About the Kohn-Dirac operator on CR manifolds Felipe Leitner (Univ. - PowerPoint PPT Presentation

About the Kohn-Dirac operator on CR manifolds Felipe Leitner (Univ. Greifswald) Jurekfest University of Warsaw – Faculty of Physics September 2019 Felipe Leitner Kohn-Dirac operator

Strictly pseudo-convex boundaries Let Ω = { ρ < 0 } ⊆ C m +1 ( m ≥ 1) be domain of holomorphy with defining C ∞ -function ρ ⇒ Levi form is positive semidefinite m ∂ρ � w ∈ T 10 ( ∂ Ω) L ( w , w ) = w j ¯ w k ≥ 0 , ∂ z j ∂ ¯ z k j , k =0

Strictly pseudo-convex boundaries Let Ω = { ρ < 0 } ⊆ C m +1 ( m ≥ 1) be domain of holomorphy with defining C ∞ -function ρ ⇒ Levi form is positive semidefinite m ∂ρ � w ∈ T 10 ( ∂ Ω) L ( w , w ) = w j ¯ w k ≥ 0 , ∂ z j ∂ ¯ z k j , k =0 If L ( w , w ) > 0 for w � = 0 then H = T ( ∂ Ω) ∩ J ( T ( ∂ Ω)) is contact on ∂ Ω ◮ ( H , J ) is CR structure on M := ∂ Ω ◮ with contact 1-form θ = d ρ ◦ J ◮ C ⊗ H = H 10 ⊕ H 01 We call directions in transverse

Abstract CR manifolds with pseudo-Hermitian form ( M 2 m +1 , H , J ) Let be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = ker θ ) characteristic vector T = T θ on M ֒ →

Abstract CR manifolds with pseudo-Hermitian form ( M 2 m +1 , H , J ) Let be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = ker θ ) characteristic vector T = T θ on M ֒ → Tanaka-Webster connection ∇ θ on H and metric connection w.r.t. g θ = 1 2 d θ ( · , J · ) on H ◮ Tor ∇ ( X , Y ) = d θ ( X , Y ) T for X , Y ∈ H ◮ Webster torsion τ ( X ) = − 1 2 ([ T , X ] + J [ T , JX ]) ◮ U ( m ) is structure group of ∇ θ ◮

Abstract CR manifolds with pseudo-Hermitian form ( M 2 m +1 , H , J ) Let be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = ker θ ) characteristic vector T = T θ on M ֒ → Tanaka-Webster connection ∇ θ on H and metric connection w.r.t. g θ = 1 2 d θ ( · , J · ) on H ◮ Tor ∇ ( X , Y ) = d θ ( X , Y ) T for X , Y ∈ H ◮ Webster torsion τ ( X ) = − 1 2 ([ T , X ] + J [ T , JX ]) ◮ U ( m ) is structure group of ∇ θ ◮ ֒ → = scal θ ρ ric 4 m · d θ Ψ-Einstein condition θ

CR spinors and Kohn-Dirac operator If c 1 ( K ) ≡ 0 mod 2, the transverse distribution ( H , g θ ) → M admits spinor bundle Σ( M ) with

CR spinors and Kohn-Dirac operator If c 1 ( K ) ≡ 0 mod 2, the transverse distribution ( H , g θ ) → M admits spinor bundle Σ( M ) with inner product ( · , · ) L 2 on sections Γ o (Σ) ◮ ∇ Σ Tanaka-Webster spinor connection ◮ Σ( M ) = � m ( µ r = m − 2 r eigenvalues of id θ r =0 Σ µ r 2 ) ◮ Kohn-Dirac operator ◮ 2 m � e i · ∇ Σ D θ Φ = e i Φ , Φ ∈ Γ(Σ) j =1 ( e 1 , · · · , e 2 m ) ONB of transverse distribution ( H , g θ )

CR spinors and Kohn-Dirac operator If c 1 ( K ) ≡ 0 mod 2, the transverse distribution ( H , g θ ) → M admits spinor bundle Σ( M ) with inner product ( · , · ) L 2 on sections Γ o (Σ) ◮ ∇ Σ Tanaka-Webster spinor connection ◮ Σ( M ) = � m ( µ r = m − 2 r eigenvalues of id θ r =0 Σ µ r 2 ) ◮ Kohn-Dirac operator ◮ 2 m � e i · ∇ Σ D θ Φ = e i Φ , Φ ∈ Γ(Σ) j =1 ( e 1 , · · · , e 2 m ) ONB of transverse distribution ( H , g θ ) Note: Eigenspinors D θ Φ = λ · Φ with λ � = 0 decompose to Φ = Φ µ r + Φ µ r +1 ∈ Γ(Σ µ r ⊕ Σ µ r +1 ) of type ( r , r + 1)

Analytic properties of D θ Let ( M 2 m +1 , H , J , θ ) be closed spin CR manifold ( m ≥ 2) D θ is formally self-adjoint ◮ D θ is not elliptic ◮ the square ◮ D 2 θ : Γ(Σ µ r ) → Γ(Σ µ r ) is hypoelliptic for the non-extremal bundles r � = 0 , m spec ( D 2 θ ) is pure point spectrum ν i ◮ with ν i ≥ 0 and ν i → ∞ (Stadtm¨ uller ’17) eigenspaces E ν i are finite-dimensional for ν i > 0 ◮ in general, the space of harmonic spinors has dim H = ∞ ◮

Example: The standard CR sphere The Euclidean ball B 1 (0) ⊆ C m +1 is convex boundary = sphere S 2 m +1 of radius 1, Ψ-Herm.form θ o = d ρ ◦ J , scal θ o ≡ 4 m ( m + 1) = const . as homogeneous space: S 2 m +1 = ˜ U ( m +1) / ˜ with ˜ U ( m ) U ( m +1) ⊆ Spin (2 m +2)

Example: The standard CR sphere The Euclidean ball B 1 (0) ⊆ C m +1 is convex boundary = sphere S 2 m +1 of radius 1, Ψ-Herm.form θ o = d ρ ◦ J , scal θ o ≡ 4 m ( m + 1) = const . as homogeneous space: S 2 m +1 = ˜ U ( m +1) / ˜ with ˜ U ( m ) U ( m +1) ⊆ Spin (2 m +2) By Frobenius reciprocity we have L 2 (Σ) ∼ � = V γ ⊗ Hom ˜ U ( m ) ( V γ , Σ) γ where V γ = ( V γ ( a , b ) , π ) ( a , b ≥ 0) are the (relevant) irreducible representations of ˜ U ( m + 1)

Spectrum of S 2 m +1 Parthasarathy-type formula scal θ o � � A ◦ π ∗ ( casimir u ( m +1) ) + · A D 2 8 θ o ( v ⊗ A ) = v ⊗ + 2 · A ◦ π ∗ ( T ) 2 + d θ o · A ◦ π ∗ ( T ) + 1 2 d θ 2 o · A for A ∈ Hom ˜ U ( m ) ( V γ ( a , b ) , Σ) with a , b ≥ 0

Spectrum of S 2 m +1 Parthasarathy-type formula scal θ o � � A ◦ π ∗ ( casimir u ( m +1) ) + · A D 2 8 θ o ( v ⊗ A ) = v ⊗ + 2 · A ◦ π ∗ ( T ) 2 + d θ o · A ◦ π ∗ ( T ) + 1 2 d θ 2 o · A for A ∈ Hom ˜ U ( m ) ( V γ ( a , b ) , Σ) with a , b ≥ 0 This gives the eigenvalues � { λ = ± 4( m − r + a )( r + 1 + b ) | a , b ≥ 0 ∧ 0 ≤ r < m } with lower bound for the non-zero eigenvalues of type ( r , r + 1): scal θ o λ 2 ≥ 4( m − r )( r + 1) · 4 m ( m + 1)

Schr¨ odinger-Lichnerowicz-type formula = Kohn-Dirac operator D θ − tr H ( ∇ Σ ◦ ∇ Σ ) ∆ Σ = spinor sub-Laplacian = ∆ 10 + ∆ 01 ( w.r.t. J on H ) − 1 i ∇ Σ 4 m ρ ric = θ · + 2 m (∆ 10 − ∆ 01 ) T

Schr¨ odinger-Lichnerowicz-type formula = Kohn-Dirac operator D θ − tr H ( ∇ Σ ◦ ∇ Σ ) ∆ Σ = spinor sub-Laplacian = ∆ 10 + ∆ 01 ( w.r.t. J on H ) − 1 i ∇ Σ 4 m ρ ric = θ · + 2 m (∆ 10 − ∆ 01 ) T Lichnerowicz-type formula (Petit ’05): scal θ D 2 ∆ Σ d θ · ∇ Σ = + − θ T 4 � 1 − id θ � � 1 + id θ � D 2 ⇒ = ∆ 10 + ∆ 01 θ 2 m 2 m � � +1 scal θ + 1 md θ · ρ ric θ 4

Lower bounds for eigenvalues Let ( M 2 m +1 , θ ), m ≥ 2, be closed with positive Ricci-form ρ ric > 0 θ Set scal min s o := 4 m ( m + 1) If Φ = Φ µ r + Φ µ r +1 is some ( r , r + 1)-eigenspinor for λ � = 0 then 2 m 2  for r = 0 , m − 1 m − 1    r = m 2 , m − 2  m ( m + 2) for   2    r = m − 1 λ 2 m ( m + 1) for ≥ s o · 2  2( r +1)( m − r ) m  0 < r < m − 2 for   m − r +1 2     2( r +1)( m − r ) m m  for 2 < r < m − 1 r

Lower bounds for eigenvalues Let ( M 2 m +1 , θ ), m ≥ 2, be closed with positive Ricci-form ρ ric > 0 θ Set scal min s o := 4 m ( m + 1) If Φ = Φ µ r + Φ µ r +1 is some ( r , r + 1)-eigenspinor for λ � = 0 then 2 m 2  for r = 0 , m − 1 m − 1    r = m 2 , m − 2  m ( m + 2) for   2    r = m − 1 λ 2 m ( m + 1) for ≥ s o · 2  2( r +1)( m − r ) m  0 < r < m − 2 for   m − r +1 2     2( r +1)( m − r ) m m  for 2 < r < m − 1 r On Ψ -Einstein spaces ( M 2 m +1 , θ ) the lower bound is given by λ 2 ≥ 4( r + 1)( m − r ) s o

Ψ-Killing spinors realizes the general lower bound for λ 2 If Φ = Φ µ r + Φ µ r +1 ⇒ Φ µ r or Φ µ r +1 satisfiy some twistor equation in the simultaneous case: ∇ Σ ∇ Σ X 10 Φ µ r = 0 , X 01 Φ µ r +1 = 0 ∇ Σ λ X 01 Φ µ r = − 2( r +1) X 01 Φ µ r +1 ∇ Σ λ X 10 Φ µ r +1 = − 2( m + r ) X 10 Φ µ r for any transversal vector X = X 10 + X 01 ∈ H

Ψ-Killing spinors realizes the general lower bound for λ 2 If Φ = Φ µ r + Φ µ r +1 ⇒ Φ µ r or Φ µ r +1 satisfiy some twistor equation in the simultaneous case: ∇ Σ ∇ Σ X 10 Φ µ r = 0 , X 01 Φ µ r +1 = 0 ∇ Σ λ X 01 Φ µ r = − 2( r +1) X 01 Φ µ r +1 ∇ Σ λ X 10 Φ µ r +1 = − 2( m + r ) X 10 Φ µ r for any transversal vector X = X 10 + X 01 ∈ H ֒ → Φ = strong (transversal) Ψ -Killing spinors Example : Such Φ exist on 3 -Sasakian spaces (with related Ψ-Hermitian form θ )

Parallel spinors We call Φ ∈ Γ(Σ) transversally parallel iff ∇ Σ X Φ = 0 ∀ X ∈ H Due to torsion R ( X , JY )Φ = d θ ( X , JY ) ∇ Σ T Φ and − 1 ∇ Σ 2 m ρ ric T Φ = θ · Φ If we modify the Tanaka-Webster connection by 1 ˆ ∇ T Φ := ∇ Σ 2 m ρ ric T Φ + θ · Φ then Φ is ˆ ∇ -parallel

Parallel spinors We call Φ ∈ Γ(Σ) transversally parallel iff ∇ Σ X Φ = 0 ∀ X ∈ H Due to torsion R ( X , JY )Φ = d θ ( X , JY ) ∇ Σ T Φ and − 1 ∇ Σ 2 m ρ ric T Φ = θ · Φ If we modify the Tanaka-Webster connection by 1 ˆ ∇ T Φ := ∇ Σ 2 m ρ ric T Φ + θ · Φ then Φ is ˆ ∇ -parallel Computing the curvature ˆ R shows: tr C ˆ θ is Ψ-Einstein iff R ( X , Y ) = 0 ∀ X , Y ∈ TM hol ( ˆ i.e. the holonomy algebra is reduced: ∇ ) ⊆ su ( m )

Characterization of Ψ-Einstein spaces Theorem: Let ( M 2 m +1 , θ ) be 1-connected. Then we have the equivalent conditions: M admits transversally parallel spinors ◮ basic holonomy Hol ( θ ) ⊆ SU ( m ) ◮ θ is Ψ-Einstein ◮ (no matter of torsion τ and sign of scal θ )