The Spinor Bundle on Loop Space & Its Fusion Product

The Spinor Bundle on Loop Space & Its Fusion Product

The spinor bundle on loop space & its fusion product Peter Kristel Greifswald University Dissertation Colloquium January 20 Overview 1. Finite dimensional Clifford algebras and spinors 2. Clifford algebras and spinors in infinite dimensions 3. The spinor bundle on loop space 4. Fusion & Locality Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality 1. Finite dimensional Clifford algebras and spinors 2. Clifford algebras and spinors in infinite dimensions 3. The spinor bundle on loop space 4. Fusion & Locality Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Finite dimensional real Clifford algebras V a finite dimensional Euclidean space Definition: Cl(V ) The Clifford algebra Cl(V ) is the algebra generated by V subject to vw + wv = 2hv; wi1; v; w 2 V: V,! Cl(V ) Lemma: Universal property If A is a unital algebra, and f : V ! A satisfies f(v)f(w) + f(w)f(v) = 2hv; wi1; v; w 2 V; then there is a unique homomorphism F : Cl(V ) ! A extending f. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality The spin group Definition: Spin(d) The spin group Spin(d) is the universal cover of SO(d). I.e. Spin(d) is a simply connected Lie group which fits in the exact sequence Z2 ,! Spin(d) SO(d): The spin group and the Clifford algebra The spin group is a subgroup of the unit group of the Clifford algebra: d × Spin(d) ⊂ Cl(R ) : d Thus, Spin(d) acts on Cl(R ) by conjugation. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Where do spin groups & Clifford algebras appear? • K-theory (Karoubi) • Index theory (Atiyah-Singer index theorem) . • Quantum mechanics of particles with spin d d Let F be a module for Cl(R ), and ej the canonical basis for R . 2 d Then spinor fields 2 L (R ; F) satisfy the Dirac equation: d X i ej@j = m : j=1 . • Quantum mechanics of strings with spin Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Spinors on Manifolds Spinor bundles Let M be a finite dimensional oriented Riemannian manifold. The Clifford algebras Cl(TxM) fit together into a bundle Cl(TM) ! M. If M is spin, then we can construct a bundle S ! M of modules for Cl(TM). In a 2005 survey Stolz & Teichner gave a blueprint for replacing M by its loop space LM. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Spinors on Manifolds Spinor bundles Let M be a finite dimensional oriented Riemannian manifold. The Clifford algebras Cl(TxM) fit together into a bundle Cl(TM) ! M. If M is spin, then we can construct a bundle S ! M of modules for Cl(TM). In a 2005 survey Stolz & Teichner gave a blueprint for replacing M by its loop space LM. • Quantum mechanics of strings with spin Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Build your own spinor bundle M some finite dimensional, oriented “spacetime” manifold. d • Pick a representation F for Cl(R ) and thus also for d Spin(d) ⊂ Cl(R ). • Reduce the structure group of SO(M) to Spin(d), call the result Spin(M). Then S := Spin(M) ×Spin(d) F is a spinor bundle. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality 1. Finite dimensional Clifford algebras and spinors 2. Clifford algebras and spinors in infinite dimensions 3. The spinor bundle on loop space 4. Fusion & Locality • Complete this algebra to a C∗-algebra. This Clifford C∗-algebra is universal. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Infinite dimensional Clifford algebras V a complex Hilbert space, now infinite dimensional. • Define a Clifford algebra as before. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Infinite dimensional Clifford algebras V a complex Hilbert space, now infinite dimensional. • Define a Clifford algebra as before. • Complete this algebra to a C∗-algebra. This Clifford C∗-algebra is universal. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Fock spaces Definition: Lagrangians A subspace L ⊂ V is Lagrangian if • V = L ⊕ L, • hv; wi = 0, for all v; w 2 L. Identify L ' L∗, through w 7! hw; •i. Definition: Fock representation The Fock space F is the Hilbert completion of the exterior algebra ΛL. • L acts on ΛL by left multiplication (creation). • L acts on ΛL by contraction (annihilation). This extends to a representation π : Cl(V ) !B(F). Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Bogoliubov transformations O(V ) is the orthogonal group. Let g 2 O(V ) Using the universal property, define the Bogoliubov transformation θg 2 Aut(Cl(V )) to be the extension of g. Question: When are π and π ◦ θg unitarily equivalent? I.e. when does there exist a U 2 U(F) such that ∗ Uπ(a)U = π(θga); a 2 Cl(V ): In this case: call θg implementable. Segal-Shale-Stinespring: Decompose g with respect to V = L ⊕ L, then θg is implementable if and only if its off-diagonal part L ! L is Hilbert-Schmidt. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality A projective representation Definition: Restricted orthogonal group The restricted orthogonal group is Ores(V ) := fg 2 O(V ) j g is implementableg: If g 2 Ores(V ), and U implements g, then so does λU, for λ 2 U(1). Hence, Ores(V ) acts projectively in F. This gives a central extension 1 ! U(1) ! Imp(V ) ! Ores(V ) ! 1: Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality 1. Finite dimensional Clifford algebras and spinors 2. Clifford algebras and spinors in infinite dimensions 3. The spinor bundle on loop space 4. Fusion & Locality Find a Hilbert space V with compatible L^Spin(d) and Cl(V ) action. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Spin structures on loop spaces Let M be a spin manifold with spin frame bundle Spin(M). Then L Spin(M) has structure group L Spin(d). The group L Spin(d) has a “basic” central extension: U(1) ! L^Spin(d) ! L Spin(d) Definition (Killingback ’87) A spin structure on the loop space LM is a lift L^Spin(M) ! L Spin(M): Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Spin structures on loop spaces Let M be a spin manifold with spin frame bundle Spin(M). Then L Spin(M) has structure group L Spin(d). The group L Spin(d) has a “basic” central extension: U(1) ! L^Spin(d) ! L Spin(d) Definition (Killingback ’87) A spin structure on the loop space LM is a lift L^Spin(M) ! L Spin(M): Find a Hilbert space V with compatible L^Spin(d) and Cl(V ) action. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality The odd spinors on the circle The Hilbert space 2 1 d Set V := L (S ; C ). An (unorthodox) orthonormal basis of V is i' i(n+1=2)' ξn;j : e 7! e ej;' 2 [0; 2π] d where n 2 Z and fejgj=1;:::;d the standard basis of C . Pointwise complex conjugation gives: ξn;j = ξ−n−1;j. 1 Let S ! S be the odd spinor bundle on the circle. The basis ξn;j 2 1 d looks more natural when V is identified with L (S ; S ⊗ C ). Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality A Lagrangian Compute for n; m > 0 and j; l = 1; :::; d, hξn;j; ξm;li = hξn;j; ξ−m−1;li = δn;−m−1 δj;l ξ’s are orthonormal = 0: n 6= −m − 1 Definition: Atiyah-Patodi-Singer (APS) Lagrangian The APS Lagrangian L ⊂ V is the closure of the span of the vectors ξn;j with n > 0 and j = 1; :::; d. Have a Clifford algebra Cl(V ), and a Fock space F corresponding to the Lagrangian L. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality The basic central extension of L Spin(d) The action of L SO(d) 2 1 d The loop group L SO(d) acts on V = L (S ; C ) pointwise. Pressley-Segal: L SO(d) ! Ores(V ). Lemma The pullback L^Spin(d) Imp(V ) L Spin(d) L SO(d) Ores(V ) is the basic central extension of L Spin(d) Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality The spinor bundle on loop space Have representations L^Spin(d) F and L Spin(d) Cl(V ). Define the Clifford bundle on loop space: L Spin(M) ×L Spin(d) Cl(V ) =: Cl(LM) ! LM. Each fibre is a Clifford algebra. Given a “spin structure on loop space”: L^Spin(M) ! L Spin(M), we Define the Spinor bundle on loop space: L^Spin(M) × F =: F(LM) ! LM. This is a bundle of rigged L^Spin(d) Hilbert spaces, where each fibre is a Fock space. The Clifford bundle acts on the Fock bundle fibrewise. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality 1. Finite dimensional Clifford algebras and spinors 2. Clifford algebras and spinors in infinite dimensions 3. The spinor bundle on loop space 4. Fusion & Locality Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Smooth loop space of a manifold Definition The space of smooth loops/paths: LM = C1(S1;M) PM = C1([0; π];M) Loops/paths that can be glued together are called compatible. Finite dimensions Infinite dimensions Spinors on loop space Fusion & Locality Interacting strings Strings interact through splitting and merging. Want to relate the fibres F(LM)γ1 and F(LM)γ2 with the fibre F(LM)γ1γ2 .

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