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

An approach to finite-dimensional realizations of 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 of “ twisted Z 2 -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 ) � � � � � � � � � � � � � Fredholm C ∗ -algebra operators 3

Fredholm operators H : separable Hilbert space (dim H = ∞ ) A Fredholm operator A : H → H  bounded linear ,  def  ⇐ ⇒ 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 Ad PU ( H ) = U ( H ) /U (1) � 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 ) ∈ H 3 ( X ; Z ) . 5

Vector bundles � ������������������������������������ � � � � � � � � � � � � � � � � � � � K ( X ) � � � � � � � � � � � � � Fredholm C ∗ -algebra operators C ( X, F ( H )) / ≃ C ( X ) ? ? ? � � � � � � � � � � � � � � � � � � � � � � � � � � K ( X ; P ) � � � � � � � Fredholm C ∗ -algebra operators Γ( P × Ad K ( H )) Γ( P × Ad F ( H )) / ≃ 6

✓ ✏ Problem Realize twisted K -theory K ( X ; P ) generally by means of finite dimensional geometric objects. ✒ ✑ δ ( P ) : finite order ⇒ ∃ answer ✓ ✏ Fact � X : compact manifold : δ ( P ) is finite order P We can define a group by means of “ twisted vector bundles ”, to which there exists an isomorphism from K ( X ; P ). ✒ ✑ K ( X ; P )   � iso  K ( { twisted vector bundles } / ∼ =) 7

Twisted vector bundle ( U , E α , φ αβ )  U = { U α } open cover of X ;   E α → U α finite rank vector bundle; φ αβ : E β | U αβ → E α | U αβ isomorphism;   φ αβ φ βγ = c αβγ φ αγ . Z 2 ( U ; U (1)) ˇ   ( c αβγ ) ∈ ↓     ∼ H 2 ( X ; U (1)) H 3 ( X ; Z ) δ ( P ) ∈ = Remark ( U , E α , φ αβ ) : rank r ⇒ r · δ ( P ) = 0. (det φ αβ )(det φ βγ ) = ( c αβγ ) r (det φ αγ ) 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 Z 2 - 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 9

Hermitian general vector bundle on X ( U , ( E α , h α ) , φ αβ )  U = { U α } open cover of X ;    E α → U α Z 2 -gr. Hermitian vector bundle;     h α : E α → E α Hermitian map of degree 1; φ αβ : E β | U αβ → E α | U αβ map of degree 0      s.t. h α φ αβ = φ αβ h β ;   1. “ φ αβ φ βα = 1”,  x ∈ ∃ V ⊂ U αβ ,  � ∀ x ∈ U αβ ; such that : ∃ µ > 0 ,        ∀ y ∈ V,       ∀ v ∈ { v ∈ ( E α ) y | h 2 �   α v = λv } ,       λ<µ         φ αβ φ βα ( v ) = v. 2. “ φ αβ φ βγ = φ αγ ”. 10

� ������� ✓ ✏ Fredholm operator A : H → H approximate ( E, h ) E = E 0 ⊕ E 1 Z 2 -gr. Herm. vector space � h : E → E Hermitian map of degree 1 ✒ ✑ ˆ  H = H ⊕ H Z 2 -graded   A ∗ � � Step 1 0 ˆ A = self-adjoint, degree 1 0  A  A 2 ) ∋ 0 : discrete ⇒ ∃ µ > 0 s.t. σ ( ˆ Step 2 A 2 ); • µ �∈ σ ( ˆ A 2 ) ∩ [0 , µ ) consists of a finite number of • σ ( ˆ eigenvalues: 0 = λ 1 < λ 2 < · · · < λ n < µ ; A 2 v = λ i v } : finite dim. • ( H , ˆ A ) λ i = { v ∈ ˆ H| ˆ A 2 ∼ ( H , ˆ A ) 0 = Ker ˆ ( = Ker A ⊕ Coker A ) 11

� ������� ˆ A ˆ ˆ H − → H � � 0 ( ˆ H , ˆ ( ˆ H , ˆ A ) 0 → A ) 0 ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ A ) λ 2 A ) λ 2 = ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ A ) λ 3 A ) λ 3 = ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ A ) λ 4 A ) λ 4 = ⊕ ⊕ . . . . . . ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ A ) λ n A ) λ n = ⊕ ⊕ ∼ complement = complement � E = ⊕ λ<µ ( ˆ H , ˆ A ) λ , Step 3 Put h = ˆ A | E . Remark family { A x : H → H} x ∈ X approximate Z 2 -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 Z 2 -graded Hermitian general vector bun- dles, into which there exists a monomor- phism from K ( X ; P ) = Γ( P × Ad F ( H )) / ≃ . ✒ ✑ • twisting ⇐ “ φ αβ φ βγ = c αβγ φ αγ ” finite dimensional • monomorphism ⇐ approximation 13