IGA Lecture V: Applications to Verlinde Formulas Eckhard Meinrenken - PowerPoint PPT Presentation

IGA Lecture V: Applications to Verlinde Formulas Eckhard Meinrenken Adelaide, September 9, 2011 Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Review We assume G compact, simple, simply connected. We consider q-Hamiltonian G -spaces, Φ: M → G : 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 V: Applications to Verlinde Formulas

Review We constructed a canonical twisted Spin c -structure, (Φ , S op ): C l( TM ) ��� A (h ∨ ) . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Review We constructed a canonical twisted Spin c -structure, (Φ , S op ): C l( TM ) ��� A (h ∨ ) . We defined a level k pre-quantization of d ω = − Φ ∗ η (Φ , E ): M × C ��� A ( k ) . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Review We constructed a canonical twisted Spin c -structure, (Φ , S op ): C l( TM ) ��� A (h ∨ ) . We defined a level k pre-quantization of d ω = − Φ ∗ η (Φ , E ): M × C ��� A ( k ) . Then (Φ , E ⊗ S op ): C l( TM ) ��� A ( k +h ∨ ) defines a push-forward in twisted K -homology 0 ( G , A ( k +h ∨ ) ) . Φ ∗ : K G 0 ( M , C l( TM )) → K G Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Review We constructed a canonical twisted Spin c -structure, (Φ , S op ): C l( TM ) ��� A (h ∨ ) . We defined a level k pre-quantization of d ω = − Φ ∗ η (Φ , E ): M × C ��� A ( k ) . Then (Φ , E ⊗ S op ): C l( TM ) ��� A ( k +h ∨ ) defines a push-forward in twisted K -homology 0 ( G , A ( k +h ∨ ) ) . Φ ∗ : K G 0 ( M , C l( TM )) → K G The l.h.s. contains the fundamental class [ M ]. The r.h.s. is the level k fusion ring R k ( G ), by the FHT theorem. We define Q ( M ) := Φ ∗ ([ M ]) ∈ R k ( G ) . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring Notation T ⊂ G maximal torus, P + = P ∩ t ∗ + dominant weights, θ ∈ P + weight of adjoint representation (highest root), · basic inner product on g ∼ = g ∗ : θ · θ = 2, ρ ∈ P + shortest weight in P ∩ int( t ∗ + ), h ∨ = 1 + ρ · θ dual Coxeter number Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) A = { ξ ∈ t + | θ · ξ ≤ 1 } is the fundamental alcove. Definition The level k weights are elements of P k = P ∩ kA . ρ ρ = θ θ G = SU(3) G = Spin(5) k = 3 k = 4 Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For λ ∈ P k define the special element t λ = exp( λ + ρ k +h ∨ ) ∈ T . Definition The level k fusion ring (Verlinde ring) is the quotient R k ( G ) = R ( G ) / I k ( G ) where I k ( G ) = { χ ∈ R ( G ) | χ ( t λ ) = 0 ∀ λ ∈ P k } . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Remark R k ( G ) is the fusion ring of level k projective representations of the loop group LG. (But we don’t need that here.) Remark For G = SU( r + 1) , the level k fusion ideal has generators χ ( k +1) ̟ 1 , . . . χ ( k + r ) ̟ 1 where ̟ 1 ∈ P + labels the defining representation. Similar descriptions exist for the compact symplectic groups. (Bouwknegt-Ridout, 2006) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Some properties of R k ( G ) = R ( G ) / I k ( G ): R k ( G ) is unital ring with involution. R k ( G ) has finite Z -basis the images τ µ of χ µ , µ ∈ P k . Thus R k ( G ) = Z [ P k ] . R k ( G ) has a trace, R k ( G ) → Z , τ �→ τ G where τ G µ = δ µ, 0 . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Notation Tensor coefficents N µ 1 µ 2 µ 3 = ( χ µ 1 χ µ 2 χ µ 3 ) G , µ i ∈ P + Level k fusion coefficents N ( k ) µ 1 µ 2 µ 3 = ( τ µ 1 τ µ 2 τ µ 3 ) G , µ i ∈ P k . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Notation Tensor coefficents N µ 1 µ 2 µ 3 = ( χ µ 1 χ µ 2 χ µ 3 ) G , µ i ∈ P + Level k fusion coefficents N ( k ) µ 1 µ 2 µ 3 = ( τ µ 1 τ µ 2 τ µ 3 ) G , µ i ∈ P k . Then N ( k ) µ 1 µ 2 µ 3 = N µ 1 µ 2 µ 3 , k >> 0 . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Example: SU(2) For G = SU(2), identify P + = { 0 , 1 , . . . } , P k = { 0 , 1 , . . . , k } . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Example: SU(2) For G = SU(2), identify P + = { 0 , 1 , . . . } , P k = { 0 , 1 , . . . , k } . Ring structure of R (SU(2)) χ l χ m = χ l + m + χ l + m − 2 + . . . + χ | l − m | . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Example: SU(2) For G = SU(2), identify P + = { 0 , 1 , . . . } , P k = { 0 , 1 , . . . , k } . Ring structure of R (SU(2)) χ l χ m = χ l + m + χ l + m − 2 + . . . + χ | l − m | . One finds I k (SU(2)) = � χ k +1 � . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Example: SU(2) For G = SU(2), identify P + = { 0 , 1 , . . . } , P k = { 0 , 1 , . . . , k } . Ring structure of R (SU(2)) χ l χ m = χ l + m + χ l + m − 2 + . . . + χ | l − m | . One finds I k (SU(2)) = � χ k +1 � . Quotient map R ( G ) → R k ( G ) is ‘signed reflection’ across indices k + 1 , 2 k + 3 , 3 k + 5 , . . . . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

Example: SU(2) For G = SU(2), identify P + = { 0 , 1 , . . . } , P k = { 0 , 1 , . . . , k } . Ring structure of R (SU(2)) χ l χ m = χ l + m + χ l + m − 2 + . . . + χ | l − m | . One finds I k (SU(2)) = � χ k +1 � . Quotient map R ( G ) → R k ( G ) is ‘signed reflection’ across indices k + 1 , 2 k + 3 , 3 k + 5 , . . . . Example Calculation of τ 3 τ 4 in R 5 (SU(2)): χ 3 χ 4 = χ 7 + χ 5 + χ 3 + χ 1 ⇒ τ 3 τ 4 = τ 3 + τ 1 since χ 7 �→ − τ 5 , χ 5 �→ τ 5 . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For general G the quotient map R ( G ) = Z [ P + ] → R k ( G ) = Z [ P k ] is ‘signed reflection’ for a shifted Stiefel diagram. 3 A Shifted affine Weyl action at level k = 3, G = SU(3) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For general G the quotient map R ( G ) = Z [ P + ] → R k ( G ) = Z [ P k ] is ‘signed reflection’ for a shifted Stiefel diagram. 3 A Shifted affine Weyl action at level k = 3, G = SU(3) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For general G the quotient map R ( G ) = Z [ P + ] → R k ( G ) = Z [ P k ] is ‘signed reflection’ for a shifted Stiefel diagram. Shifted affine Weyl action at level k = 3, G = SU(3) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For general G the quotient map R ( G ) = Z [ P + ] → R k ( G ) = Z [ P k ] is ‘signed reflection’ for a shifted Stiefel diagram. Shifted affine Weyl action at level k = 3, G = SU(3) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) For general G the quotient map R ( G ) = Z [ P + ] → R k ( G ) = Z [ P k ] is ‘signed reflection’ for a shifted Stiefel diagram. Shifted affine Weyl action at level k = 3, G = SU(3) Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Evaluation of characters at t λ = exp( λ + ρ k +h ∨ ) descends to the fusion ring: R k ( G ) → C , τ �→ τ ( t λ ) . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Evaluation of characters at t λ = exp( λ + ρ k +h ∨ ) descends to the fusion ring: R k ( G ) → C , τ �→ τ ( t λ ) . R k ( G ) ⊗ C has another basis ˜ τ µ s.t. ˜ τ µ ( t λ ) = δ λ,µ . In the new basis, τ µ ˜ ˜ τ ν = δ µ,ν ˜ τ ν . Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) Evaluation of characters at t λ = exp( λ + ρ k +h ∨ ) descends to the fusion ring: R k ( G ) → C , τ �→ τ ( t λ ) . R k ( G ) ⊗ C has another basis ˜ τ µ s.t. ˜ τ µ ( t λ ) = δ λ,µ . In the new basis, τ µ ˜ ˜ τ ν = δ µ,ν ˜ τ ν . The bases are related by the S -matrix: � S − 1 0 ,ν S ∗ τ µ = µ,ν ˜ τ ν ; ν ∈ P k here S is a symmetric, unitary matrix with S 0 ,ν > 0. Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) ⇒ Verlinde formula for level k fusion coefficients: � S µ 1 ,ν S µ 2 ,ν S µ 3 ,ν N ( k ) µ 1 µ 2 µ 3 = . S 0 ,ν ν ∈ P k Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas

The level k fusion ring (Verlinde ring) ⇒ Verlinde formula for level k fusion coefficients: � S µ 1 ,ν S µ 2 ,ν S µ 3 ,ν N ( k ) µ 1 µ 2 µ 3 = . S 0 ,ν ν ∈ P k This is one of several formulas called ‘Verlinde formulas’ – this is not the difficult one. Eckhard Meinrenken IGA Lecture V: Applications to Verlinde Formulas