VAFA-WITTEN ESTIMATES FOR COMPACT SYMMETRIC SPACES

S. GOETTE

Abstract. We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G−rk H ≤ 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.

Herzlich gave an optimal upper bound for the lowest eigenvalue of the Dirac operator on spheres with arbitrary Riemannian metrics in [9] using a method developed by Vafa and Witten in [14]. More precisely, he proved that for every metric ḡ on S^{n} that is pointwise larger than the round metric g, the first eigenvalue λ_{1}(D̄^{2}) of the Dirac operator with respect to ḡ is not larger than the first Dirac eigenvalue λ_{1}(D^{2}) of the round sphere.

Herzlich asked if there are other Riemannian manifolds with optimal Vafa-Witten bounds, in particular if the Fubini-Study metric on ℂP^{2m−1} has this property. In the present note we give positive answers to both questions by generalising Herzlich’s results to symmetric spaces G/H of compact type, where rk G − rk H ≤ 1. In particular, we improve a recent estimate by Davaux and Min-Oo for complex projective spaces in [4], see Example 6.2 below.

1. Theorem. Let M = G/H be a simply connected symmetric space of compact type with rk G − rk H ≤ 1 and assume that M is G-spin. Let g be a symmetric metric, and let D denote the corresponding Dirac operator on M. If ḡ is another metric with ḡ ≥ g on T M and D̄ is the corresponding Dirac operator, then

λ_{1}(D̄^{2}) ≤ λ_{1}(D^{2}) .

In the case of equality, we have ḡ = g.

For an arbitrary Riemannian metric ḡ such that c^{2}ḡ ≥ g for some suitable positive constant c^{2}, the theorem implies

λ_{1}(D̄^{2}) ≤ c^{2} λ_{1}(D^{2}) .

We combine the methods of [9] and [4] with related estimates in [7]. In particular, we compare D̄ to an operator D̄_{1} with nonvanishing kernel acting on the same sections as D̄_{0} = D̄ ⊗ id_{ℝ^{k}}. We use Parthasarathy’s formula to compute λ_{1}(D^{2}), and we exhibit a similar formula to estimate ∥D̄_{1} − D̄∥. Both formulas give the same value for g = ḡ.

To prove that D̄_{1} has a kernel, we use the invariance of the Fredholm index if rk H = rk G. If rk H = rk G − 1 we use the invariance of the mod-2-index as in [7]. Unfortunately, both approaches fail if rk G−rk H ≥ 2. Note that in [9], a spectral flow argument was used instead in the case rk H = rk G − 1.

In [2], Baum applied the Vafa-Witten approach to Lipschitz maps f of high degree from a closed Riemannian spin manfifold of dimension 2n to S^{2n}. We extend her result to Lipschitz maps of high Â-degree from higher dimensional closed Riemannian spin manifolds to S^{2n}. Recall that if N^{n} and M^{m} are closed oriented manifolds, [N] is the fundamental class of N and ω is a generator of H^{m}(M; ℤ), then the Â-degree is defined as

2000 Mathematics Subject Classification. 53C27; 53C35; 58J50.

Key words and phrases. Dirac operator, Vafa-Witten eigenvalue estimate, symmetric space.

Supported in part by DFG special programme “Global Differential Geometry”.

degÂ f = (Â(T N) ⌣ f^{∗}ω)[N] .

If n = m, then degÂ is the usual degree.

Let D_{N} denote the untwisted Dirac operator on N, and let 0 ≤ λ_{1}(D_{N}^{2}) ≤ λ_{2}(D_{N}^{2}) ≤ … denote the eigenvalues of D_{N}^{2}, where each eigenvalue is repeated according to its multiplicity.

2. Theorem. Let N be a closed Riemannian spin manifold and let k ∈ ℕ. Then there is no Lipschitz map N → S^{2m} of Lipschitz constant 1 with

|degÂ f| > 2^{m−1} (k − 1)

unless

λ_{k}(D_{N}^{2}) ≤ λ_{1}(D_{M}) .

If we replace the target manifold by a symmetric space G/H with rk G = rk H, then we can prove similar theorems where the degree condition on f is replaced by

⟨[N], Â(T N) ⌣ f^{∗}α⟩ > C(k − 1) ,

with α ∈ H^{∗}(M; ℤ) and C > 0 depending only on M. Alternatively, there is no 1-Lipschitz map N → M with

|degÂ f| > C(k − 1)

for some constant C depending on M, unless λ_{k}(D_{N}^{2}) ≤ c for some c depending only on M, with c > λ_{1}(D_{M}) in general. This will be discussed in section 7. In the case rk G−rk H = 1 we need a K-theoretic condition on f instead of the cohomological Â-degree condition. We may thus ask if even-dimensional spheres are the only manifolds that admit optimal Vafa-Witten bounds for Lipschitz spin maps of sufficiently large Â-degree.

1. Remark. The actual estimate and the index theoretic considerations involved in the proof of Theorem 1 are very similar to those used for the scalar curvature comparison result in [7]. Nevertheless, we need a stronger metric condition (ḡ ≥ g on T M, not just on Λ^{2}T M). This is due to the fact that the Vafa-Witten estimate is related to Gromov’s K-length, whereas scalar curvature comparison is related to K-area, see [8].

2. Remark. Note that Theorem 1 still holds if we replace D̄ by D̄′ = D̄ + A, where A is a symmetric endomorphism of the spinor bundle. This is because in the proof, we can then replace D̄_{i} by D̄′_{i} = D̄_{i} + A ⊗ id for i = 0, 1. Then D̄_{1} still has a kernel, and of course ∥D̄′_{0} − D̄′_{1}∥ = ∥D̄_{1} − D̄_{0}∥.

3. Remark. Remark 2 in particular implies that on a symmetric space,

λ_{1}(D + A) ≤ λ_{1}(D)

for all symmetric endomorphisms A of the spinor bundle. In Section 8 we will exhibit an isotropy irreducible quotient G/H = SO(5)/SO(3) of compact Lie groups with a normal metric, for which the generalised estimate above does not hold. This shows in particular that the methods of the present paper do not readily generalise to normal homogeneous spaces.

Acknowlegdements. The rigidity statement in Theorem 1 was pointed out to us by Listing. An anonymous referee helped to make this article more readable.

1. The Smallest Dirac Eigenvalue of a Symmetric Metric

Let M = G/H be symmetric quotient of compact Lie groups of equal rank with Lie algebras 𝔥 ⊂ 𝔤. We fix an Ad-invariant metric g on 𝔤 and let 𝔭 = 𝔥^{⊥}. The tangent bundle of M can be written as

T M = G ×_{H} 𝔭 ,

where H acts on 𝔭 by the restriction of the adjoint action Ad_{G}. The scalar product g induces a symmetric Riemannian metric on M that will also be denoted by g. Note that if M is symmetric, then the Levi-Civita connection on T M is precisely the reductive connection on G ×_{H} 𝔭.

Let Σ be a spinor module for the Clifford algebra of 𝔭. If we assume that M is G-spin, then the H-representation of H on 𝔭 induces an action σ: H → EndΣ. The natural metric on Σ is σ-invariant, so we obtain a G-equivariant metric g on SM. Equipped with this metric and the reductive connection, the G-equivariant vector bundle

SM = G ×_{H} Σ → M

can be identified with the spinor bundle of M, and the G-equivariant Dirac operator D acts as an essentially selfadjoint operator on L^{2}(M; SM).

Let Ĝ denote the set of equivalence classes of irreducible G-representations. We write γ: G → EndV^{γ} for all γ ∈ Ĝ. By Frobenius reciprocity and the Peter-Weyl theorem, the L^{2}-sections of SM can be decomposed G-equivariantly into a Hilbert sum

L^{2}(M; SM) = \overline{⨁_{γ∈Ĝ} V^{γ} ⊗ Hom_{H}(V^{γ}, Σ)} .  (1.1)

The Dirac operator preserves this decomposition, and we have

D|_{V^{γ}⊗Hom_{H}(V^{γ},Σ)} = id_{V^{γ}} ⊗^{γ}D

with

γD = ∑_{i=1}^{n} γ_{e_{i}}^{∗} ⊗ c_{i} ∈ End(Hom_{H}(V^{γ}, Σ)) .  (1.2)

Here e_{1}, …, e_{n} is a g-orthonormal base of 𝔭 and c_{i} denotes Clifford multiplication by e_{i}.

Let c_{G}^{γ} denote the Casimir operator of the G-action γ with respect to the metric g, which acts as a scalar on V^{γ}. If M is symmetric, then the Casimir operator c_{H}^{σ} of σ also acts as a scalar, even though σ is in general not irreducible. By Parthasarathy’s formula [12],

γD^{2} = c_{G}^{γ^{∗}} + c_{H}^{σ} .

1.3. Proposition. The smallest eigenvalue of D^{2} is given by

λ_{1}(D^{2}) = min{ c_{G}^{γ^{∗}} + c_{H}^{σ} | γ ∈ Ĝ with Hom_{H}(V^{γ}, Σ) ≠ 0 } . □

A more explicit formula for the first eigenvalue in the case rk G = rk H has recently been given in [11].

2. The Vafa-Witten Estimate

Let M = G/H and g as before and consider an arbitrary Riemannian metric ḡ on M with ḡ > g. The corresponding Dirac operator will be denoted D̄. Let λ_{1}(D̄^{2}) denote the smallest eigenvalue of D̄^{2}. As mentioned in the introduction, we will estimate λ_{1}(D̄^{2}) by comparing the related operator D̄_{0} = D̄ ⊗ id_{ℂ^{N}} to an operator D̄_{1} acting on the same space of sections.

Assume that we are given vector bundle W ⊂ V = M × ℂ^{N} → M such that the twisted Dirac operator on SM ⊗W has nonvanishing index; these bundles will be constructed in steps below and in sections 4 and 5. Let ∇^{0} be the trivial connection on V, and let ∇^{1} be another connection for which W ⊂ V is a parallel subbundle. Let D̄_{0}, D̄_{1} denote the corresponding twisted Dirac operators on SM ⊗ ℂ^{N} for the metric ḡ. Then D̄_{0} is just the direct sum of N copies of D̄, whereas D̄_{1} has nontrivial kernel. By a Rayleigh quotient argument, thus

λ_{1}(D̄^{2}) = λ_{1}(D̄_{0}^{2}) ≤ \begin{matrix}∥(D̄_{1} − D̄_{0})s∥ & {2 \atop L^{2}} \\ ∥s∥ & {2 \atop L^{2}} \end{matrix} ≤ sup_{p∈M} ∥(D̄_{1} − D̄_{0})_{p}^{2}∥_{op}  (2.1)

for some 0 ≠ s ∈ ker(D̄_{1}). Here ∥ · ∥_{L^{2}} denotes the L^{2}-norm of sections, whereas ∥ · ∥_{op} denotes the pointwise operator norm of an endomorphism. The operator D̄_{1} − D̄_{0} is of order zero, so ∥(D̄_{1} − D̄_{0})_{p}^{2}∥_{op} is well-defined and can be estimated pointwise on M.

In our current situation, let γ ∈ Ĝ be a G-representation such that

Hom_{H}(V^{γ}, σ) ≠ 0  and  λ_{1}(D^{2}) = c_{G}^{γ^{∗}} + c_{H}^{σ} ,  (2.2)

cf. Proposition 1.3. Let γ^{∗} denote the dual representation on V^{γ^{∗}} = (V^{γ})^{∗} and let

V = G ×_{H} (V^{γ^{∗}}|_{H})

denote the corresponding vector bundle over G, then V is trivialized by the map

V → M × V^{γ^{∗}}  with  [g, v] ↦ (gH, γ_{g}^{∗}v) .

To a map v: M → V^{γ} corresponds the section

gH ↦ [g, (γ^{∗})_{g}^{−1}v(g)] ∈ V_{gH} .

Now let ∇^{0} denote the trivial connection and let ∇^{1} denote the reductive connection on V. Then \overline{D}_{0}, \overline{D}_{1} are the corresponding Dirac operators on SM ⊗ V.

If we apply the reductive connection ∇^{1} on V to a section that is constant in the given trivialisation, then

∇_{[g,X]}^{1}v = [g, γ_{X}^{∗}(γ^{∗})_{g}^{−1}v] = (gH, γ_{∗ Ad_{g} X}v)

where X ∈ 𝔭 = 𝔥^{⊥} and

[g, X] = \frac{∂}{∂t} (ge^{−tX}H) ∈ T_{gH}M .

Thus we can identify T_{p}M with 𝔭 and choose a g-orthonormal frame f_{1}, …, f_{m} of 𝔥. We recall that the action σ_{∗} of 𝔥 on SM can be described in terms of Clifford multiplication and Lie brackets by

σ_{∗f_{k}} = \frac{1}{4} ∑_{i,j=1}^{n}⟨[e_{i}, e_{j}], f_{k}⟩ c_{i}c_{j} .

Let ē_{1}, …, ē_{n} be a ḡ-orthonormal base of T_{p}M. We may assume that there exist 0 < μ_{1}, …, μ_{n} ≤ 1 such that ē_{i} = μ_{i}e_{i} for all i ∈ {1, …, n}, where e_{1}, …, e_{n} is an orthonormal base with respect to g. We have to compute the operator norm of the operator

C = D̄_{1} − D̄_{0} = ∑_{i=1}^{n} c_{i}γ∗ ; ē_{i} = ∑_{i=1}^{n} μ_{i}c_{i}γ_{e_{i}}^{∗} ,

which is the difference of two Dirac operators on SM ⊗ V with respect to the metric ḡ. We note that this formula looks similar to (1.2) above. This is a special feature of symmetric spaces, which does not even generalise to normal homogeneous spaces, see Section 8 below.

Because C is selfadjoint, we have ∥Cv∥^{2} = ⟨C^{2}v, v⟩, and it suffices to estimate the eigenvalues of C^{2}. We follow the proof of Parthasarathy’s formula [12]. Using [𝔭, 𝔭] ⊂ 𝔥 we compute

C^{2} = − ∑_{i=1}^{n} μ_{i}^{2}(γ_{e_{i}}^{∗})^{2} + \frac{1}{2} ∑_{i,j=1}^{n} μ_{i}μ_{j} c_{i}c_{j} γ_{[e_{i},e_{j}]}^{∗}

= − ∑_{i=1}^{n} μ_{i}^{2}(γ_{e_{i}}^{∗})^{2} + ∑_{k=1}^{m}(γ_{f_{k}}^{∗} + \frac{1}{4} ∑_{i,j=1}^{n} μ_{i}μ_{j} ⟨[e_{i}, e_{j}], f_{k}⟩ c_{i}c_{j})^{2}

− ∑_{k=1}^{m}(γ_{f_{k}}^{∗})^{2} − \frac{1}{16} ∑_{k=1}^{m}(∑_{i,j} μ_{i}μ_{j} ⟨[e_{i}, e_{j}], f_{k}⟩ c_{i}c_{j})^{2}

≤ c_{G}^{γ^{∗}} − \frac{1}{16} ∑_{i,j,k,l=1}^{n} μ_{i}μ_{j}μ_{k}μ_{l} ⟨[e_{i}, e_{j}], [e_{k}, e_{l}]⟩ c_{i}c_{j}c_{k}c_{l} .

A term similar to the last one on the right hand side has already been estimated in [7], equation (1.11). Because [𝔭, 𝔭] ⊂ 𝔥, we have

− \frac{1}{16} ∑_{i,j,k,l=1}^{n} μ_{i}μ_{j}μ_{k}μ_{l} ⟨[e_{i}, e_{j}], [e_{k}, e_{l}]⟩ c_{i}c_{j}c_{k}c_{l} = \frac{1}{8} ∑_{i,j=1}^{n} μ_{i}^{2}μ_{j}^{2} ∥[e_{i}, e_{j}]∥^{2}

≤ \frac{1}{8} ∑_{i,j=1}^{n} ∥[e_{i}, e_{j}]∥^{2} = − ∑_{k=1}^{m}( ∑_{i,j=1}^{n}⟨[e_{i}, e_{j}], f_{k}⟩ c_{i}c_{j})^{2} = c_{H}^{σ} . (2.3)

Combining the calculations above, we have the estimate

C^{2} = (D̄_{1} − D̄_{0})^{2} ≤ c_{G}^{γ^{∗}} + c_{H}^{σ} = λ_{1}(D^{2}) .  (2.4)

3. Rigidity

We will now give a short of proof due to M. Listing that the equality λ_{1}(D̄^{2}) = λ_{1}(D^{2}) in Theorem 1 implies that ḡ = g.

Let 0 ≠ s ∈ ker(D̄_{1}) as in (2.1). If λ_{1}(D̄^{2}) = λ_{1}(D^{2}), we conclude that

∥(D̄_{1} − D̄_{0})s∥_{L^{2}}^{2} = λ_{1}(D^{2}) ∥s∥_{L^{2}}^{2} .

by (2.1) and (2.4). This implies in particular that ∥(D̄_{1} − D̄_{0})_{p}^{2}∥_{op} = λ_{1}(D^{2}) holds for all p ∈ supp(s). By [1], we know that s is nonzero on a dense open subset of M, so we have

∥(D̄_{1} − D̄_{0})_{p}^{2}∥_{op} = λ_{1}(D^{2})  (3.1)

for all p ∈ M, and we must have equality in (2.3).

Since M is of compact type, 𝔤 has trivial center. In particular, for each 1 ≤ i ≤ n there exists 1 ≤ j ≤ n such that [e_{i}, e_{j}] ≠ 0. Thus (3.1) implies by (2.3) that μ_{1} = ⋯ = μ_{n} = 1. The rigidity statement follows once we have shown that ker(D̄_{1}) ≠ 0.

4. The Equal Rank Case

We prove Theorem 1 for rk G = rk H. It remains to show that the operator D̄_{1} has a kernel.

4.1. Proposition ([12], [5]). Let M = G/K be a symmetric space with rk G = rk K + k. Then the complex spinor bundle Σ is locally induced by a representation σ of the Lie algebra 𝔨 of K, which splits as

σ = 2^{[\frac{k}{2}]} ⨁_{i=1}^{q} σ_{i} ,

where σ_{1}, …, σ_{q} are certain pairwise non-isomorphic irreducible complex representations of 𝔨.

By our choice of γ in (2.2), we have Hom_{H}(V^{γ}, Σ) ≠ 0. By Schur’s lemma, the Hrepresentations γ|_{H} and σ contain a common irreducible subrepresentation, say σ_{1} acting on Σ_{1}. Consider the vector bundle

W = G ×_{H} Σ_{1}^{∗} ,

then W is a parallel subbundle of V with respect to the reductive connection ∇^{1}. Let D_{1} be the Dirac operator acting on SM ⊗ W.

As above, we now have

L^{2}(M; SM ⊗ W) = ⨁_{γ∈Ĝ} V^{γ} ⊗ Hom_{H}(V^{γ}, Σ ⊗ Σ_{1}^{∗}) .

Again by Parthasarathy’s formula ([12], [5]), the operator D_{1}^{2} acts on V^{γ}, Σ ⊗ Σ_{1} as

γD_{1}^{2} = c_{G}^{γ} + c_{H}^{σ} − c_{H}^{σ} = c_{G}^{γ} .

Because c_{G}^{γ} = 0 iff γ is the trivial representation, the kernel of D_{1}^{2} is precisely

ker D_{1} = ker D_{1}^{2} = Hom_{H}(ℂ, Σ ⊗ Σ_{1}^{∗}) ≅ Hom_{H}(Σ_{1}, Σ) = Hom_{H}(Σ_{1}, Σ_{1})

by Schur’s Lemma. In particular dim ker D_{1} = 1, whence ind D_{1} = ±1 ≠ 0.

To complete the proof of Theorem 1 in this case, we note that the Dirac operator on \overline{SM} ⊗ W has nonzero index, and hence D̄_{1} has a kernel. The claim now follows from (2.1) and (2.4).

5. Mod-2-Indices and the Case rk H = rk G − 1

We now prove Theorem 1 for rk G = rk H + 1. We proceed similar as in [7], section 2.c. Assume first that n = dim M ≡ 1 mod 8. In this case, there exists a real vector bundle S_{ℝ}M

such that SM = S_{ℝ}M ⊗_{ℝ} ℂ, and the even part of the real Clifford algebra still acts on S_{ℝ}M. The bundle S_{ℝ}M is induced by real representation σ_{ℝ} of H. Let σ_{ℝ,i} denote its irreducible components, such that σ_{i} = σ_{ℝ,i} ⊗_{ℝ} ℂ, and let W_{ℝ} = G ×_{H} Σ_{ℝ,1}^{∗}.

Let

ω_{ℝ} = c_{1} ⋯ c_{n}

denote the real Clifford volume element, then ω_{ℝ} is parallel, anti-selfadjoint, and commutes with Dω_{ℝ}^{∗} = ω_{ℝ}. The operator

D_{ℝ,1} = ω_{ℝ} D_{1}

is anti-selfadjoint and has coefficients in the even part of the real Clifford algebra, so it acts on S_{ℝ}M. By the same reasoning as above,

ker(ω_{ℝ} D_{ℝ,1}) = Hom_{ℝ,H}(ℂ, Σ_{ℝ,1} ⊗_{ℝ} Σ_{ℝ,1}^{∗}) = Hom_{ℝ,H}(Σ_{ℝ,1}, Σ_{ℝ,1})

is one-dimensional.

With respect to the metric ḡ, we similarly construct the operator D̄_{ℝ,1} = ω̄_{ℝ} D̄_{1} acting on S̄_{ℝ}M ⊗_{ℝ} W_{ℝ}. Its restriction to \overline{S}_{ℝ}M ⊗_{ℝ} V_{ℝ} can be deformed into D_{ℝ,1} through a family of elliptic, formally anti-selfadjoint operators. Since the parity of the dimension of the kernel is preserved by such a deformation (see [10]), the operator ω̄_{ℝ} D̄_{ℝ,1} has nontrivial kernel. We regard V → M as a real vector bundle by forgetting the complex structure, then W_{ℝ} is again a parallel subbundle of V with respect to ∇^{1}, so D̄_{1} has a nontrivial kernel, too. Once again, our theorem follows from (2.1) and (2.4).

Finally, if rk G−rk H = 1 and dim M ≢ 1 mod 8, we consider N = M×S^{2m} with dim N ≡ 1 mod 8. We fix a round metric g_{0} on S^{2m} and regard N with the metrics g ⊕ g_{0} and ḡ ⊕ g_{0}. Then our main theorem holds for N, and the result for M follows because

λ_{1}(D̄_{M}^{2}) = λ_{1}(D̄_{N}^{2}) − λ_{1}(D_{S^{2m}}^{2}) ≤ λ_{1}(D_{N}^{2}) − λ_{1}(D_{S^{2m}}^{2}) = λ_{1}(D_{N}^{2}) . □

6. Examples

We consider spheres and projective spaces.

6.1. Example. For even-dimensional spheres S^{2m} = Spin(2m+1)/Spin(2m), the restriction of the 2^{m}-dimensional spinor representation σ of Spin(2m + 1) to H splits into two irreducible representations

σ|_{Spin(2m)} = σ^{+} ⊕ σ^{−} .

In [9], the bundles

V = SM = G ×_{H} Σ ⊃ W = S^{+}M = G ×_{H} Σ^{+}

were used to prove the optimality of the Vafa-Witten estimate. Our method is a direct generalisation of this approach to other symmetric spaces.

6.2. Example. The complex projective space ℂP^{n} is spin iff n is odd, so we regard the odddimensional projective space M = ℂP^{2m−1} = G/H with G = SU(2m) and H = U(2m − 1). The Dirac spectrum has been computed in [3] and [13]. To summarize, the spinor bundle splits as

SM = ⨁_{q=0}^{2m−1} Λ^{0,q}T^{∗}M ⊗ τ^{m} ,

where Λ^{0,q}T^{∗}M denotes the bundle of anti-holomorphic differential forms of degree q, and τ denotes the tautological bundle. Note that τ^{m} is a square root of the canonical bundle.

The smallest eigenvalue of D^{2} on Λ^{0,q}T^{∗}M ⊗ τ^{m} with respect to the Fubini-Study metric is given by

λ_{1}(D^{2}|_{Λ^{0,q}T^{∗}M⊗τ^{m}}) = \begin{cases}8m^{2} − 4m(q + 1) & for q < m, and \\ 8m^{2} − 4m(2m − q) & for q ≥ m . \end{cases}  (6.3)

Thus the lowest eigenvalue is attained in the middle degrees q = m − 1, m, and is given by 4m^{2} = (n + 1)^{2}. In contrast, Davaux and Min-Oo used the bundle τ^{m} for q = 0 in [4], which explains their larger upper bound 8m^{2} − 4m = 2n(n + 1).

Let us give a geometric description of the bundles Λ^{0,q}T^{∗}M ⊗ τ^{m} for q = m − 1, m. We have

T ℂP^{2m−1} = { [x, v] ∈ S^{2m−1} ⊗ ℂ^{2m} | x ≠ 0, x ⊥ v }
with  [z, v] = { (zx, zv) | z ∈ S^{1} } .

For α ∈ Λ^{0,m}T^{∗}M, we note that

α(zx, zv_{1}, …, zv_{m−1}) = z̄^{m} α(x, v_{1}, …, v_{m−1})
and  α(zv_{1}, …, zv_{m}) = z̄^{m} α(v_{1}, …, v_{m}) .

Thus inserting representatives of tangent vectors into m-forms gives a natural trivialisation

Φ = Φ^{m−1} ⊕ Φ^{m}: ℂP^{2m−1} × Λ^{0,m}ℂ^{2m} ⟶ (Λ^{0,m−1}T^{∗}M ⊕ Λ^{0,m}T^{∗}M) ⊗ τ^{m} .

7. Lipschitz maps of High Â-Degree

We give a proof of Theorem 2. Let N be a closed Riemannian spin manifolds and f: N → M a Lipschitz map of Lipschitz constant 1. Then f can be approximated by smooth (1 + ε)- Lipschitz maps for all ε > 0. We may therefore assume that f is smooth.

As in [2], [9], consider the spinor bundles Σ^{±} → S^{2n} equipped with the Dirac connection ∇^{1} with respect to the round metric g. Let ∇^{0} denote the trivial connection on Σ^{+} ⊕ Σ^{−} ≅ S^{2n} ⊗ Σ, where Σ is the spinor module of ℝ^{2n+1}.

The Chern character of the spinor bundles is given by

ch(Σ^{±}) = 2^{m−1} ± ω ,

where ω ∈ H^{2m}(S^{2m}; ℤ) is a generator. Let D_{N,1,±} be the Dirac operator on N twisted by f^{∗}Σ^{±}, and let D_{N,1} denote their direct sum. By the Atiyah-Singer index theorem, we have

ind(D_{N,1,±}) = (Â(T N) ⌣ f^{∗} ch(Σ^{±}))[N] = 2^{m−1}Â(T N)[N] ± degÂ f .

By combining both bundles, it is now easy to see that

dim ker(D_{N,1}) ≥ max(|2^{m}Â(T N)[N]|, 2 |degÂ f|) > 2^{m}(k − 1)  (7.1)

if |degÂ f| > 2^{m−1}(k − 1).

Let p ∈ N, then there exists an orthonormal frame ē_{1}, …, ē_{n} of T_{p}N and an orthonormal frame e_{1}, …, e_{2m} of T_{f(p)}S^{2m} such that

d_{p}f(ē_{k}) = \begin{cases}μ_{k} e_{k} & if k ≤ 2m, and \\ 0 & otherwise. \end{cases}

If f is (1 + ε) Lipschitz, then μ_{1}, …, μ_{2m} ≤ 1 + ε.

Let D_{N,0} denote the Dirac operator on N twisted by f^{∗}Σ, but with respect to the trivial connection ∇^{0}. Then

D_{N,1} − D_{N,0} = ∑_{i=1}^{2m} c̄_{i} γ_{∗μ_{i}e_{i}} .

Now a similar computation as in Section 2 gives

∥D_{N,1} − D_{N,0}∥^{2} ≤ (1 + ε) (c_{G}^{γ} + c_{H}^{σ}) = (1 + ε) λ_{1}(D^{2}) .  (7.2)

Combining (7.1) and (7.2) as before, we conclude at least 2^{m}(k −1) eigenvalues of D_{N,0} are not larger than (1 + ε) λ_{1}(D^{2}). Because D_{N,0} consists of 2^{m} copies of the Dirac operator D_{N}, we conclude that at least k eigenvalues of D_{N}^{2} (counted with multiplicities) are not larger than (1 + ε) λ_{1}(D^{2}). Since we can choose ε > 0 arbitrarily small, Theorem 2 is proved. □

We remarked in the Introduction that Theorem 2 does not hold unchanged for other symmetric spaces. Regard M = ℂP^{2m−1} as in Example 6.2. By (6.3), the Vafa-Witten will be optimal only if we choose one of the bundles W = Λ^{0,m−1}T^{∗}M ⊗ τ^{m} or W^{∗} = Λ^{0,m}T^{∗}M ⊗ τ^{m} as twist bundle.

To determine the Chern characters of these bundles, recall that

T′ℂP^{2m−1} ⊕ ℂ ≅ 2m τ^{−1} ,

and thus in K-theory,

[Λ^{0,q}T^{∗}M] = ∑_{i=0}^{q}(−1)^{q−i} [Λ^{i}(2m τ^{−1})] = ∑_{i=0}^{q}(−1)^{q−i} (\begin{matrix}2m \\ i \end{matrix}) [τ^{−i}] .

We know that a = c_{1}(τ) is a generator of H^{2}(ℂP^{2m−1}), and by the above, we find

ch(W) = ∑_{i=0}^{m−1}(−1)^{m−1−i} (\begin{matrix}2m \\ i \end{matrix}) e^{(m−i)a} .

Already for ℂP^{3}, the explicit classes are

ch(W) = 3 + 2a − \frac{2}{3} a^{3}  and  ch(W^{∗}) = 3 − 2a + \frac{2}{3} a^{3} .

On the other hand, we have seen that

W ⊕ W^{∗} ≅ ℂ^{( \begin{matrix}2m \\ m \end{matrix} )} .

Proceeding as above, we can prove the following result.

7.3. Proposition. Let N be a closed Riemannian spin manifold and let k ∈ ℕ. Then there is no Lipschitz map N → ℂP^{2m−1} of Lipschitz constant 1 with

|Â(T N) ⌣ f^{∗} ∑_{i=0}^{m−1}(−1)^{i} (\begin{matrix}2m \\ i \end{matrix}) (e^{(m−i)a} − 1)| > (\begin{matrix}2m − 1 \\ m − 1 \end{matrix}) (k − 1) ,

where a is a generator of H^{2}(ℂP^{2m−1}), unless

λ_{k}(D_{N}^{2}) ≤ λ_{1}(D_{M}^{2}) . □

8. A Homogeneous Counterexample

We now explain Remark 3. In general, the spectrum of a Dirac operator on a homogeneous space is hard to compute. On the Berger space M = SO(5)/SO(3) however, most of the relevant geometric structure can be described using octonions, see [6]. We will see that a straightforward adaptation of the arguments of Sections 1–5 to the Berger space is not possible.

We embed H = SO(3) in G = SO(5) via the irreducible SO(3)-representation of dimension 5. On 𝔤 = 𝔰𝔬(5), we take the scalar product ⟨A, B⟩ = −\frac{1}{2} tr(AB). As above, let 𝔭 denote the orthogonal complement of 𝔥 = 𝔰𝔬(3). Then dim 𝔭 = 7, and the isotropy action of SO(3) on 𝔭 factors as

SO(3) ⟶ G_{2} ⟶ Spin(7) ⟶ SO(7) .  (8.1)

Let [·, ·]_{𝔭} denote the projection of the Lie bracket in 𝔤. Let 𝕀 ⊂ 𝕆 denote the imaginary octonions, let ∗ denote octonion multiplication, and let ∗_{𝕀} denote octonion multiplication followed by the projection onto 𝕀.

8.2. Lemma ([6]). With a suitable isometric, G_{2}-equivariant identification of 𝔭 with the imaginary octonions, one has

[v, w]_{𝔭} = \frac{1}{\sqrt{5}} v ∗_{𝕀} w  for all  v, w ∈ 𝔭 .  □

Let Σ_{ℝ} again denote the real spinor module of 𝔭, then G_{2} also acts on Σ_{ℝ} by (8.1).

8.3. Lemma ([6]). We identify 𝔭 ≅ 𝕀 as in Lemma 8.2. With respect to a suitable isometric, G_{2}-equivariant identification Σ_{ℝ} ≅ 𝕆 and a suitable orientation of 𝔭, Clifford multiplication 𝔭 × Σ_{ℝ} → Σ_{ℝ} equals Cayley multiplication 𝕀 × 𝕆 → 𝕆 from the right. □

We fix an orthonormal base e_{1}, …, e_{7} of 𝔭 ≅ 𝕀 such that

e_{i} ∗ e_{i+1} = e_{i+3} ,

where indices are taken mod 7. Clifford multiplication with e_{i} will be abbreviated as c_{i}. As in [5] and [6], let us introduce the notation

c_{ijk} = ⟨[e_{i}, e_{j}]_{𝔭}, e_{k}⟩ = −\frac{1}{\sqrt{5}}ℜ((e_{i} ∗ e_{j}) ∗ e_{k})

and  ãd_{𝔭,X} = \frac{1}{4} ∑_{i,j=1}^{7}⟨[X, e_{i}], e_{j}⟩ c_{i}c_{j} .

Let us define

A = \frac{1}{3} ∑_{i=1}^{7} c_{i}ãd_{𝔭,e_{i}} = \frac{1}{12} ∑_{i,j,k=1}^{7} c_{ijk} c_{i}c_{j}c_{k} .

It is easy to see that A is a symmetric, G_{2}-invariant endomorphism of Σ_{ℝ}. Thus it induces a G-invariant symmetric endomorphism of S_{ℝ}M. One may compute that

A = \frac{1}{2\sqrt{5}} (\begin{matrix}7 & 0 \\ 0 & −1 \end{matrix}) : ℝ ⊕ 𝕀 → ℝ ⊕ 𝕀 .

Let D denote the real Dirac operator acting on Γ(S_{ℝ}M). We want to consider the family of operators

D^{λ} = D + (3λ − \frac{3}{2}) A .  (8.4)

The operator D^{\frac{1}{2}} is the Riemannian Dirac operator on M, whereas D̃ = D^{\frac{1}{3}} is called the reductive or cubic Dirac operator.

8.5. Proposition. For λ sufficiently close to \frac{1}{2}, the smallest eigenvalue of (D^{λ})^{2} is \frac{441}{20} λ^{2}.

In particular there exists ε > 0 sufficiently small such that

λ_{1}((D^{\frac{1}{2} +ε})^{2}) > λ_{1}(D^{2}) .

As stated in Remark 3, this implies that the Vafa-Witten technique cannot be used to prove sharp upper bounds for the first Dirac eigenvalue.

For certain questions, the reductive Dirac operator D̃ on a normal homogeneous space plays the same role as the geometric Dirac operator on a symmetric space, but here clearly λ_{1}(D̃^{2}) < λ_{1}(D^{2}), so with respect to optimal Vafa-Witten estimates, the reductive Dirac operator is even worse than the geometric Dirac operator.

Proof. The space of sections L^{2}(M; S_{ℝ}M) can be decomposed as in (1.1). As in [5], we see that D^{λ} acts on the isotypical components V^{γ} ⊗ Hom_{H}(V^{γ}, Σ_{ℝ}) by id_{V^{γ}} ⊗^{γ}D^{λ}, where

γD^{λ} = ∑_{i=1}^{7}(γ_{e_{i}}^{∗} ⊗ c_{i} + λ id_{V^{γ}} ⊗c_{i}ãd_{𝔭,e_{i}}) = ∑_{i=1}^{7} γ_{e_{i}}^{∗} ⊗ c_{i} + \frac{λ}{4} ∑_{i,j,k=1}^{7} c_{ijk} c_{i}c_{j}c_{k} .

Let us write μ = 3λ − 1, then

D^{λ} = D̃ + μ id_{V^{γ}} ⊗A .

The square of D̃ has been computed in [5] as

(^{γ}D̃)^{2} = ∥γ + ρ_{G}∥^{2} − ∥ρ_{H}∥^{2} .

If we use integers p ≥ q ≥ 0 to index the irreducible representations of SO(5), then by [6], we have the explicit formula

(^{γ_{p,q}}D̃)^{2} = ∥γ_{p,q} + ρ_{G}∥^{2} − ∥ρ_{H}∥^{2} = p^{2} + 3p + q^{2} + q + \frac{49}{20} on Hom_{H}(V^{γ_{p,q}}, Σ_{ℝ}) .

Let us now analyze all irreducible representations of SO(5). For the trivial representation, clearly

γ_{0,0}D^{λ} = 3λ^{γ_{0,0}}D̃ = 3λ A|_{ℝ} = \frac{21}{2\sqrt{5}} λ ,  so  ^{γ_{0,0}}D = ^{γ_{0,0}}D^{\frac{1}{2}} = \frac{21}{4\sqrt{5}}

The standard representation γ_{1,0} contains no SO(3)-irreducible subrepresentation isomorphic to ℝ or 𝕀, so it does not contribute to the spectrum of D^{λ}. The representation γ_{1,1} = Λ^{2}γ_{1,0} contains no trivial SO(3)-subrepresentation, but an SO(3)-subrepresentation isomorphic to 𝕀, and we find that

|λ_{1}(^{γ_{1,1}}D)| ≥ |^{γ_{1,1}}D̃| − \frac{1}{2} |A|_{𝕀}| = \frac{13}{2\sqrt{5}} − \frac{1}{4\sqrt{5}} > \frac{21}{4\sqrt{5}} = |^{γ_{0,0}}D| .

For all other representations, we have p > 2, thus

|λ_{1}(^{γ_{p,q}}D)| ≥ |^{γ_{2,0}}D̃| − \frac{1}{2} |A| = \frac{\sqrt{249}}{2\sqrt{5}} − \frac{7}{4\sqrt{5}} > \frac{21}{4\sqrt{5}} = |^{γ_{0,0}}D| .

In particular, the smallest eigenvalue of D^{λ} equals \frac{21}{2\sqrt{5}} λ for λ sufficiently close to \frac{1}{2}, and is attained by a G-invariant spinor. □

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NWF 1 Mathematik, Universität Regensburg, 93040 Regensburg, Germany

E-mail address: sebastian.goette@mathematik.uni-regensburg.de