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} ⊆ Cm+1 (m ≥ 1) be domain of holomorphy with defining C ∞-function ρ ⇒ Levi form is positive semidefinite L(w, w) =

m

• j,k=0

∂ρ ∂zj∂¯ zk wj ¯ wk ≥ 0, w ∈ T 10(∂Ω)

Strictly pseudo-convex boundaries

Let Ω = {ρ < 0} ⊆ Cm+1 (m ≥ 1) be domain of holomorphy with defining C ∞-function ρ ⇒ Levi form is positive semidefinite L(w, w) =

m

• j,k=0

∂ρ ∂zj∂¯ zk wj ¯ wk ≥ 0, w ∈ T 10(∂Ω) 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 = dρ ◦ J We call directions in C ⊗ H = H10 ⊕ H01 transverse

Abstract CR manifolds with pseudo-Hermitian form

Let (M2m+1, H, J) be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = kerθ ) ֒ → characteristic vector T = T θ on M

Abstract CR manifolds with pseudo-Hermitian form

Let (M2m+1, H, J) be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = kerθ ) ֒ → characteristic vector T = T θ on M and Tanaka-Webster connection ∇θ on H

◮

metric connection w.r.t. gθ = 1

2dθ(·, J·) on H ◮

Tor ∇(X, Y ) = dθ(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

Let (M2m+1, H, J) be strictly Ψ-convex CR manifold with adapted Ψ-Hermitian 1-form θ ( i.e. H = kerθ ) ֒ → characteristic vector T = T θ on M and Tanaka-Webster connection ∇θ on H

◮

metric connection w.r.t. gθ = 1

2dθ(·, J·) on H ◮

Tor ∇(X, Y ) = dθ(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 ֒ → ρric

θ

= scalθ 4m · dθ Ψ-Einstein condition

CR spinors and Kohn-Dirac operator

If c1(K) ≡ 0 mod 2, the transverse distribution (H, gθ) → M admits spinor bundle Σ(M) with

CR spinors and Kohn-Dirac operator

If c1(K) ≡ 0 mod 2, the transverse distribution (H, gθ) → M admits spinor bundle Σ(M) with

◮

inner product (·, ·)L2 on sections Γo(Σ)

◮

Tanaka-Webster spinor connection ∇Σ

◮

Σ(M) = m

r=0 Σµr

( µr = m − 2r eigenvalues of idθ

2 ) ◮

Kohn-Dirac operator DθΦ =

2m

• j=1

ei · ∇Σ

eiΦ,

Φ ∈ Γ(Σ) (e1, · · · , e2m) ONB of transverse distribution (H, gθ)

CR spinors and Kohn-Dirac operator

If c1(K) ≡ 0 mod 2, the transverse distribution (H, gθ) → M admits spinor bundle Σ(M) with

◮

inner product (·, ·)L2 on sections Γo(Σ)

◮

Tanaka-Webster spinor connection ∇Σ

◮

Σ(M) = m

r=0 Σµr

( µr = m − 2r eigenvalues of idθ

2 ) ◮

Kohn-Dirac operator DθΦ =

2m

• j=1

ei · ∇Σ

eiΦ,

Φ ∈ Γ(Σ) (e1, · · · , e2m) ONB of transverse distribution (H, gθ) Note: Eigenspinors DθΦ = λ · Φ with λ = 0 decompose to Φ = Φµr + Φµr+1 ∈ Γ(Σµr ⊕ Σµr+1)

• f type (r, r + 1)

Analytic properties of Dθ

Let (M2m+1, H, J, θ) be closed spin CR manifold (m ≥ 2)

◮

Dθ is formally self-adjoint

◮

Dθ is not elliptic

◮

the square D2

θ : Γ(Σµr ) → Γ(Σµr )

is hypoelliptic for the non-extremal bundles r = 0, m

◮

spec(D2

θ) is pure point spectrum νi

with νi ≥ 0 and νi → ∞ (Stadtm¨ uller ’17)

◮

eigenspaces Eνi are finite-dimensional for νi > 0

◮

in general, the space of harmonic spinors has dimH = ∞

Example: The standard CR sphere

The Euclidean ball B1(0) ⊆ Cm+1 is convex boundary = sphere S2m+1 of radius 1, Ψ-Herm.form θo = dρ ◦ J, scalθo ≡ 4m(m + 1) = const. as homogeneous space: S2m+1 = ˜ U(m+1)/˜ U(m) with ˜ U(m+1) ⊆ Spin(2m+2)

Example: The standard CR sphere

The Euclidean ball B1(0) ⊆ Cm+1 is convex boundary = sphere S2m+1 of radius 1, Ψ-Herm.form θo = dρ ◦ J, scalθo ≡ 4m(m + 1) = const. as homogeneous space: S2m+1 = ˜ U(m+1)/˜ U(m) with ˜ U(m+1) ⊆ Spin(2m+2) By Frobenius reciprocity we have L2(Σ) ∼ =

• γ

Vγ ⊗ Hom˜

U(m)(Vγ, Σ)

where Vγ = (Vγ(a,b), π) (a, b ≥ 0) are the (relevant) irreducible representations of ˜ U(m + 1)

Spectrum of S2m+1

Parthasarathy-type formula D2

θo(v⊗A) = v⊗

• A ◦ π∗(casimiru(m+1)) +

scalθo 8

· A + 2 · A ◦ π∗(T)2 + dθo · A ◦ π∗(T) + 1

2dθ2

• · A
• for A ∈ Hom˜

U(m)(Vγ(a,b), Σ) with a, b ≥ 0

Spectrum of S2m+1

Parthasarathy-type formula D2

θo(v⊗A) = v⊗

• A ◦ π∗(casimiru(m+1)) +

scalθo 8

· A + 2 · A ◦ π∗(T)2 + dθo · A ◦ π∗(T) + 1

2dθ2

• · A
• for A ∈ Hom˜

U(m)(Vγ(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): λ2 ≥ 4(m − r)(r + 1) · scalθo 4m(m + 1)

Schr¨

• dinger-Lichnerowicz-type formula

Dθ = Kohn-Dirac operator ∆Σ = − trH(∇Σ ◦ ∇Σ) spinor sub-Laplacian = ∆10 + ∆01 ( w.r.t. J on H ) ∇Σ

T

= − 1 4mρric

θ · +

i 2m (∆10 − ∆01)

Schr¨

• dinger-Lichnerowicz-type formula

Dθ = Kohn-Dirac operator ∆Σ = − trH(∇Σ ◦ ∇Σ) spinor sub-Laplacian = ∆10 + ∆01 ( w.r.t. J on H ) ∇Σ

T

= − 1 4mρric

θ · +

i 2m (∆10 − ∆01) Lichnerowicz-type formula (Petit ’05): D2

θ

= ∆Σ + scalθ 4 − dθ · ∇Σ

T

⇒ D2

θ

=

• 1 − idθ

2m

• ∆10

+

• 1 + idθ

2m

• ∆01

+1 4

• scalθ + 1

mdθ · ρric

θ

Lower bounds for eigenvalues

Let (M2m+1, θ), m ≥ 2, be closed with positive Ricci-form ρric

θ

> 0 Set so := scalmin 4m(m + 1) If Φ = Φµr + Φµr+1 is some (r, r + 1)-eigenspinor for λ = 0 then λ2 ≥ so ·                   

2m2 m−1

for r = 0, m − 1 m(m + 2) for r = m

2 , m−2 2

m(m + 1) for r = m−1

2 2(r+1)(m−r)m m−r+1

for 0 < r < m−2

2 2(r+1)(m−r)m r

for

m 2 < r < m − 1

Lower bounds for eigenvalues

Let (M2m+1, θ), m ≥ 2, be closed with positive Ricci-form ρric

θ

> 0 Set so := scalmin 4m(m + 1) If Φ = Φµr + Φµr+1 is some (r, r + 1)-eigenspinor for λ = 0 then λ2 ≥ so ·                   

2m2 m−1

for r = 0, m − 1 m(m + 2) for r = m

2 , m−2 2

m(m + 1) for r = m−1

2 2(r+1)(m−r)m m−r+1

for 0 < r < m−2

2 2(r+1)(m−r)m r

for

m 2 < r < m − 1

On Ψ-Einstein spaces (M2m+1, θ) the lower bound is given by λ2 ≥ 4(r + 1)(m − r)so

Ψ-Killing spinors

If Φ = Φµr + Φµr+1 realizes the general lower bound for λ2 ⇒ Φµr

• r Φµr+1 satisfiy some twistor equation

in the simultaneous case: ∇Σ

X10Φµr = 0,

∇Σ

X01Φµr+1 = 0

∇Σ

X01Φµr = − λ 2(r+1)X01Φµr+1

∇Σ

X10Φµr+1 = − λ 2(m+r)X10Φµr

for any transversal vector X = X10 + X01 ∈ H

Ψ-Killing spinors

If Φ = Φµr + Φµr+1 realizes the general lower bound for λ2 ⇒ Φµr

• r Φµr+1 satisfiy some twistor equation

in the simultaneous case: ∇Σ

X10Φµr = 0,

∇Σ

X01Φµr+1 = 0

∇Σ

X01Φµr = − λ 2(r+1)X01Φµr+1

∇Σ

X10Φµr+1 = − λ 2(m+r)X10Φµr

for any transversal vector X = X10 + X01 ∈ 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

XΦ

= ∀X ∈ H Due to torsion R(X, JY )Φ = dθ(X, JY )∇Σ

T Φ and

∇Σ

TΦ

= − 1 2mρric

θ · Φ

If we modify the Tanaka-Webster connection by ˆ ∇TΦ := ∇Σ

TΦ +

1 2mρric

θ · Φ

then Φ is ˆ ∇-parallel

Parallel spinors

We call Φ ∈ Γ(Σ) transversally parallel iff ∇Σ

XΦ

= ∀X ∈ H Due to torsion R(X, JY )Φ = dθ(X, JY )∇Σ

T Φ and

∇Σ

TΦ

= − 1 2mρric

θ · Φ

If we modify the Tanaka-Webster connection by ˆ ∇TΦ := ∇Σ

TΦ +

1 2mρric

θ · Φ

then Φ is ˆ ∇-parallel Computing the curvature ˆ R shows: θ is Ψ-Einstein iff trC ˆ R(X, Y ) = 0 ∀X, Y ∈ TM i.e. the holonomy algebra is reduced: hol( ˆ ∇) ⊆ su(m)

Characterization of Ψ-Einstein spaces

Theorem: Let (M2m+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 scalθ)

Characterization of Ψ-Einstein spaces

Theorem: Let (M2m+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 scalθ) Existence: (1) For M = ∂Ω ⊆ Cm+1 strictly Ψ-convex boundary

◮ volC = dz0 ∧ · · · ∧ dzm induces Ψ-Einstein structure on ∂Ω ◮ H(M) is spin with transversally parallel spinor

(2) S1-bundles L with c1(L) = [ω] ∈ H2(M, Z)

• ver compact hyperK¨

ahler manifolds have Hol(θ) = Sp(m

2 )

CR invariants

Let ˜ θ = e2σθ be another adapted Ψ-Hermitian form on (M, H, J) Then D0 := Dθ|Γ(Σ0) : Γ(Σ0) → Γ(Σ2 ⊕ Σ−2) is CR-covariant for m = even, i.e. D˜

θ(e−(m+1)σ · ˜

Φ0) = e(−m+2)σ · DθΦ0

CR invariants

Let ˜ θ = e2σθ be another adapted Ψ-Hermitian form on (M, H, J) Then D0 := Dθ|Γ(Σ0) : Γ(Σ0) → Γ(Σ2 ⊕ Σ−2) is CR-covariant for m = even, i.e. D˜

θ(e−(m+1)σ · ˜

Φ0) = e(−m+2)σ · DθΦ0 In particular, H0 = space of harmonic spinors in Γ(Σ0) is a CR invariant

Vanishing Theorems

Lichnerowicz-type formula for spinors in Γ(Σ0): D∗

0D0

= ∆Σ + scalθ 4 ⇒

◮ vanishing theorem: If scalθ > 0 on closed (M, θ) then

H0 = {0} (no harmonic spinors)

◮ vanishing of twisted Kohn-Rossi cohomology group:

H

m 2 (M,

√ K) ∼ = H0 = {0}

Vanishing Theorems

Lichnerowicz-type formula for spinors in Γ(Σ0): D∗

0D0

= ∆Σ + scalθ 4 ⇒

◮ vanishing theorem: If scalθ > 0 on closed (M, θ) then

H0 = {0} (no harmonic spinors)

◮ vanishing of twisted Kohn-Rossi cohomology group:

H

m 2 (M,

√ K) ∼ = H0 = {0}

◮ obstruction: If Kohn-Rossi group

H

m 2 (M,

√ K) = {0}, then no θ with positive scalθ > 0 exists on M Examples for obstruction: (1) compact quotients of Heisenberg groups (2) S1-bundle with basic holonomy Hol(θ) = Sp(m

2 )

SpinC-structures

Let (M2m+1, H, J) be given some SpinC-structure with determinant bundle Ldet = Eℓ , ℓ ∈ Z, where E =

m+2

√ K−1 ֒ → Dℓ Kohn-Dirac operator of weight ℓ

SpinC-structures

Let (M2m+1, H, J) be given some SpinC-structure with determinant bundle Ldet = Eℓ , ℓ ∈ Z, where E =

m+2

√ K−1 ֒ → Dℓ Kohn-Dirac operator of weight ℓ Then

◮ The µr-component of Dℓ with µr = −ℓ is CR-covariant ◮ obtain more vanishing theorems for Kohn-Rossi groups ◮ further obstructions for

ρric

θ

> 0 and scalθ > 0 in terms of Kohn-Rossi groups

More vanishing theorems

Lichnerowicz-type formula for Dℓ of weight ℓ ∈ Z: D2

ℓ

=

• 1 − idθ

2m

• ∆10

+

• 1 + idθ

2m

• ∆01

− i 2

• ℓ

m + 2 + idθ 2m

• ρric

θ

+

• 1 +

iℓdθ 2m(m + 2)

• · scalθ

4

More vanishing theorems

Lichnerowicz-type formula for Dℓ of weight ℓ ∈ Z: D2

ℓ

=

• 1 − idθ

2m

• ∆10

+

• 1 + idθ

2m

• ∆01

− i 2

• ℓ

m + 2 + idθ 2m

• ρric

θ

+

• 1 +

iℓdθ 2m(m + 2)

• · scalθ

4

• Theorem. Let M2m+1 be closed with m ≥ 2.

If |ℓ| ≤ m + 2 and ρθ > 0, then any harmonic spinor is in the extremal bundles Σm ⊕ Σ−m In particular, the qth Kohn-Rossi group H0,q(M) vanishes for q = 1, . . . , m − 1.