Gauge Transformations on Holomorphic Bundles

Gheorghe Munteanu

Abstract

In holomorphic tangent bundle T′M we define a generalization of classical gauge transformation, called complex gauge transformation, and related to it we shall study the invariant geometric objects: d-gauge tensors, nonlinear gauge connection, gauge complex derivatives.

The problem of global invariance concerning a complex Lagrangian is treated in the section related to Einstein-Yang-Mills complex equations. Finally, we shall discuss a few applications regarding infinitesimal complex gauge transformations.

Mathematics Subject Classification: 53C80, 80T13, 53C60, 53C15. Key words: gauge transformation, connection, Lagrangian, holomorphic bundle.

1 Introduction. The holomorphic bundle T′M

In many physical theories related to relativistic quantum, such as spontaneous symmetry-breaking, the Higgs theory, etc., the gauge Lagrangians which are used are complex scalar fields. Consequently, a geometrical approach of such problems requires to create a complex model for gauge geometric fields and their derivatives.

For the real case the geometric methods of fibre bundles with structural group are used for a long period of time in gauge theories ([3]). A generalization for the vectorial bundles of gauge transformation was made in [1], [7].

In a previous paper ([8]) we studied the Lagrange spaces having as base the holomorphic tangent bundle T′M, endowed with a nonlinear complex connection. Following such ideas, in the present paper we shall deal with complex gauge transformations on T′M.

Briefly, we shall introduce the basic results from [8].

Let us consider M a complex manifold, dim_{C} M = n, and (U, (z^{i})) the complex coordinates in a local map. The complexification T_{C}M of the tangent bundle T M in each z ∈ U is decomposed in the (1, 0)-vectors and their conjugates of (0, 1)-type ([5], [6]),

T_{C}M = T′M ⊕ T′′M.

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 71-80 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

The bundle π_{T} : T′M ⟶ M is holomorphic and dim_{C} T′M = 2n. With V (T′M) = {ξ ∈ T′(T′M)′ π_{T ∗}(ξ) = 0} is denoted the vertical sub-bundle, which

is also holomorphic, a local basis in V (T′M) being \{\frac{∂}{∂η^{i}} \}_{i=\overline{1,n}}.

A nonlinear complex connection, (nl.c.c.) for short, is a distribution N in T′(T′M), N : u = (z^{i}, η^{i}) → H_{u}(T′M) where H_{u}(T′M) is a supplementary sub-bundle of the vertical bundle in T′(T′M). A local basis in H_{u}(T′M) is denoted by

\{\frac{δ}{δz^{i}} = \frac{∂}{∂z^{i}} − N_{i}^{j} \frac{∂}{∂η^{j}} \}_{i,j=\overline{1,n}} ,

where N_{i}^{j} are the (nl.c.c.)-coefficients and satisfying the following rules of transformation at the change of local map

(1.1)  N_{k}^{′i} \frac{∂z^{′k}}{∂z^{j}} = \frac{∂z^{′i}}{∂z^{k}} N_{j}^{k} − \frac{∂^{2}z^{′i}}{∂z^{j}∂z^{k}} η^{k}.

From (1.1) we can easily verify that the adapted basis \{\frac{δ}{δz^{i}} \} is changed after the

rule

(1.2)  \frac{δ}{δz^{i}} = \frac{∂z^{′j}}{∂z^{i}} \frac{δ}{δz^{′j}} .

Taking the conjugate bundles \overline{H(T′M)} and \overline{V (T′M)} with the corresponding local

bases \{\frac{δ}{δ\overline{z}^{i}} \} and \{\frac{∂}{∂\overline{η}^{i}} \}, we obtain the following decomposition for the complexified

bundle:

(1.3)  T_{C}M = H(T′M) ⊕ V (T′M) ⊕ \overline{H(T′M)} ⊕ \overline{V (T′M)},

with the corresponding projectors denoted by h, v, \overline{h}, \overline{v}.

Furthermore, we shall use the following abbreviate notations: \{δ_{i}, ∂_{i}, δ\overline{i}, ∂\overline{i}\} for the adapted bases in (1.3) decomposition.

Let us consider D a derivative law on T_{C}(T′M). For a given (nl.c.c.) N, the derivative D is called N-linear complex connection, shortly: N −(l.c.c.), if D preserves the four distributions from (1.3). In the adapted basis \{δ_{i}, ∂_{i}, δ\overline{i}, ∂\overline{i}\} the N − (.c.c.)D has the following local expression:

D_{δ_{k}}δ_{j} =L_{jk}^{i}^{1} δ_{i}; D_{∂_{k}}δ_{j} =C_{jk}^{i}^{1} δ_{i}; D_{δ\overline{k}}δ_{j} =L_{j\overline{k}}^{i}^{3} δ_{i}; D_{∂\overline{k}}δ_{j} =C_{j\overline{k}}^{i}^{3} δ_{i};

D_{δ_{k}}∂_{j} =L_{jk}^{i}^{2} ∂_{i}; D_{∂_{k}}∂_{j} =C_{jk}^{i}^{2} ∂_{i}; D_{δ\overline{k}}∂_{j} =L_{j\overline{k}}^{i}^{4} ∂_{i}; D_{∂\overline{k}}∂_{j} =C_{j\overline{k}}^{i}^{4} ∂_{i};

D_{δ_{k}}δ\overline{j} =L\overline{i} ; \overline{j}k^{3} δ\overline{i}; D_{∂_{k}}δ_{j} =C\overline{i} ; \overline{j}k^{3} δ\overline{i}; D_{δ\overline{k}}δ\overline{j} =L\overline{i} ; \overline{j}\overline{k}^{1} δ\overline{i}; D_{∂\overline{k}}δ\overline{j} =C\overline{i} ; \overline{j}\overline{k}^{1} δ\overline{i};

D_{δ_{k}}∂\overline{j} =L\overline{i} ; \overline{j}k^{4} ∂\overline{i}; D_{∂_{k}}∂\overline{j} =C\overline{i} ; \overline{j}k^{4} ∂\overline{i}; D_{δ\overline{k}}∂\overline{j} =L\overline{i} ; \overline{j}\overline{k}^{2} ∂\overline{i}; D_{∂\overline{k}}∂\overline{j} =C\overline{i} ; \overline{j}\overline{k}^{2} ∂\overline{i},

with \overline{D_{X}Y} = D\overline{X} \overline{Y}. If

L_{jk}^{i}^{1}=L_{jk}^{i}^{2}= L_{jk}^{i}; L\overline{i} ; \overline{j}k^{3}=L\overline{i} ; \overline{j}k^{4}= L\overline{i} ; \overline{j}k; C_{jk}^{i}^{1}=C_{jk}^{i}^{2}= C_{jk}^{i}; C\overline{i} ; \overline{j}k^{3}=C\overline{i} ; \overline{j}k^{4}= C\overline{i} ; \overline{j}k

(and their conjugates coincide too), then the N − l.c.c. D is said to be normal, and will be denoted by M − (l.c.c).

The components of the torsion and the curvature tensors of one M − (l.c.c.), D have been calculated in [8].

A N − (l.c.c.) is decomposed in : D = D′ + D′′, where D′′ = \overline{D′} and their turn for each of them: D′ = D^{′h} + D^{′v}, D′′ = D^{′′h} + D^{′′v}, with D^{′h} = D_{δ_{k}}, D^{′v} = D_{∂_{k}},

D^{′′h} = D_{δ\overline{k}}, D^{′′v} = D_{∂\overline{k}}. A system of functions on T′M, w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s}}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}(z, η) is called

a d−complex tensor field of ( \begin{matrix}p & \overline{r} \\ q & \overline{s} \end{matrix} )-type if at change of local maps on M it well

be modified after the rule:

\begin{matrix}w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s}}^{′i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}(z′, η′) = \frac{∂z^{′i_{1}}}{∂z^{k_{1}}} ⋯ \frac{∂z^{′i_{p}}}{∂z^{k_{p}}} · \frac{∂z^{h_{1}}}{∂z^{′j_{1}}} ⋯ \frac{∂z^{h_{q}}}{∂z^{′j_{q}}} · \\ ·\frac{∂\overline{z}^{′\overline{i}_{1}}}{∂\overline{z}^{\overline{k}^{1}}} ⋯ \frac{∂\overline{z}^{′\overline{i}_{r}}}{∂\overline{z}^{\overline{k}^{r}}} · \frac{∂\overline{z}^{\overline{h}^{1}}}{∂\overline{z}^{′\overline{j}_{1}}} ⋯ \frac{∂\overline{z}^{\overline{h}^{r}}}{∂\overline{z}^{′\overline{j}_{r}}} w_{h_{1}...h_{p}\overline{h}_{1}...\overline{h}_{s}}^{k_{1...}k_{p}\overline{k}_{1}...\overline{k}_{r}}(z, η). \end{matrix}

The derivations of the d−tensor w will be expressed by ”|” and by ”||” for h−and respectively v−covariant derivative D′, and by ”\overline{|}”, ”\overline{||}” respectively for \overline{h}−and \overline{v}−covariant derivative D′′.

A metric Hermitian structure G on T_{C}(T′M) is defined by a d−complex tensor

g_{i\overline{j}}(z, η) of ( \begin{matrix}0 & \overline{0} \\ 1 & \overline{1} \end{matrix} )-type, nondegenerate, so that:

(1.6)  G = g_{i\overline{j}}dz^{i} ⊗ d\overline{z}^{j} + g_{i\overline{j}}δη^{i} ⊗ δ\overline{η}^{j},

where (dz^{i}, δη^{i}, d\overline{z}^{j}, δ\overline{η}^{j}) is the dual adapted basis.

2 Gauge complex transformations

Let us consider the holomorphic bundle π_{T} : T′M → M. Definition 2.1 A gauge transformation on complex manifold M is a pair of analytic isomorphisms Υ = (F^{0}, F^{1}) on the manifolds F^{0} : M → M and F^{1} : T′M → T′M, satisfying the property

(2.1)  π_{T} ∘ F^{1} = F^{0} ∘ π_{T} .

We can see that a gauge transformation Υ preserves the geometric structures of the manifold and the whole set determine a group structure with respect to the composition of mappings.

It is useful to obtain a local representation of complex gauge transformation Υ. Supposing that Υ = (F^{0}, F^{1}) applies the point u = (z^{i}, η^{i}) ∈ π_{T}^{−1}(U_{α}) in ũ =

(z^{i}, η^{i}) ∈ π_{T}^{−1}(U_{β}) and by adding the condition (2.1), it results Proposition 2.1 A gauge complex transformation Υ : u → ũ is locally given by a system of analytic functions:

(2.2)  z^{i} = X^{i}(z),  η^{i} = Y^{i}(z, η),

with the regularity condition det (\frac{∂X^{i}}{∂z^{j}} ) · det (\frac{∂Y^{i}}{∂η^{j}} ) ≠ 0.

At the changes of local coordinates at u and ũ it is necessary to require the conditions of global existence of the transformation Υ. If at ũ = (z^{i}, η^{i}) we consider

the local changes of the coordinates z^{′i} = z^{′i}(z); η^{′i} = \frac{∂z^{′i}}{∂z^{j}} η^{j}, then according to (2.2) we have

(2.3)  z^{′i} = X^{′i}(z);  η^{′i} = Y^{′i}(z, η),

and the inverses X^{′i}(z) = z^{′i}(X(z)), Y^{′i}(z, η) = \frac{∂z^{′i}}{∂z^{j}} Y^{j}(z, η). The global existence of Υ transformation implies:

(2.4)  X^{′i}(z′(z)) = X^{i}(z) ; Y^{′i}(z′(z), η′(z, η)) = Y^{i}(z, η),

with the regularity conditions of transformation.

Next, it is convenient to denote the following derivatives by X_{j}^{i} = \frac{∂X^{i}}{∂z^{j}}, Y_{j}^{i} = \frac{∂Y^{i}}{∂η^{j}},

their inverses by X_{j}^{i}^{∗}, Y_{j}^{i}^{∗}, and the conjugates by \overline{X_{j}^{i}}, etc.

From (2.4) we infer Proposition 2.2 At the change of local maps, the following derivatives changed by the rules

X_{k}^{′i} = \frac{∂z^{′i}}{∂z^{j}} X_{k}^{j}, Y_{k}^{′i} = \frac{∂z^{′i}}{∂z^{j}} Y_{k}^{j}, X_{k}^{i} = \frac{∂z^{i}}{∂z^{′j}} X_{k}^{′j}, Y_{k}^{i} = \frac{∂z^{i}}{∂z^{′j}} Y_{k}^{′j}.

Definition 2.2 A d−gauge complex tensor of ( \begin{matrix}p & \overline{r} \\ q & \overline{s} \end{matrix} ) type is a system of functions

w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s}}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}(z, η), which satisfies in addition to (1.5) the following rules of change

(2.5)  \begin{aligned}w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s}}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}(z̃, η̃) & = & X_{h_{1}}^{i_{1}} ⋯ X_{h_{p}}^{i_{p}}· X_{j_{1}}^{k_{1}}^{∗} · . . . · X_{j_{q}}^{k_{q}}^{∗} ·\overline{X}\begin{matrix}\overline{i} & 1 \\ \overline{h} & 1 \end{matrix} · . . . · \\ & · & \overline{X}\begin{matrix}\overline{i} & r \\ \overline{h} & r \end{matrix}· \overline{X}\begin{matrix}\overline{k} & 1 \\ \overline{j} & _{1} \end{matrix}^{∗} · . . . · \overline{X}\begin{matrix}\overline{k} & s \\ \overline{j} & _{s} \end{matrix}^{∗} · · w_{k_{1}...k_{q} \overline{k}_{1}...\overline{k}_{s}}^{h_{1...}h_{p}\overline{h}_{1}...\overline{h}_{r}}(z, η). \end{aligned}

Denote by J the natural almost tangent structure on T′M, which is defined by

J (\frac{∂}{∂z^{i}} ) = \frac{∂}{∂η^{i}}, J (\frac{∂}{∂η^{i}} ) = 0, and is globally defined. The kernel of J is just the

vertical distribution, and if N is a (nl.c.c.) then J(\frac{δ}{δz^{i}}) = \frac{∂}{∂η^{i}} . This means that the

image of the horizontal distribution is the vertical one. Definition 2.3 A nonlinear complex gauge connection (shortly (nl.c.g.c)) is a (nl.c.c) N which is preserved by the tangent map of the transformation, i.e., Υ_{∗} : T′_{u}(T′M) → T ′ ; _{u}(T′M) preserves the distributions.

If \{\frac{δ}{δz^{i}} = \frac{∂}{∂z^{i}} − Ñ_{i}^{j} \frac{∂}{∂η^{j}}; \frac{∂}{∂η^{i}} \} is the adapted basis in T ′ ; _{u}(T′M) then for Υ_{∗} to

preserve the distributions it is necessary that J(\frac{δ}{δz^{i}}) = \frac{∂}{∂η^{i}}. For this reason N is a

(nl.c.g.c.) if

(2.6)  \frac{δ}{δz^{i}} = X_{i}^{j} \frac{δ}{δz^{j}} and \frac{∂}{∂η^{i}} = X_{i}^{j} \frac{∂}{∂η^{j}} ,

this means that the elements of the adapted bases are d−gauge complex tensors. Proposition 2.3. The coefficients N_{i}^{j}(z, η) of one (nl.c.g.c.) satisfy in addition to

the condition (1.1) the next transformation law Ñ_{k}^{j}X_{j}^{k} = X_{k}^{i}N_{j}^{k} − \frac{∂Y^{i}}{∂z^{k}} . The proof results from the first relation (2.6)

Considering the conjugate Ñ_{i}^{j} one of the (nl.c.g.c.) and the corresponding adapted bases on T′′(T′M), we obtain that the extension Υ_{∗} of the tangent gauge transformation to the whole complexification T_{C}(T′M) preserves the four distributions.

3 Complex gauge derivatives

For a given (nl.c.g.c.)N, let us consider the components D^{h}, D^{v}, D_{h}, D_{v} of one N − (l.c.c.) D on T_{C}(T′M) with the local coefficients written in formula (1.4), and which at the change of local maps on M are transformed by the rules

(3.1) L_{jk}^{′i}^{α} \frac{∂z^{′j}}{∂z^{h}} \frac{∂z^{′k}}{∂z^{m}} =L_{hm}^{l}^{α} \frac{∂z^{′i}}{∂z^{l}} − \frac{∂^{2}z^{′i}}{∂z^{h}∂z^{m}}  C_{jk}^{′i}^{α} \frac{∂z^{′j}}{∂z^{h}} \frac{∂z^{′k}}{∂z^{m}} =C_{hm}^{l}^{α} \frac{∂z^{′i}}{∂z^{l}} ; α = 1, 2.

For the conjugate indices the rules are obtained from (3.1) deriving with respect to the conjugate basis.

The covariant derivatives D^{h}, D^{v}, D_{h}, D_{v} act on one complex d−tensor field w

of ( \begin{matrix}p & \overline{r} \\ q & \overline{s} \end{matrix} ) −type, determining the following d−tensor fields

w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s} |m}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}} ; w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s} ∥m}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}, w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s} \overline{|}m}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}, w_{j_{1}...j_{q} \overline{j}_{1}...\overline{j}_{s}\overline{∥}m}^{i_{1...}i_{p}\overline{i}_{1}...\overline{i}_{r}}.

Proposition 3.1 The covariant derivatives D^{h}, D^{v}, D_{h}, D_{v} of one complex d−gauge tensor field are d−complex gauge tensor fields if and only if the local coefficients of D derivative satisfy in addition to (3.1) the following rules

(3.2) L̃_{jk}^{i}^{α}=X_{j}^{h}^{∗}X_{k}^{m}^{∗} X_{l}^{i} L_{hm}^{l}^{α} − X_{j}^{h}^{∗}X_{k}^{m}^{∗} \frac{δX_{m}^{i}}{δz^{h}}  C̃_{jk}^{i}^{α}=X_{j}^{h}^{∗}X_{k}^{m}^{∗} X_{l}^{i} C_{hm}^{l}^{α}, α = 1, 2.

and analogous conditions for the conjugates indices, the X_{j}^{i}^{∗} is replaced by its conjugate

\overline{X}\overline{i} ; \overline{j}^{∗}.

Proof. For instance, starting with the first relation (1.4) D_{δ_{k}}δ_{j} =L_{jk}^{i}^{1} δ_{i}, and considering (2.6) and their conjugates it results directly the relations (3.2).

Definition 3.1 A N−linear complex connection D satisfying the conditions from Prop.3.1 is named a N−linear complex gauge connection (N − (l.c.g.c.)).

The corresponding gauge complex derivatives will be specified by D^{h}, D^{v}, D_{h}, D_{v}, replacing the short, long and conjugate bar of derivation of a d−complex gauge tensor. Definition 3.2. A N − (l.c.g.c.), D is said to be metrical if

(3.3)  D^{h}g_{i\overline{j}} = 0 ; D^{v}g_{i\overline{j}} = 0, D\overline{h}g_{i\overline{j}} = 0, D\overline{v}g_{i\overline{j}} = 0,

where g_{i\overline{j}} defines the Hermitian metric structure (1.6).

If we use [8] we obtain a N −(l.c.c.), denoted by D^{c} and called canonical connection, which is metrical and M−connection, too:

(3.4)  \begin{matrix}L_{jk}^{i}^{c}= \frac{1}{2}g^{\overline{l}i} (\frac{δg_{j\overline{l}}}{δz^{k}} + \frac{δg_{k\overline{l}}}{δz^{j}} ) , C_{jk}^{i}^{c}= \frac{1}{2}g^{\overline{l}i} (\frac{∂g_{j\overline{l}}}{∂η^{k}} + \frac{∂g_{k\overline{l}}}{∂η^{j}} ) , \\ Lī ; j̄k^{c}= \frac{1}{2}g^{\overline{i}l} (\frac{δg_{l\overline{j}}}{δz^{k}} − \frac{δg_{k\overline{j}}}{δz^{l}} ) , Cī ; j̄k^{c}= \frac{1}{2}g^{\overline{i}l} (\frac{∂g_{l\overline{j}}}{∂η^{k}} − \frac{∂g_{k\overline{j}}}{∂η^{l}} ) , \end{matrix}

and the conjugates.

Reiterating the calculus in Proposition 3.1 for the canonical connection D^{c}, using the conditions (2.6) of (nl.c.g.c.), we infer Theorem 3.2. The canonical connection D^{c} is a N − (l.c.g.c.) with the T_{jk}^{i} and S_{jk}^{i} vanishing torsions.

The connection (3.4) will be called the Miron canonical complex connection.

4 Complex Einstein-Yang-Mills equations

Let us consider N a (nl.c.g.c.) and L_{0} : T′M → R a complex Lagrangian, i.e.,

(4.1)  g_{i\overline{j}} = \frac{∂^{2}L}{∂η^{i}∂\overline{η}^{J}}

is a nondegenerated d−tensor.

From (4.1) we remark that g_{i\overline{j}} is a gauge complex tensor. The (nl.c.g.c.), N can be determined from L_{0} as is shown in [8].

In calculus L_{0} depends on the point u = (z, η) via the wave functions Φ^{A}, A = \overline{1, p}, which are gauge fields, and on their derivatives which are supposed to be with respect to the adapted basis

(4.2)  L_{0}(z, η) = L (Φ^{A}, \frac{δΦ^{A}}{δz^{i}} , \frac{δΦ^{A}}{δ\overline{z}^{i}} , \frac{∂Φ^{A}}{∂η^{i}} , \frac{∂Φ^{A}}{∂\overline{η}^{i}} ) .

The Euler equation gives the extremes of action of the L_{0} Lagrangian. But this action depends on the local maps. To remove this drawback it is useful to consider the following modified Lagrangian

(4.3)  L(z, η) = L_{0}(z, η) · |g|^{2},

where |g| = | det(g_{i\overline{j}})|.

Then the action I = ∫ L(z, η)dω does not depend on the local coordinates, and if Φ is one of the generic field Φ^{A}, then the direct calculus gives the following Proposition 4.1. The extremum of the action, δI(Φ) = 0, determines the Euler-Lagrange complex equation (4.4)

\frac{∂L}{∂Φ} − \frac{∂}{∂z^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂z^{i}} )}) − \frac{∂}{∂\overline{z}^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂\overline{z}^{i}} )}) − \frac{∂}{∂η^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂η^{i}} )}) − \frac{∂}{∂\overline{η}^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂\overline{η}^{i}} )}) = 0.

The complex (E − L) equation can be rewritten in an adapted basis \{δ_{i}, ∂_{i}, δ\overline{i}, ∂\overline{i}\} in the equivalent form

(4.5)

|g|^{2} \{\frac{∂L}{∂Φ} − \frac{δ}{δz^{i}} (\frac{∂L}{∂ (\frac{δΦ}{δz^{i}} )}) − \frac{δ}{δ\overline{z}^{i}} (\frac{∂L}{∂ (\frac{δΦ}{δ\overline{z}^{i}} )}) − \frac{∂}{∂η^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂η^{i}} )})_{|c_{1}} −

−\frac{∂}{∂\overline{η}^{i}} (\frac{∂L}{∂ (\frac{∂Φ}{∂\overline{η}^{i}} )})_{|c_{2}} + \frac{∂N_{j}^{i}}{∂η^{i}} \frac{∂L}{∂ (\frac{δΦ}{δz^{j}} )} + \frac{∂\overline{N}_{j}^{i}}{∂\overline{η}^{i}} \frac{∂L}{∂ (\frac{δΦ}{δ\overline{z}^{j}} )}\} =

= \frac{δ|g|^{2}}{δz^{i}} · \frac{∂L}{∂ (\frac{δΦ}{δz^{i}} )} + \frac{δ|g|^{2}}{δ\overline{z}^{i}} · \frac{∂L}{∂ (\frac{δΦ}{δ\overline{z}^{i}} )} + \frac{∂|g|^{2}}{∂η^{i}} · \frac{∂L}{∂ (\frac{∂Φ}{∂η^{i}} )} + \frac{∂|g|^{2}}{∂\overline{η}^{i}} · \frac{∂L}{∂ (\frac{∂Φ}{∂\overline{η}^{i}} )},

with c_{1} = \frac{δΦ}{δz^{i}}; c_{2} = \frac{δΦ}{δ\overline{z}^{i}} taking constant values.

The formula (4.5) shows that the (E.L) equation is invariant to the local change of coordinates.

Next, we shall use the notations

Φ_{A}^{i}^{h}= \frac{∂L}{∂ (\frac{δΦ}{δz^{i}} )}, Φ\overline{i} ; A^{\overline{h}}= \frac{∂L}{∂ (\frac{δΦ}{δ\overline{z}^{i}} )}, Φ_{A}^{i}^{v}= \frac{∂L}{∂ (\frac{∂Φ}{∂η^{i}} )}, Φ\overline{i} ; A^{\overline{v}}= \frac{∂L}{∂ (\frac{∂Φ}{∂\overline{η}^{i}} )},

in order to abbreviate the written form of (4.5).

Also, for a given N − (l.c.g.c.)D (for instance the (3.4) canonical connection), let us consider the h−, v−, \overline{h}−, \overline{v}−derivations of one vector X = (X^{i}),

(4.6)  \begin{matrix}X_{|k}^{i} = \frac{δX^{i}}{δz^{k}} + L_{jk}^{i}X^{j},  X_{∥k}^{i} = \frac{∂X^{i}}{∂η^{k}} + C_{jk}^{i}X^{j}, \\ X_{\overline{|}\overline{k}}^{i} = \frac{δX^{i}}{δ\overline{z}^{k}} + L_{j\overline{k}}^{i}X^{j},  X_{\overline{∥}\overline{k}}^{i} = \frac{∂X^{i}}{∂\overline{η}^{k}} + C_{j\overline{k}}^{i}X^{j}, \end{matrix}

and their conjugates. So, the formula (4.5) is written in the equivalent form

(4.7)  \frac{∂L}{∂Φ^{A}} − Φ_{A}^{i}^{h}_{|i} − Φ\overline{i} ; A^{\overline{h}}_{\overline{|}i} − Φ_{A}^{i}^{v}_{∥i} − Φ\overline{i} ; A^{\overline{v}}_{\overline{∥}i}= E_{A},

where (4.8)

E_{A} = \frac{1}{|g|^{2}} \{\frac{δ|g|^{2}}{δz^{i}} Φ_{A}^{i}^{h} +\frac{δ|g|^{2}}{δ\overline{z}^{i}} Φ\overline{i} ; A^{\overline{h}} +\frac{∂|g|^{2}}{∂η^{i}} Φ_{A}^{i}^{v} +\frac{∂|g|^{2}}{∂\overline{η}^{i}} Φ\overline{i} ; A^{\overline{v}}\} −

− (L_{jk}^{i} + \frac{∂N_{j}^{i}}{∂η^{i}} ) Φ_{A}^{i}^{h} − (L\overline{i} ; \overline{j}\overline{i} + \frac{∂\overline{N}\overline{i} ; \overline{j}}{∂\overline{η}\overline{i}} ) Φ\overline{i} ; A^{\overline{h}} −C_{ji}^{i} Φ_{A}^{i}^{v} −C\overline{i} ; \overline{j}\overline{i} Φ\overline{i} ; A^{\overline{v}}

are gauge scalars if D is a N − (l.c.g.c.).

Let us note that the invariance of the (E −L) equation is assured if the Lagrangian L is gauge invariant.

We shall analyze the gauge invariance of Lagrangian L in the particular case of gauge infinitesimal transformations.

Let us consider G a group of transformations, dim G = m, that acts on T′M,

(4.9)  ũ = Υ(u, a),  u = Υ(u, 0),

with u = (z^{i}, η^{i}) and a = (a^{1}, a^{2}, … a^{m}) ∈ G.

At an infinitesimal holomorphic of the group G, we have

(4.10)  z^{i} = z^{i}+ ξ_{λ}^{i}^{h} ε^{λ}, η^{i} = η^{i}+ ξ_{λ}^{i}^{v} ε^{λ}.

Accordingly, the gauge fields Φ^{A}(u) will be transformed by the rules

(4.11)  Φ̃^{A} = Φ^{A} + (X_{λ}Φ^{A})ε^{λ},

where X_{λ} =ξ_{λ}^{i}^{h} \frac{∂}{∂z^{i}} + ξ_{λ}^{i}^{v} \frac{∂}{∂η^{i}} are the generators of the group.

As in classical theory, let us consider a p−dimensional complex representation of the generators X_{λ} → [X_{λ}]_{B}^{A}. Then the infinitesimal transformation of the gauge fields becomes

(4.12)  Φ̃^{A} = Φ^{A} + ([X_{λ}]_{B}^{A}Φ^{B})ε^{λ}.

The gauge infinitesimal invariance condition of the Lagrangian L is that of vanishing its variation,

\frac{∂L}{∂Φ^{A}} δΦ^{A}+ Φ_{A}^{i}^{h} δ (\frac{δΦ^{A}}{δz^{i}} ) + Φ\overline{i} ; A^{\overline{h}} δ (\frac{δΦ^{A}}{δ\overline{z}^{i}} ) + Φ_{A}^{i}^{v} δ (\frac{∂Φ^{A}}{∂η^{i}} ) + Φ\overline{i} ; A^{\overline{v}} δ (\frac{∂Φ^{A}}{∂\overline{η}^{i}} ) = 0.

We propose now to discuss only the global invariance, meaning that ε^{λ} are constants, and hence the results

δ (\frac{δΦ^{A}}{δz^{i}} ) = \frac{δ}{δz^{i}} (δΦ^{A}) = ε^{λ}[X_{λ}]_{B}^{A} \frac{δΦ^{B}}{δz^{i}} , δ (\frac{∂Φ^{A}}{∂η^{i}} ) = ε^{λ}[X_{λ}]_{B}^{A} \frac{∂Φ^{B}}{∂η^{i}} ,

δ (\frac{δΦ^{A}}{δ\overline{z}^{i}} ) = \frac{δ}{δ\overline{z}^{i}} (δΦ^{A}) = ε^{λ}[X_{λ}]_{B}^{A} \frac{δΦ^{B}}{δ\overline{z}^{i}} , δ (\frac{∂Φ^{A}}{∂\overline{η}^{i}} ) = ε^{λ}[X_{λ}]_{B}^{A} \frac{∂Φ^{B}}{∂\overline{η}^{i}} .

Replacing these variations in the invariance condition of the Lagrangian, it results

(4.13)  \{\frac{∂L}{∂Φ^{A}} Φ^{B} + Φ_{A}^{i} \frac{δΦ^{B}}{δz^{i}} + Φ\overline{i} ; A \frac{δΦ^{B}}{δ\overline{z}^{i}} + Φ_{A}^{i} \frac{∂Φ^{B}}{∂η^{i}} + Φ\overline{i} ; A \frac{∂Φ^{B}}{∂\overline{η}^{i}} \} [X_{λ}]_{B}^{A} = 0.

Now, combining the formulae (4.7) and (4.13) we obtain the global invariance law of complex Lagrangian L at the gauge infinitesimal transformation (4.11)

(4.14)

\{E_{A}Φ^{B}+ Φ_{A|i}^{i}^{h} Φ^{B}+ Φ\overline{i} ; A\overline{|}\overline{i}^{\overline{h}} Φ^{B}+ Φ_{A∥i}^{i}^{v} Φ^{B}+ Φ\overline{i} ; A\overline{∥}\overline{i}^{\overline{v}} Φ^{B}+

+ Φ_{A}^{i}^{h} \frac{δΦ^{A}}{δz^{i}} + Φ\overline{i} ; A^{\overline{h}} \frac{δΦ^{A}}{δ\overline{z}^{i}} + Φ_{A}^{i}^{v} \frac{∂Φ^{A}}{∂η^{i}} + Φ\overline{i} ; A^{\overline{v}} \frac{∂Φ^{A}}{∂\overline{η}^{i}} \} [X_{λ}]_{B}^{A} = 0.

Using the complex currents

(4.15)

J_{A}^{i}^{h}= − Φ_{A}^{i}^{h} [X_{λ}]_{B}^{A}Φ^{B}, J_{A}^{i}^{v}= − Φ_{A}^{i}^{v} [X_{λ}]_{B}^{A}Φ^{B},

J\overline{i} ; A^{\overline{h}}= − Φ\overline{i} ; A^{\overline{h}} [X_{λ}]_{B}^{A}Φ^{B}, J\overline{i} ; A^{\overline{v}}= − Φ\overline{i} ; A^{\overline{v}} [X_{λ}]_{B}^{A}Φ^{B},

the global conservative law (4.13) is written in the form

(4.16)  J_{A|i}^{i}^{h} + J\overline{i} ; A\overline{|}\overline{i}^{\overline{h}} + J_{A∥i}^{i}^{v} + J\overline{i} ; A\overline{∥}\overline{i}^{\overline{v}}= E_{A}[X_{λ}]_{B}^{A}Φ^{B}.

5 Applications

Let us consider a metric Hermitian d−tensor γ_{i\overline{j}}(z) on M, and Φ : T′M → C one scalar field (A = 1). For a nonlinear complex connection N_{j}^{i}(z, η) we can consider the

one determined by the Christoffel symbols Γ_{ij}^{k} of the metric γ_{i\overline{j}}, i.e., N_{j}^{i}^{c}= Γ_{jk}^{i}η^{k} (see [8]).

In the adapted basis \{δ_{i}, ∂_{i}, δ\overline{i}, ∂\overline{i}\} we shall consider a generalization of the classical exact symmetry Lagrangian ([4]) for the complex gauge field Φ : T′M → C,

(5.1)  L = γ^{\overline{i}j}(z)\frac{δ\overline{Φ}}{δ\overline{z}^{i}} \frac{δΦ}{δz^{j}} − m^{2}\overline{Φ} · Φ − \frac{1}{4}f · (\overline{Φ} · Φ)^{2},

where m^{2} > 0 is the mass and f > 0 is the coupling constant.

The Lagrangian L is invariant to the change of local maps on T′M and is also gauge invariant with respect to the phase transformation of U_{1} group,

(5.2)  Φ → Φ̃(z, η) = e^{−igε}Φ(z, η) ; \overline{Φ} → ˜ \overline{Φ}(z, η) = e^{igε}Φ(z, η),

determined by the infinitesimal variations

(5.3)  δz^{i} = −igε, δη^{i} = igε.

The U_{1} group has one parameter ε^{1} = ε; the variations of gauge field are δΦ =

−igεΦ, and the generator of the group is X = − ig (\frac{∂}{∂z^{i}} − \frac{∂}{∂η^{i}} ) .

In classical theories, in order to study the gauge invariance, there are considered two gauge fields: Φ^{1} = Φ and Φ^{2} = \overline{Φ}. This doubles the dimension of the representation, in comparison to the present approach, where we consider only one generator.

The energy of the system, E = γ^{\overline{i}j}(z)\frac{δ\overline{Φ}}{δ\overline{z}^{i}} \frac{δΦ}{δz^{j}} + m^{2}\overline{Φ} · Φ + \frac{1}{4}f · (\overline{Φ} · Φ)^{2} is min-

imal if \frac{∂E}{∂Φ} = \frac{∂E}{∂\overline{Φ}} = 0; it results that Φ = \overline{Φ} = 0, and hence the vacuum state is

nondegenerate, preserving the exact symmetry.

Another generalization of classical case is that of the spontaneously broken symmetry, where it is considered the complex Lagrangian

(5.4)  L_{1} = γ^{\overline{i}j}(z)\frac{δ\overline{Φ}}{δ\overline{z}^{i}} \frac{δΦ}{δz^{j}} + m^{2}\overline{Φ} · Φ − \frac{1}{4}f · (\overline{Φ} · Φ)^{2},

which is also gauge invariant at (5.2) infinitesimal transformation.

The energy of the system, E_{1} = γ^{\overline{i}j}(z)\frac{δ\overline{Φ}}{δ\overline{z}^{i}} \frac{δΦ}{δz^{j}} − m^{2}\overline{Φ} · Φ + \frac{1}{4}f · (\overline{Φ} · Φ)^{2} is minimal

if |Φ| = \sqrt{2}\frac{m}{\sqrt{f}} and hence, the vacuum states are degenerated, i.e., we have in the

holomorphic bundle a spontaneously broken symmetry. Acknowledgements. A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

References

[1] G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories, D. Reidel Publ. Co., Dordecht, 1985.

[2] T. Aikou, On Complex Finsler Manifolds, Rep. of Kagoshima Univ., 24 (1991), 9-25.

[3] D. Bleecker, Gauge Theory and Variational Principles, Addison-Wesley Publ. Co.Inc., 1981.

[4] M. Chaichian, N.F. Nelipa, Introduction to Gauge Field Theories, Springer- Verlag, 1984.

[5] G. Gheorgiev, V. Oproiu, Varietăţi Diferenţiabile Finit şi Infinit Dimensionale, Ed.Acad. Romane, Vol.2, 1979.

[6] S. Kobayasi, H. Wu, Complex Diff. Geometry, DMV Seminar Band 3, Berkhause Verlag, Basel-Boston,1983.

[7] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces; Theory and Applications, Kluwer Acad. Publ., no. 59, FTPH, 1994.

[8] Gh. Munteanu, Complex Lagrange Geomety, Balkan Jour. of Diff. Geom. and Its Appl., 3 (1998), 1, 61-73.

Transilvania University of Braşov, Faculty of Science, Str. I.Maniu, 50, 2200, Braşov, România, e-mail:gh.munteanu@info.unitbv.ro