Metaplectic-c Quantization Yucong Jiang jiangyc@math.utoronto.ca - PowerPoint PPT Presentation

Metaplectic-c Quantization Yucong Jiang jiangyc@math.utoronto.ca University of Toronto CMS Winter Meeting December 9, 2019

Outline This is an ongoing project with Yael Karshon. Our approach of metaplectic-c quantization is based on Herald Hess [Hess, 1981]. Without using the metaplectic representation, this approach is different from the approach of Robinson and Rawsley [Robinson and Rawnsley, 1989]. The analog of half form bundles in mp-c case Partial connections Pairing maps and Blattner’s formula

Review of KS theory of geometric quantization The Konstant-Souriau recipe of geometric quantization: ◮ A prequantizable metaplectic manifold ( M , ω ). Then one proceeds to define partial connections, inner products and then polarized sections and quantum Hilbert spaces etc. In the mpc quantization, one combines the second and third steps into one:

Review of KS theory of geometric quantization The Konstant-Souriau recipe of geometric quantization: ◮ A prequantizable metaplectic manifold ( M , ω ). ◮ A hermitian line bundle L → M with a hermitian connection whose curvature equals to 1 i � ω . Then one proceeds to define partial connections, inner products and then polarized sections and quantum Hilbert spaces etc. In the mpc quantization, one combines the second and third steps into one:

Review of KS theory of geometric quantization The Konstant-Souriau recipe of geometric quantization: ◮ A prequantizable metaplectic manifold ( M , ω ). ◮ A hermitian line bundle L → M with a hermitian connection whose curvature equals to 1 i � ω . ◮ A polarization F . The metaplecitc structure on M enables us to define the half form bundle δ F associated to F : δ F ⊗ δ F ∼ = det( F ). Then one proceeds to define partial connections, inner products and then polarized sections and quantum Hilbert spaces etc. In the mpc quantization, one combines the second and third steps into one:

Review of KS theory of geometric quantization The Konstant-Souriau recipe of geometric quantization: ◮ A prequantizable metaplectic manifold ( M , ω ). ◮ A hermitian line bundle L → M with a hermitian connection whose curvature equals to 1 i � ω . ◮ A polarization F . The metaplecitc structure on M enables us to define the half form bundle δ F associated to F : δ F ⊗ δ F ∼ = det( F ). ◮ The quantization line bundle is defined as L ⊗ δ − 1 whose F sections are L -valued half forms normal to F . Then one proceeds to define partial connections, inner products and then polarized sections and quantum Hilbert spaces etc. In the mpc quantization, one combines the second and third steps into one:

Quantization line bundle in mpc case ◮ A metaplectic-c manifold ( M , ω ) equipped with a principal mp-c bundle ( ˜ P , ˜ γ ) and a polarization F with typical fiber F . This version of quantization line bundles coincides with the one in KS theory if we start with a metaplectic manifold.

Quantization line bundle in mpc case ◮ A metaplectic-c manifold ( M , ω ) equipped with a principal mp-c bundle ( ˜ P , ˜ γ ) and a polarization F with typical fiber F . ◮ Reduce the symplectic frame bundle P to P F which is a Sp F -principal bundle. Pull it back to ˜ P and denote the pullback by ˜ P F which is a Mp c F -bundle. This version of quantization line bundles coincides with the one in KS theory if we start with a metaplectic manifold.

Quantization line bundle in mpc case ◮ A metaplectic-c manifold ( M , ω ) equipped with a principal mp-c bundle ( ˜ P , ˜ γ ) and a polarization F with typical fiber F . ◮ Reduce the symplectic frame bundle P to P F which is a Sp F -principal bundle. Pull it back to ˜ P and denote the pullback by ˜ P F which is a Mp c F -bundle. ◮ There is a unique homomorphism χ F : Mp F → C × such that ( χ F ◦ proj ) 2 = det ◦ res , where res : Sp F → GL ( F , C ) is the F → C × by restriction map to F . Define χ c F : Mp c χ c F ([ g , z ]) = χ F ( g ) z . This version of quantization line bundles coincides with the one in KS theory if we start with a metaplectic manifold.

Quantization line bundle in mpc case ◮ A metaplectic-c manifold ( M , ω ) equipped with a principal mp-c bundle ( ˜ P , ˜ γ ) and a polarization F with typical fiber F . ◮ Reduce the symplectic frame bundle P to P F which is a Sp F -principal bundle. Pull it back to ˜ P and denote the pullback by ˜ P F which is a Mp c F -bundle. ◮ There is a unique homomorphism χ F : Mp F → C × such that ( χ F ◦ proj ) 2 = det ◦ res , where res : Sp F → GL ( F , C ) is the F → C × by restriction map to F . Define χ c F : Mp c χ c F ([ g , z ]) = χ F ( g ) z . ◮ Define the quantization line bundle as the associated line bundle to ˜ P F and χ c F : Q F := ˜ P F × χ c F C . This version of quantization line bundles coincides with the one in KS theory if we start with a metaplectic manifold.

Partial connections We want to define a F -connection on Q F . Let me explain the construction of partial connections in a simplified case: we assume F has a complement polarization G , i.e. F ⊕ G = TM C . The goal is to construct a mp c F -valued connection one form θ on ˜ P F such that the induced covariant derivative on Q F along F does not depend on the choice of G .

Sketch of the construction γ to ˜ P F via ˜ → ˜ ◮ The pullback ˜ γ F of ˜ P F ֒ P serves as the u (1)-component of θ . Lemma ∇ F , G , X ∈ F is independent of the choices of G. Hence we get a X well-defined partial connection on Q F .

Sketch of the construction γ to ˜ P F via ˜ → ˜ ◮ The pullback ˜ γ F of ˜ P F ֒ P serves as the u (1)-component of θ . ◮ Use the complement G and Bott connetions to define a symplectic connection on TM C such that ∇ TM C (Γ( F )) ⊂ Γ( F ), ∇ TM C (Γ( G )) ⊂ Γ( G ). Equivalently, we obtain a principal connection on P which can be reduced to P F . Let’s denote its pullback to ˜ P F by A F , G which serves as the sp F -component of θ . Lemma ∇ F , G , X ∈ F is independent of the choices of G. Hence we get a X well-defined partial connection on Q F .

Sketch of the construction γ to ˜ P F via ˜ → ˜ ◮ The pullback ˜ γ F of ˜ P F ֒ P serves as the u (1)-component of θ . ◮ Use the complement G and Bott connetions to define a symplectic connection on TM C such that ∇ TM C (Γ( F )) ⊂ Γ( F ), ∇ TM C (Γ( G )) ⊂ Γ( G ). Equivalently, we obtain a principal connection on P which can be reduced to P F . Let’s denote its pullback to ˜ P F by A F , G which serves as the sp F -component of θ . γ F + A F , G is an ordinary connection one form on ˜ ◮ θ F , G = ˜ P F . As a result, we obtain a covariant derivative ∇ F , G on Q F . Lemma ∇ F , G , X ∈ F is independent of the choices of G. Hence we get a X well-defined partial connection on Q F .

Pairing maps The polarization F on M 2 n we take into account satisfies the following conditions: 1. Positivity: i ω ( u , ¯ u ) ≥ 0 for all u ∈ F . 2. F ∩ ¯ F has constant rank. A pairing of polarizations ( F 1 , F 2 ) we take into account further F 2 = D C has a constant rank. satisfies F 1 ∩ ¯ Theorem (Pairing maps) There is a pairing map Q F 1 × M Q F 2 → D 1 ( TM / D ) Note that if F 1 = F 2 = F, we obtain a pairing of Q F itself.

Sketch of the proof We consider the further reduced bundle P 1 , 2 = P F 1 ∩ P F 2 consisting of symplectic frames ( e 1 , · · · , e d , u 1 , · · · , u r , f 1 , · · · , f d , iv 1 , · · · , iv r ) such that ( e 1 , · · · , e d ) ∈ F ( D ), ( e 1 , · · · , e d , u 1 , · · · , u r ) ∈ F ( F 1 ) and ( e 1 , · · · , e d , v 1 , · · · , v r ) ∈ F ( F 2 ). Then Q F 1 = ˜ P 1 , 2 × χ c F 1 C , Q F 2 = ˜ P 1 , 2 × χ c F 2 C . For ( α, β ) ∈ Q 1 × M Q 2 and e ∈ F ( D ). Lift e to ˜ e ∈ P 1 , 2 . Assume α (˜ e ) = λ and β (˜ e ) = µ . Then define � α, β � ( e ) := λ ¯ µ.

Blattner’s formula [Blattner, 1977] For X ∈ D , α ∈ Γ( F 1 ), and β ∈ Γ( F 2 ), L X � α, β � = �∇ X α, β � + � α, ∇ X β � + κ F 1 + ¯ F 2 ( X ) � α, β � , where κ is an invariant defined on a differential system associated to F 1 + ¯ F 2 . As a corollary, we have Corollary If F 1 + ¯ F 2 is integrable, then for polarized sections α ∈ Γ( F 1 ) and β ∈ Γ( F 2 ) , the function � α, β � is constant along leaves of D. As a result, � α, β � descends to a 1-density on M / D.

Blattner, R. J. (1977). The metalinear geometry of non-real polarizations. In Differential geometrical methods in mathematical physics , pages 11–45. Springer. Hess, H. (1981). On a geometric quantization scheme generalizing those of kostant-souriau and czyz. In Differential geometric methods in mathematical physics , pages 1–35. Springer. Robinson, P. L. and Rawnsley, J. H. (1989). The Metaplectic Representation, Mp c Structures and Geometric Quantization . Number 410. American Mathematical Soc.