Twisted K-theory and finite-dimensional approximation Kiyonori - PDF document

Twisted K-theory and finite-dimensional approximation Kiyonori Gomi

✓ ✏ 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 charges [Witten, Kapustin, ...] The Verlinde algebras [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 f : H → H  bounded linear ,  def  Image( f ) ⊂ H : closed , ⇐ ⇒ dimKer( f ) , dimCoker( f ) < ∞ .   F ( H ) = { Fredholm operators f : 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 ).  U ( H ) ≃ pt ,   • PU ( H ) ≃ K ( Z , 2) ,  BPU ( H ) ≃ K ( Z , 3) .  Principal PU ( H )-bundles P are classified by their Dixmier-Douady classes: δ ( P ) ∈ H 3 ( X ; Z ) . 5

Examples ∼ � H 3 ( X ; Z ) = Z , δ ( P ) = k � = 0 . X = S 3 K ( S 3 ; k ) ∼ = 0 X = S 1 × S 2 K ( S 1 × S 2 ; k ) ∼ = Z X = S 3 / Z p , ( p : prime) K ( S 3 / Z p ; k ) ∼ = Z p X = SU (3) � k odd Z k K ( SU (3); k ) ∼ = k even Z k/ 2 6

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 )) / ≃ 7

✓ ✏ Problem Realize twisted K -theory generally by means of finite dimensional geometric ob- jects. ✒ ✑ δ ( P ) : finite order ⇒ ∃ answer ✓ ✏ Fact � X : compact P : δ ( P ) is finite order We can define a group by means of “ twisted vector bundles ”, to which there exists an isomorphism from K ( X ; P ). ✒ ✑ Remark There are a number of works on twisted vector bundles. 8

Twisted vector bundle • U = { U α } : open cover of X Z 2 ( U , U (1)) : • ( z αβγ ) ∈ ˇ 2-cocycle repre- senting δ ( P ) ∈ H 3 ( X, Z ) ∼ = H 2 ( X, U (1)). ✓ ✏ twisted vector bundle ( E α , φ αβ ) � finite rank vector bundle E α → U α ⇔ φ αβ : E α | U αβ → E β | U αβ isomorphism φ αβ φ βγ = z αβγ φ αγ ✒ ✑ Remark ( E α , φ αβ ) : rank r ⇒ r · δ ( P ) = 0. (det φ αβ )(det φ βγ ) = ( z αβγ ) r (det φ αγ ) 9

§ 2 Hermitian general vector bundle M. Furuta, “ Index theorem, II ”. (Japanese) Iwanami Series in Modern Mathematics. Iwanami Shoten, Publishers, Tokyo, 2002. • to approximate Dirac-type operators; linear version of the finite dimensional ap- proximation of the Seiberg-Witten equa- tions • 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 ). ✒ ✑ 10

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 α | U αβ → E β | U αβ map of degree 0;    1. “ h α φ αβ = φ αβ h β ”,   � 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          h α φ αβ ( v ) = φ αβ h β ( v ) . 2. “ φ αβ φ βα = 1”, 3. “ φ αβ φ βγ = φ αγ ”. 11

� ������� ✓ ✏ Fredholm operator f : 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   f ∗ � � Step 1 0 ˆ f = self-adjoint, degree 1  f 0  f 2 ) ∋ 0 : discrete ⇒ ∃ µ > 0 s.t. σ ( ˆ Step 2 f 2 ); • µ �∈ σ ( ˆ f 2 ) ∩ [0 , µ ) consists of a finite number of • σ ( ˆ eigenvalues: 0 = λ 1 < λ 2 < · · · < λ n < µ ; f 2 v = λ i v } : finite dim. • ( H , ˆ f ) λ i = { v ∈ ˆ H| ˆ f 2 ∼ ( H , ˆ f ) 0 = Ker ˆ ( = Ker f ⊕ Coker f ) 12

ˆ f ˆ ˆ H − → H � � 0 H , ˆ H , ˆ ( ˆ ( ˆ f ) 0 → f ) 0 ⊕ ⊕ ∼ H , ˆ H , ˆ ( ˆ ( ˆ f ) λ 2 = f ) λ 2 ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ f ) λ 3 f ) λ 3 = ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ f ) λ 4 f ) λ 4 = ⊕ ⊕ . . . . . . ⊕ ⊕ ∼ ( ˆ H , ˆ ( ˆ H , ˆ f ) λ n f ) λ n = ⊕ ⊕ ∼ complement complement = � E = ⊕ λ<µ ( ˆ H , ˆ f ) λ , Step 3 Put h = ˆ f | E . 13

� ������� { ˆ f x : ˆ H → ˆ Remark H} x ∈ U : family f 2 • dimKer ˆ x may jump. f 2 • µ �∈ σ ( ˆ x 0 ) ( ˆ H , ˆ � ⇒ dim f x ) λ is constant near x 0 . λ<µ ✓ ✏ family { f x : H → H} x ∈ X approximate ( U , ( E α , h α ) , φ αβ ) Z 2 -gr. Herm. general vector bundle on X ✒ ✑ 14

✓ ✏ Main theorem � X : compact 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 = z αβγ φ αγ ” finite dimensional • monomorphism ⇐ approximation 15