1

Eigenvalue estimates for Dirac operators with torsion II

Prof. Dr. habil. Ilka AgricolaPhilipps-Universitat Marburg

Metz, March 2012

1

Relations between different objects on a Riemannian manifold (Mn, g):

geometricstructures

curvature, contact str.,

almost complex str. . .

solutions offield eqs.

Ric = λ · g

Einstein eq., twistor eq.,

Killing eq., parallel tensor / spinor. . .

new invariant connection∇ and its holonomy

adapted connection,

Berger’s thm if ∇ = ∇g

Hol(M ;∇) ⊂ SO(n)

N.B. ∇g := Levi-Civita connection

– joint research project with Thomas Friedrich, HU Berlin –

2

Goal:

Investigate the spectrum of the Dirac operator of an adaptedconnection with torsion on a mnfd with special geometric structurethrough a suitable twistor equation!

Plan:

• Some special geometries

• Review of spectral estimates of the Riemannian Dirac operator andRiemannian twistor spinors

• Special geometries via connections with torsion

• The square of the Dirac operator with torsion

• Twistor spinors with torsion

3

Examples of special geometries

Example 1: almost Hermitian mnfd

• (S6, gcan): S6 ⊂ R7 has an almostcomplex structure J (J2 = −id)inherited from ”cross product” on R

7.

• J is not integrable, ∇gJ 6= 0

• Problem (Hopf): Does S6 admitan (integrable) complex structure ?

x TxS2

vJ(v) := x× v

S2

for S2 ⊂ R3:

J is an example of a nearly Kahler structure: ∇gXJ(X) = 0

More generally: (M2n, g, J) almost Hermitian mnfd:J almost complex structure, g a compatible Riemannian metric.

Fact: structure group G ⊂ U(n) ⊂ SO(2n), but Hol0(∇g) = SO(2n).

Examples: twistor spaces (CP3, F1,2) with their nK str., compact complex

mnfd with b1(M) odd (6 ∃ Kahler metric) . . .

4

Example 2 – contact mnfd

• (M2n+1, g, η) contact mnfd,η: 1-form (∼= vector field)

• 〈η〉⊥ admits an almost complexstructure J compatible with g

η

J = −∇gη

TxM

〈η〉⊥

• Contact condition: η ∧ (dη)n 6= 0 ⇒ ∇gη 6= 0, i. e. contact structuresare never integrable ! (no analogue on Berger’s list)

• structure group: G ⊂ U(n) ⊂ SO(2n+ 1)

Examples: S2n+1 = SU(n+1)SU(n) , V4,2 = SO(4)

SO(2), M11 = G2

Sp(1), M31 = F4

Sp(3)

Example 3 – Mnfds with G2- or Spin(7)-structure (dim = 7, 8)

• G2 has a 7-dimensional irred. representation,

• Spin(7) has a spin representation of dimension 23 = 8.

Examples: S7 = Spin(7)G2

, MAWk,l = SU(3)

U(1)k,l, V5,2 = SO(5)

SO(3), M8 = G2

SO(4). . .

5

The square of the Riemannian Dirac operator

(Mn, g): compact Riemannian spin mnfd, Σ: spin bdle

Classical Riemannian Dirac operator Dg:

Dfn : Dg : Γ(Σ) −→ Γ(Σ), Dgψ :=∑ni=1 ei · ∇g

eiψ

Properties:

• Dg is elliptic differential operator of first order, essentially self-adjointon L2(Σ), pure point spectrum

• Schrodinger (1932), Lichnerowicz (1962): (Dg)2 = ∆ + 14Scalg

∼ ”‘root of the Laplacian”’ for Scalg = 0

6

Review of Riemannian eigenvalue estimates

SL formula ⇒ EV of (Dg)2: λ ≥ 14 Scalgmin

• optimal only for spinors with 〈∆ψ, ψ〉 = ‖∇gψ‖2 = 0, i. e. parallelspinors, and then Scalgmin = 0

• no parallel spinors if Scalgmin > 0

Thm. Optimal EV estimate: λ ≥ n

4(n− 1)Scalgmin [Friedrich, 1980]

• ”=” if there exists a Killing spinor (KS) ψ: ∇gXψ = const ·X ·ψ ∀X

Link to special geometries:

Thm. ∃ KS ⇔ n = 5 : (M, g) is Sasaki-Einstein mnfd [∈ contact str.]

⇔ n = 6 : (M, g) nearly Kahler mnfd

⇔ n = 7 : (M, g) nearly parallel G2 mnfd

[Friedrich, Kath, Grunewald. . . ]

7

Friedrich’s inequality has two alternative proofs:

• by deforming the connection ∇gXψ ∇g

Xψ + cX · ψ

• by using twistor theory: the twistor or Penrose operator:

Pψ :=n

∑

k=1

ek ⊗[

∇gekψ +

1

nek ·Dgψ

]

satisfies the identity ‖Pψ‖2 + 1

n‖Dgψ‖2 = ‖∇gψ‖2

which, together with the SL formula, yields the integral formula

∫

M

〈(Dg)2ψ, ψ〉dM =n

n− 1

∫

M

‖Pψ‖2dM+n

4(n− 1)

∫

M

Scalg‖ψ‖2dM

and Friedrich’s inequality follows, with equality iff ψ is a twistor spinor,

Pψ = 0 ⇔ ∇gXψ +

1

nX ·Dgψ = 0 ∀X

Furthermore, ψ is automatically a Killing spinor.

8

Special geometries via connections with torsion

Given a mnfd Mn with G-structure (G ⊂ SO(n)), replace ∇g by ametric connection ∇ with torsion that preserves the geometric structure!

torsion: T (X,Y, Z) := g(∇XY −∇YX − [X,Y ], Z)

Special case: require T ∈ Λ3(Mn) (⇔ same geodesics as ∇g)

⇒ g(∇XY,Z) = g(∇gXY,Z) + 1

2 T (X,Y,Z)

1) representation theory yields

- a clear answer which G-structures admit such a connection; if existent,it’s unique and called the ‘characteristic connection’

- a classification scheme for G-structures with characteristic connection:Tx ∈ Λ3(TxM)

G= V1 ⊕ . . .⊕ Vp

2) Analytic tool: Dirac operator /D of the metricconnection with torsion T/3: ‘characteristic Dirac operator’

– generalizes Dolbeault operator and Kostant’s cubic Dirac operator

9

Some characteristic connections

Example 1 – contact mnfd [Friedrich, Ivanov 2000]

A large class admits a char. connection ∇, and Hol0(∇) ⊂ U(n) ⊂SO(2n+ 1). For Sasaki manifolds, the formula is particularly simple,

g(∇cXY,Z) = g(∇g

XY,Z) + 12η ∧ dη(X,Y,Z),

and ∇T = 0 holds. [Kowalski-Wegrzynowski, 1987 for Sasaki]

Example 2 – almost Hermitian 6-mnfd [Friedrich, Ivanov 2000]

(M,g, J), J almost complex, compatible with g

intrinsic torsion ∈ W(2)1 ⊕W(16)

2 ⊕W(12)3 ⊕W(6)

4 [Gray-Hervella, ’80]

∃ a char. connection ∇ ⇔ Nijenhuis tensor g(N(X,Y ), Z) ∈ Λ3(M),

⇔ intrinsic torsion ∈ W1 ⊕W3 ⊕W4

g(∇cXY,Z) := g(∇g

XY,Z) + 12 [g(N(X,Y ), Z) + dΩ(JX, JY, JZ)]

10

N.B. Non-integrable geometries are not necessarily homogeneous. Someof those who are homogeneous fall into the following class:

Example 3 – naturally reductive homogeneous space [IA 2003]

M = G/H reductive space, g = h ⊕ m, 〈, 〉 a scalar product on m.

The PFB G→ G/H induces a metric connection ∇ with torsion

T (X,Y,Z) := −〈[X,Y ]m, Z〉,

called the ‘canonical connection’.

Dfn. M = G/H is called naturally reductive if T ∈ Λ3(M); ∇ coincidesthen with the characteristic connection.

Naturally reductive spaces have the properties ∇T = ∇R = 0direct generalisation of symmetric spaces

Thm. On a compact naturally reductive space (6= Sn, Pn, simple Liegrp), the canonical connection is unique. [Olmos-Reggiani, 2008]

11

The square of the Dirac operator with torsion

With torsion:

(M,g): mnfd with G-structure and charact. connection ∇c, torsion T ,assume ∇cT = 0 (for convenience)

/D: Dirac operator of connection with torsion T/3

generalized SL formula: [IA-Friedrich, 2003]

/D2 = ∆T +1

4Scalg +

1

8||T ||2 − 1

4T 2

[1/3 rescaling: Slebarski (1987), Bismut (1989), Kostant, Goette (1999), IA (2002)]

Applications: 1) spectral properties of /D

2) Vanishing theorems

3) harmonic analysis on homog. non symmetric spaces

[Mehdi, Zierau]

12

(non integrable)

non

homog.

homog.

naturally reductive

Parthasarathy:Kostant:

Hermitianmnfds

Hermitian symm. spacesalmost Herm. nat. red.homogeneous spaces

Kähler mnfdsalmost Herm. mnfds(nearly/almost/quasi/semi K, Hermitian, loc.conf.K etc.)

symmetric

Dolbeault op. Dolbeault op.

SL: B/K/A−F/S:

almost

(integrable)T = 0 T 6= 0

(Dg)2 = Ω + 18Scal

/D2 = Ω + const

Dg = Dg 6= /D =

(Dg)2 = ∆ + 14Scalg /D2 = ∆T + 1

4Scalg + 18||T ||2 − 1

4T2

13

Spectrum of /D

For eigenvalue estimates, the action of T on the spinor bundle needs tobe known!

Thm. Assume ∇cT = 0 and let ΣM = ⊕µΣµ be the splitting of thespinor bundle into eigenspaces of T . Then:

a) ∇c preserves the splitting of Σ, i. e. ∇cΣµ ⊂ Σµ ∀µ,

b) /D2 T = T /D2, i. e. /D2Σµ ⊂ Σµ ∀µ. [IA-Fr. 2004]

⇒ Estimate on every subbundle of Σµ

Corollary (universal estimate). The first EV λ of /D2 satisfies

λ ≥ 1

4Scalgmin +

1

8‖T‖2 − 1

4max(µ2

1, . . . , µ2k),

where µ1, . . . , µk are the eigenvalues of T .

[Idea: neglect again 〈∆Tψ,ψ〉 = ‖∇cψ‖2 ≥ 0]

14

Universal estimate:

• follows from generalized SL formula

• does not yield Friedrich’s inequality for T → 0

• optimal iff ∃ a ∇c-parallel spinor:

This sometimes happens on mnfds with Scalgmin > 0 !

Results:

• deformation techniques: yield often estimates quadratic in Scalg,require subtle case by case discussion, often restriced curvature range

[IA, Friedrich, Kassuba [PhD], 2008]

• twistor techniques: estimates always linear in Scalg, no curvaturerestriction, rather universal, leads to a twistor eq. with torsion andsometimes to a Killing eq. with torsion

– submitted – [IA, Becker-Bender [PhD], Kim 2010-11]

15

Twistors with torsion

m : TM ⊗ ΣM → ΣM : Clifford multiplication

p = projection on kerm: p(X ⊗ ψ) = X ⊗ ψ + 1n

∑ni=1 ei ⊗ eiXψ

∇s: ∇sXY := ∇g

XY + 2sT (X,Y,−)

(s = 1/4 is the ”standard” normalisation, ∇1/4 = char. conn.)

twistor operator: P s = p ∇s

Fundamental relation: ‖P sψ‖2 + 1n‖Dsψ‖2 = ‖∇sψ‖2

ψ is called s-twistor spinor ⇔ ψ ∈ kerP s ⇔ ∇sXψ + 1

nXDsψ = 0.

A priori, not clear what the right value of s might be:

different scaling in ∇[

s = 14

]

and /D[

s = 14·3

]

!

Idea: Use possible improvements of an eigenvalue estimate as a guideto the ‘right’ twistor spinor

16

Thm (twistor integral formula). Any spinor ϕ satisfies∫

M

〈/D2ϕ,ϕ〉dM =n

n− 1

∫

M

‖P sϕ‖2dM +n

4(n− 1)

∫

M

Scalg‖ϕ‖2dM

+n(n− 5)

8(n− 3)2‖T‖2

∫

‖ϕ‖2dM − n(n− 4)

4(n− 3)2

∫

M

〈T 2ϕ,ϕ〉dM,

where s = n−14(n−3).

Thm (twistor estimate). The first EV λ of /D2 satisfies (n > 3)

λ ≥ n

4(n− 1)Scalgmin +

n(n− 5)

8(n− 3)2‖T‖2 − n(n− 4)

4(n− 3)2max(µ2

1, . . . , µ2k),

where µ1, . . . , µk are the eigenvalues of T , and ”=” iff

• Scalg is constant,

• ψ is a twistor spinor for sn = n−14(n−3),

• ψ lies in Σµ corresponding to the largest eigenvalue of T 2.

17

• reduces to Friedrich’s estimate for T → 0

• estimate is good for Scalgmin dominant (compared to ‖T‖2)

Ex. (M6, g) of class W3 (”balanced”), Stab(T ) abelian

Known: µ = 0,±√

2‖T‖, no ∇c-parallel spinors

twistor estimate: λ ≥ 3

10Scalgmin −

7

12‖T‖2

universal estimate: λ ≥ 1

4Scalgmin −

3

8‖T‖2

• better than anything obtained by deformation

On the other hand:

Ex. (M5, g) Sasaki: deformation technique yielded better estimates.

18

Twistor and Killing spinors with torsion

Thm (twistor eq). ψ is an sn-twistor spinor (P snψ = 0) iff

∇cXψ +

1

nX · /Dψ +

1

2(n− 3)(X ∧ T ) · ψ = 0,

Dfn. ψ is a Killing spinor with torsion if ∇snXψ = κX ·ψ for sn = n−1

4(n−3).

⇔ ∇cψ −[

κ+µ

2(n− 3)

]

X · ψ +1

2(n− 3)(X ∧ T )ψ = 0.

In particular:

• ψ is a twistor spinor with torsion for the same value sn

• κ satisfies the quadratic eq.

n

[

κ+µ

2(n− 3)

]2

=1

4(n− 1)Scalg +

n− 5

8(n− 3)2‖T‖2 − n− 4

4(n− 3)2µ2

• Scalg = constant.

19

In general, this twistor equation cannot be reduced to a Killing equation.

. . . with one exception: n = 6

Thm. Assume ψ is a s6-twistor spinor for some µ 6= 0. Then:

• ψ is a /D eigenspinor with eigenvalue

/Dψ =1

3

[

µ− 4‖T‖2

µ

]

ψ

• the twistor equation for s6 is equivalent to the Killing equation∇sψ = λX · ψ for the same value of s.

Observation:

The Riemannian Killing / twistor eq. and their analogue with torsionbehave very differently depending on the geometry!

20

Integrability conditions & Einstein-Sasaki manifolds

Thm (curvature in spin bundle). For any spinor field ψ:

Ricc(X) · ψ = −2n

∑

k=1

ekRc(X, ek)ψ +1

2X dT · ψ.

Thm (integrability condition). Let ψ be a Killing spinor with torsionwith Killing number κ, set λ := 1

2(n−3). Then ∀X :

Ricc(X)ψ = −16sκ(X T )ψ + 4(n− 1)κ2Xψ + (1 − 12λ2)(X σT )ψ +

+2(2λ2 + λ)∑

ek(T (X, ek) T )ψ .

Cor. A 5-dimensional Einstein-Sasaki mnfd with its characteristicconnection cannot have Killing spinors with torsion.

21

Killing spinors on nearly Kahler manifolds

• (M6, g, J) 6-dimensional nearly Kahler manifold

- ∇c its characteristic connection, torsion is parallel

- Einstein, ‖T‖2 = 215Scalg

- T has EV µ = 0,±2‖T‖- ∃ 2 Riemannian KS ϕ± ∈ Σ±2‖T‖, ∇c-parallel

- univ. estimate = twistor estimate, λ ≥ 215Scalg

Thm. The following classes of spinors coincide:

• Riemannian Killing spinors • ∇c-parallel spinors

• Killing spinors with torsion • Twistor spinors with torsion

There is exactly one such spinor ϕ± in each of the subbundles Σ±2‖T‖.

22

A 5-dimensional example with Killing spinors withtorsion

• 5-dimensional Stiefel manifold M = SO(4)/SO(2), so(4) = so(2)⊕m

• Jensen metric: m = m4 ⊕ m1 (irred. components of isotropy rep.),

〈(X, a), (Y, b)〉t = 12β(X,Y ) + 2t · ab, t > 0, β = Killing form

∣

∣

m4

• t = 1/2: undeformed metric: 2 parallel spinors

• t = 2/3: Einstein-Sasaki with 2 Riemannian Killing spinors

• For general t: metric contact structure in direction m1 withcharacteristic connection ∇ satisfying ∇T = 0

• ‖T‖2 = 4t, Scalg = 8 − 2t, Ricg = diag(2 − t, 2 − t, 2 − t, 2 − t, 2t).

• Universal estimate: λ ≥ 2(1 − t) =: βuniv

• Twistor estimate: λ ≥ 52 − 25

8 t =: βtw

23

4/9

EV

βuniv

βtw

t

Result: there exist 2 twistor spinors with torsion for t = 2/5, and theseare even Killing spinors with torsion.

24

Generalisation: deformed Sasaki mnfds with Killingspinors with torsion

• (M, g, ξ, η): Sasaki mnfd, η: contact form, dimension 2n+ 1

• Tanno deformation of metrics: gt := tg + (t2 − t)η ⊗ η, again Sasakiwith ξt = 1

tξ, ηt = tη (t ∈ R∗)

• If Einstein-Sasaki: admits two Riemannian Killing spinors

Thm. Let (M, g, ξ, η) be Einstein-Sasaki, gt the Tanno deformation.Then there exists a t s.t. (M, gt, ξt, ηt) has two Killing spinors withtorsion.

– establishes existence of examples in all odd dimensions –