Representations of 2-Groups on Higher Hilbert Spaces Derek Wise - PowerPoint PPT Presentation

Representations of 2-Groups on Higher Hilbert Spaces Derek Wise Based on joint work with John Baez, Aristide Baratin, and Laurent Freidel Categorical Groups Workshop, Barcelona, June 2008

Group Representations and 2-group Representations Representation theory of a group G in a category C : • Representations are just functors ρ : G → C . • ‘ Intertwiners ’ between reps are natural transformations. Representation theory of a 2-group G in a 2-category C : • Representations are (strict) 2-functors ρ : G → C . • ‘ Intertwiners ’ are pseudonatural transformations. • ‘ 2-intertwiners ’ are modifications. Problem: What’s a good target 2-category? Kapranov–Voevodsky 2-vector spaces are “too discrete” for interesting reps of Lie 2-groups; e.g. any automorphism of Vect N is essentially described by some per- mutation of the basis 2-vectors e i = (0 , . . . , , . . . , 0) i = 1 . . . N. ❈ ���� i th place but Lie groups have too few interesting actions on finite sets!

Higher Hilbert Spaces

Toward infinite dimensional 2-Hilbert spaces Lie groups have important representations on Hilbert spaces, espe- cially L 2 spaces; we expect infinite-dimensional 2-Hilbert spaces, es- pecially “2 L 2 spaces” to be important for Lie 2-group representations. ordinary categorified Lebesgue theory Lebesgue theory Hilb ❈ + ⊕ × ⊗ 0 { 0 } 1 ❈ measurable functions ‘measurable fields of Hilbert spaces’ ‘measurable fields of operators’ � ⊕ � (‘direct integral’) So far there’s no definition of infinite dimensional 2-Hilbert spaces! A step in the right direction: “Measurable categories” [Described in work of Crane and Yetter]

First step toward L 2 ( X ): the set of measurable ❈ -valued functions on X . A possible first step toward “2 L 2 ( X )”: the category of “measurable Hilb-valued fields on X ” More precisely, this category has: • “measurable fields of Hilbert spaces” as objects • “measurable fields of linear operators” as morphisms This is the approach used by Yetter to define “measurable categories”. (The measurable fields themselves are classical analysis; they’re im- portant, e.g. for Von Neumann algebras) . . .

Measurable Fields (classical definitions) Definition 1 Let X be a measurable space. A measurable field of Hilbert spaces H on X is an assignment of a Hilbert space H x to each x ∈ X , together with a subspace M H ⊆ � x H x called the measurable sections of H , satisfying the properties: • ∀ ξ ∈ M H , the function x �→ | | ξ x | | H x is measurable. • For any η ∈ � x H x such that x �→ � η x , ξ x � H x is measurable for all ξ ∈ M H , we have η ∈ M H . • There is a sequence ξ i ∈ M H such that { ( ξ i ) x } ∞ i =1 is dense in H x for all x ∈ X . As with ordinary measurable functions, it is often useful to identify measurable fields that differ only on a set of measure zero: We say H and K are µ -equivalent if H x = K x µ - a.e. in x .

Similarly, we have: Definition 2 Let H and K be measurable fields of Hilbert spaces on X . A measurable field of bounded linear operators φ : H → K on X is an X -indexed family of bounded operators φ x : H x → K x such that ξ ∈ M H implies φ ( ξ ) ∈ M K , where φ ( ξ ) x := φ x ( ξ x ) . If φ : H → K and we consider the µ -equivalence class of H , and the ν -equivalence class of K , we can consider the ‘ √ µν -equivalence’ class of φ . Here √ µν is a measure called the geometric mean of the two measures. The important point is that the equivalence relation ‘ √ µν - a.e. ’ is the transitive closure of ‘ µ - a.e. OR ν - a.e. ’ There is also a notion of measurable field of measures assigning to each y ∈ Y a measure on X .

Higher Hilbert Spaces Given a locally compact Hausdorff Borel-measurable space X , denote by H X the category whose: • objects are measurable fields of Hilbert spaces on X • morphisms are and bounded measurable fields of linear operators on X . Morally, H X = Hilb X with some measurability restrictions. ( ∞ -dim. analogue of the Kapranov–Voevodsky 2-vector space Vect N .) Informally, we call H X a higher Hilbert space . More generally: as shown by Yetter, H X is a C ∗ -category; any C ∗ - category that is C ∗ -equivalent to some H X is a ‘higher Hilbert space’. As with 2-vector spaces, morphisms will be certain functors and nat- ural transformations whose composition laws can be given by “matrix multiplication”, but here direct sums replaced by direct integrals . . .

Direct Integrals of Hilbert spaces Given: • X — a measurable space • H — measurable field of Hilbert spaces on X • µ — a measure on X A measurable section x �→ ψ x ∈ H x is an L 2 section if � | 2 d µ ( x ) | | ψ x | x < ∞ . X The direct integral of H : � ⊕ d µ ( x ) H x X is the Hilbert space of all L 2 sections of H , with inner product � � ψ, ψ ′ � = d µ ( x ) � ψ x , ψ ′ x � x . X � ⊕ � ⊕ X d µ K ∼ If K x = H x µ - a.e. , then X d µ H in a canonical way, so = � ⊕ respects µ - a.e. classes of fields.

Direct Integrals of Operators Suppose φ : H → H ′ is a µ -essentially bounded measurable field of linear operators on X . The direct integral of φ is the linear oper- ator acting pointwise on sections: � ⊕ � ⊕ � ⊕ d µ ( x ) H ′ d µ ( x ) φ x : d µ ( x ) H x → x X X X � ⊕ � ⊕ X d µ ( x ) ψ x �→ X d µ ( x ) φ x ( ψ x ) � ⊕ X d µ ( x ) ψ x is just a cute notation for an L 2 -section. where

Morphisms Between Higher Hilbert Spaces A morphism T,t � H Y H X is: • a Y -indexed measurable family t y of measures on X ; • a t -class of measurable fields of Hilbert spaces T on Y × X , such that t is concentrated on the support of T ; that is, for each y ∈ Y , t y ( { x ∈ X : T y,x = 0 } ) = 0 . This gives a functor by H ∈ H X �→ T H ∈ H Y with � ⊕ ( T H ) y = d t y T y,x ⊗ H x , X and ( φ : H → H ′ ) �→ ( Tφ : T H → T H ′ ) with � ⊕ ( Tφ ) y = d t y ✶ T y,x ⊗ φ x X (More generally, a morphism is any functor in C ∗ Cat of the form K ∼ → H X ∼ → H X T → K ′ , w. T naturally iso. to one as above.)

Composition To compose morphisms: T,t � H Y U,u � H Z UT,ut � H Z H X H X = we first define the Z -indexed family of measures on X by � ( ut ) z = d u z ( y ) t y . Y The composite of the fields T and U is then given by a direct integral: � ⊕ ( UT ) z,x = d k z,x ( y ) U z,y ⊗ T y,z , Y where k z,x is defined by � � d( ut ) z ( x ) ( k z,x ⊗ δ x ) = d u z ( y ) ( δ y ⊗ t y ) , X Y Why this measure “ k z,x ”?

Geometry of Matrix Multiplication

Similarly. . .

� � � � � � � � � � 2-Morphisms √ T,t tt ′ -class of bounded A 2-morphism is a H X � H Y α T ′ ,t ′ → T ′ measurable fields of linear operators α y,x : T y,x − y,x on Y × X . Composition: T,t � � � α d t ′′ d t ′ d t y T ′ ,t ′ � α ′ · α � y y α ′ y,x = y,x α y,x H X � H Y d t ′′ d t ′ d t y y y α ′ � �� � = 1 if t, t ′ , t ′′ are T ′′ ,t ′′ equivalent T,t U,u H X � H Y � H Z α β T ′ ,t ′ U ′ ,u ′ � �� � � ⊕ d( u ′ t ′ ) z d u z d t y � d k ′ ( β ◦ α ) z,x ( ψ z,x ) = β z,y ⊗ α y,x ( ψ z,y,x ) z,x d u ′ d t ′ d( ut ) z Y z y

Meas The 2-category whose objects are ‘higher Hilbert spaces’, and whose morphisms and 2-morphisms are as just described, is denoted Meas . We’ll now study the representation theory of 2-groups in this 2-category . . .

Representation Theory

� � � � � � � � � � � � 2-Groups and Crossed Modules I’ll consider only strict 2-groups, and often identify a 2-group G with its crossed module ( G, H, ✄ , ∂ ). Conventions: • ∂ : H → G a group homomorphism • action ✄ of G on H ∂ ( h ) ✄ h ′ = hh ′ h − 1 , ∂ ( g ✄ h ) = g∂ ( h ) g − 1 • axioms: ∀ g ∈ G, h, h ′ ∈ H . Composition convention: g 1 g 2 g 2 g 1 = ⋆ � ⋆ � ⋆ ⋆ � ⋆ g g h = ⋆ � ⋆ ⋆ � ⋆ ∂ ( h ) g h ′ h h ′ ∂ ( h ′ h ) g ∂ ( h ′ ) ∂ ( h ) g g 2 g 1 g 1 g 2 = ⋆ � ⋆ � ⋆ ⋆ � ⋆ h 2 ( g 2 ✄ h 1 ) h 1 h 2 g ′ g ′ g ′ 2 g ′ 1 2 1

Representations in Meas Definition 3 If G is a 2-group, a representation of G in Meas is a strict 2-functor ρ : G → Meas . If ρ ( ⋆ ) = H X , we say ρ is a representation on H X . Theorem 1 Let G = ( G, H, ∂, ✄ ) be a 2-group. A representation of G on H X is specified by a measurable right G -action ✁ on X , together with a measurable field χ of group homomorphisms χ ( x ) : χ ( x ): H → ❈ ∗ satisfying: (i) x ✁ ∂ ( h ) = x for all h ∈ H, x ∈ X (ii) χ ( x )[ g ✄ h ] = χ ( x ✁ g )[ h ] for all g ∈ G , h ∈ H , x ∈ X . We call a representation unitary when χ ( x )[ H ] ⊆ U (1).