An approach to finite-dimensional realizations of twisted K-theory - - PDF document

An approach to finite-dimensional realizations

• f twisted K-theory

Kiyonori Gomi The University of Tokyo Japan

Problem in twisted K-theory

✓ ✏

Realize twisted K-theory generally by means of finite dimensional geometric ob- jects.

✒ ✑

Main theorem

✓ ✏

We can define a group by means

• f

“twisted Z2-graded Hermitian general vec- tor bundles”, into which there exists a monomorphism from twisted K-theory.

✒ ✑

Plan

§1 Twisted K-theory §2 Hermitian general vector bundle

1

§1

Twisted K-theory

Origin

• P. Donovan and M. Karoubi (1970)
• J. Rosenberg (1989)

Application D-brane charge [Witten, Kapustin, ...] The Verlinde algebra [Freed-Hopkins-Teleman] The quantum Hall effect [Carey-Hannabuss-Mathai-McCann]

2

K-theory

X : compact Vect(X) = the isomorphism classes of finite dimensional vector bundles over X Definition

✓ ✏

K(X) = K(Vect(X)) = Vect(X) × Vect(X)/∆(Vect(X))

✒ ✑

Vector bundles

• K(X)

C∗-algebra Fredholm

• perators

3

Fredholm operators

H : separable Hilbert space (dimH = ∞) A Fredholm operator A : H → H

def

⇐ ⇒

    

bounded linear, Image(A) ⊂ H : closed, dimKer(A), dimCoker(A) < ∞. F(H) = {Fredholm operators A : H → H} Fact [Atiyah, J¨ anich]

✓ ✏

X : compact C(X, F(H))/htpy iso − → K(X)

✒ ✑

4

Twisted K-theory

PU(H) = U(H)/U(1)

Ad

F(H)

Definition

✓ ✏

P → X : principal PU(H)-bundle K(X; P) = Γ(X, P ×Ad F(H))/htpy

✒ ✑

• P ∼

= X × PU(H) ⇒ K(X; P) ∼ = K(X).

• Principal PU(H)-bundles P are classified

by their Dixmier-Douady classes: δ(P) ∈ H3(X; Z).

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Vector bundles

• K(X)

C∗-algebra Fredholm

• perators

C(X) C(X, F(H))/ ≃ ? ? ?

• K(X; P)

C∗-algebra Fredholm

• perators

Γ(P ×Ad K(H)) Γ(P ×Ad F(H))/ ≃

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Problem

✓ ✏

Realize twisted K-theory K(X; P) generally by means of finite dimensional geometric

• bjects.

✒ ✑

δ(P) : finite order ⇒ ∃answer Fact

✓ ✏

• X

: compact manifold P : δ(P) is finite order We can define a group by means

• f

“twisted vector bundles”, to which there exists an isomorphism from K(X; P).

✒ ✑

K(X; P)

   iso

K({twisted vector bundles}/∼ =)

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Twisted vector bundle

(U, Eα, φαβ)

    

U = {Uα}

• pen cover of X;

Eα → Uα finite rank vector bundle; φαβ : Eβ|Uαβ → Eα|Uαβ isomorphism; φαβφβγ = cαβγφαγ.

  

(cαβγ) ∈ ˇ Z2(U; U(1)) ↓ δ(P) ∈ H2(X; U(1)) ∼ = H3(X; Z)

  

Remark (U, Eα, φαβ) : rank r ⇒ r · δ(P) = 0. (detφαβ)(detφβγ) = (cαβγ)r(detφαγ)

8

§2

Hermitian general vector bundle

• M. Furuta, “Index theorem, II”. (Japanese)

Iwanami Series in Modern Mathematics. Iwanami Shoten, Publishers, Tokyo, 2002.

• to define K(X);

Theorem[Furuta]

✓ ✏

X : compact We can define a group by means of Z2- graded Hermitian general vector bun- dles, which is isomorphic to K(X).

✒ ✑

• to approximate Dirac-type operators.

a linear version of the finite dimensional approximation of the Seiberg-Witten equa- tions

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Hermitian general vector bundle on X

(U, (Eα, hα), φαβ)

              

U = {Uα}

• pen cover of X;

Eα → Uα

Z2-gr. Hermitian vector bundle;

hα : Eα → Eα Hermitian map of degree 1; φαβ : Eβ|Uαβ → Eα|Uαβ map of degree 0 s.t. hαφαβ = φαβhβ;

• 1. “φαβφβα = 1”,

            

∀x ∈ Uαβ;

• x ∈ ∃V ⊂ Uαβ,

∃µ > 0,

such that :

    

∀y ∈ V, ∀v ∈

• λ<µ

{v ∈ (Eα)y| h2

αv = λv},

φαβφβα(v) = v.

            

• 2. “φαβφβγ = φαγ”.

10

✓ ✏

Fredholm operator A : H → H approximate

• (E, h)
• E = E0 ⊕ E1 Z2-gr. Herm. vector space

h : E → E Hermitian map of degree 1

✒ ✑

Step 1

    

ˆ H = H ⊕ H

Z2-graded

ˆ A =

• A∗

A

• self-adjoint, degree 1

Step 2 σ( ˆ A2) ∋ 0 : discrete ⇒ ∃µ > 0 s.t.

• µ ∈ σ( ˆ

A2);

• σ( ˆ

A2)∩[0, µ) consists of a finite number of eigenvalues: 0 = λ1 < λ2 < · · · < λn < µ;

• (H, ˆ

A)λi = {v ∈ ˆ H| ˆ A2v = λiv} : finite dim. ( (H, ˆ A)0 = Ker ˆ A2 ∼ = KerA ⊕ CokerA )

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ˆ H

ˆ A

− → ˆ H

• ( ˆ

H, ˆ A)0 → ( ˆ H, ˆ A)0 ⊕ ⊕ ( ˆ H, ˆ A)λ2 ∼ = ( ˆ H, ˆ A)λ2 ⊕ ⊕ ( ˆ H, ˆ A)λ3 ∼ = ( ˆ H, ˆ A)λ3 ⊕ ⊕ ( ˆ H, ˆ A)λ4 ∼ = ( ˆ H, ˆ A)λ4 ⊕ ⊕ . . . . . . ⊕ ⊕ ( ˆ H, ˆ A)λn ∼ = ( ˆ H, ˆ A)λn ⊕ ⊕ complement ∼ = complement Step 3 Put

• E = ⊕λ<µ( ˆ

H, ˆ A)λ, h = ˆ A|E. Remark family {Ax : H → H}x∈X approximate

• Z2-graded Hermitian general

vector bundle over X

12

Main theorem

✓ ✏

• X : compact manifold

P : PU(H)-bundle We can define a group by means of twisted

Z2-graded Hermitian general vector bun-

dles, into which there exists a monomor- phism from K(X; P) = Γ(P ×Ad F(H))/ ≃.

✒ ✑

• twisting ⇐ “φαβφβγ = cαβγφαγ”
• monomorphism ⇐

finite dimensional approximation

13