Catren & Devoto - Extended Connection in Yang-Mills Theory
Catren & Devoto - Extended Connection in Yang-Mills Theory
Catren & Devoto - Extended Connection in Yang-Mills Theory
I. Introduction
It is our objective in the present work to show how the main geometric structures of
Yang-Mills theory can be unified in a single geometric object, namely a connection
in an infinite dimensional principal fiber bundle. We will show how this geometric
formalism can be useful to the path integral quantization of Yang-Mills theory. Some
of the historical motivations for the study of an extended connection in Yang-Mills
theory are the following. In the beginning of the 80’s Yang Mills theory was at the
center of important mathematical developments, especially Donaldson’s theory of four
manifolds’ invariants [9] and Witten’s interpretation of this theory in terms of a topo-
logical quantum field theory [23] (for a general review on topological field theories see
Refs. [5,8]). A central aspect of these theories is the study of the topological properties of
the space A /G , where A is the configuration space of connections in a G-principal bun-
dle P → M and G the gauge group of vertical automorphisms of P. It is possible to show
that under certain hypotheses one obtains a G -principal bundle structure A → A /G [9].
94 G. Catren, J. Devoto
The non-triviality of many invariants is then intimately linked with the topological
non-triviality of this bundle. In Ref. [3] Baulieu and Singer showed that Witten’s theory
can be interpreted in terms of the gauge fixed version of a topological action through a
standard BRST procedure. To do so, the authors unify the gauge field A and the ghost field
c for Yang-Mills symmetry in an extended connection ω = A + c defined in a properly
chosen principal bundle. The curvature F of ω splits naturally as F = F + ψ + φ, where
ψ and φ are the ghost for the topological symmetry and the ghost for ghost respectively
(the necessity of a ghost for ghost is due to the dependence of the topological symme-
try on the Yang-Mills symmetry). By expanding the expressions for the curvature F
and the corresponding Bianchi identity, the BRST transformations for this topological
gauge theory are elegantly recovered. It is commonly stated that the passage from this
topological Yang-Mills theory to the ordinary (i.e. non-topological) case is mediated by
the horizontality (or flatness) conditions, i.e. by the conditions ψ = φ = 0 (see Refs.
[2], [4,22]). In this case, an extended connection ω = A + c can also be defined for an
ordinary Yang-Mills theory with a horizontal curvature of the form F = F.
pst
M / M,
where pst : M R × M → M is the projection onto the second factor. We will denote
by Ad(P) the fiber bundle P ×G G → M associated to the adjoint action of G on itself
and by ad(P) the vector bundle P ×G g → M associated to the adjoint representation
of G on g. The gauge group G is the group of vertical automorphisms of P. It can
be naturally identified with the space of sections of Ad(P). Its Lie algebra Lie(G ) is
the space of sections of ad(P). Its elements can be identified with G-equivariant maps
g : P → g.
In the case of a principal fiber bundle over a finite dimensional manifold with a
compact structure group, there are three equivalent definitions of connections [9, Chap.
2]. In what follows, we will also consider connections on infinite dimensional spaces
(see Refs.[17,18]). For the general case, we will use the following as the basic definition
[16].
π
Definition 1. Let K be a Lie group with Lie algebra k and let E − → X be a K -principal
bundle over a manifold X (both K and X can have infinite dimension). A connection on
E is an equivariant distribution H , i.e. a smooth field of vector spaces H p ⊂ T E p (with
p ∈ E) such that
1. For all p ∈ E there is a direct sum decomposition
T E p = H p ⊕ ker dπ p . (1)
2. The field is preserved by the induced action of K on T E, i.e.
H pg = Rg∗ H p ,
where Rg∗ denotes the differential of the right translation by g ∈ K .
96 G. Catren, J. Devoto
σ
y
0 / ker dπ / TE / π ∗T X / 0, (2)
ι dπ
d E = d H + dV (3)
into a horizontal and a vertical part. The horizontal part corresponds to the covariant
derivative. The vertical part is defined by the expression
ω ω
dV α(X 1 , . . . , X n ) = dα( V X 1, . . . , V X n ), (4)
where α is a (n − 1)-form. The vertical forms ∗V (E) equipped with the vertical differ-
ential dV define the vertical complex.
Let’s now suppose that it is possible to define a global section σ : X → E. This
section defines a global trivialization ϕσ : X × K → E, where ϕσ (x, g) = σ (x) · g.
This trivialization induces a distinguished connection ωσ on E such that the pullback
.
connection ω̃σ = ϕσ∗ ωσ coincides with the canonical flat connection on X × K [16].
Roughly speaking, the horizontal distribution defined by ωσ at p = σ (x) is tangent to the
section σ . The vertical complex defined by the connection ω̃σ can be naturally identified
with the de Rham complex of K . This implies that dV = d K . Since the connection form
ω̃σ is a k-valued K -invariant vertical form, it can be identified with the Maurer-Cartan
form θ MC of the group K . On the contrary, a general connection ω defines a splitting
of T E which does not coincide with the splitting induced by the section σ . In other
words, the horizontal distribution defined by ω is not tangent to σ . In fact, the pullback
connection ϕσ∗ ω at (x, g) can be written as a sum
forms AU are the so-called gauge fields [11].1 The configuration space of all connections
is an affine space modelled on the vector space 1 (M, g) consisting of 1-forms with
values on the adjoint bundle ad(P). The gauge group G acts on this configuration space
by affine transformations. We will fix a metric g on M and an invariant scalar product
tr on g. These data together with the corresponding Hodge operator ∗ induce a metric
on 1 (M, g). Hence, a metric can be defined in the spaces k (M, g), k ≥ 1, by means
of the expression
1 , 2 = tr ( 1 ∗ 2 ). (6)
M
Since the action of G on the configuration space of connections is not free, the quotient
generally is not a manifold. This problem can be solved by using framed connections
[9]. The letter A will denote the space of framed connnections of Sobolev class L l−1 2
for the metric defined in (6), where l is a fixed number bigger than 2. The group G is the
group of gauge transformations of Sobolev class L l2 . The action of G on A is free (see
Ref.[9, Sect. 5.1.1]). We will denote by B the quotient A /G . Uhlenbeck’s Coulomb
gauge fixing theorem (see Ref.[9, Sect. 2.3.3]) implies, for a generic metric g, local
triviality. Hence A → B is a G -principal bundle.
The initial geometric arena for our construction is the pullback G-principal bundle
p ∗ (P) → A × M, which is obtained by taking the pullback of the bundle P → M by
p
the projection A × M − → M. The bundle p ∗ (P) can be identified with A × P:
p ∗ (P) = A × P /P
p
A ×M / M.
The gauge group G has an action on A × M induced by its action on A . This action
is covered by the action of G on p ∗ (P) induced by its action on both A and P.
Proposition 2. The bundle p ∗ (P) → A × M induces a G-principal bundle [1]
ρ
Q = (A × P)/G −
→ B × M.
G / p ∗ (P) = A × P
q
G / Q = (A × P)/G
ρ
B × M.
1 In a covariant framework, the connection A can be regarded as the spatial part of a connection A on
P → M. In fact, since we can identify P with R × P, each connection A has a canonical decomposition
A = A(t) + A0 (t)dt, where A(t) is a time-evolving connection on P and A0 (t) is a time-evolving section
of ad(P) = P ×G g. The action of the gauge group G on A0 is induced by the natural action on associated
bundles. This action is the restriction of the action of the automorphism group of P to {t} × M.
98 G. Catren, J. Devoto
2 In Ref.[19] the authors analyze the particular case of the Coulomb connection for a SU (2) Yang-Mills
theory on S 3 × R. The authors point out that in the absence of a global section, the gauge can be consistently
fixed by means of such a connection.
Extended Connection in Yang-Mills Theory 99
ι(g) being the vertical subspace. Hence, the distribution H does not define vector
spaces complementary to the vertical subspace ι(g). However there is a reason for the
introduction of this distribution which is explained by the following lemma.
is G -invariant and induces a connection H on
Lemma 5. The distribution H
Q = (A × P)/G → B × M.
This lemma follows from the invariance of the distributions Hη and HU .
Let E be the connection on the bundle p ∗ (P) → A × M obtained as the pullback of
the connection H by the projection A × M → B × M. This pullback can be understood
either in the language of distributions or in the language of forms. Let’s consider the
diagram
p ∗ (P) = A × P
nn OOO
q nnnn OOO
nn n OOO
wnnn OOO
'
A ×P
Q= G A ×M
PPP o
PPP ooooo
PPP
PPP ooo
( wooo
B × M.
The map q : p ∗ (P) → Q induces a map q∗ : T p ∗ (P) → T Q. At each point (A, p)
the subspace of T(A, p) p ∗ (P) which defines E is q∗−1 (Hq(A, p) ). The g-valued 1-form
A ∈ 1 (A × P) ⊗ g associated to E is the pullback by q of the 1-form associated to
H . We will now identify this distribution and this 1-form.
The distribution which defines the new connection at each point (A, p) is the direct
sum
E(A, p) = T FG (A, p) ⊕ Hη (A) ⊕ H U (A)( p) . (9)
H(A, p)
100 G. Catren, J. Devoto
We will now show that the horizontal distribution defined by A is effectively given
by (9).
Lemma 6. If v ∈ T F(A, p) ⊕ H (A, p) , then A(v) = 0.
Proof.
(i) If v ∈ H U (A)( p) ⊂ H (A, p) , then
A(v) = AU
(A, p) (v) = A(v) = 0
A(v) = η(v)( p) = 0
A (v) = −AU
(A, p) (ι(τ p (g))) + η (κ(g)) ( p)
= −A(ι(τ p (g))) + η (κ(g)) ( p)
= −τ p (g) + g( p)
= 0,
Remark 7. It is the gauge fixing connection Hη that allows us to make the decompositions
(9) and (10) of the horizontal distribution E and the corresponding 1-form A. The reason
is that these kinds of decompositions require the choice of a complement to a subspace
of a vector space.
Remark 8. An important difference with the work of Baulieu and Singer for topological
Yang-Mills theory is that in Ref. [3] the connection ω is a natural connection, which
is defined by using the orthogonal complements to the orbits of G. In order to define
these orthogonal complements one uses the fact that the space A × P has a Riemannian
metric invariant under G × G (see Ref. [1] for details). In our case the connection A,
being tautological in the factor P, is not natural in the factor A , in the sense that the
gauge fixing connection η can be freely chosen. This freedom is in fact the freedom to
choose the gauge.
Extended Connection in Yang-Mills Theory 101
We will now write the explicit decomposition of both sides of Eq. (15). Let δ = δV +δ H
be the decomposition of the de Rham differential on A induced by η. On degree (0, ∗)
the decomposition is given by the following definition. Let pV and p H be the projectors
onto the factors associated with the decomposition into vertical and horizontal forms. If
we think of elements of 0 (A ) ⊗ 1 (P) ⊗G g as 1 (P) ⊗G g-valued functions on A ,
then δ has a natural decomposition δ = δV + δ H , where δV = pV ◦ δ and δ H = p H ◦ δ.
The universal family of connections AU can be interpreted as a function AU : A →
1 (P) ⊗ g. We have a splitting
G
δAU = δV AU + δ H AU ∈ 1
(A ) ⊗ 1
(P) ⊗G g,
given on each copy {A} × Lie(G ) by the covariant derivative d A : Lie(G ) → 1 (P) ⊗
Lie(G ) associated to the connection A. Recall that sections of ad(P) can be seen as
equivariant functions P → g and that the covariant derivative is d A = d ◦ π A , where d
is the exterior derivative in P and π A : T P → T P is the horizontal projection. These
constructions are equivariant, which explains the codomain in Eq. (17). The term dAU η
is by definition the extension
1 ⊗ dAU : 1
(A ) ⊗ Lie(G ) = 1
(A ) ⊗ ( 0
(P) ⊗G g) → 1
(A ) ⊗ 1
(P) ⊗G g
applied to η. Since the homomorphism 1 ⊗ dAU acts on the second factor, it preserves
vertical forms. It follows that dA U η = dη + [AU , η] is a vertical form. From these
remarks we see that the vertical summand of the left hand side of Eq. (15) is
We will consider now the right hand side of Eq. (15). We will demonstrate the
following proposition.
Proof. Since the connection A is the pullback of a connection on the G-fiber bundle
Q = (A × P)/G → A /G × M, the same is true for the curvature F. If we denote ωH
and FH for the connection and curvature forms of the distribution H on Q, then one
has
A = q ∗ ωH ,
F = q ∗ FH ,
q
where q is the projection A × P − → Q = (A × P)/G .
If X is a vector tangent to the fibers T FG of the action of G on A × P, then the
contraction ı X F is equal to ı X q ∗ FH = ı q∗ X FH . Since X has the form X = (v, −v) ∈
T FG with T FG given by (8), then q∗ X = 0. This results from the fact that the vectors
tangent to G are projected to zero when we take the quotient by the action of G . Hence,
Extended Connection in Yang-Mills Theory 103
the contraction ı X F of the curvature F with a vector X = (v, −v) tangent to the fibers
given by (8) is zero:
• ı (v,−v) F(2,0) = ı (v) F(2,0) = 0, since F(2,0) is induced by the connection η and v is
vertical for this connection.
• ı (v,−v) F(0,2) = ı −v F(0,2) = 0, since F(0,2) is induced by the connection A and −v is
tangent to the fibers of p ∗ (P) → A × P.
• ı −v F(1,1) = 0, since −v is tangent to the fibers of p ∗ (P) → A × P.
δV AU = −dAU η, (20)
δ H A = ψ.
U
(21)
The equation for the (2, 0)-form φ can also be canonically decomposed in components
belonging to the vertical and horizontal complexes. The differential δ acting on elements
of 1 (A ) ⊗ Lie(G ) = 1 (A ) ⊗ 0 (P) ⊗G g also has a decomposition δ = δV + δ H ,
where δV = δ ◦ pV and δ H = δ ◦ p H . The horizontal part δ H corresponds to the covariant
derivative with respect to the connection η. Therefore we have a splitting
δη = δV η + δ H η ∈ 2
(A ) ⊗ ( 0
(P) ⊗G g) = 2
(A ) ⊗ Lie(G ).
δ H η = φ. (22)
1
δV η = − [η, η] . (23)
2
In the next section we will show how the BRST transformations of the gauge and
ghost fields can be obtained from Eqs. (20) and (23) respectively.
104 G. Catren, J. Devoto
IV. The Relationship Between the Gauge Fixing Connection and the Ghost Field
The proposed formalism allows us to further clarify the relationship between the gauge
fixing and the ghost field. To do so, we shall first work in a local trivialization ϕσi :
Ui × G → π −1 (Ui ) defined by a local gauge fixing section σi : Ui → A over an open
subset Ui ⊂ A /G . Let η̃ be the pull-back by ϕσi of the connection form η restricted to
π −1 (Ui ). As we have seen in Sect. II, the connection form η̃ at ([A], g) ∈ Ui × G takes
the form
η̃ = adg−1 ηi + θ MC , (24)
where ηi = σi∗ η ∈ 1 (Ui ) ⊗ Lie (G ) is the local form of the connection η and
θ MC ∈ Lie (G )∗ ⊗ Lie (G ) is the Maurer-Cartan form of the gauge group G [7]. The
Maurer-Cartan form satisfies the equation
1
δG θ MC = − [θ MC , θ MC ] . (25)
2
The formal resemblance between this equation and the BRST transformation of the
ghost field c led in Ref.[6] to the identification of δ B R ST and c with the differential δG
and the Maurer-Cartan form θ MC of G respectively. Hence, Eq. (24) shows that the ghost
field can be identified with the canonical vertical part of the gauge fixing connection η
expressed in a local trivialization.
We will now show that it is possible to recover the standard BRST transformation
of the gauge field δ B R ST A = −d A c from Eq. (20). To do so, we will first suppose that
it is possible to define a global gauge fixing section σ : B → A . As we have seen in
Sect. II, the associated trivialization ϕσ induces a distinguished connection ησ such that
ϕσ∗ ησ = θ MC . Therefore, Eq. (20) yields in this trivialization
δG AU = −dAU θ MC . (26)
This equation is an extension to families of the usual BRST transformation of the
gauge field A. If a global gauge fixing section cannot be defined, then it is possible to
show that the usual BRST transformation of A is valid locally. In fact, since δV AU is
a vertical form, the substitution of the local decomposition (24) in Eq. (20) yields the
BRST transformation (26). We can thus conclude that the usual BRST transformation of
A given by (26) is only valid in a local trivialization of A → A /G . Therefore, Eq. (20)
can be considered the globally valid BRST transformation of the gauge field A. In fact,
we will now show that Eq. (20) plays the same role as the usual BRST transformation
of the gauge field. To do so, we have to take into account that the BRST transformation
δ B R ST A = −d A c defines a general infinitesimal gauge transformation of A [6]. In order
to obtain a particular gauge transformation from this general expression, it is necessary
to choose an element ξ ∈ Lie (G ). In doing so, the usual gauge transformation of A is
recovered
δ A = (δ B R ST A)(ξ ) = −d A (c(ξ )) = −d A ξ. (27)
Let’s now consider Eq. (20). According to the definition of connections, η(ξ ) = ξ ,
where ξ is the fundamental vector field in T A corresponding to ξ ∈ Lie (G ). Therefore,
Eq. (20) yields
Remark 10. Contrary to what is commonly done in order to reobtain the BRST
transformations for the ordinary (non-topological) Yang-Mills case, it has not been nec-
essary to impose the horizontality conditions φ = ψ = 0 on the extended curvature F
(see for example Refs.[2,4,22]).
where S is the canonical action, D A is the Feynman measure on the space of paths
P([A0 ], [A1 ]) = {γ : [0, 1] → A /G | γ (i) = [Ai ], i = 0, 1},
in A /G and Dπ is a Feynmann measure in the space of moments. The canonical
Yang-Mills’s action is given by the expression
S = dt d 3 x Ȧak πak − H0 πak , Bak − Aa0 φa , (29)
with πak = Fak0 and Bka = 21 εkmn F amn (where Fmna are the field strengths). The Yang-
Mills Hamiltonian H0 πa , Ba is
k k
1
H0 πak , Bak = πak πka + Bak Bka , (30)
2
and the functions φa are
φa = −∂k πak + f ab πc Ak .
c k b
(31)
The pairs (Aak , πak ) are the canonical variables of the theory. The temporal component
Aa0 is not a dynamical variable, but the Lagrange multiplier for the generalized Gauss
constraint φa ≈ 0.
The geometry of the quotient space A /G is generally quite complicated. The usual
approach is to replace the integral (28) with an integral over the space of paths in the
affine space A . To do so, one must pick two elements Ai ∈ π −1 [Ai ] in the fibres
[Ai ] ∈ A /G (i = 0, 1). Then one replaces the integral (28) by
A0 | A1 = exp{i S}DA Dπ, (32)
T ∗ P (A0 , A1 )
where the integral is now defined on the cotangent bundle of the following space of paths
in A
P(A0 , A1 ) = {γ : [0, 1] → A | γ (0) = A0 , γ (1) = A1 }.
The problem with this approach is that it introduces an infinite volume in the path
integral, which corresponds to the integration over unphysical degrees of freedom. The
projection π : A → A /G induces a projection
π : P(A0 , A1 ) → P([A0 ], [A1 ]).
Extended Connection in Yang-Mills Theory 107
V.2. Generalized gauge fixing. We will now consider in which sense the connection η
can be used to fix the gauge. This gauge fixing will be globally well-defined, even if there
is a Gribov’s obstruction. We shall begin by considering paths in A such that the initial
condition A0 is fixed and the final condition is defined only up to a gauge transformation
(see Ref.[19, p. 123]). This means that the final condition can be any element of the final
fiber π −1 [A1 ]. The corresponding space of paths is
P(A0 , π −1 [A1 ]) = {γ : [0, 1] → A | γ (0) = A0 , π(γ (1)) = [A1 ]}.
The relevant path group is now
PG = {g(t) : [0, 1] → G | g(0) = idG }.
This group acts on P(A0 , π −1 [A1 ]). This actions defines the projection
π : P(A0 , π −1 [A1 ]) → P([A0 ], [A1 ]).
It is easy to show that the action of the path group PG on P(A0 , π −1 [A1 ]) is free.
We will not need to assume that it is a principal bundle.
The gauge fixing by means of the connection η is defined by taking parallel transports
along paths in A /G of the initial condition A0 ∈ π −1 [A0 ] (as has already been suggested
in Ref.[19]). This procedure defines a section σ η of the projection π ,
ση
u
π / P([A0 ], [A1 ]).
P(A0 , π −1 [A1 ]) (33)
108 G. Catren, J. Devoto
The section σ η sends each path γ ∈ P([A0 ], [A1 ]) to its η-horizontal lift
γ = σ η γ ∈ P(A0 , π −1 [A1 ]) starting at A0 .3 A path γ ∈ P(A0 , π −1 [A1 ]) is
in the image of σ η if and only if the tangent vectors to γ at each A ∈ A belong to the
horizontal subspaces Hη (A) defined by η. Recalling that Hη (A) = Ker η(A), this local
condition leads to the gauge fixing equation
In local bundle coordinates this condition defines a non-linear ordinary equation. The
explicit form of this equation is given in (57) (see Ref.[18] for details). In the case of
the Coulomb connection defined in (7), Eq. (34) becomes
Remark 11. The generalized gauge fixing can be also be defined as the null space of
a certain functional as follows. The Lie algebra Lie(G ) of the gauge group G can
be identified with the sections of the adjoint bundle ad(P). The invariant metric on g
induces a metric < , >ad(P) on ad(P). Using this metric we define a G -invariant metric
on Lie(G ) = (ad(P)) by
σ1 , σ2 Lie(G ) = < σ1 ( p), σ2 ( p) >ad(P) d x. (36)
M
This is a positive functional and the image of the section ση is the null space of F.
We will now proceed to implement the proposed generalized gauge fixing at the level of
the path integral. To do so, we will show that the usual Faddeev-Popov method can also
3 The theorem of existence of parallel transport has been extended to infinite dimensions in Ref. [17,
Theorem 39.1]. It can be shown that under suitable assumptions the parallel transport depends smoothly on
the path. We will therefore assume that the section σ η is smooth and that its image is a smooth submanifold
of P(A0 , π −1 [A1 ]). By definition, this submanifold is transversal to the action of PG .
Extended Connection in Yang-Mills Theory 109
be used with this generalized gauge fixing. We will then start by introducing our gauge
fixing condition at the level of the transition amplitude
A0 | π −1 [A1 ] = exp{i S}DA Dπ. (38)
T ∗ P (A0 , π −1 [A1 ])
The first possible form of the gauge fixing condition is δ(F(γ )), where δ is the Dirac
delta function on R and F(γ ) the functional (37). This form is mathematically consistent
and does not require any product of distributions. This gauge fixing condition has the
direct exponential representation
δ(F(γ )) = dλeiλF (γ )
iλ η(γ̇ (t))2Lie(G ) dt
= dλe γ
iλ η(γ̇ (t))2g d xdt
= dλe γ M .
The second form is based on the elementary observation that the integral of a con-
tinuous positive function is zero if and only if the function is zero at all points. One can
then define the gauge fixing condition
N
δ(η(γ̇ )) = lim δLie(G ) (η(γ̇ (tk ))) (39)
N
k=1
N
M
= lim δg(η(γ̇ (tk ))(x j )),
N ,M
k=1 j=1
where δLie(G ) is the delta function on Lie(G ) defined as an infinite product of the Dirac
delta δg on g. If Ta is a fixed basis of g, we can write δg(η(γ̇ (tk ))(x j )) in terms of
δ(η(γ̇ (tk ))(x j )a ), where now the delta function is the usual delta function on R3 .
As usual we define the element −1 [γ ] as
−1 [γ ] = D g δ(η(γ ˙g )), (40)
PG
Proof.
−1 [γ g] = ˙ )) =
D g δ(η(γ gg ˙ ))
D(gg )δ(η(γ gg
PG PG
Roughly speaking, the element [γ ] corresponds to the determinant of the oper-
ator which measures the gauge fixing condition’s variation under infinitesimal gauge
transformations.
It can be shown that the element [γ ] is never zero. The local gauge fixing con-
dition η(γ̇ (t)) = 0 induces a well-defined section σ η of the PG -projection π :
P(A0 , π −1 [A1 ]) → P([A0 ], [A1 ]) in the space of paths. Since the sub-manifold defined
by the image of σ η is by definition transversal to the action of PG , an infinitesi-
mal gauge transformation of the gauge fixing condition η(γ̇ (t)) = 0 will be always
non-trivial. In this way we can argue that the element [γ ] is never zero. This fact
ensures that the connection η induces a well-defined global gauge fixing, even when the
topology of the fiber bundle A → A /G is not trivial.
By inserting (41) in (38) we obtain
A0 | π −1 [A1 ] = DA Dπ [γ ] D g δ(η(γ ˙g ))) exp{i S} .
T ∗ P (A0 , π −1 [A1 ]) PG
where we have used that DA Dπ , −1 [γ ] and S are gauge invariant. In this way we
have isolated the infinite volume of the path group PG .
We will now follow the common procedure for finding the new terms in the action
coming from the Dirac delta δ(η(γ̇ )) and the element [γ ].
In order to find an integral representation of the gauge condition’s Dirac delta δ(η(γ̇ ))
we will start by finding the integral representation of the Dirac delta function δLie(G ) on
Lie(G ) used in (39). If ξ ∈ Lie(G ), the Dirac delta δLie(G ) (ξ ) defined as
M
δLie(G ) (ξ ) = lim δg(ξ(x j )),
M
j=1
can be expressed in terms of the integral representations of the Dirac delta δg(ξ(x j )) on
g as
M
M
λ(x j ),ξ(x j )
δLie(G ) (ξ ) = lim dλ(x j )ei j=1 g
,
M
j=1
= i
Dλe d x λ(x),ξ(x) g ,
= i
Dλe λ,ξ Lie(G ) ,
Extended Connection in Yang-Mills Theory 111
where λ is a section of ad(P) = P ×G g and Dλ = lim M M dλ(x j ). The Dirac delta
j=1
δ(η(γ̇ )) of the gauge fixing condition can then be expressed as
N
δ(η(γ̇ )) = lim δLie(G ) (η(γ̇ (tk )))
N
k=1
N
N
= lim k ei
Dλ k=1 λk ,η(γ̇ (tk )) Lie(G )
N
k=1
i λ,η(γ̇ (t)) Lie(G ) dt
= Dλe γ
i λ,η(γ̇ (t)) g d xdt
= Dλe γ M ,
Let’s now compute explicitly the element [γ ]. Let X be an element of the Lie
algebra Lie(PG ) identified with the tangent space of PG at the identity element.
Given a path γ (t) ∈ P(A0 , π −1 [A1 ]), one must calculate the variation of η (γ̇ ) under
an infinitesimal gauge transformation defined by X ∈ Lie(PG ). Let u → ku be the
uniparametric subgroup of PG generated by X by means of the exponential map exp :
Lie(PG ) → PG . We have then X = du d
ku |u=0 ∈ Lie(PG ).
Remark 13. The gauge fixing condition has a natural interpretation in terms of the geom-
etry of P(A0 , π −1 [A1 ]). The tangent space Tγ P(A0 , π −1 [A1 ]) to P(A0 , π −1 [A1 ]) at
a path γ can be identified with the sections of the pullback γ ∗ (T A). The connection η
induces a map from Tγ P(A0 , π −1 [A1 ]) to the Lie algebra of PG . The tangent field γ̇
represents a marked point in Tγ P(A0 , π −1 [A1 ]). The gauge fixing condition means that
we will only consider paths such that the connection η vanishes on the marked point γ̇ .
Using the description of the tangent spaces to path spaces given in the previous remark
we can identify X with a map t → X t with 0 ≤ t ≤ 1 and X t ∈ Lie(G ). In order to find
an expression for the vectors X t one must take into account that ku describes a family
of paths in the gauge group G parameterized by t. This means that ku = gu (t) ⊂ G for
u fixed. To emphasize this, let us write ku (t). For a given t, the vector X t ∈ Lie(G ) is
then given by X t = du d
ku (t)u=0 . If γ ∈ P(A0 , π −1 [A1 ]) we must compute
d d
ηγ (t)ku (t) Rk (t) γ (t) |u=0 . (42)
du dt u
At a fixed time t0 the time derivative in (42) is equal to
d d
Rk (t) γ (t)|t=t0 = Rku (t0 )ku (t0 )−1 ku (t) γ (t)|t=t0
dt u dt
d
= Rk (t ) γ (t)|t=t0
dt u 0
d
+ Rku (t0 )−1 ku (t) Rku (t0 ) γ (t0 ) |t=t0
dt
= d Rku (t0 ) (γ̇ (t0 )) + ι(γ (t0 )ku (t0 )) (X u ), (43)
112 G. Catren, J. Devoto
ηγ (t0 )ku (t0 ) (d Rku (t0 ) (γ̇ (t0 ))) = Ad(ku−1 (t0 ))ηγ (t0 ) (γ̇ (t0 )). (44)
The equality (44) follows from one of the connection’s defining properties (see (54)
in Appendix A). The second term is
where we have used Eq. (54). The infinitesimal variation defined by X ∈ Lie(PG ) is
given by the sum of
d
Ad(ku (t0 )−1 )ηγ (t0 ) (γ̇ (t0 )) |u=0 = Ad(−X t0 )ηγ (t0 ) (γ̇ (t0 ))
du
= −X t0 , ηγ (t0 ) (γ̇ (t0 ))
and d
du X u |u=0 . Let’s now calculate this last term:
d d −1 d
X u |u=0 = ku (t0 ) ku (t) |u=0,t=t0
du du dt
−2 dku (t0 ) dku (t) −1 d dku (t)
= −ku (t0 ) + ku (t0 ) |u=0,t=t0
du dt dt du
dku (t0 ) dk0 (t)
= −k0 (t0 )−2 |u=0 |t=t0
du dt
d dku (t)
+k0 (t0 )−1 |u=0
dt du t=t0
= Ẋ t (t0 ),
This expression defines a linear endomorphism Mγ in Lie(PG ) for each path γ (t).
Equivalently, it defines a linear endomorphism Mγ (t) in Lie(G ) for each t.
4 If we use the usual form for the gauge transformation of connections (49) the time derivative in (42) can
be expressed as
d
ad(ku−1 (t0 ))γ̇ (t0 ) + ad ku−1 (t) γ (t0 ) + ku−1 (t)dku (t) t=t0 .
dt
The first term is equal to the differential of the action d Rku (t0 ) (γ̇ (t0 )) (see (53)). The second term is equal to
the infinitesimal symmetry ι(γ (t0 )ku (t0 )) (X u ) (see (52)).
Extended Connection in Yang-Mills Theory 113
In order to find the exponential representation of the element γ , we will introduce
a Grassmann algebra generated by the anticommuting variables c and c̄. By following
the common procedure, we can express the element γ as
[γ ] = D c̄Dce c̄t ,Mγ (t)ct g d xdt ,
3
where
N
N
M
Dc = lim t = lim
Dc dctk (x j ),
k
N N ,M
k=1 k=1 j=1
and the same for D c̄ (see Ref.[24] for a precise definition of c and c̄).
By gathering all the pieces together, the path integral takes the form
−1
A0 |π [A1 ] = D g DA Dπ DλDcD c̄ exp{i Sg f }, (46)
PG
where Sg f is the gauge fixed action
Sg f = d 4 x Ȧk π k − H0 − A0 φ + λ, η(γ̇ ) g − i c̄, Mγ c g .
By explicitly introducing the indices of the Lie algebra g, the endomorphisms Mγ (t)(x)
can be expressed as
The last term can be recast as +i ċa c̄a . Therefore, this term can be interpreted as the
kinetic term corresponding to the new pair of canonical variables (c, i c̄).
VII. Conclusions
The principal aim of this work was to study the quantization of
Yang-Mills theory by using an extended connection A defined in a properly chosen
principal bundle. This connection unifies the three fundamental geometric objects of
Yang-Mills theory, namely the gauge field, the gauge fixing and the ghost field. This
unification is an extension of the known fact that the gauge and ghost fields can be
assembled together as ω = A + c [3].
The first step in the unification process was to generalize the gauge fixing procedure
by replacing the usual gauge fixing section σ with a gauge fixing connection η in the
G -principal bundle A → A /G . We have then shown that the connection η also encodes
the ghost field of the BRST complex. In fact, the ghost field can either be considered
the canonical vertical part of η in a local trivialization or the universal connection in the
gauge group’s Weil algebra. The unification process continues by demonstrating that the
universal family of gauge fields AU and the gauge fixing connection η can be unified in
the single extended connection A = AU +η on the G-principal bundle A × P → A × M.
114 G. Catren, J. Devoto
In this way, we have shown that the extended connection A encodes the gauge field, the
gauge fixing and the ghost field. A significant result is that it is possible to derive the
BRST transformations of the relevant fields without imposing the usual horizontality or
flatness conditions on the extended curvature F = φ + ψ + FU ([2,4,22]). In other words,
it is not necessary to assume that φ = ψ = 0. Moreover, the proposed formalism allows
us to show that the standard BRST transformation of the gauge field A is only valid in a
local trivialization of the fiber bundle A → A /G . In fact, Eq. (20) can be considered
the corresponding global generalization.
We then applied this geometric formalism to the path integral quantization of
Yang-Mills theory. Rather than selecting a fixed representant for each [A] ∈ A /G
by means of a section σ , the gauge fixing connection η allows us to parallel trans-
port any initial condition A0 ∈ A belonging to the orbit [A0 ]. A significant advan-
tage of this procedure is that one can always define a section σ η of the projection
P(A0 , π −1 [A1 ]) → P([A0 ], [A1 ]) in the space of paths, even when it is not possible to
define a global section σ : A /G → A in the space of fields. Since the path integral is
not an integral in the space of fields A , but rather an integral in the space of paths in A ,
such a section σ η suffices for eliminating the infinite volume of the group of paths PG .
Hence, this generalized gauge fixing procedure is globally well-defined even when the
topology of the fiber bundle A → A /G is not trivial (Gribov’s obstruction).
We then used the standard Faddeev-Popov method in order to introduce the gener-
alized gauge fixing defined by η at the level of the path integral. The corresponding
gauge fixed extended action Sg f was thereby obtained. We have thus shown that the
Faddeev-Popov method can be used even when there is a Gribov’s obstruction.
Acknowledgement. We wish to thank Marc Henneaux for his helpful comments and the University of Buenos
Aires for its financial support (Projects No. X103 and X193).
A. The Geometry of η
In this appendix we will review some geometric properties of the connection η. The
gauge group G consists of diffeomeorphisms ϕ : P → P such that π ϕ = π and
ϕ( pg) = ϕ( p)g (with g ∈ G). Each element ϕ ∈ G can be associated to a map
g : P → G by ϕ( p) = pg( p). This map g satisfies g( ph) = h −1 g( p)h = ad(h −1 )g( p).
This description of the elements of G also allows one to describe the elements of the
Lie algebra Lie(G ). These elements consist of maps g : P → g such that g( ph) =
Ad(h −1 )g( p).
The gauge group G acts on the right on A via the pullback of connections. If ϕ :
P → P is an element of G and A ∈ A , then the action is given by:
Rϕ A = ϕ ∗ A. (48)
The first term in the right hand side of Eq. (49) denotes the composition
Ap adg−1 ( p)
T p P −→ g −−−−−→ g. (50)
Extended Connection in Yang-Mills Theory 115
The second part g−1 dg is the pullback of the Maurer-Cartan form ω defined by the
map g : P → G.
The action (48) induces two constructions of interest. The first one is the infinitesimal
action. This is a morphism of Lie algebras ι : Lie(G ) → (T P), where (T P) is the
Lie algebra of the vector fields on P. We will identify elements of a Lie algebra with
tangent vectors at the identity. If ζ ∈ Lie(G ) and ϕs is a curve such that
d
ϕs s=0 = ζ,
ds
and A ∈ A , then we define
d
ιAζ = Rϕ As=0 . (51)
ds s
In terms of the explicit description of Eq. (49) we obtain
d
ι A ζ = Ad(ζ )A + g−1 dgs=0 . (52)
ds
The second action is the differential of the action of G . Since A is an affine space,
there is a natural identification T A = A ⊕ A . An element V ∈ T A A can then be
identified with a connection. Let As be a curve in A such that ds d
As |s=0 = V . Then we
define
d
d Rϕ V = ad(g)As s=0 . (53)
ds
Remark 14. The infinitesimal action and the differential of the action are geometric
intrinsic constructions. They only depend on the vectors ζ and V and not on the particular
curves used to compute them.
The connection η has the following properties
η(ι(ζ )) = ζ, (54)
η(d Rϕ V ) = ad(g−1 )η(V ). (55)
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