Motivation Spin and String structures The String Group Geometric String Structures String Connections via the Caloron Correpondence Christian Becker, Potsdam work in progress, joint with C. Wockel, Hamburg Infinite-dimensional Riemannian geometry ESI Wien, 2015
Motivation Spin and String structures The String Group Geometric String Structures Outline Motivation Spin and String Structures A Smooth Model for the String Group The Caloron Correspondence and Geometric String Structures
Motivation Spin and String structures The String Group Geometric String Structures Witten conjecture M smooth manifold, L M := { S 1 → M smooth } free loop space Conjecture (Witten, 1984) D (hypothetical) U (1)-equivariant Dirac operator on L M : ind ( D ) = w [ M ] . Here w is a topological invariant, nowadays called Witten genus, computable in local terms Many difficulties to overcome: • construction of spinor bundle Σ L M • construction of Dirac operator D : Γ(Σ L M ) → Γ(Σ L M ) • analytic properties of D like Fredholmness • methods to prove an index theorem
� � � � Motivation Spin and String structures The String Group Geometric String Structures Classical Dirac operator M comp. or. Riem. n -manifold, n ≥ 3, ∇ Levi-Civita connection F SO M → M or. orthon. frame bundle: principal SO n -bundle Spin n → SO n universal covering Y → M Spin structure, i.e. principal Spin n -bundle, such that Y × Spin n Y � F SO M � M F SO M × SO n � ∇ Spin connection (uniquely determined by ∇ ) ̺ : Spin n → GL (Σ) Spin repr., Σ M := Y × ̺ Σ spinor bundle Dirac operator i =1 e i · � σ �→ Σ n D : Γ(Σ M ) → Γ(Σ M ) , ∇ e i σ.
� � � � Motivation Spin and String structures The String Group Geometric String Structures Spin structure on loop space M compact oriented manifold, L M free loop space G compact connected Lie group, L G loop group Y → M principal G -bundle, e.g. frame bundle or Spin structure L Y → L M principal L G -bundle � U (1) � � � L G � 1 1 L G universal central extension Definition (Killingback, Coquereaux/Pilch) A Spin Structure on L Y → L M is a lift bundle L Y × � � � L G L Y � L Y � L M L Y × L G
� � Motivation Spin and String structures The String Group Geometric String Structures Obstruction exact sequence in ˇ Cech cohomology H 1 ( L M ; U (1)) → ˇ ˇ H 1 ( L M ; � L G ) → ˇ H 1 ( L M ; L G ) → ˇ H 2 ( L M ; U (1)) transition functions L M ⊃ U αβ → L G : cycle in ˇ Z 1 ( L M ; L G ) obstruction to lift to cycle U αβ → � L G : cohomology class q ∈ ˇ H 2 ( L M ; U (1)) ∼ = H 3 ( L M ; Z ) Actually, q = τ ( p ), where p ∈ H 4 ( M ; Z ) F ◦ ev ∗ : H ∗ ( M ; Z ) → H ∗− 1 ( L M ; Z ) transgression, where ffl τ := ev L M × S 1 M L M q = 0 sufficient for exist. of Spin structure on L Y → L M For G = Spin n , we have q = 1 2 p 1 .
Motivation Spin and String structures The String Group Geometric String Structures Lifting Problems M comp. n -manifold univ . cov . conn . comp hom . equiv . − − − − − − → SO n − − − − − − − → O n − − − − − − − → GL n Spin n FM → M frame bundle: principal GL n -bundle F O M → M orthon. frame bundle: lift to O n -bundle: no obstr. F SO M or. orth. frame bundle: lift to SO n -bundle: w 1 ∈ H 1 ( M ; Z 2 ) Y → M Spin structure: lift to Spin n -bundle: w 2 ∈ H 2 ( M ; Z 2 ) Further lifts? Homotopy groups of GL n : k 0 1 2 3 4 5 6 7 π k ( GL n ) 0 0 0 0 Z 2 Z 2 Z Z conn . comp hom . equiv . kill π 3 univ . cov . String n − − − → Spin n − − − − − − → SO n − − − − − − − → O n − − − − − − − → GL n i.e. π 3 ( String n ) = 0 and String n → Spin n group homomorphism inducing isomorphisms on π k for k � = 3.
Motivation Spin and String structures The String Group Geometric String Structures String structures M compact n -manifold, Y → M principal Spin n -bundle Definition (Stolz/Teichner, 2004) A String structure on Y → M is a reduction to a principal String n -bundle. Obstruction: 1 2 p 1 ∈ H 4 ( M ; Z ). Definition/Theorem (Redden, 2006/2011) A String class on Y → M is a cohomology class H ∈ H 3 ( Y ; Z ), such that for any x ∈ M , H | Y x = H 0 = 1 ∈ H 3 ( Spin n ; Z ) ∼ = Z 1:1 String classes ← → { String structures } /isom.
Motivation Spin and String structures The String Group Geometric String Structures Models for String n Theorem (Cartan, 1936) G compact simple, simply connected Lie group. Then π 2 ( G ) = 0 and π 3 ( G ) ∼ = Z . Thus String n cannot be finite dimensional Lie group! Problem: construct nice models of String n • topological group models: Stolz(1996), Stolz/Teichner (2004) • Lie 2-group models: Henriques (2008), Schommer-Pries (2010) • Fr´ echet-Lie group model: Nikolaus/Sachse/Wockel (2011) From Cartan’s theorem and Hurewicz isomorphism: π 3 ( Spin n ) ∼ = H 3 ( Spin n ; Z ) ∼ = Z . Thus String n → Spin n needs to kill H 3 ( X ; Z ).
Motivation Spin and String structures The String Group Geometric String Structures PU ( H )-bundles Geometric realization of H 3 ( X ; Z ): H separable Hilbert space. Kuipers theorem: U ( H ) contractible homotopy exact sequence of the U (1)-bundle U ( H ) → PU ( H ) ∼ = π i +1 U ( H ) − → π i +1 PU ( H ) − − → π i U (1) − → π i U ( H ) EPU ( H ) → BPU ( H ) universal principal PU ( H )-bundle: ∼ = π i +2 EPU ( H ) − → π i +2 BPU ( H ) − − → π i +1 PU ( H ) − → π i +1 EPU ( H ) � Z i = 1 Thus π i +2 BPU ( H ) ∼ = π i +1 PU ( H ) ∼ = π i U (1) ∼ i � = 1. = { 0 } Thus BPU ( H ) is a K ( Z , 3). 1:1 1:1 → { PU ( H )-bundles � H 3 ( M ; Z ) ← → [ M , BPU ( H )] ← P → M } /isom.
Motivation Spin and String structures The String Group Geometric String Structures A Smooth Model for the String Group G = Spin n (or G compact, simple, simply connected Lie group) H 0 = 1 ∈ H 3 ( Spin n ; Z ) ∼ = Z Q → Spin n principal PU ( H )-bundle representing H 0 Aut ( Q ) ⊂ Diff ( Q ) automorphism group of Q p → Diff Q ( G ) → 1 1 → Gau ( Q ) → Aut ( Q ) − G ⊂ Diff Q ( G ) by left translations: L g : G → G , g �→ g · h Definition (Nikolaus/Sachse/Wockel) String G := { γ ∈ Aut ( Q ) | p ( γ ) ∈ G ⊂ Diff Q ( G ) } . String G is a Fr´ echet Lie group, π 3 ( String G ) = 0 p 1 → Gau ( Q ) → String G − → G → 1 p induces isomorphisms on π i for i � = 3.
Motivation Spin and String structures The String Group Geometric String Structures U ( H )-bundles once again M comp. Riem. n -manifold, Y → M principal Spin n -bundle Aim: lift ( Y , ∇ ) → M to String n -bundle ( P , ∇ ) → M recall: 1:1 String classes H ∈ H 3 ( Y ; Z ) { String-structures } /isom. ← → H 3 ( Spin n ; Z ) ∋ 1 = H 0 = H | Y x , repr. by PU ( H )-bundle Q → Spin n Definition PU ( H )-bundle � P → Y of type Q → Spin n : ⇔ ∀ x ∈ M : � P | Y x ∼ = Q 1:1 String classes ← → { PU ( H )-bundles of type Q → Spin n } Goal: construct String struct. P → M from PU ( H )-bundle � P → Y endow both sides with (suitable) connections
� Motivation Spin and String structures The String Group Geometric String Structures Caloron Correspondence Two categories of bundles: Definition Bun PU ( H ) ( Y → M ):= { PU ( H )-bdl � P → Y of type Q → Spin n } [ Q ] ψ Bun String n [ Y → M ] ( M ):= { String n -bdl P → M | P × String n Spin n − → Y } Caloron Correspondence: equivalence of categories � C Bun PU ( H ) � Bun String n ( Y → M ) [ Y → M ] ( M ) [ Q ] C Caloron Correspondence over arbitrary fiber bundles Y → M : Hekmati/Murray/Vozzo (2011) Similarly: Caloron Correspondence with connections
� Motivation Spin and String structures The String Group Geometric String Structures Construction of the correspondence Theorem (Hekmati/Murray/Vozzo 2011; B./Wockel 2015) Equivalence of categories � C � Bun String n Bun PU ( H ) ( Y → M ) [ Y → M ] ( M ) [ Q ] C Construction: � � C ( � � F : Q → � P ) := P bundle map cov. frame f : Spin n → Y x → M String n ⊂ Aut ( Q ) acts by ( F , γ ) �→ F ◦ γ ev C ( � � P ) × String n Spin n − − → Y , [ F , g ] �→ f ( g ) Reverse construction: id × π ψ C ( P , ψ ) := P × String n Q − − − − → P × String n Spin n − → Y
� Motivation Spin and String structures The String Group Geometric String Structures String connections Endow both sides of the correspondence with (suitable notion of) connections: Theorem (Hekmati/Murray/Vozzo 2011; B./Wockel 2015) Equivalence of categories � C � Bun String n , ∇ ,ξ Bun PU ( H ) , ∇ ( Y → M ) ( M ) [ Q ] [ Y → M ] C In particular: get String n -connections on String structure P → M from Spin n - and PU ( H )-connections.
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