Appendix: Equivalences between 2-Stacks

Appendix 131 (a) Let F :X ^// Y and G:Y ^// Z be Γ-equivariant anafunctors. If Γ-equivariant structures onF andGcorrespond toΓ₁-actions under the isomorphism of Lemma 3.8.2, then the Γ-equivariant structure on the compositeF◦G corresponds to the Γ₁-action defined above.

(b) The isomorphism of Lemma 3.8.2 identifies the trivial Γ-equivariant structure on the identity anafunctor id : X ^// X with the Γ₁-action R₁ : X₁×Γ₁ ^// X₁ on its total space X.

132 Four Equivalent Versions of Non-Abelian Gerbes Since F is fully faithful on 2-morphisms, we may choose the unique 2-morphism G(ψ) : G(g) ⁺³ G(g⁰) such that F(G(ψ)) = ˜ψ. In order to define the compositor of G we look at 1-morphisms g₁₂ : Y₁ ^// Y₂ and g₂₃ : Y₂ ^// Y₃. We consider the 2-morphism

F(G(g₂₃)◦G(g₁₂))

c⁻¹_G(g

12),G(g23) +3F(G_˜g23)◦F(G_g˜12)

η_g⁻¹_˜

23◦η⁻¹_˜g

12

((ξ_Y3 ◦g₂₃)◦ξ¯_Y2)◦((ξ_Y2 ◦g₁₂)◦ξ¯_Y1)

a,iY2

(ξ_Y3 ◦(g₂₃◦g₁₂))◦ξ¯_{Y1 η}

g23^◦g12

+3F(G(g23◦g12));

its unique preimage under the 2-functor F is the compositor c_g12,g23 :G(g₂₃)◦G(g₁₂) ⁺³G(g₂₃◦g₁₂).

In order to define the unitor of G we consider an object Y ∈ D and look at the 2-morphism

F(G(id_Y))

η⁻¹

idgY +3(ξY ◦idY)◦ξ¯Y

l_ξY,j_Y⁻¹

+3id_F(G(Y)) ^u

−1

G(Y) +3F(id_G(Y)).

Its unique preimage under the 2-functor F is the unitor u_Y :G(id_Y) ⁺³ id_G(Y). The second step is to verify the axioms of a 2-functor. This is simple but extremely tedious and can only be left as an exercise. The third step is to construct the pseudonatural transformation

a : id_D ⁺³ F ◦G.

Its component at an object Y inD is the 1-morphism a(Y) :=ξY : Y ^// F(G(Y)).

Its component at a 1-morphism g :Y₁ ^// Y₂ is the 2-morphism a(g) defined by a(Y₂)◦g ξ_Y2 ◦g

id◦l⁻¹

ξY2◦g

(ξ_Y2 ◦g)◦id

a,i⁻¹_Y

2

((ξ_Y2 ◦g)◦ξ¯_Y1)◦ξ_Y1

η˜g◦id

F(G_˜g)◦ξ_Y1 F(G(g))◦a(Y1).

There are two axioms a pseudonatural transformation has to satisfy, and their proofs are again left as an exercise. It is easy to see that a is a pseudonatural equivalence,

Appendix 133 with an inverse transformation given by ¯a(Y) := ¯ξ_Y. The fourth and last step is to construct the pseudonatural transformation

b:G◦F ⁺³ idC.

Its component at an object X isb(X) :=Gξ¯F(X) :G(F(X)) ^// X. Its component at a 1-morphism f :X₂ ^// X₂ is the 2-morphism

b(f) :b(X2)◦G(F(f)) ⁺³ f◦b(X1) given as the unique preimage under F of the 2-morphism

F(b(X2)◦G(F(f))) ^{c−1 +3}F(b(X2))◦F(G(F(f)))

η⁻¹_¯

ξF(X2)

◦η⁻¹

F(f)

ξ¯_F(X2)◦((ξ_F(X2)◦F(f))◦ξ_F(X1))

a,iF(X2),r

F(f)◦ξ¯_F(X1)

id_F(f)◦η¯ ξF(X1)

F(f)◦F(b(X₁)) _c ⁺³F(f◦b(X₁)).

The proofs of the axioms are again left for the reader, and again it is easy to see that b is a pseudonatural equivalence with an inverse transformation given by ¯b(X) :=

Gξ_F(X).

As a consequence of Lemma 3.8.4 we obtain the certainly well-known

Corollary 3.8.5. Let F : C ^// D be essentially surjective, and an equivalence on all Hom-categories. Then, F is an equivalence of bicategories.

Since we work with 2-stacks over manifolds, we need the following “stacky” ex-tension of Lemma 3.8.4. For a pre-2-stack C, we denote by C_M the 2-category it associates to a smooth manifoldM, and byψ^∗ :CN // CM the 2-functor it associates to a smooth map ψ :M ^// N. The pseudonatural equivalences ψ^∗ ◦ϕ^∗ ∼= (ϕ◦ψ)^∗ will be suppressed from the notation in the following. If C and D are pre-2-stacks, a 1-morphism F :C ^// Dassociates 2-functorsFM :CM // DM to a smooth manifold M, pseudonatural equivalences

F_ψ :ψ^∗◦F_N ^// F_M ◦ψ^∗

to smooth mapsψ :M ^// N, and certain modificationsFψ,ϕthat control the relation between F_ψ and F_ϕ for composable maps ψ and ϕ.

Lemma 3.8.6. Suppose C and D are pre-2-stacks over smooth manifolds, and F : C ^// D is a 1-morphism. Suppose that for every smooth manifold M

134 Four Equivalent Versions of Non-Abelian Gerbes 1. the assumptions of Lemma 3.8.4 for the 2-functor F_M are satisfied and

2. the data (G_Y, ξ_Y) and (G_g, η_g) is chosen for all objects Y and 1-morphisms g in D_M.

Then, there is a 1-morphism G : D ^// C of pre-2-stacks together with 2-isomor-phisms

a:F ◦G ⁺³ idD and b:G◦F ⁺³ idC

such that for every smooth manifold M the 2-functor G_M and the pseudonatural transformations a_M and b_M are the ones of Lemma 3.8.4. In particular, F is an equivalence of pre-2-stacks.

For the proof one constructs the required pseudonatural equivalencesG_ψ and the modifications G_ψ,ϕ from the given ones, F_ψ and F_ψ,ϕ, respectively, in a similar way as explained in the proof of Lemma 3.8.4. Since these constructions are straightfor-ward to do but would consume many pages, and the statement of the lemma is not too surprising and certainly well-known to many people, we decided to leave these constructions for the interested reader.

Chapter 4

A Smooth Model for the String Group

In this chapter we discuss an example of a Lie 2-group that can be used as structure 2-group for our general theory of 2-bundles (resp. non-abelian gerbes). The string-2-group can be used to define string-structures, like the spin-group is used to define spin-structures. These string-structures are needed to cancel certain anomalies in supersymmetric sigma models as mentioned in the introduction. In fact we not only construct a string-2-group model but also a model as an infinite dimensional Lie group. Therefore we have to adapt our general setting (Lie groups, Lie groupoids, bundles, 2-bundles, ...) to the infinite dimensional setting. This is why we repeat some definitions with special emphasis on the infinite dimensional context.

4.1 Recent and new models

String structures and the string group play an important role in algebraic topology [Hen08b, Lur09a, BN09], string theory [Kil87, FM06] and geometry [Wit88, Sto96].

The group

String

is defined to be a 3-connected cover of the spin group or, more generally of any simple simply connected compact Lie groupG[ST04]. This definition fixes only its homotopy type and makes abstract homotopy theoretic constructions possible. But for geometric applications these models are not very well suited, one is rather interested in concrete models that carry, for instance, topological or even Lie-group structures.

There is a direct cohomological argument showing that

String

_G cannot be a finite CW-complex or a finite-dimensional manifold (see Corollary 4.3.3), so the best thing one can hope for is a topological group or an infinite-dimensional Lie group. There have been various constructions of models of

String

_G ^{as A∞}-spaces or topological groups, but the question whether an infinite-dimensional Lie group model is also possible has been open so far. One of the main contributions of the present chapter is to give an affirmative answer to this question and provide an explicit Lie group

135

136 A Smooth Model for the String Group model, based on a topological construction of Stolz [Sto96].

Something that is not directly apparent from the setting of the problem is that string group models as Lie 2-groups are something more natural to expect when taking the perspective of string theory or higher homotopy theory into account.

However, the notion of a Lie 2-group model deserves a thorough clarification itself.

We discuss this notion carefully by establishing the relevant homotopy theoretic facts about infinite-dimensional Lie 2-groups and promote our Lie group model

String

_G

to such a Lie 2-group model STRINGG.

Before we outline our construction let us briefly summarize the existing ones. One model for

String

_G is the pullback of the path fibrationP K(Z,3) ^// K(Z,3) along a characteristic map u: G ^// K(Z,3). This is a standard construction of the White-head tower and leads to a model of

String

_G as a space. Since this construction also works for a characteristic map BG ^// K(Z,4), each 3-connected cover is homotopy equivalent to a loop space and thus admits anA_∞-structure. Taking a functorial con-struction of the Whitehead tower one even obtains a model as a topological group.

Unfortunately, these models are not very tractable.

There are more geometric constructions of

String

_G, for instance the one by Stolz in [Sto96]. The model given there has as an input the basic principal P U(H)-bundle P overG, whereHis a separable Hilbert space. Stolz then defines a model for

String

_G

as a topological group together with a homomorphism

String

_G ^{// G}whose kernel is the group of continuous gauge transformations of the bundle P. Our constructions will be based on this idea. In [ST04] Stolz and Teichner construct a model for

String

_G as an extension of G by P U(H). It is a natural idea to equip this model with a smooth structure. But this does not work since this extension is constructed as a pushout along a positive energy representation of the loop group of G which is not smooth.

We now come to Lie 2-group models. One construction has been given by Hen-riques [Hen08a], based on work of Getzler [Get09]. Its basic idea is to apply a general integration procedure forL∞-algebras to the string Lie 2-algebra. To make this con-struction work one has to weaken the naive notion of a Lie 2-group and besides that work in the category of Banach spaces. Similarly, the model of Schommer-Pries [SP10] realizes

String

_G as a stacky Lie 2-group, but it has the advantage of being finite-dimensional. This model is constructed from a cocycle in Segal’s Cohomology for G [Seg70].

A common thing about the above Lie 2-group models is that they are not strict, i.e., not associative on the nose but only up to an additional coherence. This com-plication is not present in the strict 2-group model of Baez, Crans, Schreiber and Stevenson from [BCSS07]. It is constructed from a crossed moduledΩG ^// P_eG, built out of the level one Kac-Moody central extension dΩG of the loop group of G and its path space P_eG. The price to pay is that the model is infinite dimensional, but the strictness makes the corresponding bundle theory more tractable (as explained

Recent and new models 137 in chapter 3.

Summarizing, quite some effort has been made in constructing models for

String

_G

that are as close as possible to finite-dimensional Lie groups. However, one of the most natural questions, namely whether there exists an infinite-dimensionalLie group model for

String

_G is still open. We answer this question by the following result.

Let P ^// G be a basic smooth principal P U(H)-bundle. Basic here means that [P] ∈ [G, BP U(H)] ∼= H³(G,Z) = Z is a generator. In Section 4.2 we review the fact that

Gau

^(P) is a Lie group modeled on the infinite-dimensional space of vertical vector fields on P. The main result of Section 4.3 is then

Theorem (Theorem 4.3.6). LetG be a simple, simply connected and compact Lie group, then there exists a smooth string group model

String

_G ^turning

Gau

^{(P) //}

String

_G ^{// G}

into an extension of Lie groups. It is uniquely determined up to isomorphism by this property.

From now on

String

_G will always refer to this particular model. The proof of the theorem is based on [Sto96] and [Woc08]. We also show that

String

_G is metrizable and Fr´echet. This metrizability makes the homotopy theory that we use in the sequel work due to results of Palais [Pal66].

In Section 4.4 we introduce the concept of Lie 2-group models culminating in Definition 4.4.10. An important construction in this context is the geometric real-ization that produces topological groups from Lie 2-groups. We show that geometric realization is well-behaved under mild technical conditions, such as metrizability.

In Section 4.5 we then construct a central extension U(1) ^//

Gau

d^{(P) //}

Gau

^(P)

with contractible

Gau

d^(P). We define an action of

String

_G ^on

Gau

^d(P) such that

Gau

d^{(P) //}

String

_G is a smooth crossed module. Crossed modules are a source for Lie 2-groups (Example 4.4.3) and in that way we obtain a Lie 2-group STRING_G. Theorem(Theorem 4.5.6). STRINGGis a Lie 2-group model in the sense of Definition 4.4.10.

The proof of this theorem relies on a comparison of the model

String

_G ^{with the}

geometric realization of STRING_G. Moreover, this direct comparison allows to derive a comparison between the corresponding bundle theories and string structures, see Section 4.6. This explicit comparison is a distinct feature of our 2-group model that is not available for the other 2-group models.

In an appendix we have collected some elementary facts about infinite dimensional manifolds and Lie groups. A second appendix gives a useful characterization of smooth weak equivalences between Lie 2-groups.

138 A Smooth Model for the String Group

Im Dokument Higher Categorical Structures in Geometry - General Theory and Applications to Quantum Field Theory (Seite 153-160)