Eigenvalue estimates for Dirac operators with torsion II

Eigenvalue estimates for Dirac operators with torsion II Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Metz, March 2012 1

Relations between different objects on a Riemannian manifold (M n, g): geometric structures Ric = λ g curvature, contact str., almost complex str... adapted connection, Berger s thm if = g solutions of field eqs. new invariant connection and its holonomy Einstein eq., twistor eq., Killing eq., parallel tensor / spinor... N.B. g := Levi-Civita connection Hol(M; ) SO(n) joint research project with Thomas Friedrich, HU Berlin 1

Goal: Investigate the spectrum of the Dirac operator of an adapted connection with torsion on a mnfd with special geometric structure through a suitable twistor equation! Plan: Some special geometries Review of spectral estimates of the Riemannian Dirac operator and Riemannian twistor spinors Special geometries via connections with torsion The square of the Dirac operator with torsion Twistor spinors with torsion 2

Examples of special geometries Example 1: almost Hermitian mnfd (S 6,g can ): S 6 R 7 has an almost complex structure J (J 2 = id) inherited from cross product on R 7. J is not integrable, g J 0 Problem (Hopf): Does S 6 admit an (integrable) complex structure? for S 2 R 3 : J(v) := x v v x T x S 2 S 2 J is an example of a nearly Kähler structure: g X J(X) = 0 More generally: (M 2n,g, J) almost Hermitian mnfd: J almost complex structure, g a compatible Riemannian metric. Fact: structure group G U(n) SO(2n), but Hol 0 ( g ) = SO(2n). Examples: twistor spaces (CP 3,F 1,2 ) with their nk str., compact complex mnfd with b 1 (M) odd ( Kähler metric)... 3

Example 2 contact mnfd (M 2n+1,g, η) contact mnfd, η: 1-form ( = vector field) η admits an almost complex structure J compatible with g η T x M J = g η η Contact condition: η (dη) n 0 g η 0, i. e. contact structures are never integrable! (no analogue on Berger s list) structure group: G U(n) SO(2n + 1) Examples: S 2n+1 = SU(n+1) SU(n), V 4,2 = SO(4) SO(2), M11 = G 2 Sp(1), M31 = F 4 Sp(3) Example 3 Mnfds with G 2 - or Spin(7)-structure (dim = 7,8) G 2 has a 7-dimensional irred. representation, Spin(7) has a spin representation of dimension 2 3 = 8. Examples: S 7 = Spin(7) G 2, Mk,l AW = SU(3) U(1) k,l, V 5,2 = SO(5) SO(3), M8 = G 2 SO(4)... 4

The square of the Riemannian Dirac operator (M n, g): compact Riemannian spin mnfd, Σ: spin bdle Classical Riemannian Dirac operator D g : Dfn : D g : Γ(Σ) Γ(Σ), D g ψ := n i=1 e i g e i ψ Properties: D g is elliptic differential operator of first order, essentially self-adjoint on L 2 (Σ), pure point spectrum Schrödinger (1932), Lichnerowicz (1962): (D g ) 2 = + 1 4 Scalg root of the Laplacian for Scal g = 0 5

Review of Riemannian eigenvalue estimates SL formula EV of (D g ) 2 : λ 1 4 Scalg min optimal only for spinors with ψ, ψ = g ψ 2 = 0, i. e. parallel spinors, and then Scal g min = 0 no parallel spinors if Scal g min > 0 Thm. Optimal EV estimate: λ n 4(n 1) Scalg min [Friedrich, 1980] = if there exists a Killing spinor (KS) ψ: g Xψ = 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 Kähler mnfd n = 7 : (M, g) nearly parallel G 2 mnfd [Friedrich, Kath, Grunewald... ] 6

Friedrich s inequality has two alternative proofs: by deforming the connection g X ψ g X ψ + cx ψ by using twistor theory: the twistor or Penrose operator: Pψ := n e k k=1 [ g e k ψ + 1 ] n e k D g ψ satisfies the identity Pψ 2 + 1 n Dg ψ 2 = g ψ 2 which, together with the SL formula, yields the integral formula (D g ) 2 ψ, ψ dm = n Pψ 2 dm+ n Scal g ψ 2 dm n 1 4(n 1) M M and Friedrich s inequality follows, with equality iff ψ is a twistor spinor, Pψ = 0 g X ψ + 1 n X Dg ψ = 0 X M Furthermore, ψ is automatically a Killing spinor. 7

Special geometries via connections with torsion Given a mnfd M n with G-structure (G SO(n)), replace g by a metric connection with torsion that preserves the geometric structure! torsion: T(X,Y, Z) := g( X Y Y X [X,Y ], Z) Special case: require T Λ 3 (M n ) ( same geodesics as g ) g( X Y,Z) = g( g X Y,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: T x Λ 3 (T x M) G = V 1... V p 2) Analytic tool: Dirac operator /D of the metric connection with torsion T/3: characteristic Dirac operator generalizes Dolbeault operator and Kostant s cubic Dirac operator 8

Some characteristic connections Example 1 contact mnfd [Friedrich, Ivanov 2000] A large class admits a char. connection, and Hol 0 ( ) U(n) SO(2n + 1). For Sasaki manifolds, the formula is particularly simple, g( c X Y,Z) = g( g X Y,Z) + 1 2 η 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) Λ 3 (M), intrinsic torsion W 1 W 3 W 4 g( c X Y,Z) := g( g X Y,Z) + 1 2 [g(n(x, Y ),Z) + dω(jx,jy,jz)] 9

N.B. Non-integrable geometries are not necessarily homogeneous. Some of 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 called the canonical connection. T(X,Y,Z) := [X,Y ] m,z, Dfn. M = G/H is called naturally reductive if T Λ 3 (M); coincides then with the characteristic connection. Naturally reductive spaces have the properties T = R = 0 direct generalisation of symmetric spaces Thm. On a compact naturally reductive space ( S n, P n, simple Lie grp), the canonical connection is unique. [Olmos-Reggiani, 2008] 10

The square of the Dirac operator with torsion With torsion: (M,g): mnfd with G-structure and charact. connection c, torsion T, assume c T = 0 (for convenience) /D: Dirac operator of connection with torsion T/3 generalized SL formula: [IA-Friedrich, 2003] /D 2 = T + 1 4 Scalg + 1 8 T 2 1 4 T 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] 11

SL: B/K/A F/S: (D g ) 2 = + 1 4 Scalg /D 2 = T + 1 4 Scalg + 1 8 T 2 1 4 T 2 non homog. Kähler mnfds almost Herm. mnfds (nearly/almost/quasi/semi K, Hermitian, loc.conf.k etc.) D g = D g /D = Dolbeault op. Dolbeault op. Hermitian symm. spaces almost Herm. nat. red. homogeneous spaces homog. Parthasarathy: Kostant: (D g ) 2 = Ω + 1 8 Scal /D 2 = Ω + const almost Hermitian mnfds symmetric naturally reductive 12 T = 0 (integrable) T 0 (non integrable)

Spectrum of /D For eigenvalue estimates, the action of T on the spinor bundle needs to be known! Thm. Assume c T = 0 and let ΣM = µ Σ µ be the splitting of the spinor bundle into eigenspaces of T. Then: a) c preserves the splitting of Σ, i. e. c Σ µ Σ µ µ, b) /D 2 T = T /D 2, i.e. /D 2 Σ µ Σ µ µ. [IA-Fr. 2004] Estimate on every subbundle of Σ µ Corollary (universal estimate). The first EV λ of /D 2 satisfies λ 1 4 Scalg min + 1 8 T 2 1 4 max(µ2 1,...,µ 2 k), where µ 1,...,µ k are the eigenvalues of T. [Idea: neglect again T ψ, ψ = c ψ 2 0] 13

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 Scal g min > 0! Results: deformation techniques: yield often estimates quadratic in Scal g, require subtle case by case discussion, often restriced curvature range [IA, Friedrich, Kassuba [PhD], 2008] twistor techniques: estimates always linear in Scal g, no curvature restriction, rather universal, leads to a twistor eq. with torsion and sometimes to a Killing eq. with torsion submitted [IA, Becker-Bender [PhD], Kim 2010-11] 14

Twistors with torsion m : TM ΣM ΣM: Clifford multiplication p = projection on kerm: p(x ψ) = X ψ + 1 n n i=1 e i e i Xψ s : s X Y := 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 s ψ 2 + 1 n Ds ψ 2 = s ψ 2 ψ is called s-twistor spinor ψ kerp s s X ψ + 1 n XDs ψ = 0. A priori, not clear what the right value of s might be: different scaling in [ s = 1 4] and /D [ s = 1 4 3]! Idea: Use possible improvements of an eigenvalue estimate as a guide to the right twistor spinor 15

Thm (twistor integral formula). Any spinor ϕ satisfies /D 2 n ϕ,ϕ dm = P s ϕ 2 n dm + n 1 4(n 1) M where s = n 1 4(n 3). + M n(n 5) 2 8(n 3) 2 T ϕ 2 dm M n(n 4) 4(n 3) 2 Thm (twistor estimate). The first EV λ of /D 2 satisfies (n > 3) λ n 4(n 1) Scalg min + n(n 5) 8(n 3) 2 T 2 where µ 1,...,µ k are the eigenvalues of T, and = iff Scal g is constant, ψ is a twistor spinor for s n = n 1 4(n 3), Scal g ϕ 2 dm M n(n 4) 4(n 3) 2 max(µ2 1,...,µ 2 k), T 2 ϕ,ϕ dm, ψ lies in Σ µ corresponding to the largest eigenvalue of T 2. 16

reduces to Friedrich s estimate for T 0 estimate is good for Scal g min dominant (compared to T 2 ) Ex. (M 6,g) of class W 3 ( balanced ), Stab(T) abelian Known: µ = 0, ± 2 T, no c -parallel spinors twistor estimate: λ 3 10 Scalg min 7 T 2 12 universal estimate: λ 1 4 Scalg min 3 T 2 8 better than anything obtained by deformation On the other hand: Ex. (M 5,g) Sasaki: deformation technique yielded better estimates. 17

Twistor and Killing spinors with torsion Thm (twistor eq). ψ is an s n -twistor spinor (P s n ψ = 0) iff c Xψ + 1 n X /Dψ + 1 (X T) ψ = 0, 2(n 3) Dfn. ψ is a Killing spinor with torsion if s n X ψ = κx ψ for s n = n 1 [ ] c µ 1 ψ κ + X ψ + (X T)ψ = 0. 2(n 3) 2(n 3) In particular: ψ is a twistor spinor with torsion for the same value s n 4(n 3). κ satisfies the quadratic eq. [ ] 2 µ 1 n κ + = 2(n 3) 4(n 1) Scalg + n 5 8(n 3) 2 T 2 n 4 4(n 3) 2µ2 Scal g = constant. 18

In general, this twistor equation cannot be reduced to a Killing equation.... with one exception: n = 6 Thm. Assume ψ is a s 6 -twistor spinor for some µ 0. Then: ψ is a /D eigenspinor with eigenvalue /Dψ = 1 3 [µ 4 ] T 2 ψ µ the twistor equation for s 6 is equivalent to the Killing equation s ψ = λx ψ for the same value of s. Observation: The Riemannian Killing / twistor eq. and their analogue with torsion behave very differently depending on the geometry! 19

Integrability conditions & Einstein-Sasaki manifolds Thm (curvature in spin bundle). For any spinor field ψ: Ric c (X) ψ = 2 n k=1 e k R c (X, e k )ψ + 1 X dt ψ. 2 Thm (integrability condition). Let ψ be a Killing spinor with torsion with Killing number κ, set λ := 1 2(n 3). Then X: Ric c (X)ψ = 16sκ(X T)ψ + 4(n 1)κ 2 Xψ + (1 12λ 2 )(X σ T )ψ + +2(2λ 2 + λ) e k (T(X,e k ) T)ψ. Cor. A 5-dimensional Einstein-Sasaki mnfd with its characteristic connection cannot have Killing spinors with torsion. 20

Killing spinors on nearly Kähler manifolds (M 6,g, J) 6-dimensional nearly Kähler manifold - c its characteristic connection, torsion is parallel - Einstein, T 2 = 2 15 Scalg - T has EV µ = 0, ±2 T - 2 Riemannian KS ϕ ± Σ ±2 T, c -parallel - univ. estimate = twistor estimate, λ 2 15 Scalg Thm. The following classes of spinors coincide: Riemannian Killing spinors Killing spinors with torsion c -parallel spinors Twistor spinors with torsion There is exactly one such spinor ϕ ± in each of the subbundles Σ ±2 T. 21

A 5-dimensional example with Killing spinors with torsion 5-dimensional Stiefel manifold M = SO(4)/SO(2), so(4) = so(2) m Jensen metric: m = m 4 m 1 (irred. components of isotropy rep.), (X,a),(Y,b) t = 1 2 β(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 m 1 with characteristic connection satisfying T = 0 T 2 = 4t, Scal g = 8 2t, Ric g = diag(2 t, 2 t, 2 t,2 t, 2t). Universal estimate: Twistor estimate: λ 2(1 t) =: β univ λ 5 2 25 8 t =: β tw 22

EV β univ 4/9 β tw t Result: there exist 2 twistor spinors with torsion for t = 2/5, and these are even Killing spinors with torsion. 23

Generalisation: deformed Sasaki mnfds with Killing spinors with torsion (M, g, ξ, η): Sasaki mnfd, η: contact form, dimension 2n + 1 Tanno deformation of metrics: g t := tg + (t 2 t)η η, again Sasaki with ξ t = 1 t ξ, η t = tη (t R ) If Einstein-Sasaki: admits two Riemannian Killing spinors Thm. Let (M, g, ξ, η) be Einstein-Sasaki, g t the Tanno deformation. Then there exists a t s.t. (M, g t,ξ t,η t ) has two Killing spinors with torsion. establishes existence of examples in all odd dimensions 24