DIRAC-HARMONIC MAPS FROM INDEX THEORY

BERND AMMANN AND NICOLAS GINOUX

Abstract. We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

Contents

1. Introduction  1
2. Previously known results  3
3. Notation and Preliminaries  6
4. Index theory of Dirac-harmonic maps  7
5. Variational formulas for the fermionic energy functional  8
6. The graded case  10
7. The case m ∈ 4ℕ  11
8. Minimality  11
9. Non-minimality  14
10. Examples of maps with non-trivial index  16
Appendix A. Bubbling-off for harmonic maps  23
References  25

1. Introduction

Dirac-harmonic maps are stationary points of the fermionic analogue of the energy functional. The associated Euler-Lagrange equation is a coupled system consisting of a map f : M → N and a spinor on M twisted by f^{∗}T N, such that the (twisted) spinor is in the kernel of the (twisted) Dirac operator. The goal of this article is to use index theoretical tools for providing such Dirac-harmonic maps.

1.1. Dirac-harmonic maps. In mathematical physics a (non-linear) sigmamodel consists of Riemannian manifolds M and N. For simplicity we always assume M and N to be compact. The classical or bosonic energy of a map f : M → N is defined as \frac{1}{2} ∫_{M} |df|^{2}. Stationary points of this functional are called harmonic maps. Harmonic maps from M to N are interpreted as solutions of this sigma-model. The term “non-linear” indicates that the target manifold N is not a vector space and will be omitted from now on. A super-symmetric sigma-model (see [13, Chapter 3], [15]) also consists of Riemannian manifolds M and N, but in addition to above one assumes that M carries a fixed spin structure. A fermionic energy functional can now be defined on pairs (f, Φ), where f is again a map M → N and Φ is a spinor on M twisted by f^{∗}T N. In the literature a linear and a non-linear versions of this functional are considered, see Subsection 3.2 for details. In our article and in most of the mathematical literature one considers the fermionic energy functional

Date: October 27, 2015.

2010 Mathematics Subject Classification. 58E20 (Primary), 53C43, 58J20, 53C27 (Secondary).

Key words and phrases. Dirac harmonic maps, index theory.

L(f, Φ) = \frac{1}{2} ∫_{M} (|df|^{2} + ⟨Φ, D /^{f}Φ⟩) dv^{M},

where D /^{f} is the Dirac operator acting on spinors twisted by f^{∗}T N. Stationary points (f_{0}, Φ_{0}) of this fermionic energy functional are called Dirac-harmonic maps. They are characterized by the equations D /^{f_{0}}Φ_{0} = 0 and tr_{g}(∇df_{0}) = \frac{V_{Φ_{0}}}{2} , see Proposition 5.1 below. They are interpreted as solutions of the super-symmetric sigma-model. Regularity and existence questions for Dirac-harmonic maps have been the subject of many articles recently where important results have been obtained, see Section 2.

1.2. Trivial Dirac-harmonic maps. A harmonic map together with a spinor that vanishes everywhere provides a Dirac-harmonic map. In the present article, such Dirac-harmonic maps are called spinor-trivial Diracharmonic maps. Another class of obvious solutions consists of constant maps f together with a spinor Φ lying in the kernel of the (untwisted) Dirac operator, see Subsection 2.3. Such Dirac-harmonic maps will be called map-trivial. However, the dimension of the kernel depends in a subtle way on the conformal structure of M. For the sake of clarity we give a small review on known results about the kernel in Subsection 2.3.

1.3. Main result.

Definition 1.1. A Dirac-harmonic map will be called uncoupled if f is harmonic, otherwise it is called coupled.

Most of the existence results for Dirac-harmonic maps in the literature actually yield uncoupled Dirac-harmonic maps. Coupled Dirac-harmonic maps are discussed in [21, Thm. 3], see Subsection 2.2 for details. Our result provides criteria for the existence of sufficiently many non-trivial, uncoupled Dirac-harmonic maps.

Theorem 1.2. For M and N as above consider a homotopy class [f] of maps f : M^{m} → N^{n} such that the KO_{m}(pt)-valued index α(M, [f]) is nontrivial. Assume that f_{0} ∈ [f] is a harmonic map. Then there is a real vector space V of real dimension a_{m} such that all (f_{0}, Φ), Φ ∈ V, are Diracharmonic maps, where

a_{m} := \begin{cases}2 & if m ≡ 0 (8) \\ 2 & if m ≡ 1 (8) \\ 4 & if m ≡ 2 (8) \\ 4 & if m ≡ 4 (8). \end{cases}

This theorem is an immediate consequence of Theorem 9.1. It is mainly interesting in dimensions 1 and 2 modulo 8. In dimensions divisible by 4 the statement directly follows from the Atiyah-Singer-index theorem and the grading techniques of Sections 6 and 7. In dimension 1 modulo 8, these grading techniques would also yield a similar result, but the space of Diracharmonic maps obtained this way would have only real dimension 1 instead of 2. In order to derive such results we have structured the article as follows. We start by giving an overview about known results in Section 2 in order to compare them to our results. It is logically independent from the rest. Having introduced the notations and the fermionic energy in Section 3, we recall some well-known facts about the index of twisted Dirac operators in Section 4 and the variational formula in Section 5. This immediately leads to the first existence result using grading techniques (Section 6) and the result in dimensions m divisible by 4 (Section 7). In order to get better results in dimensions 8k+1 and 8k+2 we study the “minimal” and “non-minimal case” in Sections 8 and 9. Finally, we discuss in Section 10 numerous examples to which the above theorem applies.

2. Previously known results

To compare our results with the existing literature we briefly recall several known facts about Dirac-harmonic maps. Results about regularity, investigations about the necessity for the mapping component f to be harmonic, removal of singularity theorems and blow-up analysis of Dirac-harmonic maps are developed in [9, 8, 38, 33, 39, 40]. We do not give details, as these issues are far from our topic. We now summarize existence results for Dirac-harmonic maps. Not many concrete examples are known. Apart from the trivial examples in subsections 1.2 and 2.3, almost all examples occur in dimension m = 2 and can be divided into uncoupled and coupled solutions. Before we describe them, recall first that, in dimension m = 2, the existence of a Dirac-harmonic map only depends on the conformal class of the metric chosen on M: a pair (f_{0}, Φ_{0}) is Dirac-harmonic on (M^{2}, g) if and only if (f_{0}, e^{− \frac{u}{2}} Φ_{0}) is Diracharmonic on (M^{2}, e^{2u}g), whatever u ∈ C^{∞}(M, ℝ) is. From the point of view of string theory, the case m = 2 is of central importance as it describes the evolution of strings in space-times.

2.1. Existence of uncoupled Dirac-harmonic maps. The first existence result, which appears in [9, Prop. 2.2] for M = N = 𝕊^{2} and then in [21, Thm. 2] for general surfaces M, is based on an explicit construction involving a harmonic map and so-called twistor-spinors on the source manifold. Twistor-spinors are sections of the spinor bundle ΣM of M lying in the kernel of the Penrose operator P : C^{∞}(M, ΣM) ⟶ C^{∞}(M, T^{∗}M ⊗ ΣM), P_{X}φ := ∇_{X}φ + \frac{1}{m}X · D / φ (here D / denotes the untwisted Dirac operator on M). The construction goes as follows. Let M be any m-dimensional (non necessarily compact) spin manifold and f : M ⟶ N be a smooth map. For an arbitrary φ ∈ C^{∞}(M, ΣM) consider the smooth section Φ := ∑_{j=1}^{m} e_{j} ·φ⊗df(e_{j}) of ΣM ⊗f^{∗}T N, where (e_{j})_{1≤j≤m} is a local orthonormal basis of T M. Note that this (pointwise) definition does not depend on the choice of the local orthonormal basis (e_{j})_{1≤j≤m} and that the vanishing of Φ is equivalent to that of φ as soon as f is an immersion. Then a short computation shows that

D /^{f}Φ = −2 ∑_{j=1}^{m} P_{e_{j}}φ ⊗ df(e_{j}) + \frac{2 − m}{m} ∑_{j=1}^{m} e_{j} · D / φ ⊗ df(e_{j}) − φ ⊗ tr_{g}(∇df).

Therefore, if φ is a twistor-spinor, m = 2 and f is harmonic, then D /^{f}Φ = 0. Moreover, if m = 2, then one calculates that the vector field V_{Φ} defined in Proposition 5.1 vanishes. Therefore, (f, Φ) is a Dirac-harmonic map as soon as f is harmonic and the spinor φ involved in the definition of Φ is a twistor-spinor. This construction has the obvious drawback that the only closed surfaces admitting non-zero twistor-spinors are the 2-torus 𝕋^{2} with its non-bounding spin structure and the 2-sphere 𝕊^{2}. In particular no example on a hyperbolic surface can be given using this approach. However, let us mention that all Dirac-harmonic maps from 𝕊^{2} to 𝕊^{2} are of that form [37, Thm. 1.2]. Vanishing results in case the spinor part Φ has the form above are considered in [26].

2.2. Existence of coupled Dirac-harmonic maps. Up to the knowledge of the authors, only two examples of coupled Dirac-harmonic maps are known. Both are constructed explicitly considering particular isometric but non-minimal immersions from special non-compact source manifolds into the hyperbolic space. The first one deals with an explicit non-minimal isoparametric immersion from a hyperbolic surface of revolution into the 3-dimensional hyperbolic space [21, Thm. 3]. The second one is provided by the totally umbilical (but non-totally geodesic) embedding of hyperbolic hyperplanes of sectional curvature −\frac{4}{m+2} into the m + 1(≥ 4)-dimensional hyperbolic space of sectional curvature −1. This example can be handled with the twistor-spinor-ansatz of [21] and is carried out in [3].

2.3. More on trivial Dirac-harmonic maps. In this subsection we will explain a product construction for producing examples of Dirac-harmonic maps, and then we will summarize the knowledge about map-trivial ones. The main objective of the article is to derive existence of Dirac-harmonic maps which do not fall in any of these trivial categories: spinor-trivial one, map-trivial ones, products of them. To understand the product construction, assume that f_{1} : M → N_{1} and f_{2} : M → N_{2} are smooth maps, so that they are the components of the map (f_{1}, f_{2}) : M → N_{1} × N_{2}. Similarly, let Φ_{j} ∈ Γ(ΣM ⊗ f_{j}^{∗}T N_{j}), j = 1, 2, and let (Φ_{1}, Φ_{2}) ∈ Γ(ΣM ⊗ (f_{1}^{∗}T N_{1} ⊕ f_{2}^{∗}T N_{2}) denote their sum. Let L (resp. L_{i}) be the fermionic energy functional for N = N_{1} × N_{2} (resp. N = N_{i}). Then L((f_{1}, f_{2}), (Φ_{1}, Φ_{2})) = L_{1}(f_{1}, Φ_{1}) + L_{2}(f_{2}, Φ_{2}). Thus, the map ((f_{1}, f_{2}), (Φ_{1}, Φ_{2})) is Dirac-harmonic if and only if both (f_{1}, Φ_{1}) and (f_{2}, Φ_{2}) are Dirac-harmonic. So we obtain new examples of Dirac-harmonic maps by taking products of map-trivial Dirac harmonic maps with spinor-trivial maps. The pair (f, Φ) is a map-trivial Dirac-harmonic map if and only if the following holds:

(1) f is constant, say f ≡ x ∈ N,

(2) Φ ∈ Γ(ΣM ⊗T_{x}N) ≅ Γ(ΣM)⊗ℝ^{n}, n = dim N, is a harmonic spinor, i.e. D /^{f}Φ = 0 and D /^{f} = D / ⊗ id_{ℝ^{n}} where D / : Γ(ΣM) → Γ(ΣM) is the untwisted Dirac operator.

Thus, to determine the map-trivial Dirac-harmonic maps, one has to determine the kernel of the untwisted Dirac operator. The dimension of the kernel is invariant under conformal change of the metric, but it depends on the conformal class. Important progress was obtained in particular in [19], [4], [5] and [6]. The KO_{m}(pt)-valued index obviously gives a lower bound on the dimension of the kernel. We say, that a Riemannian metric g is D / -minimal if this lower bound is attained. At first one knows that generic metrics are D / -minimal.

Theorem 2.1 ([2]). Let M be a connected compact spin manifold. Then the set of D / -minimal Riemannian metrics is open and dense in the set of all Riemannian metrics in the C^{k}-topology for any k ∈ {1, 2, 3, …} ∪ {∞}.

The theorem was already proved in [6] for simply connected manifolds of dimension at least 5. There is also a stronger version of the theorem in [1] which states that any metric can be perturbed in an arbitrarily small neighborhood to a D / -minimal one. In the case m = 2 the complex dimension of the kernel of D / is at most g + 1 where g is the genus of the surface M [19, Prop. 2.3]. This upper bound determines the dimension of the kernel of D / in the case g ≤ 2, α(M) = 0 and in the case g ≤ 4, α(M) = 1. The dimension of the kernel can also be explicitly determined for hyperelliptic surfaces [7]. In the same article the dimension of the kernel is calculated for special non-hyperelliptic surfaces of genus 4 and 6. However, for general surfaces of sufficiently large genus, the dimension of the kernel is unknown. In contrast to m = 2, it is conjectured that for m ≥ 3 there is no topological quantity that yields an upper bound on dim ker D /.

Conjecture 2.2. [4, p. 941] On any non-empty compact spin manifold of dimension m ≥ 3, there is a sequence of Riemannian metrics g_{i} with

lim_{i→∞} dim ker D /^{g_{i}} = ∞.

The conjecture is known only in few cases as e.g. the 3-sphere S^{3} where Hitchin [19] has shown that a sequence of Berger metrics provides such a sequence of metrics. In particular, the conjecture would include that any manifold with m ≥ 3 carries a non-D / -minimal metric. The latter statement is partially known to be true. It was shown in [19] for m ≡ 7, 0, 1 (8) and in [4] for m ≡ 3, 7 (8) that any compact m-dimensional spin manifold carries a metric with ker D / ≠ 0. A similar statement is known for all spheres of dimension m ≥ 4 [31] and [12]. However it is still unknown whether an arbitrary compact spin manifold of dimension m ≡ 2, 4, 5, 6 (8) admits a non-D / -minimal metric.

3. Notation and Preliminaries

3.1. Notation and Conventions. The Riemannian manifolds M and N are always assumed to be compact without boundary, m = dim M, n = dim N. We assume that M is spin and — unless stated otherwise — we always assume that one of the spin structures on M is fixed. The (untwisted) spinor bundle on M is denoted by ΣM, the fiber over x ∈ M is denoted by Σ_{x}M. In this article we consider complex spinors, i. e. , ΣM is a complex vector bundle of rank 2^{[ \frac{m}{2} ]}; similar statements also hold for real spinors. The untwisted spinor bundle carries a Hermitian metric, a metric connection and a Clifford multiplication. For an introduction to these structures we refer to textbooks on spin geometry such as [23, 16, 18]. These structures allow to define an (untwisted) Dirac operator D / : C^{∞}(M, ΣM) ⟶ C^{∞}(M, ΣM). If E is a real vector bundle with given metric and metric connection, we define the twisted spinor bundle as ΣM ⊗ E, the tensor bundle being taken over ℝ. The twisted spinor bundle carries similar structures as the untwisted one. Sections of ΣM ⊗E will be called spinors or spinors twisted by E. One uses the structures on ΣM ⊗ E to define the twisted Dirac operator

D /^{E} : C^{∞}(M, ΣM ⊗ E) ⟶ C^{∞}(M, ΣM ⊗ E).

It is a first order elliptic and selfadjoint differential operator. Its spectrum is thus discrete, real and of finite multiplicity. In this article E will often be obtained as E = f^{∗}T N where f : M → N is smooth. Here E is equipped with the pull-back of the metric and the Levi-Civita connection on T N. In this case we simply write D /^{f} for D /^{f^{∗}T N}. Our convention for curvature tensors is R_{X,Y}^{N} = [∇_{X}^{N}, ∇_{Y}^{N}] − ∇_{[X,Y]}^{N}, and R is then considered as an element in Hom(T N ⊗ T N ⊗ T N, T N) through R(X ⊗ Y ⊗ Z) = R(X, Y)Z. The sphere S^{n} with its standard metric (of constant sectional curvature 1) will be denoted as 𝕊^{n}. The complex projective space ℂP^{n} will be always equipped with its Fubini-Study metric of holomorphic sectional curvature 4. The tautological bundle on ℂP^{n} is denoted by γ_{n}.

3.2. Linear and non-linear energy functional. As mentioned in the introduction, the fermionic energy functional is discussed in a linear and non-linear version in the literature. We will give some details now. For a concise introduction see [15] or [13, Chapter 3]. The most appropriate functional from the physical point of view would be Grassmann-algebra-valued. However, as this formalism is a bit overloaded for analytical purposes, one usually works with simplified models. One often considers the following fermionic energy functional

L_{c}(f, Φ) := \frac{1}{2} ∫_{M}(|df|^{2} + ⟨Φ, D /^{f}Φ⟩ + \frac{1}{6}⟨Φ, R(Φ, Φ)Φ⟩) dv^{M}.

Here R is a section of Hom_{ℂ}((ΣM ⊗_{ℝ} T N) ⊗_{ℂ} (\overline{ΣM} ⊗_{ℝ} T N) ⊗_{ℂ} (ΣM ⊗_{ℝ} T N), ΣM ⊗_{ℝ} T N) obtained by tensoring the Riemann curvature tensor R ∈ Hom_{ℝ}(T N ⊗_{ℝ}T N ⊗_{ℝ}T N, T N) with the spinorial contraction map id⊗⟨ . ⟩ ∈ Hom_{ℂ}(ΣM ⊗_{ℂ} \overline{ΣM} ⊗_{ℂ} ΣM, ΣM), id ⊗ ⟨ . ⟩(φ_{1}, φ_{2}, φ_{3}) = ⟨φ_{2}, φ_{3}⟩φ_{1}. This functional is often called the fermionic energy functional with curvature term, and stationary points of this functional are called Dirac-harmonic maps with curvature term. However this functional makes analytical considerations involved, and thus only few analytical articles include this curvature term, e. g. [11] and [10]. In [20] interesting techniques are developed which might turn helpful to find solutions of the non-linear equation. As the curvature term is of fourth order in Φ whereas the dominating term is quadratic on Φ, it seems acceptable from the view point of physical applications to neglect the curvature term. The fermionic energy functional thus obtained

L(f, Φ) := \frac{1}{2} ∫_{M}(|df|^{2} + ⟨Φ, D /^{f}Φ⟩) dv^{M} ∈ ℝ,

is analytically much easier to study. A pair (f_{0}, Φ_{0}) is a stationary point of L if and only if it satisfies D /^{f_{0}}Φ_{0} = 0 and tr_{g}(∇df_{0}) = \frac{V_{Φ_{0}}}{2} , see Proposition 5.1 below. Stationary points of this functional are called Dirac-harmonic maps. Starting with [9], this functional has been intensively studied in the literature, see Subsection 2. The method developed in the present article applies to this linear version.

4. Index theory of Dirac-harmonic maps

Let M be a closed m-dimensional Riemannian spin manifold with spin structure denoted by χ. Let E ⟶ M be a Riemannian (real) vector bundle with metric connection. Then one can associate to the twisted Dirac operator D /^{E} : C^{∞}(M, ΣM ⊗ E) ⟶ C^{∞}(M, ΣM ⊗ E) an index α(M, χ, E) ∈ KO_{m}(pt) (see e. g. [23, p.151]). Using the isomorphism [23, p.141]

KO_{m}(pt) ≅ \{ \begin{matrix}ℤ & if m ≡ 0 (4) \\ ℤ_{2} & if m ≡ 1, 2 (8) \\ 0 & otherwise, \end{matrix}  (1)

the index α(M, χ, E) will be identified either with an integer or an element in the group ℤ_{2} of integers modulo 2. We also say that α(M, χ, E) is the α-genus of E ⟶ M. The α-genus can be easily determined out of ker D /^{E} using the following formula [23, Thm. II.7.13]:

α(M, χ, E) = \{ \begin{matrix}\{ch(E) · Â(T M)\}[M] & if m ≡ 0 (8). \\ [dim_{ℂ}(ker(D /^{E}))]_{ℤ_{2}} & if m ≡ 1 (8) \\ [\frac{dim_{ℂ}(ker(D /^{E}))}{2}]_{ℤ_{2}} & if m ≡ 2 (8) \\ \frac{1}{2}\{ch(E) · Â(T M)\}[M] & if m ≡ 4 (8) \end{matrix}

As usual ch(E) is to be understood as the Chern character of E ⊗_{ℝ} ℂ. In dimensions 1 and 2 modulo 8 the α-genus depends on the spin structure χ on M, however we often simply write α(M, E), when χ is clear from the context. We will assume from now on and until the end of Section 9 that M comes with a fixed orientation and spin structure, so we omit the notation χ in those sections. The number α(M, E) is a spin-bordism-invariant, where a spin-bordism for manifold with vector bundles means that the restriction of a vector bundle to the boundary of the spin manifold must coincide with the vector bundle given on the boundary: namely, [23, Thm. II.7.14] implies that α defines a map Ω_{∗}^{Spin}(BO) ⟶ KO_{∗}(pt). Note in particular that α(M, E) does depend neither on the metric nor on the connection chosen on E ⟶ M. In case that E is the trivial real line bundle \underline{ℝ} := ℝ × M ⟶ M, the number α(M, E) =: α(M) is the classical α-genus of M.

Definition 4.1. Let M be a closed m-dimensional Riemannian spin manifold and f : M → N be a smooth map into an n-dimensional Riemannian manifold. The α-genus of f is α(M, f) := α(M, f^{∗}T N).

The spin-bordism-invariance of the α-genus has an important consequence for α(M, f). We first make a

Definition 4.2. With the notations of Definition 4.1, two maps f_{1} : M_{1} → N and f_{2} : M_{2} → N are spin-bordant in N, if there is a spin manifold W of dimension m+1 together with a map F : W → N such that ∂W = −M_{1}⨿M_{2} (in the sense of manifolds with spin structure) and such that F_{|_{M_{j}}} = f_{j} for both j = 1, 2.

Obviously, given any smooth spin-bordant maps f_{j} : M_{j} ⟶ N, j = 1, 2, the pull-back bundles f_{j}^{∗}(T N) ⟶ M_{j} are spin-bordant as vector bundles. Therefore, we obtain the

Proposition 4.3. Assume that f_{1} : M_{1} → N and f_{2} : M_{2} → N are spinbordant in N. Then α(M_{1}, f_{1}) = α(M_{2}, f_{2}).

Since a homotopy between maps f_{j} : M ⟶ N defines a spin-bordism between them, we deduce the

Corollary 4.4. Assume that f_{1} : M → N and f_{2} : M → N are homotopic maps. Then α(M, f_{1}) = α(M, f_{2}).

Because of this homotopy invariance we also write α(M, [f]) for α(M, f), where [f] is the homotopy class of f.

5. Variational formulas for the fermionic energy functional

In the following, we denote by h the Riemannian metric on N. If f : M → N is a smooth map, then M carries the twisted spinor bundle ΣM ⊗ f^{∗}T N. For given x ∈ M and vectors X ∈ T_{x}M and Y ∈ T_{f(x)}N, we denote by X · ⊗R_{Y,df(X)}^{N} the linear endomorphism obtained by tensorizing the Clifford multiplication by X (acting on Σ_{x}M) with the curvature R_{Y,df(X)}^{N} ∈ End(T_{f(x)}N). It is the real tensor product of a skew-hermitian endomorphism with a skew-symmetric endomorphism, and thus a hermitian endomorphism. In particular, the expression ⟨X · ⊗R_{Y,df(X)}^{N}Φ_{0}, Φ_{0}⟩ is real. It is easy to notice that, if a local orthonormal basis (e_{j})_{1≤j≤m} of T M is chosen, then the pointwise (hermitian) endomorphism ∑_{j=1}^{m} e_{j} · ⊗R_{Y,df_{0}(e_{j})}^{N} of ΣM ⊗ f^{∗}T N does not depend on the choice of the basis (e_{j})_{1≤j≤m} and hence defines a global section of End(ΣM ⊗ f^{∗}T N). We recall the following formulas for the first variation of L [9, Prop. 2.1].

Proposition 5.1. For (f_{0}, Φ_{0}) ∈ C^{∞}(M, N) × C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N) and some ε > 0 let (f_{t}, Φ_{t})_{t∈]−ε,ε[} be a variation of (f_{0}, Φ_{0}) which is differentiable at t = 0. Denote by D /_{t} := D /^{f_{t}} for all t ∈] − ε, ε[. Then

\frac{d}{dt}⟨D /_{t}Φ_{t}, Φ_{t}⟩_{|_{t=0}} = h(V_{Φ_{0}}, \frac{∂f}{∂t} (0)) + ⟨D /_{0} \frac{∂Φ}{∂t}(0), Φ_{0}⟩ + ⟨D /_{0}Φ_{0}, \frac{∂Φ}{∂t}(0)⟩ (2)

where the section V_{Φ_{0}} of f_{0}^{∗}T N is defined by the pointwise relation h(V_{Φ_{0}}, Y) = ∑_{j=1}^{m}⟨e_{j} · ⊗R_{Y,df_{0}(e_{j})}^{N}Φ_{0}, Φ_{0}⟩ for all Y ∈ T N and every choice of local orthonormal frame (e_{j})_{1≤j≤m} of T M. In particular,

\frac{d}{dt}L(f_{t}, Φ_{t})_{|_{t=0}} = ∫_{M} ℜe(⟨D /_{0}Φ_{0}, \frac{∂Φ}{∂t}(0)⟩)−h (tr_{g}(∇df_{0}) − \frac{V_{Φ_{0}}}{2} , \frac{∂f}{∂t} (0)) dv^{M}.

The differentiability of t ↦ Φ_{t} at t = 0 is to be understood as follows. Denote by f :] − ε, ε[×M ⟶ N, (t, x) ↦ f(t, x) := f_{t}(x) the variation above. The metric and the Levi-Civita connection on T N induce a metric and a metric connection on the pull-back bundle f^{∗}T N ⟶] − ε, ε[×M. For every t ∈] − ε, ε[ let β_{t} : C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N) ⟶ C^{∞}(M, ΣM ⊗ f_{t}^{∗}T N) be the unitary and parallel isomorphism induced by the parallel transport along the curves [0, t] →] − ε, ε[×M, s ↦ (s, x), where x runs in M. Then we require the map t ↦ Φ̂_{t} := β_{t}^{−1} ∘ Φ_{t} ∈ C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N) to be differentiable at t = 0 in the following sense: the map ] − ε, ε[⟶ C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N), t ↦ Φ̂_{t}, has a derivative at t = 0 which is at least continuous on M. Here we consider the topology induced by the H^{1,2}-norm on C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N). In that case, we denote by \frac{∂Φ}{∂t}(0) := \frac{∂Φ̂}{∂t}(0) ∈ C^{1}(M, ΣM ⊗ f_{0}^{∗}T N). Note that, if D̂ /_{t} := β_{t}^{−1}∘D /_{t}∘β_{t} : C^{∞}(M, ΣM ⊗f_{0}^{∗}T N) ⟶ C^{∞}(M, ΣM ⊗f_{0}^{∗}T N), then ⟨D̂ /_{t}Φ̂_{t}, Φ̂_{t}⟩ = ⟨D /_{t}Φ_{t}, Φ_{t}⟩ since β_{t} is unitary.

As a straightforward consequence of Proposition 5.1, we obtain:

Corollary 5.2. Let f_{0} ∈ C^{∞}(M, N) be a harmonic map and Φ_{0} ∈ ker(D /^{f_{0}}). Assume the existence, for every smooth variation (f_{t})_{t∈]−ε,ε[} of f_{0}, of a variation (Φ_{t})_{t∈]−ε,ε[} of Φ_{0} which is differentiable at t = 0 and such that

\frac{d}{dt} (D /^{f_{t}}Φ_{t}, Φ_{t})_{L^{2}} _{|_{t=0}} = 0. Then the pair (f_{0}, Φ_{0}) is an uncoupled Diracharmonic map.

Proof. Only V_{Φ_{0}} = 0 has to be proved, which follows from integrating (2) and using the self-adjointness of D /^{f_{0}}. □

Corollary 5.2 is the corner stone for the proof of our main result, because it implies the following: to obtain the pair (f_{0}, Φ_{0}) (with f_{0} harmonic) as a Dirac-harmonic map, it suffices to construct, for any smooth variation (f_{t})_{t} of f_{0}, a sufficiently smooth one-parameter family of twisted spinors (Φ_{t})_{t} with (D /^{f_{t}}Φ_{t}, Φ_{t})_{L^{2}} = 0 for all sufficiently small t. And this can be achieved either with considerations about the ℤ_{2}-grading of the twisted spinor bundle (see Corollary 6.1) or with a non-trivial α-genus (see Proposition 8.2).

6. The graded case

Let f ∈ C^{∞}(M, N) be any map. In this section we consider the situation where the map bundle ΣM admits an orthogonal and parallel ℤ_{2}-grading G ∈ End(ΣM), G^{2} = id, anti-commuting with the Clifford-multiplication by tangent vectors. Such a grading naturally exists in even dimensions and in dimension m ≡ 1 (8) and induces a grading of ΣM ⊗ f^{∗}T N . In even dimension m, it is given by the Clifford action of the so-called complex volume form of T M, namely G(φ) = i^{\frac{m}{2}} e_{1}·. . .·e_{m}·φ for a positively oriented orthonormal local frame. One easily checks that the definition of G does not depend on the choice of the local frame, and thus G is globally defined. The spinor bundle then decomposes into two complex subbundles Σ_{+}M and Σ_{−}M associated to the +1 and −1-eigenvalue of G respectively. As G is Hermitian and parallel, the decomposition ΣM = Σ_{+}M ⊕ Σ_{−}M is orthogonal in the complex sense and parallel. In dimension m ≡ 1 (8), the grading G is provided by a real structure on the complex spinor representation, see e. g. [16, Sec. 1.7]. This map is complex anti-linear, involutive and anticommutes with the Clifford action by tangent vectors. It is still self-adjoint in the real sense, i. e. ℜe(⟨G(φ), ψ⟩) = ℜe(⟨φ, G(ψ)⟩) for all spinors φ, ψ. Thus the real structure induces a realorthogonal and parallel decomposition ΣM = Σ_{+}M ⊕ Σ_{−}M into real subbundles associated to the eigenvalues ±1. We also have ΣM = Σ_{+}M ⊗_{ℝ} ℂ.

Since f^{∗}T N is considered as a real vector bundle, we obtain a ℤ_{2}-grading G ⊗ id on the tensor product ΣM ⊗ f^{∗}T N anticommuting with the Clifford multiplication by tangent vectors. In particular G ⊗ id anti-commutes with D /^{f} which hence splits into D / _{+}^{f} + D / _{−}^{f}, where

D / _{±}^{f} : C^{∞}(M, Σ_{±}M ⊗ f^{∗}T N) ⟶ C^{∞}(M, Σ_{∓}M ⊗ f^{∗}T N).

From the orthogonality of the splitting, we see that ℜe(⟨D /^{f}Φ_{+}, Φ_{+}⟩) = ℜe(⟨D /^{f}Φ_{−}, Φ_{−}⟩) = 0 for all Φ_{±} ∈ C^{∞}(M, Σ_{±}M ⊗ f^{∗}T N). On the other hand, (D /^{f}Φ_{±}, Φ_{±})_{L^{2}} is real as D /^{f} is self-adjoint. Thus (D /^{f}Φ_{+}, Φ_{+})_{L^{2}} = (D /^{f}Φ_{−}, Φ_{−})_{L^{2}} = 0. Therefore, Corollary 5.2 implies:

Corollary 6.1. Assume m is even or m ≡ 1 (8). Let f_{0} ∈ C^{∞}(M^{m}, N) be a harmonic map. Split Φ_{0} ∈ ker(D /^{f_{0}}) into Φ_{0} = Φ_{0}^{+} + Φ_{0}^{−}. Then the pairs (f_{0}, Φ_{0}^{+}) and (f_{0}, Φ_{0}^{−}) are Dirac-harmonic.

Remark 6.2. An alternative proof of Corollary 6.1 is obtained by showing V_{G⊗id(Φ_{0})} = −V_{Φ_{0}} for all spinors φ_{0}. If Φ_{0} = ±G ⊗ id(Φ_{0}), then using V_{−Φ_{0}} = V_{Φ_{0}}, we obtain V_{Φ_{0}} = 0 and hence (f_{0}, Φ_{0}) is Dirac-harmonic.

Corollary 6.1 is of central importance, since it will shows the existence of non-trivial Dirac-harmonic maps in the case where the underlying map f_{0} is not perturbation-minimal, see Section 9.

7. The case m ∈ 4ℕ

As noticed above, if we assume m to be even, the Dirac operator D^{f} is odd with respect to the above grading. Restriction to sections of Σ_{+}M thus yields an operator

D / _{+}^{f} : C^{∞}(M, Σ_{+}M) → C^{∞}(M, Σ_{−}M).

It is a Fredholm operator and the Atiyah-Singer index theorem yields its (complex) index:

ind(D / _{+}^{f}) = {ch(f^{∗}T N) · Â(T M)}[M].

This implies

ind(D / _{+}^{f}) = dim_{ℂ} ker(D / _{+}^{f})−dim_{ℂ} ker(D / _{−}^{f}) = \{ \begin{matrix}α(M, f) & if m ≡ 0 & (8) \\ 0 & if m ≡ 2, 6 & (8) \\ 2 α(M, f) & if m ≡ 4 & (8) \end{matrix}

This follows from the definition of α if m is a multiple of 4 and from the fact that the real (case m ≡ 6 (8)) or quaternionic (case m ≡ 2 (8)) structure on ΣM exchanges ker D / _{+}^{f} and ker D / _{−}^{f} if m ≡ 2, 6 (8). Alternatively, the statement in the case m ≡ 6 (8) can be deduced from the index theorem. We now restrict to the case that m is a multiple of 4. Corollary 6.1 yields the following

Corollary 7.1. Let m ∈ 4ℕ. Assume f_{0} ∈ C^{∞}(M^{m}, N) to be a harmonic map with ind(D / _{+}^{f_{0}}) ≠ 0. Set ε := sign(ind(D / _{+}^{f_{0}})) ∈ {+, −}. Then

{(f_{0}, Φ_{0}^{ε}) | Φ_{0}^{ε} ∈ ker(D / _{ε}^{f_{0}})}

is a linear space of Dirac-harmonic maps of complex dimension at least |ind(D / _{+}^{f_{0}})|. This complex dimension is even for m ≡ 4 (8).

8. Minimality

The present and the following sections provide results for any dimension m, but are mainly interesting if m ≡ 1, 2 (8).

Definition 8.1. A smooth map f_{0} : M ⟶ N is called perturbationminimal if and only if dim(ker(D /^{f^{∗}T N})) ≥ dim(ker(D /^{f_{0}^{∗}T N})) for all f in a C^{∞}-neighbourhood of f_{0}.

Obviously, any homotopy class of maps from M to N contains perturbation-minimal maps.

Proposition 8.2. Let f_{0} : M → N be perturbation-minimal and harmonic. Then for any Φ_{0} ∈ ker(D /^{f_{0}}), the pair (f_{0}, Φ_{0}) is an uncoupled Dirac-har-monic map.

Proof. Since f_{0} is harmonic and D /^{f_{0}}Φ_{0} = 0, it suffices by Corollary 5.2 to show that, for every smooth variation (f_{t})_{t∈]−ε,ε[} of f_{0}, there exists a variation (Φ_{t})_{t} of Φ_{0}, differentiable at t = 0, such that D /^{f_{t}}Φ_{t} = 0 for all t close enough to 0. To that extent we fix such a variation (f_{t})_{t∈]−ε,ε[} of f_{0} and consider, for each t ∈]−ε, ε[, the L^{2}-orthogonal projection π_{t} : L^{2}(M, ΣM ⊗ f_{0}^{∗}(T N)) ⟶ ker(D̂ /_{t}) ⊂ C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N), where D̂ /_{t} := β_{t}^{−1} ∘ D /_{t} ∘ β_{t} and D /_{t} := D /^{f_{t}}, see Proposition 5.1. Claim A: For every Ψ_{0} ∈ ker(D /_{0}) one has ∥π_{t}(Ψ_{0})−Ψ_{0}∥_{L^{2}} = O(t) as t → 0. Proof of Claim A. By assumption, one has dim(ker(D̂ /_{t})) ≥ dim(ker(D̂ /_{0})) = dim(ker(D /_{0})) for all t small enough. Since the dimension of the kernel is always upper semi-continuous in the parameter, this already implies the equality dim(ker(D̂ /_{t})) = dim(ker(D /_{0})) =: k for all t in a sufficiently small neighbourhood of 0. Let now {λ_{j}(D̂ /_{t})}_{j=1}^{∞} denote the spectrum of D̂ /_{t}, where 0 ≤ |λ_{1}| ≤ … ≤ |λ_{j}| ≤ |λ_{j+1}| ≤ . . .. Since the map t ↦ f_{t} is smooth, the spectrum of D̂ /_{t} depends continuously on t (each eigenvalue is continuous in t) [34, Sec. 9.3], in particular t ↦ λ_{k+1}(D̂ /_{t}) is continuous. But the condition dim(ker(D̂ /_{t})) = k forces λ_{k+1}(D̂ /_{t}) to be positive for all small enough t, therefore there exists λ_{0} > 0 such that |λ_{k+1}(D̂ /_{t})| ≥ λ_{0} for all small t. The min-max principle then implies that, for every Ψ ∈ ker(D̂ /_{t})^{⊥} ∩ H^{1,2}(M),

∥D̂ /_{t}Ψ∥_{L^{2}} ≥ |λ_{k+1}(D̂ /_{t})| · ∥Ψ∥_{L^{2}} ≥ λ_{0} · ∥Ψ∥_{L^{2}}.

Putting Ψ = π_{t}(Ψ_{0}) − Ψ_{0} ∈ ker(π_{t}) ∩ H^{1,2}(M) = ker(D̂ /_{t})^{⊥} ∩ H^{1,2}(M), we obtain

λ_{0} · ∥π_{t}(Ψ_{0}) − Ψ_{0}∥_{L^{2}} ≤ ∥D̂ /_{t}(π_{t}(Ψ_{0}) − Ψ_{0})∥_{L^{2}}
= ∥(D̂ /_{0} − D̂ /_{t})(Ψ_{0})∥_{L^{2}}
≤ ∥D̂ /_{0} − D̂ /_{t}∥_{op} · ∥Ψ_{0}∥_{H^{1,2}},

where ∥ · ∥_{op} denotes the operator norm for bounded linear operators from H^{1,2}(M) into L^{2}(M). Since by construction all operators D̂ /_{t} have the same principal symbol and depend smoothly on the parameter t, it is easy to see that ∥D̂ /_{0} − D̂ /_{t}∥_{op} = O(t) and hence ∥π_{t}(Ψ_{0}) − Ψ_{0}∥_{L^{2}} = O(t) as t goes to 0, which was to be shown. ✓

Claim B: For every Ψ_{0} ∈ ker(D /_{0}), the limit lim_{t→0} \frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} exists in H^{1,2} and is actually smooth on M. Proof of Claim B. For obvious reasons (see (2) above), we use the short notation \frac{d}{dt} D̂ /_{t}_{|_{t=0}} := ∑_{j=1}^{m} e_{j} · ⊗R_{\frac{∂f}{∂t} (0),df_{0}(e_{j})}^{N} ∈ C^{∞}(M, End(ΣM ⊗ f_{0}^{∗}T N)).

We first show that \frac{d}{dt} D̂ /_{t}_{|_{t=0}}Ψ_{0} ∈ ker(D /_{0})^{⊥} ∩ H^{1,2}(M) (as in Claim A, the orthogonal complement is considered in L^{2}). Let X_{0} ∈ ker(D /_{0}), then for every small t ≠ 0 one has (\frac{d}{dt} D̂ /_{t}_{|_{t=0}}Ψ_{0}, X_{0})_{L^{2}} =lim_{t→0} (\frac{1}{t} D̂ /_{t}Ψ_{0}, X_{0})_{L^{2}}, where the

convergence of the zero-order-operator \frac{D /_{t}−D /_{0}}{t} to \frac{d}{dt} D̂ /_{t}_{|_{t=0}} is to be understood in the C^{0}-sense (and not in the operator norm ∥ · ∥_{op} above). Now

(\frac{1}{t}D̂ /_{t}Ψ_{0}, X_{0})_{L^{2}} = (\frac{1}{t} · Ψ_{0}, D̂ /_{t}X_{0})_{L^{2}}
= (Ψ_{0} − π_{t}(Ψ_{0}), \frac{1}{t}D̂ /_{t}X_{0})_{L^{2}}

because of D̂ /_{t}(π_{t}(Ψ_{0})) = 0. But \frac{1}{t} D̂ /_{t}X_{0} ⟶_{t→0} \frac{d}{dt} D̂ /_{t}_{|_{t=0}}X_{0} (which is well-

defined and lies in C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N)) and Ψ_{0} − π_{t}(Ψ_{0}) ⟶_{t→0} 0 in L^{2} by

Claim A, therefore (\frac{1}{t} D̂ /_{t}Ψ_{0}, X_{0})_{L^{2}} ⟶_{t→0} 0, and this implies

(\frac{d}{dt}D̂ /_{t}_{|_{t=0}}Ψ_{0}, X_{0})_{L^{2}} = 0.

Since this holds for every X_{0} ∈ ker(D /_{0}), we obtain \frac{d}{dt} D̂ /_{t}_{|_{t=0}}Ψ_{0} ∈ ker(D /_{0})^{⊥}. Elliptic regularity yields the smoothness of Ψ_{0} and thus \frac{d}{dt} D̂ /_{t}_{|_{t=0}}Ψ_{0} is smooth as well. Hence there exists a unique Θ_{0} ∈ ker(D /_{0})^{⊥} such that D /_{0}Θ_{0} = −\frac{d}{dt} D̂ /_{t}_{|_{t=0}}Ψ_{0}. Note that, because of the ellipticity of D /_{0}, the section Θ_{0} has to be smooth. We show that lim_{t→0} \frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} = Θ_{0} in H^{1,2}. By Claim A, we have, for every t ≠ 0,

D /_{0}(\frac{π_{t}(Ψ_{0}) − Ψ_{0}}{t} − Θ_{0}) = \frac{(D /_{0} − D /_{t})}{t}(π_{t}(Ψ_{0})) − D /_{0}Θ_{0}
⟶_{t→0}^{L^{2}} −\frac{d}{dt}D̂ /_{t}_{|_{t=0}}Ψ_{0} + \frac{d}{dt}D̂ /_{t}_{|_{t=0}}Ψ_{0},

that is, ∥D /_{0}(\frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} − Θ_{0})∥_{L^{2}} ⟶_{t→0} 0. Since D /_{0} is elliptic, it remains to

show that ∥\frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} − Θ_{0}∥_{L^{2}} ⟶_{t→0} 0. Pick any X_{0} ∈ ker(D /_{0}), then for any t ≠ 0,

(\frac{π_{t}(Ψ_{0}) − Ψ_{0}}{t} − Θ_{0}, X_{0})_{L^{2}} = (\frac{π_{t}(Ψ_{0}) − Ψ_{0}}{t} , X_{0})_{L^{2}}
= (\frac{π_{t}(Ψ_{0}) − Ψ_{0}}{t} , X_{0} − π_{t}(X_{0}))_{L^{2}}.

Since by Claim A both \frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} remains L^{2}-bounded near t = 0 and X_{0} − π_{t}(X_{0}) ⟶_{t→0} 0 in L^{2}, we deduce that (\frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} −Θ_{0}, X_{0})_{L^{2}} ⟶_{t→0} 0. This holds

for any X_{0} in the finite-dimensional space ker(D /_{0}), therefore ∥\frac{π_{t}(Ψ_{0})−Ψ_{0}}{t} − Θ_{0}∥_{L^{2}} ⟶_{t→0} 0, which yields the result. ✓

It follows from Claim B that, setting Φ̂_{t} := π_{t}(Φ_{0}), then Φ̂_{t} ∈ C^{∞}(M, ΣM ⊗ f_{0}^{∗}T N) is a solution of D̂ /_{t}Φ̂_{t} = 0 for every t with Φ̂_{t}_{|_{t=0}} = Φ_{0} and the

map ] − ε, ε[⟶ H^{1,2}(M), t ↦ Φ̂_{t} is differentiable at t = 0 with \frac{∂Φ̂}{∂t}(0) ∈ C^{∞}(M, ΣM ⊗f_{0}^{∗}T N). Therefore, Φ_{t} := β_{t}∘Φ̂_{t} fulfills the conditions required above and the proposition is proved. □

9. Non-minimality

Assume that a harmonic map f_{0} : M → N is given with α(M, f_{0}) ≠ 0, thus in particular ker(D /^{f_{0}}) ≠ 0. In the previous section we have seen that, if f_{0} is perturbation-minimal, then for any Φ ∈ ker(D /^{f_{0}}) the pair (f_{0}, Φ) is an uncoupled Dirac-harmonic map. Hence we have obtained a linear space of uncoupled Dirac-harmonic maps of real dimension 2 dim_{ℂ} ker(D /^{f_{0}}). Although we have no proof for the moment, it seems that the perturbationminimal case is the generic one. In this section we study the case where f_{0} is not perturbation-minimal. Obviously, the proof of the perturbation-minimal setting cannot be easily adapted since the dimension of ker(D /_{t}) does no longer remain constant for small t. However, using the grading arguments explained in Section 6, we will obtain a space of uncoupled, nontrivial Dirac-harmonic maps of even larger dimension. Most of the spaces — but not all — are actually complex vector spaces. Nevertheless, for homogeneity of presentation we only use real dimensions in the theorem below. To that extent we define, for any m ≡ 0, 1, 2, 4 (8), the integers b_{m} and d_{m} by the following formulas in which the minimum runs over all compact Riemannian spin manifolds M of dimension m, all spin structures on M, all compact Riemannian manifolds N and all smooth maps f : M → N with non-trivial α-index.

b_{m} := min { dim_{ℝ}(ker(D /^{f})) | f ∈ C^{∞}(M^{m}, N), α(M, f) ≠ 0}

and

d_{m} := min { max_{ε∈{±}} { dim_{ℝ}(ker(D / _{ε}^{f}))} | f ∈ C^{∞}(M^{m}, N), α(M, f) ≠ 0

and f non-perturbation-minimal}.

Both b_{m} and d_{m} are positive integers because of the assumption α(M, f) ≠ 0 for all admissible M and f (and such exist in each dimension under consideration). From now on assume f_{0} ∈ C^{∞}(M, N) to be a harmonic map. By Proposition 8.2, the number b_{m} is a lower bound for the real dimension of the space of Φ_{0}’s such that (f_{0}, Φ_{0}) is a Dirac-harmonic map for a given perturbationminimal and harmonic map f_{0} ∈ C^{∞}(M^{m}, N). In case the given map f_{0} ∈ C^{∞}(M^{m}, N) is harmonic but no more perturbation-minimal, the number d_{m} is a lower bound for the real dimension of the space of Φ_{0}’s such that (f_{0}, Φ_{0}) is Dirac-harmonic (Corollary 6.1). The main result of this section provides a lower bound for d_{m}. For the sake of completeness, we include the corresponding lower bound for b_{m}.

Theorem 9.1. With the above notations, b_{m} and d_{m} are bounded below by the following integers:

\begin{matrix}m mod 8 & 0 & 1 & 2 & 4 \\ b_{m} ≥ & 2 & 2 & 4 & 4 \\ d_{m} ≥ & 4 & 3 & 6 & 8 \end{matrix}

In particular, b_{m} ≥ a_{m} and d_{m} ≥ a_{m} where again a_{m} = 2 for m ≡ 0, 1 (8) and a_{m} = 4 for m ≡ 2, 4 (8).

Roughly speaking, the theorem says that in the non-perturbation-minimal case the space of Dirac-harmonic maps we obtain is larger than in the perturbation-minimal one. Note that Theorem 9.1 implies Theorem 1.2.

Proof. We handle the four cases separately.

• Case m ≡ 0 (8): If α(M, f_{0}) ≠ 0, then |α(M, f_{0})| ≥ 1, thus

| dim_{ℂ} ker D / _{+}^{f_{0}} + dim_{ℂ} ker D / _{−}^{f_{0}}| ≥ |ind(D / _{+}^{f_{0}})| ≥ 1,

which implies b_{m} ≥ 2. On the other hand we have

max_{ε∈{±}}(dim_{ℂ} ker(D / _{ε}^{f_{0}})) = min_{ε∈{±}}(dim_{ℂ} ker(D / _{ε}^{f_{0}})) + |ind(D / _{+}^{f_{0}})|

Thus, if f_{0} is not perturbation-minimal, i. e. if ker(D / _{ε}^{f_{0}})) ≠ {0}, then

d_{m} = 2 min{ max_{ε∈{±}} (dim_{ℂ} ker(D / _{ε}^{f_{0}}))} ≥ 4.

• Case m ≡ 4 (8): If α(M, f_{0}) ≠ 0, then |α(M, f_{0})| ≥ 1 as well. Recall that, in these dimensions there exists a quaternionic structure on the (twisted) spinor bundle commuting with both the ℤ_{2}-grading and the Dirac operator D /^{f_{0}}, see e. g. [16, Sec. 1.7]. This quaternionic structure turns the vector spaces Σ_{±}M ⊗ f_{0}^{∗}T N into quaternionic spaces. Thus the discussion is analogous to above, but all real dimensions are divisible by 4 instead of 2. We obtain b_{m} ≥ 4 and d_{m} ≥ 8.

• Case m ≡ 1 (8): If α(M, f_{0}) ≠ 0, then α(M, f_{0}) = 1 ∈ ℤ_{2}, so that dim_{ℂ} ker(D /^{f_{0}}) ≥ 1 in case f_{0} is perturbation-minimal. Hence b_{m} ≥ 2. If f_{0} is not perturbation-minimal, then for any ε ∈ {+, −} we have dim_{ℝ} ker(D / _{ε}^{f_{0}}) = dim_{ℂ} ker(D /^{f_{0}}) ≥ 3. This shows d_{m} ≥ 3.

• Case m ≡ 2 (8): If α(M, f_{0}) ≠ 0, then α(M, f_{0}) = 1 ∈ ℤ_{2}. Recall that, in these dimensions there exists a quaternionic structure on the (twisted) spinor bundle anti-commuting with the ℤ_{2}-grading G ⊗ id and commuting with the Dirac operator D /^{f_{0}}, see again e. g. [16, Sec. 1.7]. Thus ker(D /^{f_{0}}) is a quaternionic vector spaces and thus dim_{ℂ} ker(D /^{f_{0}}) ≥ 2 in the perturbation-minimal case, implying b_{m} ≥ 4. As the quaternionic structure anti-commutes with G ⊗ id, we also have

dim_{ℝ} ker(D / _{ε}^{f_{0}}) = dim_{ℂ} ker(D /^{f_{0}})

for ε = + and ε = −. If f_{0} is not perturbation-minimal, then dim_{ℂ} ker(D /^{f_{0}}) ≥ 6. It follows d_{m} ≥ 6. □

Remark 9.2. For a given Dirac-harmonic map (f_{0}, Φ_{0}) with harmonic mapping-component f_{0}, the pair (f_{0}, λΦ_{0}) obviously remains Dirac-harmonic for every λ ∈ ℂ. In particular the space of Dirac-harmonic maps (f_{0}, Φ_{0}) with fixed harmonic mapping-component f_{0} is a complex cone. Therefore, in dimension m ≡ 1 (8) (which is the only one where the space of Φ_{0}’s making (f_{0}, Φ_{0}) Dirac-harmonic is a priori only real) and for non-perturbation-minimal harmonic maps f_{0}, we actually obtain a complex cone of real dimension at least 4 of Φ_{0}’s such that (f_{0}, Φ_{0}) is Dirac-harmonic.

10. Examples of maps with non-trivial index

In this section, we still assume that M is a compact spin manifold but do not fix the spin structure since e. g. in Corollary 10.3 we want to choose it such that the index does not vanish. As a consequence the spin structure χ will no longer be suppressed in α(M, χ, E).

10.1. The case m = 2. In this subsection we assume M to be a closed orientable connected surface. Such a surface always carries a spin structure χ. In general this spin structure is not unique; the space of all spin structures (up to isomorphism) is an affine ℤ_{2}-space, modeled on the ℤ_{2}-vector space H^{1}(M, ℤ_{2}). Thus χ + α is a spin structure on M for every α ∈ H^{1}(M, ℤ_{2}). We first compute the α-genus in terms of simpler bordism invariants of surfaces which we use to produce new examples of Dirac-harmonic maps.

Proposition 10.1. Let E ⟶ M be a real vector bundle of rank k over a closed connected oriented surface M with spin structure χ. Let w_{1}(E) ∈ H^{1}(M, ℤ_{2}) and w_{2}(E) ∈ H^{2}(M, ℤ_{2}) be the first and second Stiefel-Whitney classes of E ⟶ M respectively. Then

α(M, χ, E) = (k + 1) · α(M, χ) + α(M, χ + w_{1}(E)) + w_{2}(E)[M]. (3)

Proof. The proof consists of two steps. Claim 1: The statement holds in case that E ⟶ M is orientable. Proof of Claim 1. Since any orientable real line bundle is trivial, we may assume k ≥ 2. First, we show that it suffices to handle the case where M = ℂP^{1}. Let indeed D^{2} ⊂ M be any embedded closed 2-disc in M. Then

the loop ∂D^{2} lies in the commutator subgroup [π_{1}(M\ D^{∘}^{2}), π_{1}(M\ D^{∘}^{2})] of

π_{1}(M\ D^{∘} ^{2}). More precisely, M\ D^{∘} ^{2} deformation retracts onto the wedge sum of 2g circles, where g is the genus of M. Since any orientable real vector bundle over the circle is trivial, it is also trivial on the wedge sum of circles.

This in turn implies that E_{|_{M\^{∘} D^{2}}} ⟶ M\ D^{∘} ^{2} is trivial. It follows that M

together with E ⟶ M can be seen as the connected sum M♯ ℂP^{1}, where the first factor M carries the trivial vector bundle \underline{ℝ^{k}} := ℝ^{k} × M ⟶ M and ℂP^{1} carries some vector bundle F ⟶ ℂP^{1} such that E is obtained by gluing \underline{ℝ^{k}} together with F. Here one should pay attention to the fact that, when performing the connecting sum, the bundles on both factors have to be trivialized so that their trivializations coincide on ∂D^{2}. Now as in classical surgery theory (without bundles), any surgery between manifolds with bundles provides a spin-bordism for vector bundles. The invariance of α under spin-bordism then gives α(M, χ, E) = α(M, χ, \underline{ℝ^{k}}) + α(ℂP^{1}, F) (there is only one spin structure on ℂP^{1}, so we omit the notation χ in that case). Obviously one has α(M, χ, \underline{ℝ^{k}}) = kα(M, χ, \underline{ℝ}) = kα(M, χ) - in particular (3) holds for trivial vector bundles (of any rank), since w_{2}(\underline{ℝ^{k}}) = 0. Moreover, the Stiefel-Whitney number w_{2}(E)[M] is also a spin-bordism invariant, a fact which is analogous to Pontrjagin’s theorem [25, Thm. 4.9 p.52] and which can be elementarily proved in just the same way. Therefore, w_{2}(E)[M] = w_{2}(\underline{ℝ^{k}})[M]+w_{2}(F)[ℂP^{1}] = w_{2}(F)[ℂP^{1}]. Hence we are reduced to showing α(ℂP^{1}, F) = w_{2}(F)[ℂP^{1}], which is exactly (3) for M = ℂP^{1} since α(ℂP^{1}) = 0 (the existence of a metric with positive scalar curvature implying the vanishing of the kernel of the untwisted Dirac operator). Since ℂP^{1} is simply-connected, each vector bundle over ℂP^{1} is orientable. If k ≥ 3, then there are only two isomorphism-classes of k-ranked real vector bundles over ℂP^{1}, whereas those isomorphism classes stand in one-to-one correspondence with the integral powers of the tautological complex line bundle γ_{1} ⟶ ℂP^{1} if k = 2, see e. g. [32, Thm. 18.5]. Actually, it suffices to show (3) for the tautological complex line bundle E = γ_{1}. Consider indeed l · γ_{1} := ⨁_{j=1}^{l} γ_{1} ⟶ ℂP^{1} for any l ∈ ℕ, l ≥ 1. Then l · γ_{1} is a 2l-ranked real vector bundle over ℂP^{1} and is non-trivial (its total Chern-class is 1 + l · a ≠ 1, where a ∈ H^{2}(ℂP^{1}, ℤ) ≅ ℤ is the generator given by the tautological bundle). Moreover, l · γ_{1} represents up to isomorphism the only non-trivial 2l-ranked real vector bundle over ℂP^{1} if l ≥ 2. Since trivially α(ℂP^{1}, l·γ_{1}) = l · α(ℂP^{1}, γ_{1}) and w_{2}(l · γ_{1}) = l · w_{2}(γ_{1}), we are reduced to the case where l = 1. In case k = 2l+1 is odd, the bundle l·γ_{1}⊕\underline{ℝ} ⟶ ℂP^{1} is again k-ranked and non-trivial, so it is up to isomorphism the only non-trivial k-ranked real vector bundle over ℂP^{1}. As noticed above, the spin Dirac operator on ℂP^{1} has trivial kernel, so that α(ℂP^{1}, l · γ_{1} ⊕ \underline{ℝ}) = α(ℂP^{1}, l · γ_{1}) = l · α(ℂP^{1}, γ_{1}) and, as is well-known, w_{2}(l · γ_{1} ⊕ \underline{ℝ}) = w_{2}(l · γ_{1}) = l · w_{2}(γ_{1}), so that again we are reduced to the case where E = γ_{1}. Note that the case k = 2 can be deduced from the case k ≥ 3 by adding a trivial real line bundle: as we have seen above, α(M, χ, E⊕\underline{ℝ}) = α(M, χ, E)+α(M, χ) and w_{2}(E⊕\underline{ℝ}) = w_{2}(E). It remains to show α(ℂP^{1}, γ_{1}) = 1 = w_{2}(γ_{1})[ℂP^{1}]. On the one hand, since γ_{1} is a complex line bundle, w_{2}(γ_{1}) = [c_{1}(γ_{1})]_{ℤ_{2}}. From c_{1}(γ_{1})[ℂP^{1}] = −1, we obtain w_{2}(γ_{1})[ℂP^{1}] = [c_{1}(γ_{1})[ℂP^{1}]]_{ℤ_{2}} = 1. On the other hand, it follows from [17, Thm. 4.5] that dim_{ℂ}(ker(D /^{γ_{1}}) = 2 (beware that we tensorize over ℝ, so that ΣT ℂP^{1} ⊗_{ℝ} γ_{1} = {ΣT ℂP^{1} ⊗_{ℂ} γ_{1}} ⊕ {ΣT ℂP^{1} ⊗_{ℂ} γ_{1}^{−1}}). Therefore α(ℂP^{1}, γ_{1}) = 1, which concludes the proof of Claim 1. ✓ Claim 2: The statement holds in general. Proof of Claim 2. By definition of the determinant bundle Λ^{k}E → M, the bundle E ⊕ Λ^{k}E → M is orientable. Alternatively, this follows from w_{1}(E) = w_{1}(Λ^{k}E). Moreover, the spinor bundle associated to the spin structure χ+w_{1}(E) on M is ΣM ⊗_{ℝ} Λ^{k}E, in particular α(M, χ, E ⊕Λ^{k}E) = α(M, χ, E) + α(M, χ, Λ^{k}E) = α(M, χ, E) + α(M, χ + w_{1}(E)). We apply the proposition in the orientable case for E ⊕ Λ^{k}E instead of E and obtain

α(M, χ, E) + α(M, χ + w_{1}(E)) = (k + 1)α(M, χ) + w_{2}(E ⊕ Λ^{k}E).

Now we calculate w_{2}(E ⊕ Λ^{k}E) = w_{2}(E) + w_{1}(E) ∪ w_{1}(Λ^{k}E) + w_{2}(Λ^{k}E). As Λ^{k}E is of real rank 1 we have w_{2}(Λ^{k}E) = 0. According to Lemma 10.2 below w_{1}(E) ∪ w_{1}(E) = 0. Thus w_{2}(E ⊕ Λ^{k}E) = w_{2}(E) which shows Claim 2 and concludes the proof of Proposition 10.1. □

Lemma 10.2. Let M be an orientable surface and β ∈ H^{1}(M, ℤ_{2}). Then β ∪ β = 0.

Proof. As M is orientable, the homology group H_{k}(M, ℤ) is a finitely generated free ℤ-module for k = 0, 1, 2, and thus the universal coefficient theorem implies that tensoring the coefficients with ℤ_{2} yields isomorphisms mod 2 : H^{k}(M, ℤ) ⊗ ℤ_{2} ⟶^{≅} H^{k}(M, ℤ_{2}). These isomorphisms are compatible with the ∪-products on H^{∗}(M, ℤ) and on H^{∗}(M, ℤ_{2}). Let \overline{β} ∈ H^{1}(M, ℤ) such that \overline{β} mod 2 = β. As the cup-product on H^{∗}(M, ℤ) is skew-symmetric \overline{β} ∪ \overline{β} = 0. Thus

β ∪ β = (\overline{β} mod 2) ∪ (\overline{β} mod 2) = (\overline{β} ∪ \overline{β}) mod 2 = 0.
□

From now on and unless explicitly mentioned N will be orientable.

Corollary 10.3. Let f : M → N be a smooth map from a closed orientable connected surface M of positive genus to an odd-dimensional orientable manifold N. Then M admits a spin structure χ such that the α-genus α(M, χ, f) ≠ 0. In particular, each harmonic map in the homotopy class of f provides a 4-dimensional space of Dirac-harmonic maps.

Proof. For any spin structure χ on M the identity (3) reads α(M, χ, f) = α(M, χ)+w_{2}(f^{∗}T N)[M] because of N being odd-dimensional and orientable. The term w_{2}(f^{∗}T N)[M] does not depend on the spin structure χ on M. Since the genus of M is at least one, there exists for any x ∈ ℤ_{2} at least one spin structure with α-genus x. Therefore, whatever w_{2}(f^{∗}T N)[M] is, α(M, χ, f) can be made equal to 1 for at least one spin structure on M. The homotopy invariance of the α-genus (Corollary 4.4) concludes the proof. □

Example 10.4. The choice of spin structure χ for Corollary 10.3 depends on f through the value w_{2}(f^{∗}T N)[M]. Choose for instance E to be an orientable but non-spin real vector bundle of rank 3 over M := 𝕋^{2}, e. g. let E := (γ_{1}♯ \underline{ℝ^{2}}) ⊕ \underline{ℝ} ⟶ ℂP^{1}♯ 𝕋^{2} = 𝕋^{2} where γ_{1} is the tautological bundle over ℂP^{1} which is glued together with the trivial line bundle to give a bundle over the connected sum ℂP^{1}♯ 𝕋^{2}. Let N be the total space of E, take f_{1} : M → N to be the embedding by the zero-section and consider f_{2} : M → N defined by f_{2}(z_{1}, z_{2}) := f_{1}(z_{1}^{2}, z_{2}) for all (z_{1}, z_{2}) ∈ 𝕊^{1} ×𝕊^{1} = 𝕋^{2}. Then w_{2}(f_{1}^{∗}T N)[M] = 1 but w_{2}(f_{2}^{∗}T N)[M] = 0.

Another straightforward consequence of Proposition 10.1 is the following

Corollary 10.5. Let M be a compact orientable connected surface with a bounding spin structure χ, and assume that f : M → N is given. If N is orientable and ⟨w_{2}(T N), f_{∗}[M]⟩ ≠ 0 (in particular N is non-spin), then α(M, χ, f) ≠ 0.

Next we apply Proposition 10.1 and Corollary 10.3 to produce new examples of Dirac-harmonic maps. Those examples are constructed out of harmonic maps from surfaces with spin structures such that the corresponding genus α(M, χ, f) does not vanish.

Recall a result from the celebrated Sacks-Uhlenbeck paper [29], proven simultaneously by L. Lemaire [24]. Given a closed orientable surface M and a closed n-dimensional Riemannian manifold N, there exists in each homotopy class of maps from M to N an energy-minimizing (in particular harmonic) map as soon as π_{2}(N) = 0; if π_{2}(N) ≠ 0 then there exists a system of generators for π_{2}(N) as π_{1}(N)-module, each containing an energy-minimizing map from 𝕊^{2} = ℂP^{1} to N.

Case π_{2}(N) = 0: We obtain α(𝕊^{2}, f) = 0 for every map f : 𝕊^{2} ⟶ N since f^{∗}T N ⟶ 𝕊^{2} is trivial, therefore we cannot apply our main result for M = 𝕊^{2}. If the genus of M is at least 1 and n is odd, then Corollary 10.3 already shows the existence of a spin structure χ on M for which α(M, χ, f) ≠ 0, whatever f : M ⟶ N is. Thus we obtain in that case the existence of a 4-dimensional space of Dirac-harmonic maps of the form (f_{0}, Φ) where f_{0} is a harmonic map in the homotopy class of f. This enhances the previous example [21, Thm. 2] (based on twistor-spinors) for either genus at least 2 or genus 1 and bounding spin structures on 𝕋^{2}, since in both situations no non-trivial twistor-spinor is available. If n is even, then α(M, χ, f) = w_{2}(f^{∗}T N)[M]. For n = 2 and N closed one has w_{2}(f^{∗}T N)[M] = 0 since N is spin, hence our main result cannot be applied. For n ≥ 4 the number α(M, χ, f) ≠ 0 as soon as f^{∗}T N ⟶ M is non-trivial, since a k(≥ 3)-ranked real vector bundle over a surface is trivial if and only if its first and second Stiefel-Whitney classes vanish, see e. g. [25, Sec. 12] (in particular N must be non-spin). Note that, if n ≥ 3 and N is non-spin, then there exists at least one M and one map f such that f^{∗}T N ⟶ M is non-trivial [23, Prop. II.1.12], thus providing a non-trivial Dirac-harmonic map. Case π_{2}(N) ≠ 0: We obtain α(𝕊^{2}, f) = w_{2}(f^{∗}T N)[𝕊^{2}] for any map f : 𝕊^{2} ⟶ N. For n = 2 and N non-orientable (otherwise N is spin) the degree of any map f : 𝕊^{2} ⟶ N has to be even since 𝕊^{2} is simply-connected, therefore w_{2}(f^{∗}T N)[𝕊^{2}] = deg(f) · w_{2}(T N)[N] = 0, so that nothing can be said. For n ≥ 3 the α-genus vanishes if and only if f^{∗}T N ⟶ 𝕊^{2} is trivial. As an example for non-vanishing, consider the standard embedding ℂP^{1} ⟶^{f} ℂP^{2}, [z_{1} : z_{2}] ⟼ [z_{1} : z_{2} : 0]. Choosing the Fubini-Study metrics of holomorphic sectional curvature 4 on both ℂP^{1} and ℂP^{2}, the map f becomes isometric and totally geodesic, in particular harmonic. Moreover, the pull-back bundle f^{∗}T ℂP^{2} can be identified with γ_{1}^{−2} ⊕ γ_{1}^{−1}, in particular w_{2}(f^{∗}T ℂP^{2}) = w_{2}(γ_{1}) ≠ 0, so that α(𝕊^{2}, f) = 1. Therefore there exists a non-zero Φ ∈ C^{∞}(𝕊^{2}, Σ𝕊^{2} ⊗ f^{∗}T ℂP^{2}) such that (f, Φ) is Dirac-harmonic. Actually the space of such Φ’s is at least (real) 12-dimensional and Φ may even be chosen not to come from any twistor-spinor on 𝕊^{2}, see [3]. Note that, if one changes f into f̂:= ι ∘ π, where ι : ℝP^{2} ⟶ ℂP^{2} is a totally geodesic embedding and π : 𝕊^{2} ⟶ ℝP^{2} is the canonical projection, then f̂ is harmonic but this time f̂^{∗}T ℂP^{2} becomes trivial, hence no non-trivial Dirac-harmonic map can be found using our methods.

In an analogous way, existence results for harmonic maps by e. g. Eells-Sampson [14] (see also [30]) or Y.L. Xin [35] (see also [36, Thm 5.11]) give Dirac-harmonic maps provided the corresponding α-genus does not vanish. We summarize the results in the following two theorems, where we assume all surfaces to be connected. Note that if we say that there is a 4-dimensional space of Dirac-harmonic maps, we do not exclude the possibility that there is even a space of higher dimension. We distinguish both cases where the homotopy class of the underlying map can be prescribed (Theorem 10.6) from the one where it cannot (Theorem 10.7).

Theorem 10.6. Let M be a closed orientable surface of genus g and N be a closed n(≥ 3)-dimensional orientable Riemannian manifold. Let f : M ⟶ N be an arbitrary smooth map.

i) Assume π_{2}(N) = 0, g > 0 and n odd. Then there exist a harmonic map f_{0} homotopic to f and at least one spin structure χ on M such that there is a 4-dimensional space of Dirac-harmonic maps of the form (f_{0}, Φ).

ii) Assume π_{2}(N) = 0, g > 0, n even, and f^{∗}T N ⟶ M non-trivial. Then there is a harmonic map f_{0} homotopic to f such that for any spin structure on N there is a 4-dimensional space of Diracharmonic maps of the form (f_{0}, Φ).

Note in particular that the situation where the target manifold N has nonpositive sectional curvature or, more generally, no focal points, is covered by Theorem 10.6.i) and ii), since then π_{2}(N) = 0.

In the case π_{2}(N) ≠ 0, bubbling-off can happen, and in general one cannot prescribe a homotopy class [f] ∈ [M, N]. It only can be prescribed modulo π_{2}(N). However similar statements follow in this case, summarized in Theorem 10.7. In this theorem two modifications of the spin condition appear that shall be discussed now. The second Stiefel-Whitney class of T N, denoted by w_{2}(T N) ∈ H^{2}(N, ℤ_{2}) = Hom(H_{2}(N, ℤ_{2}), ℤ_{2}), can be composed with the map mod 2 : H_{2}(N, ℤ) → H_{2}(N, ℤ_{2}) and then with the Hurewicz map h : π_{2}(N) → H_{2}(N, ℤ). Then w_{2}(T N) = 0 if and only if N is spin, and it is an elementary exercise to show that w_{2}(T N) ∘ mod 2 ∘ h = 0 if and only if the universal covering Ñ of N is spin. If w_{2}(T N) ∘ mod 2 vanishes, then w_{2}(T N) factors over the Bockstein map β : H_{2}(N, ℤ_{2}) → H_{1}(N, ℤ), i.e. there is a homomorphism w : im(β) → ℤ_{2} with w_{2}(T N) = w ∘ β. Choose a complement Γ of the 2-torsion subgroup of H_{1}(N, ℤ), and let Γ̂ ⊂ π_{1}(N) be its preimage under the Hurewicz map π_{1}(N) → H_{1}(N, ℤ). Let N̂ → N be the covering associated to Γ̂. This is a finite covering and the number of sheets is the order of the 2-torsion group of H_{1}(N, ℤ), thus a power of 2. Since im(β) is included in the 2-torsion

subgroup of H_{1}(N, ℤ), the composition H_{2}(N̂, ℤ) → H_{2}(N, ℤ) →^{w_{2}(T N)} ℤ_{2} vanishes, and it follows that N̂ is spin. This provides a necessary condition for the vanishing of w_{2}(T N) ∘ mod 2, but it is unclear to us whether this is sufficient as well.

Theorem 10.7. Let N be a closed n(≥ 3)-dimensional orientable Riemannian manifold.

iii) Assume that the universal covering Ñ of N is not spin. Then there is a non-constant harmonic map f_{0} : 𝕊^{2} → N such that f_{0}^{∗}T N is non-trivial. In particular, there is a 4-dimensional space of Diracharmonic maps of the form (f_{0}, Φ).

iv) Assume that Ñ is spin, even-dimensional and that w_{2}(T N) defines a non-trivial map H_{2}(N, ℤ) → ℤ_{2}. Then there is an oriented closed surface M of genus g ≥ 1 and a non-constant harmonic map f_{0} : M → N such that for any spin structure on M there is a 4-dimen-sional space of Dirac-harmonic maps of the form (f_{0}, Φ).

v) Assume that Ñ is spin, odd-dimensional and that w_{2}(T N) defines a non-trivial map H_{2}(N, ℤ) → ℤ_{2}. Then there is an oriented closed surface M of genus g ≥ 1, a non-constant harmonic map f_{0} : M → N, and at least 3 spin structures on M such that there is a 4-dimensional space of Dirac-harmonic maps of the form (f_{0}, Φ). vi) Assume that w_{2}(T N) defines a trivial map H_{2}(N, ℤ) → ℤ_{2} (in particular Ñ is spin) and that n is odd. Then for every oriented closed surface M of genus g ≥ 1 and for every smooth map f : M → N with f_{∗}[M] ∉ h(π_{2}(N)), there exists a non-constant harmonic map f_{0} : M → N and at least one spin structure on M such that there is a 4-dimensional space of Dirac-harmonic maps of the form (f_{0}, Φ).

vii) Assume that w_{2}(T N) defines a trivial map H_{2}(N, ℤ) → ℤ_{2}, and that N is not spin. Then there is a non-orientable closed surface \overline{M} on which there is a non-constant harmonic map f̄_{0} : \overline{M} → N with ⟨\overline{f}_{0}^{∗}w_{2}(T N), [\overline{M}]⟩ ≠ 0.

vii − α) Assume that \overline{M} is not diffeomorphic to ℝP^{2} and that n is odd. The orientation covering M̂ → \overline{M} defines a non-constant harmonic map f̂_{0} : M̂ → N. Then there is at least one spin structure on M̂ for which there is a 4-dimensional space of Diracharmonic maps of the form (f̂_{0}, Φ).

vii − β) Assume \overline{M} = ℝP^{2}. Then the induced map π_{1}(ℝP^{2}) → π_{1}(N) is injective. Thus π_{1}(N) has an element of order 2.

In vii) we used [\overline{M}] for the ℤ_{2}-fundamental class of \overline{M}. Note that, in the

cases where either H_{2}(N, ℤ) →^{w_{2}(T N)} ℤ_{2} vanishes and n is even or \overline{M} = ℝP^{2} (case vii−β)) we cannot deduce the existence of non-trivial Dirac-harmonic maps.

Proof. We prove all cases separately. iii) Choose any smooth map f : 𝕊^{2} → N with ⟨f^{∗}w_{2}(T N), [𝕊^{2}]⟩ ≠ 0. Applying Sacks-Uhlenbeck’s method (Theorem A.3), one obtains a harmonic map f_{0} : ∐_{j=1}^{l} 𝕊^{2} → N which is spin bordant to f, in particular ⟨f^{∗}w_{2}(T N), [𝕊^{2}]⟩ = ⟨f_{0}^{∗}w_{2}(T N), [∐_{j=1}^{l} 𝕊^{2}]⟩ = ∑_{j=1}^{l}⟨f_{0j}^{∗}w_{2}(T N), [𝕊^{2}]⟩, where f_{0j} : 𝕊^{2} → N denotes the (harmonic) map induced by f_{0} on the j^{th} connected component of ∐_{j=1}^{l} 𝕊^{2}. In particular there is at least one j ∈ {1, …, l} with ⟨f_{0j}^{∗}w_{2}(T N), [𝕊^{2}]⟩ ≠ 0, which already implies that f_{0j} is non-constant. Therefore, the harmonic map f_{0j} : 𝕊^{2} → N satisfies α(𝕊^{2}, f_{0j}) = ⟨f_{0j}^{∗}w_{2}(T N), [𝕊^{2}]⟩ ≠ 0 and hence provides a 4-dimensional space of Dirac-harmonic maps.

iv) Since H_{2}(N, ℤ) →^{w_{2}(T N)} ℤ_{2} does not vanish, there exists a closed orientable surface M and a smooth map f : M → N such that ⟨f^{∗}w_{2}(T N), [M]⟩ ≠ 0. Necessarily M is of positive genus because of Ñ being spin. Theorem A.3 yields a harmonic map f_{0} : M ⨿ ∐_{j=1}^{l} 𝕊^{2} → N which is bordant to f, in particular ⟨f^{∗}w_{2}(T N), [M]⟩ = ⟨f_{0}^{∗}w_{2}(T N), [M ⨿ ∐_{j=1}^{l} 𝕊^{2}]⟩ = ⟨f_{00}^{∗}w_{2}(T N), [M]⟩ + ∑_{j=1}^{l}⟨f_{0j}^{∗}w_{2}(T N), [𝕊^{2}]⟩ , where f_{00} : M → N and f_{0j} : 𝕊^{2} → N denote the maps induced on each connected component. Again, since Ñ is spin, each ⟨f_{0j}^{∗}w_{2}(T N), [𝕊^{2}]⟩ vanish, so that ⟨f_{00}^{∗}w_{2}(T N), [M]⟩ ≠ 0, in particular f_{00} is a non-trivial harmonic map. Because of n being even, we obtain α(M, χ, f_{00}) = ⟨f_{00}^{∗}w_{2}(T N), [M]⟩ ≠ 0 for every spin structure χ on M, which shows the result in that case.

v) The only difference with v) lies in the formula α(M, χ, f_{00}) = α(M, χ) + ⟨f_{00}^{∗}w_{2}(T N), [M]⟩. Choosing those spin structures χ with α(M, χ) = 0 (and there are at least 3 of those) one obtains the statement. vi) For any closed orientable surface M of genus g ≥ 1 and any smooth map f : M → N with f_{∗}[M] ∉ h(π_{2}(N)), Theorem A.3 yields a harmonic map f_{0} : M → N which has to satisfy ⟨f_{0}^{∗}w_{2}(T N), [M]⟩ = 0 because of the vanishing of the map H_{2}(N, ℤ) → ℤ_{2}. However, because only 2-spheres have possibly popped off M, the map f_{0} also satisfies f_{0} ∗ [M] ∉ h(π_{2}(N)), in particular f_{0} cannot be constant. Since n is odd and g ≥ 1, there exists at least one spin structure χ on M with α(M, χ) = 1, in particular α(M, χ, f_{0}) = α(M, χ) = 1 and the statement follows.

vii) In case H_{2}(N, ℤ) →^{w_{2}(T N)} ℤ_{2} vanishes but N itself is non-spin, there exists a (necessarily) non-orientable closed surface \overline{M} and a smooth map \overline{f} : \overline{M} → N with ⟨\overline{f}^{∗}w_{2}(T N), [\overline{M}]⟩ ≠ 0. Theorem A.2 gives a harmonic map \overline{f}_{0} : \overline{M} → N with ⟨\overline{f}_{0}^{∗}w_{2}(T N), [\overline{M}]⟩ = ⟨\overline{f}^{∗}w_{2}(T N), [\overline{M}]⟩ ≠ 0 (recall that none of the bubbles contribute to the second Stiefel-Whitney number because of Ñ being spin). In particular \overline{f}_{0} cannot be constant. Obviously, the induced map f̂_{0} on M̂ remains harmonic and non-constant. Beware however that, since M̂ → \overline{M} is two-fold, ⟨f̂_{0}^{∗}w_{2}(T N), [M̂]⟩ = 2·⟨\overline{f}_{0}^{∗}w_{2}(T N), [\overline{M}]⟩ = 0, so that α(M̂, χ, f̂_{0}) = n · α(M̂, χ) for any spin structure χ on M̂. In case n is odd and M̂ is not diffeomorphic to 𝕊^{2} (that is, \overline{M} is not diffeomorphic to ℝP^{2}), there exists at least one spin structure χ on M̂ with α(M̂, χ) = 1, hence α(M̂, χ, f̂_{0}) = 1, which proves vii − α). In case \overline{M} = ℝP^{2}, assume the group homomorphim π(ℝP^{2}) → π_{1}(N) induced by \overline{f}_{0} were trivial. Then \overline{f}_{0} could lift to a smooth map f̃_{0} : ℝP^{2} → Ñ through the universal cover Ñ → N. Since by assumption Ñ is spin one would have 0 = ⟨f̃_{0}^{∗}w_{2}(T Ñ), [ℝP^{2}]⟩ = ⟨\overline{f}_{0}^{∗}w_{2}(T N), [ℝP^{2}]⟩, contradiction. Therefore π(ℝP^{2}) → π_{1}(N) is injective. This shows vii − β) and concludes the proof. □

10.2. The case m ≥ 3. In higher dimensions, existence results for harmonic maps still provide non-trivial examples of Dirac-harmonic maps. Consider for instance the situation where the closed target manifold N has non-positive sectional curvature. Then there exists an energy-minimizing (hence harmonic) map in every homotopy class of smooth maps from any closed manifold M into N [14, 30]. As an application, pick any closed connected Riemannian spin manifold N with non-positive sectional curvature and dimension n ≡ 1 (8). Let N′ be any n-dimensional closed Riemannian spin manifold with non-vanishing α-genus. In dimension n = m = 9 the Riemannian product of a Bott manifold with an 𝕊^{1} with non-bounding spin structure gives an example of such a manifold. Define the map f :

N ⨿ N′ → N, f_{|_{N}} := id_{N} and f_{|_{N′}} := cst. Obviously, the restricted map f_{|_{N′}} being constant, the vector bundle f_{|_{N′}}^{∗} T N is trivial and hence

α(N ⨿ N′, f) = α(N, f_{|_{N}}) + α(N′, f_{|_{N′}})
= α(N, id_{N}) + nα(N′)
= α(N, id_{N}) + α(N′)
= α(N, id_{N}) + 1.

On the other hand, since disks are contractible and N is connected, the map f induces a (smooth) map f^{♯} : M := N♯N′ → N which is spin bordant to f. By the spin-bordism-invariance of the α-genus, α(M, f^{♯}) = α(N, id_{N}) + 1. Therefore, either α(N, id_{N}) = 1 or α(M, f^{♯}) = 1. A 2-dimensional space of Dirac-harmonic maps is provided by (N, id_{N}) in the first case and by (M, f^{♯}) in the second case.

Appendix A. Bubbling-off for harmonic maps

In their celebrated article [29] Sacks and Uhlenbeck showed the existence of harmonic maps under certain conditions. In particular, they explained that bubbling-off effects play an important role. However, the conclusions one may draw out of the proofs of their article are stronger than the statements of their theorem. In particular, certain relations to bordisms hold, see Theorems A.1 to A.3. As these theorems are used in our article, we will explain them and give other references that finally yield complete proofs. We also need harmonic maps defined on non-orientable surfaces, and modifications of the results of [29] also hold in this non-orientable case. In order to facilitate the comparison to [29] we adapt to their notation to a large extend. One considers maps s : M → N from a closed Riemannian surface M with metric g to a closed Riemannian manifold N with metric h.

A.1. Non-orientability of M. As already said above, in [29] it is claimed that M should be orientable. This assumption can be easily removed. Orientability of M is needed to define the quadratic differential ϕ in [29] and to prove [29, Lemma 1.5] which states ∂ϕ = 0 for harmonic maps. One easily verifies that the only parts in [29] using orientation are parts of the introduction, the definition of ϕ, Lemma 1.5 and Theorem 1.6. As a consequence, in all existence results of [29] the “orientability of M”- assumption can be removed. Theorem 1.6 provides an interpretation for harmonic maps with the special property ϕ = 0. By slightly exchanging notation, this also works for non-orientable surfaces, as we will explain now. On Riemann surfaces quadratic differentials are in natural bijection to tracefree symmetric 2-tensors, and ∂ then turns into the divergence. Thus the condition ∂ϕ = 0 is equivalent to saying that the g-trace-free part of s^{∗}h has vanishing divergence. This is usually written as div(s^{∗}h)_{0} = 0. Lemma 1.5 then says: If s is harmonic, then (s^{∗}h)_{0} is divergence free. Theorem 1.6 then states: If s is harmonic and if s^{∗}h_{0} = 0, then s is a (conformal) branched immersion. Since those conditions are independent on the orientability of M, both proofs also work in the non-orientable case.

A.2. Bubbling-off and bordisms. We recall further notations from [29]. For α ≥ 1 they define the energy functional E_{α}(s) := ∫_{M}(1 + |ds|^{2})^{α} dv^{M}. The map s_{0} : M → N is harmonic if and only if it is a stationary point of the energy functional E_{1}. One can try to fix a homotopy class [M, N] of maps M → N and to minimize E_{1} in this class. Unfortunately the direct method in the calculus of variation fails. The reason for this is that E_{1} is conformally invariant in the following sense: if g_{1} and g_{2} are conformal metrics with vol(M, g_{1}) = vol(M, g_{2}) then the functional E_{1} defined with respect to g_{1} coincides with the functional E_{1} defined for g_{2}. The problem is avoided if one minimizes E_{α} for α > 1 instead. For α > 1 Sacks and Uhlenbeck showed that any homotopy class [M, N] contains a smooth map s_{α} minimizing E_{α}. It remains to discuss whether and in which sense the functions s_{α} converge to a harmonic map s : M → N for α → 1. An interesting bubbling-off phenomenon appears in [29]. As one often passes to subsequences, we will take a sequence α_{i} > 1, i ∈ ℕ, converging to 1, and substitute s_{i} := s_{α_{i}}. The following theorem can be deduced from the proofs in [29]:

Theorem A.1. Assume that M is a closed surface, and that n = dim N ≥ 3. Further let s_{i} : M → N be stationary points of E_{α_{i}} where α_{i} > 1, i ∈ ℕ, is a sequence converging to 1, and assume that E_{α_{i}}(s_{i}) converges. Then after passing to a subsequence of indices i there is a harmonic map s : M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2} → N for some ℓ ∈ ℕ ∪ {0} such that s_{i} converges to s in a certain sense.

The convergence of s_{i} to s is as follows: there are finitely many poins x_{1},…, x_{l} such that s_{i} converges to s in C^{1}(U \ {x_{1}, …, x_{l}}), see [29, Theorem 4.4]. In x_{j} spheres may bubble-off. We work in isothermic coordinates around x_{j}, i.e. x_{i} ≅ 0 is the base point, and the coordinates are defined on the neighborhood U(x_{i}) ≅ B_{2r}(0). Then there are sequences y_{i} → 0 and ε_{i} → 0 such that w ↦ s_{i}(ε_{i}w + y_{i}), defined on a ball of radius rε_{i}^{−1}, converges to a map s̃: ℝ^{2} → N in the C^{1}-topology. This map s̃ is harmonic. Conformally identifying ℝ^{2} with 𝕊^{2}\{p} one obtains a harmonic map 𝕊^{2}\{p} → N, which can be extended to a harmonic map 𝕊^{2} → N, see [29, Theorem 4.6]. This is the restriction of the map s above to one of the spheres 𝕊^{2}. The situation is even slightly more involved as spheres can bubble-off simultaneously on different growth scales in the same point x_{j}. In other words it might happen that on a given bubble, a finite numbers of new bubbles emerge. And this bubbles-on-bubbles-phenomenon can be iterated to bubbles on bubbles on bubbles, etc.. However, for a given manifold N and a given upper bound on the energy (which trivially exists as we consider minimizing maps, see also [29, Prop. 2.4]), a finite number of iterations suffices. This leads to the notion of bubble tree. The precise way in which the maps s_{α} converge to such a bubble tree was analyzed by Parker in [27]. We also recommend [28] for a more informal introduction to bubble trees. This precise description of the bubbling-off procedure also implies the following theorem.

Theorem A.2. With the assumptions of the previous theorem, the subsequence can be chosen such that there is a 3-dimensional manifold W with boundary ∂W = M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2} ⨿ M with smooth maps F_{i}, F : W → N extending s_{i} and s.

We will now describe the manifold W. We assume for simplicity that M is connected, otherwise one should consider each connected component separately. The manifold W is obtained by attaching ℓ copies of a 1-handle to W_{0} := (M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2}) × [0, 1]. Such an attachment is carried out inductively as follows, see e.g. [22, VI.6] for details. We combine two maps \overline{B_{1}(0)} → M and \overline{B_{1}(0)} → S^{2} which are diffeomorphisms onto balls in M and 𝕊^{2}, to a map G : \overline{B_{1}(0)} ⨿ \overline{B_{1}(0)} → (M ⨿ 𝕊^{2}) × \{1\} ⊂ ∂W_{0}. We define W_{1} := W_{0} ∪_{G} ([−1, 1]×\overline{B_{1}(0)}), where the subscript G indicates that we identify each point in \{−1, 1\}×\overline{B_{1}(0)}) = \overline{B_{1}(0)}⨿\overline{B_{1}(0)} with its image under G in ∂W_{0}. Strictly speaking W_{1} is a smooth manifold with boundary and a corner, but this corner can be smoothed out by slightly changing the differentiable structure close to the corner. Then W_{1} is a manifold with boundary, the boundary consists of two pieces: one piece is (M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2}) × {0} ⊂ W_{0} ⊂ W_{1}. The remaining piece is diffeomorphic to M ⨿ ∐_{j=1}^{ℓ−1} 𝕊^{2}. Several choices are done here, but they do not alter the diffeomorphism type of W_{1}. By attaching a further 1-handle we obtain a manifold W_{2} with boundary (M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2}) ⨿ (M ⨿ ∐_{j=1}^{ℓ−2} 𝕊^{2}), and finally after ℓ such attachments, we get W = W_{ℓ} with boundary (M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2}) ⨿ M. If M is orientable, we even get more structure on W. In this case M also admits a spin structure, and this defines an orientation and spin structure on M × [0, 1]. Orientations can then be chosen on the spheres 𝕊^{2} such that the induced orientation on W_{0} extends to W. Similarly the spin structure on M yields a unique spin structure on W. The manifold W then has the boundary

∂W = −(M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2}) ⨿ M.

Here the minus sign indicates that this piece of the boundary carries the opposite orientation, and all pieces of the boundary carry the spin structures induced from W. Such a W is called a spin-bordism from M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2} to M. For orientable surfaces we have thus strengthened the previous theorem:

Theorem A.3. With the hypotheses of the previous theorems, assume furthermore that M carries a fixed orientation and spin structure. Then the 3-dimensional manifold W carries an orientation and spin structure, such that it is a spin bordism from M ⨿ ∐_{j=1}^{ℓ} 𝕊^{2} to M.

Acknowledgements. The authors want to thank Guofang Wang, Jürgen Jost, Ulrich Bunke, Frank Pfäffle, and Helmut Abels for interesting discussions and interesting talks related to this article. We also thank the referees for their helpful comments.

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Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg

E-mail address: bernd.ammann@mathematik.uni-regensburg.de

Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg

E-mail address: nicolas.ginoux@mathematik.uni-regensburg.de