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, LM := {S1 → M smooth} free loop space

Conjecture (Witten, 1984)

D (hypothetical) U(1)-equivariant Dirac operator on LM: 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 ΣLM
• construction of Dirac operator D : Γ(ΣLM) → Γ(ΣLM)
• 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 SOM → M or. orthon. frame bundle: principal SOn-bundle Spinn → SOn universal covering Y → M Spin structure, i.e. principal Spinn-bundle, such that Y × Spinn

• Y
• F SOM × SOn

F SOM M

• ∇ Spin connection (uniquely determined by ∇)

̺ : Spinn → GL(Σ) Spin repr., ΣM := Y ×̺ Σ spinor bundle

Dirac operator

D : Γ(ΣM) → Γ(ΣM), σ → Σn

i=1ei ·

∇eiσ.

Motivation Spin and String structures The String Group Geometric String Structures

Spin structure on loop space

M compact oriented manifold, LM free loop space G compact connected Lie group, LG loop group Y → M principal G-bundle, e.g. frame bundle or Spin structure LY → LM principal LG-bundle 1

U(1)

LG

LG 1

universal central extension

Definition (Killingback, Coquereaux/Pilch)

A Spin Structure on LY → LM is a lift bundle

• LY ×

LG

• LY
• LY × LG

LY LM

Motivation Spin and String structures The String Group Geometric String Structures

Obstruction

exact sequence in ˇ Cech cohomology ˇ H1(LM; U(1)) → ˇ H1(LM; LG) → ˇ H1(LM; LG) → ˇ H2(LM; U(1)) transition functions LM ⊃ Uαβ → LG: cycle in ˇ Z 1(LM; LG)

• bstruction to lift to cycle Uαβ →

LG: cohomology class q ∈ ˇ H2(LM; U(1)) ∼ = H3(LM; Z) Actually, q = τ(p), where p ∈ H4(M; Z) τ := ffl

F ◦ ev∗ : H∗(M; Z) → H∗−1(LM; Z) transgression, where

LM × S1

ev

• M

LM q = 0 sufficient for exist. of Spin structure on LY → LM For G = Spinn, we have q = 1

2p1.

Motivation Spin and String structures The String Group Geometric String Structures

Lifting Problems

M comp. n-manifold Spinn

• univ. cov.

− − − − − − → SOn

• conn. comp

− − − − − − − → On

• hom. equiv.

− − − − − − − → GLn FM → M frame bundle: principal GLn-bundle F OM → M orthon. frame bundle: lift to On-bundle: no obstr. F SOM or. orth. frame bundle: lift to SOn-bundle: w1 ∈ H1(M; Z2) Y → M Spin structure: lift to Spinn-bundle: w2 ∈ H2(M; Z2) Further lifts? Homotopy groups of GLn: k 1 2 3 4 5 6 7 πk(GLn) Z2 Z2 Z Z Stringn

kill π3

− − − → Spinn

• univ. cov.

− − − − − − → SOn

• conn. comp

− − − − − − − → On

• hom. equiv.

− − − − − − − → GLn i.e. π3(Stringn) = 0 and Stringn → Spinn group homomorphism inducing isomorphisms on πk for k = 3.

Motivation Spin and String structures The String Group Geometric String Structures

String structures

M compact n-manifold, Y → M principal Spinn-bundle

Definition (Stolz/Teichner, 2004)

A String structure on Y → M is a reduction to a principal Stringn-bundle. Obstruction: 1

2p1 ∈ H4(M; Z).

Definition/Theorem (Redden, 2006/2011)

A String class on Y → M is a cohomology class H ∈ H3(Y ; Z), such that for any x ∈ M, H|Yx = H0 = 1 ∈ H3(Spinn; Z) ∼ = Z String classes

1:1

← → {String structures}/isom.

Motivation Spin and String structures The String Group Geometric String Structures

Models for Stringn

Theorem (Cartan, 1936)

G compact simple, simply connected Lie group. Then π2(G) = 0 and π3(G) ∼ = Z. Thus Stringn cannot be finite dimensional Lie group! Problem: construct nice models of Stringn

• 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: π3(Spinn) ∼ = H3(Spinn; Z) ∼ = Z. Thus Stringn → Spinn needs to kill H3(X; Z).

Motivation Spin and String structures The String Group Geometric String Structures

PU(H)-bundles

Geometric realization of H3(X; Z): H separable Hilbert space. Kuipers theorem: U(H) contractible homotopy exact sequence of the U(1)-bundle U(H) → PU(H) πi+1U(H) − → πi+1PU(H)

∼ =

− − → πiU(1) − → πiU(H) EPU(H) → BPU(H) universal principal PU(H)-bundle: πi+2EPU(H) − → πi+2BPU(H)

∼ =

− − → πi+1PU(H) − → πi+1EPU(H) Thus πi+2BPU(H) ∼ = πi+1PU(H) ∼ = πiU(1) ∼ =

• Z

i = 1 {0} i = 1. Thus BPU(H) is a K(Z, 3). H3(M; Z)

1:1

← → [M, BPU(H)]

1:1

← → {PU(H)-bundles P → M}/isom.

Motivation Spin and String structures The String Group Geometric String Structures

A Smooth Model for the String Group

G = Spinn (or G compact, simple, simply connected Lie group) H0 = 1 ∈ H3(Spinn; Z) ∼ = Z Q → Spinn principal PU(H)-bundle representing H0 Aut(Q) ⊂ Diff (Q) automorphism group of Q 1 → Gau(Q) → Aut(Q)

p

− → Diff Q(G) → 1 G ⊂ Diff Q(G) by left translations: Lg : G → G, g → g · h

Definition (Nikolaus/Sachse/Wockel)

StringG := {γ ∈ Aut(Q) | p(γ) ∈ G ⊂ Diff Q(G)}. StringG is a Fr´ echet Lie group, π3(StringG) = 0 1 → Gau(Q) → StringG

p

− → G → 1 p induces isomorphisms on πi 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 Spinn-bundle Aim: lift (Y , ∇) → M to Stringn-bundle (P, ∇) → M recall: {String-structures}/isom.

1:1

← → String classes H ∈ H3(Y ; Z) H3(Spinn; Z) ∋ 1 = H0 = H|Yx, repr. by PU(H)-bundle Q → Spinn

Definition

PU(H)-bundle P → Y of type Q → Spinn :⇔ ∀ x ∈ M: P|Yx ∼ = Q String classes

1:1

← → {PU(H)-bundles of type Q → Spinn} 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

BunPU(H)

[Q]

(Y → M):={PU(H)-bdl P → Y of type Q → Spinn} BunStringn

[Y →M](M):={Stringn-bdl P → M | P ×Stringn Spinn ψ

− → Y } Caloron Correspondence: equivalence of categories BunPU(H)

[Q]

(Y → M)

• C

BunStringn

[Y →M](M) 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 BunPU(H)

[Q]

(Y → M)

• C

BunStringn

[Y →M](M) C

• Construction:
• C(

P) :=

• F : Q →

P bundle map cov. frame f : Spinn → Yx

• → M

Stringn ⊂ Aut(Q) acts by (F, γ) → F ◦ γ

• C(

P) ×Stringn Spinn

ev

− − → Y , [F, g] → f (g) Reverse construction: C(P, ψ) := P ×Stringn Q

id ×π

− − − − → P ×Stringn Spinn

ψ

− → 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 BunPU(H),∇

[Q]

(Y → M)

• C

BunStringn,∇,ξ

[Y →M]

(M)

C

• In particular: get Stringn-connections on String structure P → M

from Spinn- and PU(H)-connections.