IGA Lecture III: Twisted Spin c structures Eckhard Meinrenken Adelaide, September 7, 2011 Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Review: Spin c -structures ( V , B ) a finite-dimensional Euclidean vector space, C l( V ) complex Clifford algebra: generators v ∈ V , relations vv ′ + v ′ v = 2 B ( v , v ′ ) . Then C l( V ) is a finite-dimensional C ∗ -algebra. Similarly, for a finite rank Euclidean vector bundle V → X with fiber metric B define a complex Clifford bundle C l( V ) → X . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Let V → X be a Euclidean vector bundle, rank( V ) even. Definition A Spin c -structure on V is a Z 2 -graded Hermitian vector bundle S → X with a ∗ -isomorphism ̺ : C l( V ) → End(S) . S is called the spinor module. Remarks The definition is equivalent to an orientation on V together with a lift of the structure group from SO( n ) to Spin c ( n ). (Connes, Plymen.) If V has odd rank, one defines a Spin c -structure on V to be a Spin c -structure on V ⊕ R . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Let V → X be a Euclidean vector bundle. Example Suppose J ∈ Γ(O( V )) is a complex structure, J 2 = − id V . Get V C = V + ⊕ V − . Then S = ∧ ( V + ) √ 2( ǫ ( v + ) + ι ( v − )) defines a Spin c -structure on V , with ̺ ( v ) = for v ∈ V . Example Suppose ω ∈ Γ( ∧ 2 V ∗ ) is symplectic; let R ω be the corresponding skew-adjoint endomorphism. Then J ω = R ω | R ω | ∈ Γ(O( V )) is a complex structure, defining a Spin c -structure on V . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Spin c -structures Basic properties Any two Spin c -structure S , S ′ on V differ by a line bundle: S ′ = S ⊗ L ↔ L = Hom C l (S , S ′ ) . Obstructions to existence of Spin c -structure: W 3 ( V ) ∈ H 3 ( X , Z ) , w 1 ( V ) ∈ H 1 ( X , Z 2 ) . Example The dual S ∗ of a spinor module is again a spinor module. Get a line bundle K S = Hom C l (S , S ∗ ) called the canonical line bundle for S. Note K S ⊗ L = K S ⊗ L − 2 . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Spin c -structures If M is a manifold with a smooth Spin c -structure S, one defines the Spin c -Dirac operator ∂ : Γ(S) ∇ ̺ / − → Γ( TM ⊗ S) − → Γ(S) . If L → M is a line bundle, denote by / ∂ L the Spin c -Dirac operator for S ⊗ L . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Quantization of Hamiltonian G -spaces (in a nutshell) Hamiltonian G -space Φ: M → g ∗ 1 ι ( ξ M ) ω = − d � Φ , ξ � , 2 d ω = 0, 3 ker( ω ) = 0. 1. Pick G -invariant Riemannian metric on M ⇒ ω determines a Spin c -structure. 2. Assume ( M , ω, Φ) pre-quantizable; pick a pre-quantum line bundle L → M . 3. Define Q ( M ) := index G ( / ∂ L ) ∈ R ( G ) . For q-Hamiltonian spaces already Step 1 fails, since ω may be degenerate. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Review: q-Hamiltonian G -spaces Let G be a compact Lie group, and · an invariant inner product on g = Lie( G ). Definition A q-Hamiltonian G -space ( M , ω, Φ) is a G -manifold M , with ω ∈ Ω 2 ( M ) G and Φ ∈ C ∞ ( M , G ) G , satisfying 2 Φ ∗ ( θ L + θ R ) · ξ , 1 ι ( ξ M ) ω = − 1 2 d ω = − Φ ∗ η , 3 ker( ω ) ∩ ker(dΦ) = 0. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
For q-Hamiltonian spaces already Step 1 fails: Problems: There is no notion of ‘compatible almost complex structure’ In general, q-Hamiltonian G -spaces need not even admit Spin c -structures. Example = S 4 (does not admit G = Spin(5) has a conjugacy class C ∼ almost complex structure). G = Spin(2 k + 1) , k > 2 has a conjugacy class not admitting a Spin c -structure. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
However, we will show that q-Hamiltonian spaces carry ‘ twisted ’ Spin c -structures. The definition of the twisted Spin c -structures involves Dixmier-Douady bundles Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Dixmier-Douady theory Notation: H separable complex Hilbert space, possibly dim H < ∞ , B ( H ) bounded linear operators, K ( H ) compact operators (= B fin ( H )) Fact: Aut( K ( H )) = PU( H ) (strong topology). Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Dixmier-Douady theory Definition A DD-bundle A → X is a Z 2 -graded bundle of ∗ -algebras modeled on K ( H ), (for H a Z 2 -graded Hilbert space), with structure group the even part of PU( H ). Theorem (Dixmier-Douady) The obstruction to writing A = K ( E ) , with E a Z 2 -graded bundle of Hilbert spaces, is a class DD( A ) ∈ H 3 ( X , Z ) × H 1 ( X , Z 2 ) . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Dixmier-Douady theory Hence, the trivially graded DD bundles give a ‘realization’ of H 3 ( X , Z ), similar to line bundles ‘realizing’ H 2 ( X , Z ). Remark This framework is not convenient for a Chern-Weil theory. A more differential-geometric realization is given by the theory of bundle gerbes. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Dixmier-Douady theory Definition Let A 1 → X 1 , A 2 → X 2 be DD -bundles. A Morita morphism (Φ , E ): A 1 ��� A 2 is a map Φ: X 1 → X 2 together with a Z 2 -graded bundle E → X 1 of bimodules Φ ∗ A 2 � E � A 1 , locally modeled on K ( H 2 ) � K ( H 1 , H 2 ) � K ( H 1 ). Remark (Φ , E ): A 1 ��� A 2 exists if and only if DD( A 1 ) = Φ ∗ DD( A 2 ) . Any two Morita bimodules E , E ′ differ by a line bundle: E ′ = E ⊗ L ↔ L = Hom Φ ∗ A 2 −A 1 ( E , E ′ ) . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Dixmier-Douady theory Example V → X Euclidean vector bundle of even rank ⇒ C l( V ) is a DD-bundle. A Morita trivialization ( p , S op ): C l( V ) ��� C is a Spin c -structure. The DD-class is given by DD(S) = ( W 3 ( V ) , w 1 ( V )) ∈ H 3 ( X , Z ) × H 1 ( X , Z 2 ) . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
From Dirac structures to DD bundles Review of linear Dirac structures A Dirac structure on vector space V is a Lagrangian subspace E ⊂ V = V ⊕ V ∗ . For Θ: V 1 → V 2 and ω ∈ ∧ 2 V ∗ 1 write � v 2 = Θ( v 1 ) v 1 + µ 1 ∼ (Θ ,ω ) v 2 + µ 2 ⇔ µ 1 = Θ ∗ ( µ 2 ) + ω ( v 1 , · ) It defines a Dirac morphism (Θ , ω ): ( V 1 , E 1 ) ��� ( V 2 , E 2 ) if every element of E 2 is related to a unique element of E 1 . The definitions extend to vector bundles V → X . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
From Dirac structures to DD bundles Example Hamiltonian G -spaces are described as G -equivariant Dirac morphisms (Φ , ω ): ( T M , TM ) ��� ( T g ∗ , E g ∗ ) . q-Hamiltonian G -spaces are described as G -equivariant Dirac morphisms (Φ , ω ): ( T M , TM ) ��� ( T G η , E G ) . There is a multiplication morphism (Mult G , ς ): ( T G η , E G ) × ( T G η , E G ) ��� ( T G η , E G ) . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
The Dirac-Dixmier-Douady functor Theorem (Alekseev-M, 2010) There is a functor from Dirac structures on vector bundles V → X to DD-bundles: E �→ A E . Furthermore, there are canonical Morita isomorphisms C l( V ) ��� A V , C ��� A V ∗ N.B.: We identify two Morita morphisms E , E ′ : A 1 ��� A 2 if they are related by a trivial line bundle. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Example The Cartan Dirac structure ( T G η , E G ) defines a DD-bundle A Spin := A E G → G . The ‘multiplication morphism’ for the Cartan Dirac structure gives a morphism Mult ∗ : A Spin × A Spin ��� A Spin . Example A q-Hamiltonian G -space ( M , ω, Φ) defines a Dirac morphism (dΦ , ω ): ( T M , TM ) ��� ( T G η , E G ) . Hence we get a Morita morphism C l( TM ) ��� A TM ��� A E G = A Spin , a ‘twisted Spin c -structure’. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Construction of the DDD functor E �→ A E Outline 1 From E ⊂ V , construct family of skew-adjoint operators D x , x ∈ X acting on real Hilbert spaces V x . 2 From D = { D x } , construct family of ‘polarizations’ of V x . 3 From the polarization, construct DD -bundle A → X . Inspired by and/or similar to: Carey-Mickelsson-Murray 1997, Lott 2002, Atiyah-Segal 2004, Freed-Hopkins-Teleman 2005, Bouwknegt-Mathai-Wu 2011. Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Step 1: Constructing { D x , x ∈ X } Assume X = pt, so V is a vector space. Choice of Euclidean metric B identifies Lag( V ) ∼ = O( V ) . Here A ∈ O( V ) corresponds to E = { (( A − I ) v , 1 2 ( A + I ) v ) ∈ V = V ⊕ V ∗ | v ∈ V } . Define skew-adjoint operator D E = ∂ ∂ t on V = L 2 ([0 , 1] , V ), with domain dom( D E ) = { f : f (1) = − Af (0) } . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
Step 1: Constructing { D x , x ∈ X } Example E = V ∗ corresponds to A = I , and f (1) = − Af (0) are anti-periodic boundary conditions. Note ker( D E ) = 0. Example E = V corresponds to A = − I , and f (1) = − Af (0) are periodic boundary conditions. Note ker( D E ) = V . Note that in general, ker( D E ) = ker( A + I ) = E ∩ V . Eckhard Meinrenken IGA Lecture III: Twisted Spin c structures
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