Representation of a gauge group as motions of a Hilbert space

This is a survey article based on the author’s Master thesis 1 on aﬃne representations of a gauge group. Most of the results presented here are well-known to diﬀerential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation oﬀers a useful self-contained introduction to the subject. In the ﬁrst part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second part is about the deﬁnition of the gauge group, G P , associated to a principal bundle, P . In the third part we present the construction of the Hilbert space where the representation takes place. Finally, in the fourth part, we show the construction of the representation and the way in which this representation goes to the set of connections associated to P


Group of Motions of a Hilbert Space
In this section we want to study the group of motions of a Hilbert space H considering it as a Banach Lie group.A general account of the theory of infinite dimensional Lie groups and their structure can be found in [13], [14], [16], [20] and [21].The group of motions of a Hilbert space is defined as the semi-direct product of the group of translations of H and the Hilbert group Hilb(H) associated to H, (if H is real Hilb(H) is the orthogonal group O(H), if H is complex Hilb(H) is the unitary group U (H)).
Let H be a Hilbert space over R or C. We consider L(H, H), the set of all bounded linear operators defined on H.It is well known that L(H, H) is a Banach Lie algebra with norm and Lie bracket given by the following equations: with A, B ∈ L(H, H) and x ∈ H. Let GL(H) be the set of all invertible elements of L(H, H), from elementary functional analysis we know that it is a group.Since multiplication and inversion operations are smooth functions in GL(H) we have that it is a Banach Lie group, its exponential function exp : L(H, H) → GL(H) is given by the series that converges for all A ∈ L(H, H).Furthermore if A ∈ L(H, H) is close to the identity I, the series (−1) n+1 (A − I) n n converges, see [15], chapter VII, section 2.
If A ∈ Sk(H), Sk(H) the skew-symmetric algebra of H, then exp(A) ∈ Hilb(H) and exp is an analytic diffeomorphism of the open unit ball of 0 ∈ Sk(H) onto the open unit ball at I ∈ Hilb(H).Since Hilb(H) is a closed submanifold of GL(H), the last results allow us to consider U (H) as a Banach Lie Group with associated Banach Lie algebra Sk(H).
Semi-direct Product of Lie Groups and its Lie Algebra.The general construction of the semi-direct product of two Lie groups and its corresponding Lie algebra can be found in [5], [11] and [17], in the infinite dimensional case of regular Lie groups it can be found in [13], section 38.9.Here we consider the semi-direct product in the particular case when H = (H, +), the additive group of the Hilbert space H, and G = (GL(H), •), where '•' denotes the composition of linear operators.Let η : GL(H) → Aut(H), A → A, where Aut(H) is the group of automorphism of H as an additive group.η can be thought of as an action η : GL(H) × H → H, (A, x) → Ax.From the definition of the norm we have that Ax ≤ A x for any A ∈ L(H, H) and for all x ∈ H, thus η is continuous and the semidirect product is a Lie group.It is called the affine group, its elements are the functions (x, A) : H → H that are composition of invertible linear maps and translations, if w ∈ H (x, A)w = Aw + x.
The group operation in Aff(H) is given by and it corresponds to the well known composition of affine transformations.

Definition 1.1:
The group of motions M (H) of a Hilbert space H is defined as the semi-direct product H Hilb(H), where Hilb(H) is the Hilbert group of H.
M (H) is a subgroup of the affine group, it is called the group of motions of H because its elements can be interpreted as the physical motions of rigid objects lying in H; if there is a rigid object in H the classical way in which it can move is by rotations, translations and compositions of these two; these are precisely the elements of M (H).Every element of M (H) is an isometry of H, but it is important to note that the whole set of isometries is not M (H).
In the following part we describe the Lie algebra associated to Aff(H).In order to do that we identify H with its Lie algebra, h = T 0 (H), since H is an additive commutative Lie group its Lie algebra has a trivial Lie bracket, see [22], chapter 1, section 2.
In the general case, let G and H be Lie groups with Lie algebras g and h, respectively and let e ∈ H denote the neutral element in H.The semi-direct product of G and H is determined by an action η : G → Aut(H).Let g ∈ G, the derivative of η(g) : H → H at e ∈ H, T η(g) e : T e (H) → T e (H), is an automorphism of h.If we denote T η(g) e by τ g , we have that τ : In this way, τ is a homomorphism of groups, where Aut(h) is considered as: Aut(h) is a group with the usual composition of linear operators.Therefore we have a representation of the group G on the vector space h.To this representation there corresponds a linear representation of the Lie algebra g in h given by the formula: for X, X 1 , X 2 ∈ g and for Y 1 , Y 2 ∈ h, we have that τX is a derivation.Thus τ : g → Der h, where Der h is the set of all derivations of h, is a Lie algebra homomorphism.
The semi-direct product of Lie algebras can be defined in the following way.Let g and h be Lie algebras, and let π : g → Der h, be a Lie algebra homomorphism, the vector space h × g becomes a Lie algebra with structure given by the equation: This Lie algebra is called the semi-direct product of the Lie algebras h and g by the homomorphism π and it is denoted by h × π g.

Proposition 1.2:
With the notation introduced above we have that the Lie algebra given by h × τ g is the Lie algebra associated to the Lie group H × η G. See [17], appendix 5, section 5, and [11], chapter I, section 12.
In our particular case equations (1.2) and (1.3) take the form for y, y 1 , y 2 ∈ H and A, A 1 , A 2 ∈ L(H, H).The Lie bracket in H × L(H, H), given by equation (1.4), is now

The gauge group associated to a principal bundle
The theory of gauge groups has been widely studied and there are many references related to this topic, see for example [3], pages 539 and 546, [4], section 5.6, [13], chapter IX section 44, and [18], section 5.In this section we consider M a compact Riemannian manifold without boundary and G a compact semi-simple Lie group with Lie algebra g.

Principal Bundles and Connections
Let M be a manifold and G be a Lie group with identity e.We consider here only the finite dimensional case.The theory presented here can be found, among others, in [12], sections II.5 and III.1, [4], chapter 4, [18], sections 2 and 3, and [19], chapter 3.

Definition 2.1:
A differentiable principal fiber bundle over M with group G consists of a differentiable manifold P and a right action Θ of G on P that satisfy the following conditions: 1.The action of G on P is free, that is Θ(p, g) = p for some p ∈ P , implies that g = e.
2. M is the quotient space P/G and π : P → M is differentiable.
3. P is locally trivial.This is, for each x ∈ M there exist U a neighborhood of x, U ∈ V x , and ψ : and a ∈ G.
A principal bundle is denoted by P (M, G, π) or just by P (M, G) or P .The manifold P is called the total space, M is the base space, G is the structural group and π is the projection.The fiber at x ∈ M , π −1 (x), is denoted by Example 2.2: The bundle of orthonormal linear frames on R 2 .We consider the set: and the projection π : acts on the right of P as follows: if , the action is given by The action is also transitive because given u 1 , u 2 ∈ P , we take A = (u −1 2 )u 1 and we have that trivializations are given by: where u y = (v 1 v 2 ).P is a manifold of dimension 3.
The Lie algebra g can be considered either as the tangent space to G at the identity e or as the space of left invariant vector fields over G.If we consider g as the space of left invariant vector fields, then exp(tX) is the one-parametric group associated to the left invariant vector field X, that is, exp(0) = e and d dt exp(tX) t=0 = X.Definition 2.3: For A ∈ g and p ∈ P we define the fundamental vector field over P corresponding to A as: p is tangent to the fibre in each p ∈ P , that is, π * (A * ) = 0.If A is not zero then A * is also different from zero.Thus, we have that the function σ : When a principal bundle is given it carries a notion of verticality in its tangent space but there is not an intrinsic notion of horizontality associated to it.To give a connection on P is to give a horizontal subspace of the tangent space in each point of P .

Definition 2.4:
A connection Γ on P is an assignment of a subspace H u of T u (P ) to each u ∈ P such that 3. H u depends differentiably on u.
It is well known that to have a connection on P is equivalent to have a connection form on P .See [12], chapter II, section 1.The correspondence between a connection Γ and its connection form ω is given as follows: Given Γ a connection on P , for each u ∈ P , X ∈ T u (P ), ω(X) = A, where A is the unique vector in g such that A * u = X v , X v the vertical component of X. Conversely, given a connection form the distribution of horizontal spaces is defined by H u = ker ω u ⊆ T u (P ).
Example 2.6: Let P = R×S 1 be a trivial principal bundle over R with fibre and group S 1 , S 1 acts on itself by the usual multiplication.The Lie algebra associated to S 1 is iR and the exponential function is given by exp : iR → S 1 , iλ → e iλ .The tangent space to P at the point (x, z) is given by Then a tangent vector at (x, z) is given by X (x,z) = (r, izs) with r, s ∈ R. From equation (2.1) it follows that the vertical space at p = (x, z), determined by the structure of the principal bundle, is V (x,z) = ziR.Given a vector A in the Lie algebra of S 1 , A = ia with a ∈ R, the corresponding fundamental vector field in P is given by Now we want to define a nontrivial connection on P .We can do it by defining first a distribution of horizontal spaces at each point.Let X = (r, zis) be a tangent vector to P at (x, z), let us define the corresponding projections, clearly p h and p v are projections and ) .We also have to check that the distribution is right invariant, that is, that it satisfies R a * (H (x,z) ) = H (x,za) for all a ∈ S 1 .Let a ∈ S 1 and X ∈ T (x,z) P , then R a * (X h ) = R a * ((r, zir)) = (r, (za)ir) ∈ H (x,za) .
The connection form ω associated with the distribution is given by since (i(s − r)) * (x,z) = (0, zi(s − r)) , the connection is well defined.In a similar way other connections on P can be obtained.
Given two connections ω, ω , their difference ω − ω is a horizontal equivariant 1-form with values in g, where by horizontal we mean (ω −ω )(X) = 0, for all X vertical.

The Group Bundle and the Adjoint Bundle
Let P (M, G) be a principal bundle.In order to construct the desired representation and to define the gauge group associated to P we need to define the group and the adjoint bundles.These bundles are associated bundles to P , see [12], and are defined as follows: Definition 2.7: Consider the left action of G on itself given by γ a (g) = a g a −1 , for a, g ∈ G.This gives rise to the right action of G on P × G, Θ : The group bundle G is the bundle (P, M, G, π G , G) associated to P .It looks locally like M × G and its elements are the equivalence classes: The set of smooth section of G is denoted by Γ(G), that is Definition 2.8: Consider the left action of G on g given by Ad : G × g → g, (a, X) → Ad(a)X.This gives rise to the action of G on the right of P × g given by Θ((u, X), a) = (u • a, Ad(a −1 )(X)) = (u • a, (R a ) * (X)) The adjoint bundle P (g) is the bundle (P, M, g, π P (g) , G) associated to P .It looks locally like M × g and its elements are the equivalence classes: The set of all smooth sections of P (g) over M is denoted by Γ(P (g)) or by A 0 (g).

Example 2.9:
The isomorphism in the last case is given by the map [(x, a, X)] → (x, Ad(a)X), for x ∈ M , a ∈ G and X ∈ g.

Description of the Gauge Group
In this section we will construct the gauge group associated to a principal bundle P .As we mentioned before, some good references for this topic are [3], [4], [13], and [18].
Let Aut(P ) be the group of all equivariant diffeomorphisms of P , that is, where diff(P ) denotes the set of all diffeomorphism of P as a manifold.Now we define: Definition 2.10: The gauge group, G P , of P is the group If we consider the correspondence between Aut(P ) and diff(M ) given by the following commutative diagram: where ρ : Aut(P ) → diff(M ), we have that Let C ∞ G (P, G) be given by Then we have the following proposition: Proposition 2.11: The groups G P , C ∞ G (P, G), and Γ(G) are isomorphic as groups.
Proof: Let α ∈ G P and p ∈ P .Since α(p) ∈ P π(p) , there exists g ∈ G such that α(p) = p • g.Let us define f : P → G by f (p) = g, where g satisfies α(p) = p • g. f is well defined because the action of G on P is free and transitive in the fibres.For α ∈ G P and g ∈ G, On the other hand, To see that σ is well defined, let q ∈ P such that π(q) = x, then q = p • g, and for every h ∈ G.But for every q ∈ P x , there exists a unique h ∈ G such that q = p • h.Thus, we can define We need to check now that these are group isomorphisms.Let α, β ∈ G P , f, g ∈ C ∞ G (P, G), and σ, ς ∈ Γ(G) be in the correspondence described above, that is, for p ∈ P , x ∈ M , π(p) = x, we have α(p) = p • f (p), σ(x) = [p, f (p)] and β(p) = p • g(p), ς(x) = [p, g(p)].Since the group operation in G P is given by composition we have Thus the isomorphisms are group isomorphisms, and the proof is complete.

Example 2.12:
We want to consider Γ(G) with a topology that makes it into a Lie group, then we use the isomorphisms given in Proposition 2.11 to endow G P and C ∞ G (P, G) with topologies that allow us to consider these three spaces as the same.For this, let S be the set [23] page 55 and [4] page 77).The set S forms a subbasis for a topology in C k (M, G), this topology is called the C k -Whitney topology.Now we assume that M is compact and we take the C ∞ -Whitney topology over Γ(G) is the coarsest topology among all topologies on Γ(G) for which the canonical injections: are continuous, k = 0, 1, . . .

Construction of the Hilbert Space of the Representation
Let us consider the set of p-forms on M with values in P (g).
Since P (g) is a bundle of Lie algebras, A 0 (g) becomes a Fréchet Lie algebra under pointwise operations, a Fréchet Lie algebra is a Lie algebra that also is Fréchet space, (we refer the reader to [13], [21], [25] and [26] for Fréchet structures).Let σ 1 , σ 2 ∈ A 0 (g), for x ∈ M , without lost generality we can consider We can consider the elements of A p (g) to be of the form γ⊗σ, with γ ∈ Ω p (M ) and σ ∈ A 0 (g) but is important to keep in mind that their real form is Using the facts that M is a Riemannian compact manifold and that G is a compact semi-simple Lie group we can define an inner product in A 1 (g) in the following way: In the Lie algebra g the Killing form is given by the equation: with X, Y ∈ g; since G is semi-simple, the Killing form is negative-definite; then the equation gives an inner product on g, see for example [6], sections XXI.5 and XXI.6.We can lift this inner product to P (g) fibrewise putting, for it follows from the invariance of Killing form that last equation determines a well defined inner product in P (g).Let ς 1 , ς 2 ∈ A 1 (g) be given by ς in the following way: where the first term in the product of the right hand is the inner product in the cotangent space of M induced by the Riemannian metric of M .We can extend this operation to the whole space A 1 (g) × A 1 (g) by bilinearly and then define an inner product in A 1 (g) by the equation: ) where µ(x) is the measure in M induced by the Riemannian metric.Thus A 1 (g) has a well defined inner product and it can be completed to a Hilbert space, this is the space that we will use as representation space of the affine representation of the gauge group G P , we will denote the completation of A 1 (g) by H. 2) be a trivial principal bundle over R with fibre and group SU(2), SU(2) is a semi-simple Lie group and its Lie algebra is su (2).A basis of su(2), as real vector space, is:

Example 3.1:
For X, Y ∈ su(2), with X = 3 i=1 x i A i and Y = 3 i=1 y i A i , the Killing form is given by the equation: is the trace of the matrix whose k-th column, k = 1, 2, 3, is the vector whose components are the coefficients of (ad in the basis {A 1 , A 2 , A 3 }.An easy computation shows that B(A i , A j ) = −8δ ij .Thus the killing form in su(2) has the form: It is clearly negative definite and gives rise to an inner product in A 1 (su( 2)).
Since in this case P is a trivial principal bundle, we have that

Construction of the Representation
The representation we are going to construct arose as a preliminary step in the construction of the energy representation of the group of mappings of a Riemannian manifold into a compact semi-simple Lie group studied in [8] and [2].Definition 4.1: Let G be a group, an affine representation of G in a vector space E is a homomorphism Ψ : G → Aff(E), i.e. an element of Hom(G, Aff(E)).
We want to construct an affine representation of the gauge group G P in H, the completation of A 1 (g) respect to the inner product given by equation (3.3).We have a natural action of the gauge group on each set A p (g), the description of this action is the following: Consider the map G × M P (g) → P (g) given by ([p, g], [p, X]) → [p, Ad(g)X].The corresponding action of Γ(G) on A 0 (g) = Γ(P (g)), by the Ω-lemma (see [1], page 101), is given by γ(t) be such that X = d dt γ(t) t=0 and γ(0 If X is any vector in P , we get the decomposition of X into its horizontal and vertical parts, X = X h + X v , therefore the statement above also works for X.

Proposition 4.4:
The operator d ω satisfies the equations: Proof: Let p ∈ P , and X ∈ T p (P ) For the second equation we take f = f and g = f for all g ∈ G. Let Y ∈ X (P ) and q ∈ P with π(q) = x, we define β : T (P ) → g by β q (Y ) = X(π(q), π * q (Y )).
We need to check that β is horizontal and equivariant.If Y is vertical, π * (Y ) = 0.Then, X(π(q), π * q (Y )) = X(x, π * q (0)) = 0. Thus β p (Y ) = 0, i.e., β is horizontal.We have to see now that Then the correspondence between the horizontal and equivariant elements β, β ∈ Γ(Λ 1 (P ) ⊗ g), and the 1-forms β, β ∈ A 1 (g), is one to one and onto.Thus, we can consider δ ω (f ), the right logarithmic differential of f ∈ C ∞ G (P, G) (i.e.f in the gauge group), as an element in the Hilbert space H and we can finally define the affine representation of the gauge group: Let ω be a fixed connection form over P , consider the action of C ∞ G (P, G) on A 1 (g) given by Let us check that the action is well defined: on the other hand we have Then, from Proposition 4.6 we have that δ ω (f , then the action of f ∈ G P on β is obtained extending the action of f on A 1 (g) by continuity.Then the affine representation of G P in H is given by: H) and we get that the representation is in fact on the group of motions of H, M (H).
The affine representation of the gauge group in the Hilbert space H is interesting because it provides the first step for the construction of a unitary representation Ψ of the gauge group in the symmetric Fock space F s (H) associated to H.The representation Ψ is obtained as the composition of the affine representation described in this paper and a representation of the group of motions in F s (H).Although the description of Ψ is not the purpose of this paper, let us say something about it: The representation of G P in F s (H) is equivalent to a representation of G P in a space L 2 µ (H) where µ is a special measure, see for example [9] and [10].As we already said, these kind of representations arose from physics and from the study of current groups.The study of the irreducibility of these representations involves a lot of work in measure theory and functional analysis.In the case when the gauge group is of the form C ∞ (M, G) this study was developed by Gelfand, Graev and Vershik ( [8] and [9]), Wallach ([27]), Pressley ([24]) and Albeverio ( [2]), among others, during the 70s and the 80s and the results depend strongly on the dimension of the base manifold M : if dim M ≥ 4, then the representation is irreducible, see [9], but if we consider loop groups, that is, if M = S 1 , then the representation is highly reducible, see [24], chapter 9.The question about the representation of the gauge group in the general case (when G P is not of the form C ∞ (M, G)) is still open and we think that the geometry involved in the representation of G P in M (H) could help to solve this problem.

Representation of the gauge group in the set of connections.
The representation of the gauge group in M (H) described above can also be interpreted as an affine representation of the gauge group in the space of connections C, thinking of C as an affine space.This interpretation is important when Yang-Mills theories are considered and it has implications in this area, see [3], section 3, [7], and [18], section 6; other applications to physics can be found in [4] and [19].Now we will describe the action of the gauge group on the set of connections: As we mentioned above, given two connections their difference is a horizontal, equivariant 1-form over P with values in the Lie algebra g.Then for a fixed connection ω, it follows from theorem 4.7 that C is isomorphic to the affine space ω +A 1 (g), and the action of G P on C Θ : is given by the following equations: