The Fundamental Gerbe of a Compact Lie Group
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1 The Fundamental Gerbe of a Compact Lie Group Christoph Schweigert Department of Mathematics, University of Hamburg and Center for Mathematical Physics, Hamburg Joint work with Thomas Nikolaus Sophus Lie Seminar, Göttingen, July 2009 Transparencies available at Christoph Schweigert, Fundamental Gerbe p.1/15
2 Hermitian line bundles 1 Setting: M a smooth manifold with open cover (U α α I. Hermitian line bundle, locally U α (C,, s α U α Christoph Schweigert, Fundamental Gerbe p.2/15
3 Hermitian line bundles 1 Setting: M a smooth manifold with open cover (U α α I. Hermitian line bundle, locally U α (C,, s α U α 2 Gluing: transition functions g αβ : U α U β U(1 Identify for x U α U β (x, α, z (x, β, g αβ (x z Total space of bundle L := α I U α C/ M π Christoph Schweigert, Fundamental Gerbe p.2/15
4 Hermitian line bundles 1 Setting: M a smooth manifold with open cover (U α α I. Hermitian line bundle, locally U α (C,, s α U α 2 Gluing: transition functions g αβ : U α U β U(1 Identify for x U α U β (x, α, z (x, β, g αβ (x z Total space of bundle L := α I U α C/ M π 3 Connection: Local section s α : U α U α C Ds α := ds α + A α s α with A α Ω 1 (U α Condition: A α A β = dlog g αβ on U α U β Curvature: da α = F Uα with F Ω 2 (M Christoph Schweigert, Fundamental Gerbe p.2/15
5 Hermitian line bundles 1 Setting: M a smooth manifold with open cover (U α α I. Hermitian line bundle, locally U α (C,, s α U α 2 Gluing: transition functions g αβ : U α U β U(1 Identify for x U α U β (x, α, z (x, β, g αβ (x z Total space of bundle L := α I U α C/ M π 3 Connection: Local section s α : U α U α C Ds α := ds α + A α s α with A α Ω 1 (U α Condition: A α A β = dlog g αβ on U α U β Curvature: da α = F Uα with F Ω 2 (M 4 Holonomy for a smooth map Φ : S 1 M hol L (Φ = b E exp( b Φ A α(b v b g±1 α(vα(b (Φ(v Christoph Schweigert, Fundamental Gerbe p.2/15
6 More on hermitian line bundles with connection 1 Remarks Bundles on M form a category Bun (M Subcategory Buntriv (M Objects L ω with ω Ω 1 (M Morphism ω 1 ω 2 is η C (M, U(1 s.t. dlog η = ω 2 ω 1 Pullbacks: f L := M f π L M Functor f : Bun (M Bun (M f L M π Christoph Schweigert, Fundamental Gerbe p.3/15
7 More on hermitian line bundles with connection 1 Remarks Bundles on M form a category Bun (M Subcategory Buntriv (M Objects L ω with ω Ω 1 (M Morphism ω 1 ω 2 is η C (M, U(1 s.t. dlog η = ω 2 ω 1 Pullbacks: f L := M f π L M Functor f : Bun (M Bun (M f L M π 2 Holonomy revisited: L S 1 Φ M π Φ L is trivial Choose trivialization T : Φ L L ω with ω Ω 1 (S 1 Holonomy: hol L (Φ := exp( S 1 ω Well-defined Christoph Schweigert, Fundamental Gerbe p.3/15
8 More on hermitian line bundles with connection 1 Remarks Bundles on M form a category Bun (M Subcategory Buntriv (M Objects L ω with ω Ω 1 (M Morphism ω 1 ω 2 is η C (M, U(1 s.t. dlog η = ω 2 ω 1 Pullbacks: f L := M f π L M Functor f : Bun (M Bun (M f L M π 2 Holonomy revisited: L S 1 Φ M π Φ L is trivial Choose trivialization T : Φ L L ω with ω Ω 1 (S 1 Holonomy: hol L (Φ := exp( S 1 ω Well-defined 3 Pullbacks 2-functor Man opp Cat Isomorphism of line bundles f (g L (g f L Presheaf in categories Sheaf for Grothendieck topology given by Surjective submersions 4 Rationale: Existence of local sections Y π s U M U Note: π s U = id U dπ ds U = id Christoph Schweigert, Fundamental Gerbe p.3/15
9 Descent of bundles for a covering 1 Pullback gives for a line bundle L M and a covering Y := α I U α M Y [3] Y [2] := Y M Y 1 Y f M π f L L Christoph Schweigert, Fundamental Gerbe p.4/15
10 Descent of bundles for a covering 1 Pullback gives for a line bundle L M and a covering Y := α I U α M Y [3] Y [2] := Y M Y 1 Y f M π f L L 2 Structure Data Simplicial manifold Object f L =: L on Y Compatibility condition on Y [3] : Equality f = f 1 Morphism on Y [2] φ : 1 L = 1 f L 0 f L = 0 L 1 1 L = 2 1 L 2 φ 2 0 L = 0 1 L 0 φ 0 0 L = 1 0 L 1 φ Christoph Schweigert, Fundamental Gerbe p.4/15
11 Descent of bundles for a covering 1 Pullback gives for a line bundle L M and a covering Y := α I U α M Y [3] Y [2] := Y M Y 1 Y f M π f L L 2 Structure Data Simplicial manifold Object f L =: L on Y Compatibility condition on Y [3] : Equality f = f 1 Morphism on Y [2] φ : 1 L = 1 f L 0 f L = 0 L 1 1 L = 2 1 L 2 φ 2 0 L = 0 1 L 0 φ 0 0 L = 1 0 L 1 φ 3 Category of descent data Desc(Y f M functor ι f : Bun(M Desc(Y f M Equivalence of categories for Bun (M Not equivalence of categories for Buntriv (M Christoph Schweigert, Fundamental Gerbe p.4/15
12 Bundle gerbes with connection 1 Category Grbtriv (M of trivial gerbes: Surface holonomy = Wess-Zumino term for σ-model Objects I ω with ω Ω 2 (M 1-Morphism ω η ω is η Ω 1 (M such that dη = ω ω 2-Morphism ω η η ω are φ C (M, U(1 such that 1 i dlog φ = η η Christoph Schweigert, Fundamental Gerbe p.5/15
13 Bundle gerbes with connection 1 Category Grbtriv (M of trivial gerbes: Surface holonomy = Wess-Zumino term for σ-model Objects I ω with ω Ω 2 (M 1-Morphism ω η ω is η Ω 1 (M such that dη = ω ω 2-Morphism ω η η ω are φ C (M, U(1 such that 1 i dlog φ = η η 2 Complete category Hom(I ω, I ω under descent: Hom(I ω, I ω := Bun ω ω (M with curv(l = ω ω Christoph Schweigert, Fundamental Gerbe p.5/15
14 Bundle gerbes with connection 1 Category Grbtriv (M of trivial gerbes: Surface holonomy = Wess-Zumino term for σ-model Objects I ω with ω Ω 2 (M 1-Morphism ω η ω is η Ω 1 (M such that dη = ω ω η 2-Morphism ω η ω are φ C (M, U(1 such that 1 i dlog φ = η η 2 Complete category Hom(I ω, I ω under descent: Hom(I ω, I ω := Bun ω ω (M with curv(l = ω ω 3 Complete the 2-category under descent: Y [4] Y [3] Y [2] := Y M Y Y f 1 M on Y : a trivial gerbe I ω = local two-form = B-field on Y [2] : a 1-morphism = line bundle L with connection on Y [2] curv(l = 1 ω 0 ω on Y [3] : a 2-morphism = morphism of line bundles μ : 2 L 0 L 1 L on Y [4] : compatibility condition: associativity of μ Christoph Schweigert, Fundamental Gerbe p.5/15
15 Bundle gerbes with connection 1 Category Grbtriv (M of trivial gerbes: Surface holonomy = Wess-Zumino term for σ-model Objects I ω with ω Ω 2 (M 1-Morphism ω η ω is η Ω 1 (M such that dη = ω ω η 2-Morphism ω η ω are φ C (M, U(1 such that 1 i dlog φ = η η 2 Complete category Hom(I ω, I ω under descent: Hom(I ω, I ω := Bun ω ω (M with curv(l = ω ω 3 Complete the 2-category under descent: Y [4] Y [3] Y [2] := Y M Y Y f 1 M on Y : a trivial gerbe I ω = local two-form = B-field on Y [2] : a 1-morphism = line bundle L with connection on Y [2] curv(l = 1 ω 0 ω on Y [3] : a 2-morphism = morphism of line bundles μ : 2 L 0 L 1 L on Y [4] : compatibility condition: associativity of μ = Topology classified by Dixmier-Douady class in H 2 (M, U(1 = H 3 (M, Z 3-form curvature H Ω 3 (M with dω = π H Christoph Schweigert, Fundamental Gerbe p.5/15
16 Surface holonomy 1 Definition Σ closed oriented surface, smooth map Φ : Σ M Choose trivialization and define holonomy T : Φ G I ω hol G (Φ := exp(i Σ ω Christoph Schweigert, Fundamental Gerbe p.6/15
17 Surface holonomy 1 Definition Σ closed oriented surface, smooth map Φ : Σ M Choose trivialization and define holonomy T : Φ G I ω hol G (Φ := exp(i Σ ω 2 Comments: (i Well-defined: other trivialization: Then T : Φ G I ω T T : I ω Iω is in Hom(I ω, I ω = Bun ω ω(m. Thus Σ (ω ω 2πZ Christoph Schweigert, Fundamental Gerbe p.6/15
18 Surface holonomy 1 Definition Σ closed oriented surface, smooth map Φ : Σ M Choose trivialization and define holonomy T : Φ G I ω hol G (Φ := exp(i Σ ω 2 Comments: (i Well-defined: other trivialization: Then T : Φ G I ω T T : I ω Iω is in Hom(I ω, I ω = Bun ω ω(m. Thus Σ (ω ω 2πZ (ii Aharonov-Bohm effect: Holonomy is observable in quantum theory Example: H 2 (Spin(4n/Z 2 Z 2, U(1 = Hom(Z 2, U(1 = Z 2 Two gerbes with same curvature 3-form on Spin(4n/Z 2 Z 2 Christoph Schweigert, Fundamental Gerbe p.6/15
19 1 Lie groupoids Equivariance: from Lie groupoids to simplicial manifolds A Lie groupoid is a groupoid internal in Man s, t : Γ 1 Γ 0 ι : Γ 0 Γ 1, composition Γ 1 Γ0 Γ 1 Γ 1, inverse Γ 1 Γ 1 Christoph Schweigert, Fundamental Gerbe p.7/15
20 1 Lie groupoids Equivariance: from Lie groupoids to simplicial manifolds A Lie groupoid is a groupoid internal in Man s, t : Γ 1 Γ 0 ι : Γ 0 Γ 1, composition Γ 1 Γ0 Γ 1 Γ 1, inverse Γ 1 Γ 1 2 Examples: 1 Lie groupoid BG for any Lie group G: G Composition = Multiplication 2 Action groupoid M//G: G M s t M s t pt pt ι G M G M with s(g, m = m and t(g, m = g.m. 3 Čech groupoid Č(Y for a cover Y = i IU α M: Y M Y Composition (y 1, y 2 (y 2, y 3 = (y 1, y 3 s t Y Christoph Schweigert, Fundamental Gerbe p.7/15
21 1 Lie groupoids Equivariance: from Lie groupoids to simplicial manifolds A Lie groupoid is a groupoid internal in Man s, t : Γ 1 Γ 0 ι : Γ 0 Γ 1, composition Γ 1 Γ0 Γ 1 Γ 1, inverse Γ 1 Γ 1 2 Examples: 1 Lie groupoid BG for any Lie group G: G Composition = Multiplication 2 Action groupoid M//G: G M s t M s t pt pt ι G M G M with s(g, m = m and t(g, m = g.m. 3 Čech groupoid Č(Y for a cover Y = i IU α M: Y M Y Composition (y 1, y 2 (y 2, y 3 = (y 1, y 3 s t Y 3 From Lie groupoids to simplicial manifolds: 1 Nerve of the Lie groupoid Simplicial manifold 3 Γ 2 2 Γ 1 Γ 0 with Γ n := Γ 1 Γ0 Γ 1 Γ0... Γ0 Γ 1 1 }{{} n 2 Lie functor Π : (Λ 1 Λ 0 (Γ 1 Γ 0 gives simplicial map Λ Γ Christoph Schweigert, Fundamental Gerbe p.7/15
22 Equivariant objects on simplicial manifolds 1 Equivariant objects X a presheaf in bicategories, Γ a simplical manifold A Γ -equivariant object of X consists of (O1 An object G of X(Γ 0 (O2 A 1-isomorphism P : 0 G 1 G in X( Γ 1 (O3 A 2-isomorphism μ: 2 P 0 P 1 P in X( Γ 2 (O4 A coherence condition on 2-morphisms in X ( Γ 3 2 μ (id 0 μ = 1 μ ( 3 μ id Christoph Schweigert, Fundamental Gerbe p.8/15
23 Equivariant objects on simplicial manifolds 1 Equivariant objects X a presheaf in bicategories, Γ a simplical manifold A Γ -equivariant object of X consists of (O1 An object G of X(Γ 0 (O2 A 1-isomorphism P : 0 G 1 G in X( Γ 1 (O3 A 2-isomorphism μ: 2 P 0 P 1 P in X( Γ 2 (O4 A coherence condition on 2-morphisms in X ( Γ 3 2 μ (id 0 μ = 1 μ ( 3 μ id 2 Remark: Bicategory X(Γ = holim i Δ X(Γ i is a homotopy limit In particular: Desc X (Y M = holim i Δ X(Y [i+1] Christoph Schweigert, Fundamental Gerbe p.8/15
24 Equivariant objects on simplicial manifolds 1 Equivariant objects X a presheaf in bicategories, Γ a simplical manifold A Γ -equivariant object of X consists of (O1 An object G of X(Γ 0 (O2 A 1-isomorphism P : 0 G 1 G in X( Γ 1 (O3 A 2-isomorphism μ: 2 P 0 P 1 P in X( Γ 2 (O4 A coherence condition on 2-morphisms in X ( Γ 3 2 μ (id 0 μ = 1 μ ( 3 μ id 2 Remark: Bicategory X(Γ = holim i Δ X(Γ i is a homotopy limit In particular: Desc X (Y M = holim i Δ X(Y [i+1] 3 Simplicial coverings: (i A simplicial map Π : Λ Γ is called a covering, if all maps Π i : Λ i Γ i are coverings in the Grothendieck topology (ii A Lie functor Π : (Λ 1 Λ 0 (Γ 1 Γ 0 is called a covering, if the simplicial map Π : Λ Γ is a covering of the nerves Example: Lie group G acting on a manifold M with G-invariant cover (U α Π : α I U α // G M// G Christoph Schweigert, Fundamental Gerbe p.8/15
25 Equivariant descent 1 Simplicial objects in simplicial manifolds A covering Π : Λ Γ of simplicial manifolds gives the simplicial manifold Λ Γ Λ := ( 3 Λ 2 Γ2 Λ 2 2 Λ 1 Γ1 Λ 1 1 Λ 0 Γ0 Λ 0 Simplicial maps δ 0, δ 1 : Λ [2] Λ etc give augmented simplicial object in simpl. mfds. ( Λ [ ] := ( δ 0 Λ [3] δ 3 δ 0 Λ [2] δ 2 δ 0 δ 1 Λ Γ Christoph Schweigert, Fundamental Gerbe p.9/15
26 Equivariant descent 1 Simplicial objects in simplicial manifolds A covering Π : Λ Γ of simplicial manifolds gives the simplicial manifold Λ Γ Λ := ( 3 Λ 2 Γ2 Λ 2 2 Λ 1 Γ1 Λ 1 1 Λ 0 Γ0 Λ 0 Simplicial maps δ 0, δ 1 : Λ [2] Λ etc give augmented simplicial object in simpl. mfds. ( Λ [ ] := ( δ 0 Λ [3] δ 3 δ 0 Λ [2] δ 2 δ 0 δ 1 Λ Γ 2 In detail Horizontally: nerves of Čech groupoids Λ [3] 2 Λ [2] 2 Λ 2 Γ 2 Λ [3] 1 Λ [2] 1 Λ 1 Γ 1 Λ [3] 0 Λ [2] 0 Λ 0 Γ 0 Christoph Schweigert, Fundamental Gerbe p.9/15
27 Equivariant descent 1 Simplicial objects in simplicial manifolds A covering Π : Λ Γ of simplicial manifolds gives the simplicial manifold Λ Γ Λ := ( 3 Λ 2 Γ2 Λ 2 2 Λ 1 Γ1 Λ 1 1 Λ 0 Γ0 Λ 0 Simplicial maps δ 0, δ 1 : Λ [2] Λ etc give augmented simplicial object in simpl. mfds. ( Λ [ ] := ( δ 0 Λ [3] δ 3 δ 0 Λ [2] δ 2 δ 0 δ 1 Λ Γ 2 Main theorem about equivariant descent X a 2-stack and Π : Λ Γ a covering of simplicial manifolds. Then there is an equivalence of bicategories ( X (Γ holim X (Λ δ 0 ( X Λ [2] δ 0 ( X Λ [3] δ 0. δ 1 δ 2 δ 3 Christoph Schweigert, Fundamental Gerbe p.9/15
28 Equivariant gerbes on compact Lie groups Known facts: G a compact connected Lie group The Dixmier-Douady class gives a bijection [Brylinski 2000] dd : π 0 (Grb(M//G HG 3 (M, Z G a compact connected Lie group with adjoint action HG 3 (G, Z = Z. A fundamental gerbe is an element of the isomorphism class of the generator. Neglect connection data ( pseudo connection: Grb (M//G Grb(M//G is not surjection Christoph Schweigert, Fundamental Gerbe p.10/15
29 Equivariant gerbes on compact Lie groups Known facts: G a compact connected Lie group The Dixmier-Douady class gives a bijection [Brylinski 2000] dd : π 0 (Grb(M//G HG 3 (M, Z G a compact connected Lie group with adjoint action HG 3 (G, Z = Z. A fundamental gerbe is an element of the isomorphism class of the generator. Neglect connection data ( pseudo connection: Grb (M//G Grb(M//G is not surjection Plan: Equivariant gerbes on a point Equivariant gerbes on equivariantly contractible spaces Equivariant gerbes around G-orbits Equivariant gerbes on compact Lie groups with respect to the adjoint action Christoph Schweigert, Fundamental Gerbe p.10/15
30 Equivariant gerbes on a point Gerbes on pt//g Simplicial manifold G G 2 3 G 1 pt with, 2 : G G G projections on components and 1 : G G G multiplication Christoph Schweigert, Fundamental Gerbe p.11/15
31 Gerbes on pt//g Equivariant gerbes on a point Simplicial manifold 3 G G 2 G 1 pt with, 2 : G G G projections on components and 1 : G G G multiplication Gerbe on BG (O1 A (trivial gerbe over the point (O2 An endomorphism of the trivial gerbe on G, i.e. a bundle π : P G (O3 A morphism μ : 0 P 2 P 1 P over G G Since 0 P 2 P = P U(1 P and since pullback 1 P is a fibre product P U(1 P μ P with μ respecting the U(1-action π π π G G (O4 A diagram expressing associativity G Christoph Schweigert, Fundamental Gerbe p.11/15
32 Gerbes on pt//g Equivariant gerbes on a point Simplicial manifold 3 G G 2 G 1 pt with, 2 : G G G projections on components and 1 : G G G multiplication Gerbe on BG (O1 A (trivial gerbe over the point (O2 An endomorphism of the trivial gerbe on G, i.e. a bundle π : P G (O3 A morphism μ : 0 P 2 P 1 P over G G Since 0 P 2 P = P U(1 P and since pullback 1 P is a fibre product P U(1 P μ P with μ respecting the U(1-action π π π G G (O4 A diagram expressing associativity Proposition G A bundle gerbe on BG is a central extension U(1 P G 1-morphisms=morphisms of central extensions, 2-automorphisms = elements ξ U(1 Christoph Schweigert, Fundamental Gerbe p.11/15
33 Equivariant gerbes on contractible manifolds and orbits Equivariant Gerbes on equivariantly contractible manifolds G a compact connected Lie group, M a G-equivariantly contractible space. The 2-category Grb(M//G is given up to equivalence by (O1 Objects are U(1-central extensions of G (O2 1-Morphisms are morphisms of central extensions (O3 2-Automorphisms are smooth maps M U(1. Christoph Schweigert, Fundamental Gerbe p.12/15
34 Equivariant gerbes on contractible manifolds and orbits Equivariant Gerbes on equivariantly contractible manifolds G a compact connected Lie group, M a G-equivariantly contractible space. The 2-category Grb(M//G is given up to equivalence by (O1 Objects are U(1-central extensions of G (O2 1-Morphisms are morphisms of central extensions (O3 2-Automorphisms are smooth maps M U(1. Reformulation M, G as above and G the universal cover of G. Then Grb(M//G is given by (O1 Objects G ρ are group homomorphisms ρ : π 1 (G U(1 s.t. P = G ρ U(1 (O2 1-Morphisms G ρ G μ are morphisms G U(1 whose restriction to π 1 (G G equals ρ μ 1 (O3 2-Automorphisms are smooth maps M U(1 Christoph Schweigert, Fundamental Gerbe p.12/15
35 Equivariant gerbes on contractible manifolds and orbits Equivariant Gerbes on equivariantly contractible manifolds G a compact connected Lie group, M a G-equivariantly contractible space. The 2-category Grb(M//G is given up to equivalence by (O1 Objects are U(1-central extensions of G (O2 1-Morphisms are morphisms of central extensions (O3 2-Automorphisms are smooth maps M U(1. Reformulation M, G as above and G the universal cover of G. Then Grb(M//G is given by (O1 Objects G ρ are group homomorphisms ρ : π 1 (G U(1 s.t. P = G ρ U(1 (O2 1-Morphisms G ρ G μ are morphisms G U(1 whose restriction to π 1 (G G equals ρ μ 1 (O3 2-Automorphisms are smooth maps M U(1 Gerbes around orbits Given a G action on M, consider orbit Gx M Find G-equivariant tubular neighbourhood U of orbit Gx with slice S Then Grb(U//G is given by (O1 Objects are U(1-central extension of G x = Stab(x (O2 1-Morphisms are morphisms of central extension (O3 2-Automorphisms are smooth maps S U(1 Christoph Schweigert, Fundamental Gerbe p.12/15
36 Equivariant covers for a compact Lie group Notation Choose maximal torus T G t = Lie(T with r := dim t Choose simple roots Positive Weyl chamber C Weyl alcove A := {c C α 0 (c 1} with α 0 highest root A simplex with vertices μ α, α = 0,... r. H α affine hyperplane opposite to μ α Christoph Schweigert, Fundamental Gerbe p.13/15
37 Equivariant covers for a compact Lie group Notation Choose maximal torus T G t = Lie(T with r := dim t Choose simple roots Positive Weyl chamber C Weyl alcove A := {c C α 0 (c 1} with α 0 highest root A simplex with vertices μ α, α = 0,... r. H α affine hyperplane opposite to μ α Facts Weyl alcove parametrizes conjugacy classes: q : G A Open subsets A α := A \ H α give G-invariant open cover U α := q 1 (A α of G Intersections U αβ := U α U β = q 1 (A α A β = q 1 (A αβ Christoph Schweigert, Fundamental Gerbe p.13/15
38 Equivariant covers for a compact Lie group Notation Choose maximal torus T G t = Lie(T with r := dim t Choose simple roots Positive Weyl chamber C Weyl alcove A := {c C α 0 (c 1} with α 0 highest root A simplex with vertices μ α, α = 0,... r. H α affine hyperplane opposite to μ α Facts Weyl alcove parametrizes conjugacy classes: q : G A Open subsets A α := A \ H α give G-invariant open cover U α := q 1 (A α of G Intersections U αβ := U α U β = q 1 (A α A β = q 1 (A αβ More generally For I {0, 1,..., r} consider the interior F I of the face of A spanned by the vertices (μ α α I. All group elements exp(x with x F I have the same stabilizer G I. Then S I := ad GI (exp(a I is a G-equivariant slice for any orbit in exp(a I and contracts to exp(x for any x F I Christoph Schweigert, Fundamental Gerbe p.13/15
39 The fundamental gerbe on a compact Lie group Equivariant descent implies the following equivalence of bicategories of gerbes: Grb(G//G = ( holim Grb (U α //G = α δ0 δ1 Grb ( U αβ //G δ0 ( Grb Uαβγ //G δ0 αβ δ2 δ3 holim ( Grb (Sα //G α δ 0 Grb ( Sαβ //G αβ δ 0 Grb ( Sαβγ //G αβγ δ 0 δ 1 δ 2 δ 3 Christoph Schweigert, Fundamental Gerbe p.14/15
40 The fundamental gerbe on a compact Lie group Equivariant descent implies the following equivalence of bicategories of gerbes: Grb(G//G = ( holim Grb (U α //G = α δ0 δ1 Grb ( U αβ //G δ0 ( Grb Uαβγ //G δ0 αβ δ2 δ3 holim ( Grb (Sα //G α δ 0 Grb ( Sαβ //G αβ δ 0 Grb ( Sαβγ //G αβγ δ 0 δ 1 δ 2 δ 3 All centralizers G I are connected Central extensions of G I A G-equivariant gerbe amounts to (O1 For each α = 0,..., r a central extension U(1 P α G α (O2 For each pair α, β a morphism φ αβ : P Gαβ α P Gαβ β (O3 For each triple (α, β, γ the equality of morphisms Φ αβ Φ βγ = Φ αγ on G αβγ as well as functions g αβγ : S αβγ U(1 (O4 Cocycle condition g αβδ g βγδ = g αγδ g αβγ on S αβγδ. Christoph Schweigert, Fundamental Gerbe p.14/15
41 The fundamental gerbe on G in terms of Lie theoretic data Choose a maximal torus T α of G α and consider the linear form Lie(T α R t μ α, t Exponentiate and restrict Λ α := ker(exp G α U(1 descends to character ρ α : π 1 (G α = Λ α /coroot(g α U(1 Christoph Schweigert, Fundamental Gerbe p.15/15
42 The fundamental gerbe on G in terms of Lie theoretic data Choose a maximal torus T α of G α and consider the linear form Lie(T α R t μ α, t Exponentiate and restrict Λ α := ker(exp G α U(1 descends to character ρ α : π 1 (G α = Λ α /coroot(g α U(1 Similarly, μ α μ β fix under action of G αβ gives a character χ αβ U(1 with χ αβ π1 (G αβ = ρ α ρ 1 β. Christoph Schweigert, Fundamental Gerbe p.15/15
43 The fundamental gerbe on G in terms of Lie theoretic data Choose a maximal torus T α of G α and consider the linear form Lie(T α R t μ α, t Exponentiate and restrict Λ α := ker(exp G α U(1 descends to character ρ α : π 1 (G α = Λ α /coroot(g α U(1 Similarly, μ α μ β fix under action of G αβ gives a character χ αβ U(1 with χ αβ π1 (G αβ = ρ α ρ 1 β. The basic gerbe amounts to the following Lie theoretic data: (O1 Central extensions G ρ α of G α given by ρ α. (O2 Morphisms χ αβ : G ρ α G ρ β on G αβ (O3 Obeying the equality χ αβ χ βγ = χ αγ (O4 Functions g αβγ : S αβγ U(1 constant to identity Christoph Schweigert, Fundamental Gerbe p.15/15
44 The fundamental gerbe on G in terms of Lie theoretic data Choose a maximal torus T α of G α and consider the linear form Lie(T α R t μ α, t Exponentiate and restrict Λ α := ker(exp G α U(1 descends to character ρ α : π 1 (G α = Λ α /coroot(g α U(1 Similarly, μ α μ β fix under action of G αβ gives a character χ αβ U(1 with χ αβ π1 (G αβ = ρ α ρ 1 β. The basic gerbe amounts to the following Lie theoretic data: (O1 Central extensions G ρ α of G α given by ρ α. (O2 Morphisms χ αβ : G ρ α G ρ β on G αβ (O3 Obeying the equality χ αβ χ βγ = χ αγ (O4 Functions g αβγ : S αβγ U(1 constant to identity Outlook (Equivariant gerbe modules D-branes (Equivariant gerbe bimodules topological defects Verlinde formula Christoph Schweigert, Fundamental Gerbe p.15/15
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