Integrating the central extension of the gauge algebra

Definition 5.1.4. The continuous cocycle ω_κ,A from the preceding lemma is called covariant cocycle.

Remark 5.1.5. Lemma 5.1.3 implies that the class [ω_κ,A]∈H_c²(gau(P),z_M(Y)) is independent of the choice of the connection 1-formA. Thus, the equivalence class of the central extension

gau(P)\_ω

κ,A =z_M(Y)⊕_ωκ,Agau(P) with [(x, η),(y, µ)] = (ω_κ,A(η, µ),[η, µ]) (cf. Remark A.2.2) does not depend on the choice ofAbut only on the bundleP and onκ.

Now the question arises how exhaustive the constructed central extension of gau(P) is, i.e., for which spaces it is universal.

Remark 5.1.6. It has been shown in [Ma02] that the central extension ofgau(P) from Remark 5.1.5 is universal in the case of a trivial bundle, finite-dimensional and semisimple k and the universal invariant bilinear form k×k→V(k), since then gau(P)∼=C^∞(M,k) and the cocycle (5.1) is universal.

For non-trivial bundles it is not know to the author whether the central extension of the gauge algebra is universal. The arguments from [Ma02] do not carry over directly, because they use heavily the fact thatkembeds as a subalgebra intoC^∞(M,k). This is not true forC^∞(P,k)^K and causes the main problem.

Problem 5.1.7. For which bundles (beside trivial ones) and for which locally convex spaces is the central extension of Remark 5.1.5 universal?

The period homomorphism encodes a crucial part of the information on the integrability of the cocycleω.

Remark 5.2.2. Let Gbe a connected Lie group with Lie algebra gand z be a s.c.l.c. space.

Let Γ⊆zbe a discrete subgroup and Z:=z/Γ be the corresponding quotient Lie group. Then we define

I:H_c²(g,z)→Hom(π₂(G), Z)×Hom(π₁(G),Lin(g,z))

as follows. For the first component we takeI1([ω]) :=qZ◦per_ω, whereqZ:z→Zis the quotient map and per_ω:π₂(G)→z is the period map of ω. To define I₂([ω]), for eachx∈g, we write X_r for the right invariant vector field onGwithX_r(e) =xand Ω for the left invariantz-valued closed 2-from on G with Ω(e) =ω. Then i_Xr(Ω) is a closed z-valued 1-from ([Ne02a, Lemma 3.11]) to which we associate a homomorphismπ₁(G)→zvia

I2([ω])([α])(x) :=

Z

α

iX_r(Ω).

for a smooth representativeα∈C_∗^∞(S¹, K). We refer to [Ne02a, Section 7] for arguments showing that I is well-defined, i.e., that the right hand side depends only on the cohomology class ofω and the homotopy class ofα.

Theorem 5.2.3. ([Ne02a, Theorem 7.12]) Let Gbe a connected Lie group,zbe a s.c.l.c. space, Γ⊆z be a discrete subgroup and ω∈Z_c²(g,z) be a continuous Lie algebra cocycle. Then the central extension of Lie algebrasz,→bg:=z⊕_ωggintegrates, in the sense of Remark A.2.6, to a central extension of Lie groups Z ,→GbGwithZe=z/Γ, if and only ifI([ω]) = 0.

As we will see later on, the hard part is to check whether I₁ vanishes. By choosing Z appropriately this can always be achieved as long as the image of the period homomorphism is discrete.

Proposition 5.2.4. Let G be a connected Lie group, z be a s.c.l.c. space andω∈Z_c²(g,z). If per_ω:π₂(G)→z is the associated period homomorphism and the period group Π_ω:= im(per_ω) is discrete, then I1([ω])from Remark 5.2.2 vanishes if we takeΓ = Πω.

Proof. In this case, ker(q_Z) = im(per_ω) and thusI₁([ω]) =q_Z◦per_ω vanishes.

In the case that the period group is discrete, one still has to check thatI₂vanishes in order to show that the central extension, determined byω, integrates. This is always the case ifGis simply connected, but in general, the condition thatI₂vanishes seems to be as hard to check as the vanishing of I₁. However, there is an equivalent condition, which makes life easier (at least in the case that we consider here).

Proposition 5.2.5. ([Ne02a, Proposition 7.6]) Let G be a connected Lie group, z be a s.c.l.c.

space andω∈Z_c²(g,z). Then the adjoint action ofgonz⊕_ωg, given by (x,(z, y))7→(ω(x, y),[x, y]),

integrates to a smooth action ofGif and only if I₂([ω]) = 0.

We now return to our particular cocycleω_κ,A. The invariant formsκ:k×k→Y that we will mostly work with are the universal ones, which we introduce now.

Definition 5.2.6. If kis a locally convex Lie algebra and Y is a locally convex space, then a continuous, symmetric andk-invariant bilinear form κ:k×k→Y is calleduniversal if for each k-invariant symmetric bilinear mapf :k×k→Z factors through a unique continuous linear map fe:Y →Z satisfyingf =fe◦κ.

We collect some facts on universal forms that we use in the sequel. In particular, ifkis finite-dimensional and simple, then the universal form coincides with the well-known Cartan–Killing form.

Remark 5.2.7. If k is finite-dimensional, then a universal, continuous, symmetric k-invariant bilinear form can be obtained as follows. Denote byV(k) the quotientS²(k)/k.S²(k), whereS²(k) is the universal symmetric product, wherekacts on byx.(y∨z)7→[x, y]∨z+y∨[x, z]. Then

κ:k×k→V(k), (x, y)7→[x∨y],

is universal. We shall frequently denote by κ:k×k→V(k) the universal form of k and con-siderV as a covariant functor form the category of dimensional) Lie algebras to (finite-dimensional) vector spaces.

We collect some facts the universal form κ:k×k→V(k)∼=Rⁿ. The facts used below can be found in the standard literature on (semi-) simple complex and real Lie algebras, e.g., [Ja62], [He78], [Wa01] or [On04]. Note that n≥1 ifk is semi-simple, because then theCartan–Killing form

κCK :k×k→R, (x, y)7→tr(ad(x)◦ad(y))

is a symmetric and invariant bilinear form which is non-degenerate by Cartan’s Criterion.

Furthermore, since κCK is non-degenerate, for each otherR-valued invariant symmetric bi-linear form κ⁰ we find a unique A∈End(k) such that κCK(A.x, y) =κ(x, y) for all x, y∈k.

Moreover, we have

κCK(A.[x, y], z) =κ⁰([x, y], z) =κ⁰(x,[y, z]) =κCK(A.x,[y, z]) =κCK([A.x, y], z) for allx, y, z∈k, which impliesA.[x, y] = [A.x, y]. Taking kas a module over itself, this implies thatA is a module map, i.e.,A∈End_k(k). Thus

κ(x, y) = (κ_CK(A₁.x, y), . . . κ_CK(A_n.x, y)) forAi∈Endk(k) and we see that V(k)∼= Endk(k) for uniqueness reasons.

If k is semi-simple with the simple factors k1, . . . ,kn, then κ is clearly the direct sum of κ₁, . . . , κ_n, whereκ_i:ki×ki→V(k_i) is the universal form ofki. This reduces the determination ofκto the case where kis simple, so letkbe a real simple Lie algebra from now on. From the classification of simple real Lie algebras, it follows that k is either the restriction of a complex simple Lie algebra to real scalars, or k is a real form of a simple complex Lie algebra. In the first case we have that the complexificationk^C:=k⊗_RCis not simple as a complex Lie algebra and in the second case that k^C is simple as a complex Lie algebra. We shall treat these cases separately.

If kis the restriction of a complex simple Lie algebra to real scalars, then the module maps which are also complex linear, are precisely Endk(k) =C·1by Schur’s Lemma. If ∈Endk(k) is complex anti-linear, then we deduce from

i[x, A.y] =A.(i[x, y]) =−iA.([x, y]) =−i[x, A.y]

that it vanishes. By decomposing eachA∈End_k(k) in its complex linear and complex anti-linear part we see that this impliesV(k)∼=C, and the two components of the universal fromκare the real and imaginary part of the Cartan–Killing form ofk^C.

Ifk^C is simple as a complex Lie algebra, then we have Endk(k^C)∼= Endk(k)⊗_RCand by the same argument as above

1·C= End_kC(k^C)∼= Endk(k^C)∼= (Endk(k))⊗_RC,

which implies V(k)∼= Endk(k)∼=R. In this case, κ_CK is the universal invariant bilinear form.

This is particular the case if k is a compact Lie algebra, i.e., if κ_CK is negative definite or, equivalently, if each Lie groupKwith L(K) =kis compact.

In the case of a finite-dimensional trivial principal K-bundle over S¹ and universal κ, the image of the period homomorphism is known to be discrete. As we will see later on, this is the generic case for all finite-dimensional bundles.

Proposition 5.2.8. If K is a finite-dimensional Lie group, PK is the trivial bun-dle over S¹ with canonical connection 1-form A, κ:k×k→V(k) is universal and ωK :=ωκ,A∈Z_c²(gau(PK),z_S1(Y))is the cocycle from Remark 5.1.5, then the associated period groupim(per_ωK) =: Πω_K is discrete.

Proof. We have Gau(P)∼=C^∞(M, K) and gau(P)∼=C^∞(M,k), because PK is trivial. Then Ad(P) is also trivial andf 7→df is the covariant derivative induced by the canonical connection 1-form on P. Therefore, ω_K coincides with the cocycle (f, g)7→[κ(f, dg)] in [MN03, Theorem II.9], whereK is assumed to be connected. Sinceπ₂(K) =π₂(K₀) is trivial,

π2(C^∞(S¹, K))∼=π2(C_∗^∞(S¹, K)) =π2(C_∗^∞(S¹, K0))∼=π2(C^∞(S¹, K0))

and L(C^∞(S¹, K))∼=C^∞(S¹,k)∼=L(C^∞(S¹, K0)), the image of per_ωK is not affected by the missing assumption onK of being connected and [MN03, Theorem II.9] yields the assertion.

We now turn to the computation of the image of the period homomorphism in the non-trivial case. As indicated before, bundles overS¹ play a key role in this computation, because we can reduce the situation for arbitrary bundles to the case of bundles overS¹by choosing appropriate curvesα:S¹→M and pull back the bundles alongα.

One of the fundamental ideas in bundle theory is that pulling back bundles along homotopic maps does not change the (equivalence class of) the pull-back bundles. We shall adopt this idea and will show that pulling back bundles along homotopic mapsα1, α2:S¹→M will not change the (image of) the period homomorphism of the pull-back bundles. This will be the crucial observation to make the whole reduction process to bundles overS¹work.

Remark 5.2.9. For the entire section we fix a system of representatives (ki)_i∈π0(K) for the groupπ0(K) :=K/K0of connected components ofKwithk[e]=e. Forα∈C^∞(S¹, M), we get from Remark B.2.9 thatα^∗(P) is equivalent toPk for somek∈Kand that [k]∈π₀(K) depends only on the homotopy class ofα. We thus obtain a homomorphismϕ:π₁(M)→π₀(K) (which is the connecting homomorphism in the long exact homotopy sequence of P), which satisfies α^∗(P)∼=P_kϕ([α]) and we set P_[α]:=P_kϕ([α]). Furthermore, for each [α]∈π₁(M) this yields a bundle mapα_P :P_[α]→P coveringα.

The connection 1-formAonP induces a connection 1-formAαonP_[α] by pulling backAto a connection 1-formα_P^∗(A) onP_[α]. Then the induced covariant derivative∇^α satisfies

∇^αη(Xp) =∇^A(η◦α_P).T α_P(Xp) (5.5) forη∈C^∞(P_[α],k)^K andXp∈TpP_[α]. We denote the corresponding cocycle by ωκ,α. Further-more, ifαand α⁰ are homotopic, then α^∗_P(A) andα^0∗_P(A) are two different connection 1-forms onP[α]and thusωκ,α−ωκ,α⁰ is a coboundary. Since the period homomorphism of a coboundary vanishes (cf. [Ne02a, Remark 5.9]) and

per_ω

κ,α−per_ω

κ,α0 = per_ω

κ,α−ω_κ,α0 = 0,

this implies that per_ωκ,α :π2(Gau(P_[α]))→zM(Y) depends only on the homotopy class ofαand we thus denote it per_ω

κ,[α].

We now take the mappings between the gauge groups into account that we get from pulling back bundles.

Remark 5.2.10. If P is a smooth principal K-bundle over the compact manifoldM (possibly with corners) andf :N →M is smooth, then the induced bundle mapf_P :f^∗(P)→P induces in turn a map f_Gau: Gau(P)→Gau(f^∗(P)), given by γ7→γ◦f_P if we identify Gau(P) with C^∞(P, K)^K and Gau(f^∗(P)) withC^∞(f^∗(P), K)^K.

Correspondingly, we have a homomorphism f_gau:gau(P)→gau(f^∗(P)), η7→η◦f_P, which is a morphism of topological Lie algebras by Lemma 2.2.24. It follows thatf_Gauis a morphism of Lie groups, because Gau(P) is locally exponential if K is so, and f_Gau makes the following diagram commutative

Gau(P) −−−−→^fGau Gau(f^∗(P))

exp

x





exp

x



 gau(P) −−−−→^fgau gau(f^∗(P))

We are now able to describe what happens to the period homomorphism when pulling bundles back along smooth curves. The formula derived in the next lemma will be the crucial one to make the reduction to bundles overS¹work.

Lemma 5.2.11. If α∈C^∞(S¹, M)andλ_αdenotes the linear map from Remark 5.1.2, then λα◦per_ωκ,A =λid_S1◦per_ωκ,[α]◦π2(αGau), (5.6) whereα_Gau is the induced mapGau(P)→Gau(P_[α])from Remark 5.2.10.

Proof. We identify Gau(P) and Gau(P_[α]) withC^∞(P, K)^K andC^∞(P_[α], K)^K. Thenα_Gau is given byf 7→f◦α_P, whereα_P :P_[α]→P is the induced bundle map.

Denote by Ω_P and Ω_α the left invariant closed 2-forms on Gau(P) and Gau(α^∗(P)) with Ω_P(e) =ω_κ,aand Ω_α(e) =ω_κ,α. Thenλ_α◦Ω_P is also left invariant, as well asα^∗_Gau(Ω_α) since

λ^∗_γ(α^∗_Gau(Ωα)) = (αGau◦λγ)^∗(Ωα)

= (λ_αGau(γ)◦α_Gau)^∗(Ω_α) =α^∗_Gau(λ^∗_α

Gau(γ)(Ω_α)) =α^∗_Gau(Ω_α).

Thusα^∗_Gau(Ωα) is determined by its values onTeGau(P)∼=gau(P), where it is given by (η, µ)7→[κ_∗(α_gau(e).η,∇^Aα_gau(e).µ)] = [κ_∗(η◦α_P,∇^A(µ◦α_P))].

Sinceλid_S1◦α^∗_Gau(Ωα) is also left invariant, we haveλ_S¹◦α^∗_Gau(Ωα) =λα◦ΩP, because Z

S¹

(α^∗_Gau(Ωα))(e)(η, ν) = Z

S¹

κ_∗(η◦α_P,∇^α(µ◦α_P)(∂t))dt

=i)

Z

S¹

κ_∗(η,∇^Aµ)(dα(t).∂t)dt= Z

α

κ_∗(η,∇^Aµ) = Z

α

Ω_P(e)(η, µ) (5.7) forη, µ∈gau(P), wherei) holds due to (5.5). Forβ∈C^∞(S²,Gau(P)) we thus have

λ_id

S1 per_ω

κ,[α] π₂(α_Gau)([β])

=λ_id

S1(per_ω

κ,[α]([α_Gau◦β])) =λ_id

S1

Z

αGau◦β

Ω_α

= λ_id

S1

Z

β

α^∗_Gau(Ω_α)

= Z

β

λ_id

S1◦α^∗_Gau(Ω_α) = Z

β

λ_α◦Ω_P =λ_αZ

β

Ω_P

=λ_α(per_P,κ([β])).

We are now quite close to our aim of showing that pulling back bundles along homotopic maps α₁, α₂:S¹→M does not change the image of the period group. In view of (5.6), it remains to show thatπ₂(α_1,Gau) =π₂(α_2,Gau), which follows from the next lemma.

Lemma 5.2.12. If α₁, α₂:S¹→M are homotopic, then P[α₁]=P[α₂]=:P[α] and the induced bundle maps α_1,P :P_[α]→P andα_2,P :P_[α]→P are also homotopic.

Proof. Recall from Remark B.2.9 that a representative k∈K for a bundle over S¹ may be obtained as follows. We identify S¹ with [0,1]/{0,1} and denote by q: [0,1]→S¹ the corre-sponding quotient map. Then there exists a lift Q: [0,1]→P with Q(0) =Q(1)·k and k is a representative of the bundle.

Now identify [0,1]×S¹ with the quotient [0,1]²/∼with (x, y)∼(x⁰, y⁰) :⇔

x=x⁰ if y, y⁰∈ {0,1}

x=x⁰ andy=y⁰ esle

and denote byq⁰: [0,1]²→[0,1]×S¹the corresponding quotient map. LetF : [0,1]×S¹→M be a homotopy with F(0,·) =α₁ and F(1,·) =α₂. Then there exists a lift Q⁰: [0,1]²→P of q⁰, because [0,1]² is contractible, and we have Q⁰(t,0) =Q⁰(t,1)·k(t) for somek: [0,1]→K.

Furthermore,k_t:=k(t) representsF(t,·)^∗(P) by its construction, i.e.,P_kt∼=F(t,·)^∗(P). Finally, k_tdepends continuously ont, because

k(t) =k_σ(Q⁰(t,1))⁻¹·k_σ(Q⁰(t,0))

for an arbitrary sectionσ:U →P for a trivialising neighbourhoodU ofF(t,{0,1}) (cf. Remark B.1.5).

From the identificationPk_t ∼=F(t,·)^∗(P) we get bundle maps (Ft)_P :Pk_t →P. Furthermore, letR: [0,1]²→Kbe such thatR|_[0,1]×{0}≡k_[α]and R|_(t,1)=kt. This induces continuous maps (Rt)P :Pk_[α] →Pkt and

[0,1]×P_k[α]→P, (t, p)7→(F_t)_P (R_t)_P(p) is a homotopy betweenα1,P andα2,P.

In order to perform the reduction, we have to know how π₂(f_Gau) looks in two very special cases.

Lemma 5.2.13. Forα∈C^∞(S¹, M), letP[α] be the bundle overS¹ represented by kϕ([α])∈K as in Remark 5.2.9. Iff ∈C^∞(S¹,S¹)is homotopic to the identity, then P[α]=P[α◦f] and

π2(fGau) :π2(Gau(P_[α]))→π2(Gau(P_[α◦f])) =π2(Gau(P_[α])) is the identity map.

On the other hand, if Kis finite-dimensional andf ∈C^∞(S¹,S¹)is homotopic to a constant map, then P_[α◦f] =P_[e] is the trivial bundle and

π2(fGau) :π2(Gau(P[α]))→π2(Gau(P[α◦f])) =π2(Gau(P[e])) vanishes.

Proof. Lemma 5.2.12 tells us that homotopic maps between the base spaces induce homotopic maps between the gauge groups since they are given by pull-backs of the corresponding bundle maps. If f is homotopic to id_S1 we may thus assume that f = id_S1, and thenπ₂(f_Gau) is the identity, becausef_Gau is so. Accordingly, in the case thatf is homotopic to the constant map, we may assume that f ≡m₀ and thus P_[α◦f] =P_[e]. In this case f_P has values in one single fibre and thus

fGau: Gau(P_[α])→Gau(P_[e])∼=C^∞(M, K) takes values inK≤C^∞(M, K) and sinceπ₂(K) vanishes so doesπ₂(f_Gau).

One crucial step in the reduction is to show that the image of the period homomorphism is contained in the subspaceH_dR¹ (M, Y) ofz_M(Y), which is well accessible.

Remark 5.2.14. Let M be a closed finite-dimensional manifold and Y be a Fr´echet space.

Since an element β∈Ω¹(M, Y) is an exact form if and only if all integrals R

αβ vanish for α∈C^∞(S¹, M), the linear mapsλ_αseparate the points ofzM(Y).

A 1-formβ∈Ω¹(M, Y) is closed if and only if for all pairs of homotopic pathsα, α⁰ we have R

αβ=R

α⁰β. Therefore, the subspace H_dR¹ (M, Y)⊆Ω¹(M, Y) is the annihilator of the linear mapsλ_α−λ_α⁰ for [α] = [α⁰] inπ₁(M). In particular,H_dR¹ (M, Y) is a closed subspace ofz_M(Y).

Moreover, we have for [β]∈z_M(Y) that [β]∈H_dR¹ (M, Y) if and only ifλ_α([β]) only depends on the homotopy class ofα.

We still have to choose our curvesα:S¹→M in a way that the image of the period homomor-phism of the pull-back bundles carries all information on the image of the period homomorhomomor-phism onP. This choice is the last thing we have to do before we can prove the Reduction Theorem.

This choice makes the spaceH_dR¹ (M, Y) accessible.

Remark 5.2.15. IfM is a closed finite-dimensional manifold andY is a Fr´echet space, then the de Rham isomorphism and the Universal Coefficient Theorem (cf. [Br93, Theorem V.7.2]) yield

H_dR¹ (M, Y)∼=H¹(M, Y)∼= Hom(H1(M), Y),

because H0(M) is free. If M is compact, denote by r the rank of the finitely generated free abelian group

H₁(M)/tor(H₁(M))

and consider a basis given by the smooth representatives [α1], . . .[αr]. Since H0(M) is free, the Universal Coefficient Theorem and Huber’s Theorem (cf. [Hu61] or [Br93, Corollary VII.13.16]) imply

Hom(π1(M),Z)∼= Hom(π1(M)/[π1(M), π1(M)],Z)∼= Hom(H1(M),Z)∼=H¹(M,Z)∼= [M,S¹].

In particular, there exist mapsf1, . . . , fr∈C^∞(M,S) such that [fi◦αj] =δij ∈π1(S¹), and, in virtue of [Ne02a, Theorem A.3.7], we can assume thefi to be smooth. Since we chose theαi to build a basis ofH1(M)/tor(H1(M)) and each homomorphism from tor(H1(M)) toY vanishes, we eventually obtain an isomorphism

Φ :H_dR¹ (M, Y)∼= Hom(H1(M), Y)→Y^r,[β]7→Z

α_i

β

i=1,...,r, (5.8) whose inverse is given by Φ⁻¹(y₁, . . . , y_r)7→Pr

i=1[δ^l(f_i)·y_i].

Theorem 5.2.16 (Reduction Theorem). The period groupΠP,κ:= im(per_ωκ,A)is contained in the subspace H_dR¹ (M, Y) of zM(Y). If K is finite-dimensional, r denotes the rank of H1(M)/tor(H1(M))andα1, . . . , αr∈C^∞(S¹, M)andf1, . . . , fr∈C^∞(M,S¹) are chosen as in Remark 5.2.15, then

Π_P,κ∼=

r

M

i=1

[δ^l(fi)]·im(λ_idS1◦per_ω

κ,[αi])∼=

r

M

i=1

Π_P[αi],κ. (5.9) In particular,Π_P,κ is discrete if and only ifΠ_P[

αi],κ is discrete fori= 1, . . . , r.

Proof. Remark 5.2.9, Lemma 5.2.11 and Lemma 5.2.12 imply that forα∈C^∞(S¹, M) λ_α(per_ω

κ,A([β])) =λ_id

S1(per_ω

κ,[α]([α_Gau◦β]))∈z_M(Y)

depends only on the homotopy class ofα. Consequently, per_ω

κ,A([β]) is an element ofH_dR¹ (M, Y) by Remark 5.2.14, establishing the first assertion.

In order to show (5.9), we evaluate λ_α on per_ω

κ,A(π₂(f_Gau)) for α∈C^∞(S¹, M) and f ∈C^∞(M,S¹):

λα◦per_ωκ,A◦π2(fGau) =λid

S1◦per_ω

κ,[α]◦π2(αGau)◦π2(fGau)

=λid

S1◦per_ω

κ,[α]◦π2((α◦f)Gau).

We thus obtain with Lemma 5.2.13

λα_i◦per_ωκ,A◦π2(fj,Gau) =δij·λid

S1◦per_ω

κ,[αi]. (5.10)

Applying Φ⁻¹ to (5.10), we thus obtain im per_ω

κ,A◦π₂(f_i,Gau)

= [δ^l(f_i)]·im(λ_id

S1◦per_ω

κ,[αi]) and hence

ΠP,κ⊇

r

M

i=1

[δ^l(fi)]·im(λid_S1◦per_ωκ,[

αi])∼=

r

M

i=1

ΠP_[αi],κ. On the other hand,λ_αi◦per_ω

κ,A =λ_S1◦per_ω

κ,[αi]◦π₂(α_i,Gau) implies directly ΠP,κ⊆

r

M

i=1

[δ^l(fi)]·im(λid

S1◦per_ω

κ,[αi])∼=

r

M

i=1

ΠP_[αi],κ.

In the case of a connected structure group, the pull-back bundles over S¹ are trivial and we thus have the discreteness of the period group that we are aiming for.

Corollary 5.2.17. If K is finite-dimensional and connected, then the period group Π_P,κ:= im(per_ω

κ,A) is discrete if and only if Π_PK,κ= Π_S1,κ is discrete for the trivial bundle PK overS¹. Moreover, if κ:k×k→V(k) is universal, thenΠω_κ := Π_S1,κ is discrete.

Proof. Since principal bundles overS¹are trivial for connected structure groups (cf. Proposition B.2.8), eachP[α_i] in the preceding theorem is in fact trivial and the first assertion follows. Since inner automorphisms induce the identity on V(k) by its construction,K=K₀ acts trivially on V(k), because it is generated exp(k). Thus κis K-invariant and the second is assertion follows from Proposition 5.2.8.

At first glance it does not seem to be a hard restriction to require K to be connected. But since only trivial bundles overS¹ arise in this way, one needs to consider also bundles with non-connected structure groups in order to obtain interesting generalisations of loop groups, e.g., twisted affine Kac–Moody groups (cf. Section 5.4).

Proposition 5.2.18. If K is finite-dimensional, k∈K, and Pk is the smooth principal K-bundle overS¹from Remark B.2.9, then the period groupΠ_Pk,κ:= im(per_ω

κ,Ak)equals the period groupΠ_S1,κ:= im(per_ω

κ,Ae)of the trivial bundle, whereA_k andA_e are the canonical connection 1-forms. Furthermore, ifκ:k×k→V(k) is universal andK-invariant, thenΠ_Pk,κ is discrete.

Proof. We identify Gau(Pk) with the twisted loop group

C_k^∞(S¹, K) ={f ∈C^∞(R, K) :f(x+ 1) =k⁻¹·f(x)·k for all x∈R},

and consider the evaluation fibration ev_k :C_k^∞(S¹, K)→K,f 7→f(0). Then we have homotopy equivalences

ker(ev_k) ={f ∈C_k^∞(S¹, K) :f(Z) ={e}}

'{f ∈C_k^∞(S¹, K) :f(Z+ [−ε, ε]) ={e}}=:C_k,ε^∞(S¹, K)

∼={f ∈C_e^∞(S¹, K) :f(Z+ [−ε, ε]) ={e}}=:C_e,ε^∞(S¹, K)

'{f ∈C_e^∞(S¹, K) :f(Z) ={e}}= ker(eve) (5.11) for 0< ε < ¹₂. Here the isomorphism ψ:C_k,ε^∞(S¹, K)→C_e,ε^∞(S¹, K) is given by first restricting f ∈C_k,ε^∞(S¹, K) to [0,1] and then extend f|_[0,1] tofb:R→Kby definingfbto be constant of the Z-translates ofx∈[0,1]. This implies in particular

fb

_[0,1]= f|_[0,1] and fb

Z+[−ε,ε]= f|_Z+[−ε,ε]

and thus that fb is smooth. Now these homotopy equivalences induce an isomorphism Ψ :π2(ker(evk))−^∼=→π2(ker(eve))

Now we have that the inclusions ι_k: ker(ev_k),→C_k^∞(S¹, K) induce surjective mapsπ₂(ι_k), becauseπ₂(K) = 0.

We abbreviateωk :=ωκ,A_k andωe:=ωκ,A_e, whereAeis the canonical connection onPeand Ak is the canonical connection on Pk (cf. Lemma B.3.14). We then have the following diagram

π2(ker(evk)) −−−−→^π2(ιk) π2(C_k^∞(S¹, K)) −−−−→^perk z_S1(Y)

λ_id

S1

−−−−→ Y

Ψ



 y

π2(ker(eve)) −−−−→^πe(ιe) π2(C^∞(S¹, K)) −−−−→^pere z_S1(Y)

λ_id

S1

−−−−→ Y,

(5.12)

which we claim to be commutative. Ifβ∈C_∗^∞(S²,(ker(evk))), then we may assume w.l.o.g. that β takes values in C_k,ε^∞(S¹, K), due to the homotopy equivalences (5.11). This implies that the restriction of per_k([β]) to [0,1] coincides with the restriction of per_e(Ψ([β])) to [0,1], because

ψ(f)|_[0,1]=f|_[0,1]. Since the computation of λ_id

S1(per_k([β])) and λ_id

S1(per_e([β])) involves only the values on [0,1], we deduce that (5.12) is commutative.

Thus ΠP_k,κ equals Π_S1,κ, because π2(ιk) and π2(ιe) are surjective. If we choose κ:k×k→V(k) to be universal, then Π_S¹,κ= ΠP_k,κ is discrete by Proposition 5.2.8.

The following corollary we will need later on when discussing Kac–Moody groups. There we will also encounter examples of interesting formsκ, which are k-invariant, but notK-invariant and give an outline of possible generalisations.

Corollary 5.2.19. If K is finite-dimensional, K₀ is compact and k is simple over R and κ:k×k→Ris the Cartan–Killing form, then λ_id

S1(Π_Pk,κ)∼=Z.

Proof. First note that by Remark 5.2.7 the Cartan–Killing form is universal if kis simple and compact. Sinceλ_id¹

S(ΠP_k,κ) =λid_S1(Π_S¹,κ) by Proposition 5.2.18, this follows from the explicit description of Π_S1,κin [MN03, Remark II.10] (where the period group is identified with a subset ofY^r by Φ from (5.8)).

Note that the previous corollary does not generalise to simplek, because then the universal form neednot beK-invariant (cf. Example 5.4.13).

Im Dokument Infinite-Dimensional Lie Theory for Gauge Groups (Seite 81-90)