Critical principal connections and gauge-invariance

Vol. 13 (1978) REPORTS ON MATHEMATICAL PHYSICS No. 3

CRITICAL PRINCIPAL CONNECTIONS AND

PEDRO L. GARCIA

Departamento de Matemkticas, Universidad de

GAUGE-INVARIANCE”

Salamanca, Spain

(Received September 18, 1976)

Let p: P + X be a principal G-bundle over an oriented manifold. As suggested by the classical Yang-Mills field theory, a certain class of variational problems for connections on P is defined. Global equations are derived for the corresponding critical connections and for the Jacobi fields along a critical connection, which are a very adequate tool for the study of the symplectic theory associated with the said variational problems.

1.

The classical theory of Yang-Mills fields suggests the study of the following type of variational problems in differential geometry.

Let p: P -+ X be a principal bundle with the structural group G which has the Lie algebra 3. As it is known, a connection on P can be defined by a splitting 6: T(X) -+ Tc(P)

of the exact sequence of vector bundles on X:

0 --) AdP : T,(P) -r, T(X) -+ 0

where T,(P) is the vector bundle of G-invariant vector fields on P, AdP is the subbundle of TG(P) defined by the G-invariant vector fields that are tangent to the fibres of P, and T(X) is the tangent bundle of X. These connections on P can be identified with the global sections of the affine bundle z: E -+ X defined as follows: x E X being given, let E, be the set of homomorphisms 0; : T,(X) + T,(P), such that pz * ox = 1, let E = U E,, and

XEX

let z be the natural projection of E onto X. z: E + X has a unique affine bundle structure such that, for every connection c on P, the mapping 0: Horn (T(X), Ad P) --f E defined by h, H O(X)+&, is an affine bundle isomorphism on X. Now, if we suppose that the manifold X is orientable and endowed with a volume element 7, then it is possible to speak about connections on P that are “critical” with respect to a variational problem defined on the bundlejlz: J’E -+ X of the l-jets of local sections of E, by a Lagrangian density 87, where 2’: J’E --f R is an arbitrary differentiable function.

In particular, suggested always by the Yang-Mills theory, we are interested in La- grangians 2’ that are “gauge-invariant” in the following sense.

* Presented at the Symposium on Methods of Differential Geometry in Physics and Mechanics, June 1976, Warsaw.

[3371

338 P. L. GARCIA

Every element s of the Lie module I’ of sections of the bundle AdP of Lie R-algebras,

defines a one-parameter group tf of n-vertical automorphisms in the natural way:

ZtCr‘x = (T,+t[CTx, S], cx E E

where [o., , sl E Horn (T(X), AdI’) is given by [ax, s]& = [c&Q, s]. If D, is the infini-

tesimal generator of rt, one can prove [2] that the map s +, D, is a homomorphism from

the real Lie algebra r into the real Lie algebra of n-vertical vector fields on E. Thus, we

have obtained a real Lie algebra {D,}, s E r, of n-vertical vector fields on E.

We say that 9 is gauge-invariant when (j1D,)9 = 0 for every s E r, where j’D, is

the l-jet extension of the vector field D,. For example, if G = U(1) ( i.e. a unitary l-dimensional group), then E is the affine

bundle corresponding to the cotangent bundle T*(X) of X, I’ coincides with the abelian

real Lie algebra of functions f: X -+ R, and the one-parameter group rt corresponding

to f is given by:

r,w, = C&+t(df),, w, E F(X).

In particular, if X is a Riemannian manifold with metric tensor g, the Lagrangian 9

defined by:

_Y(j:w) = $ * gx(do, dco)

is gauge-invariant in the above sense.

When (X, g) is the Minkowski’s space-time, one has the classical theory of free electro-

magnetic field.

Now we have three questions of a general character that arise from these variational

problems.

(a) To characterize the type of Lagrangians satisfying the above condition of gauge-

invariance.

(b) To try to find global Euler equations for the corresponding “critical connections”

and for the “Jacobi fields along a critical connection”.

(c) To study the corresponding symplectic structure in the sense of the general theory

of classical fields.

As concerns the question (a), a theorem proved by Utiyama [4], of which we have

given a geometrical version [2], characterizes this type of Lagrangians in terms of the

“curvature of a connection”. In [3], where we deal with the question (c) when studying

the reducibility of the symplectic structure for this type of fields, the question (b) is essentially

posed when the need for adequate global Euler equations is felt. In the particular case

of free electromagnetic field, the global version adcc, = 0 of Maxwell’s equations is very suitable for this purpose.

Thus in a natural way there arises the question of finding some type of Maxwellian

equations for the general case. This is the aim of the present note. We shall show that an

adequate application of the general formalism developed in [1] will allow us to obtain

a satisfactory solution of this problem.

CRITICAL PRINCIPAL CONNECTIONS AND GAUGE-INVARIANCE 339

2.

As it is known, a “classical field” is given by a variational problem geometrically defined by a fiber bundle z: E -+ X on an orientable manifold X endowed with a volume element 9, and by a real differentiable function 2’ (Lagrangian) defined on the manifold J’E of the fiber bundle j’li: J’E + X of the l-jets of local sections of E.

In [l] we associate with such a structure three differential forms 19, 52 and 0; the structure form of the l-jet fiber bundle, and the Legendre and Cartan forms corresponding to the Lagrangian 2, from them the geometrical theory of the given variational problem can be intrinsically developed.

The structure form 19, intrinsically introduced from the notion of a “vertical differential of a section” in the fiber bundle 7~: E -+ X, is a l-form on the manifold J’E valued in the module A4 of sections of the induced vector bundle q*T”(E), where T”(E) is the vertical tangent bundle of E and q is the canonical projection from J’E onto E. Schemati- cally :

q”T”(E) T”(E)

I I J'(E) 1-E

The expression for 8 in a system of natural local coordinates (XiZjpij) on JIE and in a

a local basis q*-- of the module M is: &j

The Legendre form Q corresponding to the Lagrangian 2 is an (n - I)-form on J’E, n = dim_%‘, valued in the module M* dual to M. It is introduced, in the way similar to that employed in symplectic geometry to define the Hamiltonian vector field corresponding to a function, as follows:

~2 is the unique M*-valued (n- I)-form on J’(E) such that:

iv * d8 z -d_Y’(modQ:), i’p * 17 = !2

where the differential of the M-valued l-form 0 is taken according to an arbitrary derivation law in M, ‘$J is an M*-valued I-contravariant tensor on J’(E), a: is the module of ordinary l-forms on J’(E) which are null on the fibres of q: J’(E) + E, and the interior products are taken with respect to the natural bilinear products of corresponding. modules.

340 I’. L. GARCIA

We must precise the following: the first equation defines 9 locally and non-uniquely,

but, when carrying out the interior product, iv . q with any local $3, the local (n- 1)-form

thus obtained is unique; this allows us to globalize it in order to obtain 52.

The local expression for 52 with respect to the system of natural local coordinates

(.Kizjpij) on J’(E) and the local basis q*dZj of M* is:

where J = 11 a a

__ ax, .‘. ax, .

Finally, the Cartan form

n-form on J’(E) defined by :

where the exterior product is taken with respect to the bilineal product defined for M

. A&l . . . Adx,, 0 q*dzj

0 corresponding to the Lagrangian 9 is the ordinary

0 = erl.Q-9rj

and M* by the duality notion.

The idea of geometrization of the calculus of variations that we have proposed in

[l] consists in the formulation of all concepts and manipulations of the theory in terms

of differential forms and operations of differential calculus on the manifold J’(E) with

values in the modulus M, W, Hom(M, M) etc. Naturally, in order to introduce such

a differential calculus, it is necessary to give a derivation law on M. Once this has been

done, some of the introduced notions, for example the forms of Legendre and Cartan, do not depend on the chosen derivation law; nevertheless, some others, like the Euler

equations that we wish to propose here, do depend on it. Thus, it is an essential question

in this formalism how to chose a derivation law on M. In our case, where Z: E -+ X is the affine bundle corresponding to the vector bundle

Hom(T(X), AdP), the vertical tangent bundle 7’“(E) coincides with the induced vector

bundle

n*Hom(T(X), AdP) = Hom(n*T(X), z*AdP).

By taking advantage of this fact, we can take the derivation law on M induced by a natural

connection on T”(E) that we define as follows:

We have seen in [2] that the principal G-bundle SP, induced from p: P -+ X by the

projection ti: E -+ X, has a canonical connection, which defines a derivation law on the

sections of its associated vector bundle n*Ad P. Now, if we take on the connection n* T(X) induced by a linear connection on X, we can define in the usual way a derivation law on

the sections of T”(E) = Hom (n*T(X), n*AdP). We shall suppose that the chosen linear

connection on X is torsionless.

In the system of natural local coordinates (xi Aij) on E, this derivation law is given by:


I yiTi&& =O where 6, is the Kronecker’s symbol, c$ are the structural constants of the Lie algebra $9 of the group G, and _rf,,, are the coefficients of the linear connection on X.

Now, taking our differential calculus with the derivation law induced on A4 by the one just introduced on T”(E), we have the following:

LEMMA (fundamental).

d0 = -0Aa?Q.

Proof: d@ = d(bl!2-2’~) = d0&2-BAdn-d_Yfl~. Now the problem reduces to proving the equality :

d&lQ--d._YAq = 0.

Let U be an open neighbourhood of an arbitrary point in J’E with natural local coordinates

(X1 AjPilj). If @ is any solution in U of the equation iv. dtl = - dY(moda:), one has !2 = iv . 4, and then our equality becomes:

(ipd8+d9)ilq = 0 (*)

for i@(d&ly) = 0, because d&Iv = 0 as it can be shown after an easy local calculation. In order to verify (*) it is now necessary and sufficient that i$Jdi?+d8 be zero on the

a vector fields __

%Jilj

and &. It is true for the first one, by the definition of q. Thus, IJ

it only remains to prove that

(i‘$ . dB+d2’) -$ = 0. ii

Taking for S$ the local solution

one gets:

a9 a9 =-----_- a4

-+ c r’ adp ihk @ilk im Em G’

- (1) -

By Utiyama’s theorem (1) = 0 and ~ aPimj +ap,=

condition ri,,, = J’j,;, (2) = 0 too, which ends the proof.

(2)

0. Then, by the symmetry

342 P. L. GARCIA

Once this formula has been given, all basic results of the variational calculus (for- mulas for first and second variations, Euler equations, Noether’s theorems, etc.) can be easily derived as in [l]. In particular we have the fundamental results):

THEOREM (characterization of critical connections). A connection CJ: X -+ E is critical if and only {f its l-jet extension sati.$es the condition

dJllji,tx, = 0.

This is the global version of the Euler equations that we propose and would like to comment.

3.

Given a connection a: X -+ E on a principal bundle P, Qjj~~(x> defines a 1-contravariant (n- I)-covariant tensor Q(a) on the manifold X with values in the set of sections of the bundle AdP. Q(a) is hemisymmetric in its covariant indices.

In terms of this tensor? the above equation for critical connections takes the form:

dvQ(a) = 0

where dv is the “covariant exterior differential” for AdP-valued tensors on X, defined by the linear connection we have chosen on X and by the connection defined by a on AdP.

In the case of free electromagnetic field, Q(a) is the interior product iF2v (with respect to the first indices in F2 and q) of the contravariant electromagnetic field F2 and the volume element 77 ; and dviFZq coincides with the Maxwell’s equations.

The Yang-Mills field equations [4] can be globally formulated in a similar way. The consideration of these physical examples suggests, in turn, to deal with the follow-

ing problem of possible geometrical interest. Let p: P --f X be the bundle of linear references of a Riemannian oriented manifold

with the metric tensor g, and the Riemannian volume element,q. In this case, the sections of the affine bundle Z: E --f X can be identified with linear connections on X. The Lagrangians introduced for electromagnetic Yang-Mills fields suggest, for this case, the definition of the gauge-invariant Lagrangian:

(.i.:V) = #W?’ * bw:,1

where R(V):,, is the curvature tensor of the linear connection V, R(V):*l is the contraction of R(V):,, with the discriminant A4 of the contravariant metric g2 (i.e. .14(rr)i~j~~~k) = g’(eJioh)* g2(~j~~)-g2(~i~~)~ g2(Wjmh)), and the - means contraction of the contra- and covariant indices of R(V):*’ with the corresponding co- and contravariant ones of

R(V):, I . In this case Q(V) coincides with the interior product i(f&)f,’ . q relative to the first

two contravariant indices of (Rv)):*l and the first two covariant indices of 7, in such a way

that the criticality condition for a linear connection V is


where c&,,~, means “covariant exterior differential” for AdP-valued tensors on X (that is, tensors on X taking their values in the set of mixed tensors T: on X) defined by the auxiliary linear connection V0 taken in the general theory and by the connection defined by V on AdP.

If V = V,,, the first member of the above equation becomes the exterior covariant differential &I, (i(Rv,):” * 91) relative to the linear connection V,,. Thus, if this differential is zero, the auxiliary linear connection V0 will be critical. For example this is the case, when V,, is the Riemannian connection of g, with an antiparallel curvature tensor.

4.

Finally, the “Jacobi fields along a critical connection” can be introduced, as in [l], in the following way:

Let 0: X + E be a given connection and let D, be a z-vertical vector field on E defined along o(X). Then we define the l-jet extension j’(f),) of D, as the unique .i’(n)-vertical vector field defined alongj’a(X) that satisfies Lj1(D,,81jIb(x) = 0. We say that D, is a Jacobi

field along the critical connection (T when:

Once (T is fixed, we deal with a linear equation for D#, that can be thought of on the manifold X in a analogous way to the one used for the Euler equations.

Indeed, all fields D,,, among which the solutions of Jacobi’s equations can be found can be identified via 0 with the sections of the bundle Hom(T(X), AdP), on which the afhne bundle ~2: E --f X is based.

On the other hand, using the commutation formula for the Lie derivation and the exterior differential, we get:

LjlCDn) dQljIn(x) = dLjl~~,~~Ij~,~x,+ijl(D~)~~~ljI,(x)

where K is the curvature 2-form of the derivation law on M with respect to which all our calculations are being made.

Then, a critical connection u: X + E and a section Db: X + Hom(T(X), AdP) are

given, Ljl(D,@ljl,tx) and ij’(D~)X~Qljt,cx) define, respectively the Ad P-valued tensors Q’(D,,) and L?“(Db) on X, in terms of which the Jacobi equation becomes:

ddX(D,)+%'(D,) = 0

where dv is the above introduced differential. When G is Abelian, the curvature 2-form K is zero, and L&‘(D,) = 0; so the Jacobi

equation takes the homogeneous form dvf2L(D,) = 0. This is the case of free electromagnetic field, where, by identifying the D, with the sections of Hom(T(X), AdP) = T*(X), we recover the Maxwell’s equations.

344 P. L. GARCIA

REFERENCES

[l] Garcia, P.: Symp. Math. 14 (19741, 219. [2] -: Gauge algebras, curvature and symplectic structure, to appear in J. Differential Geometry. [31 -: Reducibi1it.v of the symplectic structure of classicalfields with garage-symmetry, Conference on Differ-

ential Geometrical Methods in Mathematical Physics, Bonn, July 1975. [41 Utiyama, R.: Whys. Rev. 101 (19.56), 1597.

Critical principal connections and gauge-invariance

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