that the choice of a framing of a knot C ⊂ R 3 is equivalent to the specification of a trivialisation [100] of a tubular neighbourhood N of C. The space N ⊂ R 3 is a solid torus, in which C is the core of N and C f ⊂ ∂N. Let us define the standard solid torus V as the product V = S 1 × D 2 , where the two-dimensional disc D 2 is represented by the unit disc in the complex plane with coordinates {re iθ } in which 0 ≤ r ≤ 1 and 0 < θ ≤ 2π. Let {e iφ , re iθ } be coordinates of V ; the standard longitude λ of V is the curve on the boundary ∂V of coordinates {e iφ , 1} with 0 < φ ≤ 2π. A framing for C is a homeomorphism f : V → N, and the image of λ is precisely the knot C f . Up to ambient isotopy, the homeomorphism f : V → N is uniquely specified by the linking number of C and C f . This means that, in the quantum field theory context of the BF or CS theories, the whole dependence of W C on the framing is given precisely by the sum of all the perturbative contributions which are proportional to the linking number ℓk(C, C f ).
The linking number ℓk(C, C f ) is given the integral along C and C f of the corresponding Gauss density which appears in the expression (2.12) of the components of the propagator for the connection. The propagator corresponds to the two-point function of the connection fields that, in the BF and CS theories, receives no loop corrections (see Section 4 and [98,95]). When the components of the connection are coupled with classical sources J a µ (x) and K a µ (x), the set of the corresponding Feynman diagrams is described by the generating functional and, since the two-point function is connected, the sum of all the contributions containing the linking number ℓk(C, C f ) is precisey the exponential of the two-point function [87,88,104]. This means that, by neglecting the commutators between the generators J a and P a , the entire framing dependence of W C is given by the overall multiplicative factor which is just the exponential of ℓk(C, C f ) multiplied by the quadratic Casimir operator which is defined by the two-point function of the connection framing factor = e −i ℓk(C,C f )[P J−(g/2)P 2 ] . (8.21) Let us now take into account the fact that the generators {J a , P b } do not generally commute. The holonomy h C is defined by means of the path-ordered exponential and, in the perturbative expansion (7.2) of h C in powers of the fields, the path-ordering determines the precise position of the J a and P b operators in the product of the group generators along the knot C. Let us consider the Feynman diagrams -contributing to W C -in which a AA propagator connects two points of the knot C. There are only two possibilities: (a) the associated group generators are placed in consecutive positions in the path-ordering, or (b) the associated generators are nonconsecutive.
α β Figure 12. Part of a diagram with one propagator associated with two consecutive generators.
In the case (a), sketched in Figure 12, the two-point function is proportional to the contraction T α T β which is equal to the Casimir operators JP or P 2 , which commute with all the remaining generators and therefore behave as classical numbers (or classical sources).
In case (b), depicted in Figure 13, the generators T α and T β which are associated with the propagator are nonconsecutive, and one has, for instance, the sequence T α T σ T γ T β ; this product can be written as The first term on the r.h.s. of expression (8.22) contains the quadratic Casimir operator entering T α T β (which is equal to JP or P P ) and, when one combines all the terms of this type with the terms coming from case (a), one gets precisely the exponentiation shown in equation (8.21).
Since the set of all the perturbative contributions to W C takes the form of a sum of knot invariants, if one extract the knot invariant ℓk(C, C f ) the remaining terms necessarily represent knot invariants. Thus the remaining contributions, which contain the commutator appearing in expression (8.22), combine to produce knot invariants, which necessarily are not proportional to the linking number ℓk(C, C f ) because they do not contain the complete line integral along C and C f of the Gauss density. Therefore the framing dependence of W C is given by an overall factor which is precisely the exponential of ℓk(C, C f ) multiplied by the quadratic Casimir operator which is defined by the two-point function of the connection. In the CS theory, the quadratic Casimir operator is exactly T b T b = c 2 (R), whereas in the BF theory the two points function gives the combination [ P J − (g/2)P P ] of Casimir operators.

Conclusions
The gauge theory of topological type which is usually called the BF theory is a superrenormalizable quantum field theory in R 3 . We have described the structure of the Feynman diagrams which enter the perturbative expansion of the correlation functions of the connection, the corresponding generating functional has been computed and the relationship with the Chern-Simons theory has been produced. We have presented the complete renormalization of the BF theory, which involves the two-points function and three-points function of the connection. By means of the renormalization procedure in the space of coordinates -which is in complete agreement with the renormalization procedure in momentum space-one finds that, as in the case of the Chern-Simons theory, the two-points function of the connection does not receive loop corrections and therefore the bare propagator coincides with the dressed propagator. We have defined gauge invariant observables by means of appropriately normalized traces of the holonomies which are associated with oriented, framed and coloured knots in R 3 . The colour of a knot is specified by a given unitary irreducible representation of the structure group ISU(2). We have described the unitary ISU(2) representations with Casimir operators P 2 = Λ 2 and JP = rΛ -with r = 0, 1/2and the ISU(2) conjugacy classes have been determined. It has been shown that the expectation value of a knot holonomy is a function of the Casimir operators of the gauge group, so the expectation value of the normalized trace of knot holonomies are well defined and are gauge invariant.
The perturbative computation of the observables has been successfully achieved up to the third order in powers of . The knot invariants that we have found at first and second order correspond to the knot invariants that also appear in the Chern-Simons theory. Whereas the BF and CS knot invariants differ at the third order of perturbation theory. We have shown that the entire framing dependence of the knot observables is completely determined by an overall multiplicative factor which is the exponential of the linking number between the knot and its framing multiplied by the combination of the quadratic Casimir operators which is determined by the two point function of the connection.
In the present article, we have described the fundamentals of the perturbative approach to the BF theory in the case of structure group ISU (2). The extensions to more complicated groups appear to be quite natural. In particular, our results admit rather simple generalizations to the case of gauge group ISO(2, 1), which is related to a gravitational model in (2 + 1) dimensions.