String Connections via the Caloron Correpondence Christian Becker, - PowerPoint PPT Presentation

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.