Harmonic spinors and local deformations of the metric
Bernd Ammann, Mattias Dahl, Emmanuel Humbert
设（M，g）是紧致黎曼自旋流形. Atiyah-Singer指数定理给出了Dirac算子核维数的一个下界。证明了在任意小的开集上，通过改变黎曼度量g，可以得到这个界。
Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.
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Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds
我们考虑Sasakian流形和3-Sasakian流形上的一些自然的无穷小Einstein形变。其中一些是Killing旋量的无穷小变形，还有一些是实际Killing旋量变形的积分。特别是，在3-Sasakian 7流形上，这些产生无穷小的爱因斯坦变形，保持2，1，或没有3个独立的Killing旋量。复曲面3-Sasakian流形提供了可积变形精确保持2个Killing旋量的非平凡例子。因此，Killing旋量的维数在爱因斯坦变形下不保持，而只是上半连续的。
Manifolds admitting Killing spinors are Einstein manifolds. Thus, a deformation of a Killing spinor entails a deformation of Einstein metrics. In this paper, we study infinitesimal deformations of Killing spinors on nearly parallel $\mathrm{G}_2$-manifolds. Since there is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$-structures and Killing spinors on 7-dimensional spin manifolds, our results imply that infinitesimal deformations of nearly parallel $\mathrm{G}_2$-structures are examined in terms of Killing spinors. Applying the same technique, we identify that the space of …
In this note, using the spinorial description of $SU(3)$ and $G_2$-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique section induced by a $G$-structure in consideration.
We prove the existence of singular harmonic ${\bf Z}_2$ spinors on $3$-manifolds with $b_1 > 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg-Witten equation with two spinors. The existence of singular harmonic ${\bf Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce regarding Donaldson and Segal's proposal for counting $G_2$-instantons.
Left-invariant harmonic spinors on three-dimensional Lie groups
研究三维李群上左不变调和旋量的存在性，通过修正并专门化旋量狄拉克算子公式，探讨李群结构方程的约束条件及对应度量的识别。
The universal moduli space of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors
本文为非退化$\boldsymbol{\frac{\boldsymbol{Z}}{\boldsymbol{2}Z}}$调和旋子的通用模空间赋予驯服Fréchet流形结构，证明其投影映射为一致Fredholm，在统一框架下重现了Donaldson等人的早期工作。
The zero loci of Z/2 harmonic spinors in dimension 2, 3 and 4
设X是一个黎曼流形，X上的Z/2旋量由一个数据集定义，该数据集由X上的一个闭集Z、X-Z上的一个实线丛以及实线丛与旋量丛的张量积在X-Z上的无处零截面组成.集合Z和旋量共同受到以下条件的约束：旋量的范数必须作为在Z上消失的连续函数延伸到Z上。特别地，旋量范数的消失轨迹是实线丛被定义的集合的补集，因此旋量被定义。当Z/2旋量服从X-Z上的一阶Dirac方程时，称之为调和旋量。这本专着分析了结构的一套Z为Z/2谐波旋量的流形上的二维，三维或四维。
Spinor Fields on the Surface of Revolution and their Integrable Deformations via the mKdV-Hierarchy
旋量领域的旋转表面共形地沉浸到3维空间被认为是在旋量表示的框架内的表面。证明了在旋转曲面的情况下，一个与修正的Veselov-Novikov族有关的线性问题（二维Dirac方程）可化为一个著名的Zakharov-Shabat系统.在单孤子解的情况下，利用线性Bargmann势给出了旋量场的显式形式，并用Zakharov-Shabat系统的Jost函数表示。它还表明，旋转旋转曲面上的旋量场的可积变形定义的修改Korteweg-de Vries层次。
Conformal properties of harmonic spinors and lightlike geodesics in signature $(1,1)$
本文研究Lorentz曲面上的调和扭量方程，即二维可定向流形上的度量为$(1,1)$的调和扭量方程。我们将研究这些方程的解的性质，并试图将调和旋量和扭量旋量空间的共形不变维数与洛伦兹度规给出的自然共形不变量联系起来。我们将介绍半共形平面的概念，并建立一个完整的分类可能的尺寸为这个家庭。
Almost harmonic spinors
我们表明，任何封闭的自旋流形不同构的两个球承认一个序列的体积一黎曼度量的最小非零狄拉克特征值趋于零。作为应用，我们比较了Dirac谱和共形体积.
In my previous paper, I prove the existence of the Kuranishi structure for the moduli space $\mathfrak{M}$ of zero loci of $\mathbb{Z}/2$-harmonic spinors on a 3-manifold. So a nature question we can ask is to compute the virtual dimension for this moduli space $\mathfrak{M}_{g_0}:=\mathfrak{M}\cap\{g=g_0\}$. In this paper, I will first prove that $v-dim(\mathfrak{M}_{g_0})=0$. Secondly, I will generalize this formula on 4-manifolds by using a special type of index developed by Jochen Bruning, Robert Seeley, and Fangyun Yang.
Surgery and Harmonic Spinors
设M是具有固定自旋结构χ的紧致流形. Atiyah-Singer指标定理意味着对于M上的任何度量g，狄拉克算子的核的维数由一个仅依赖于M和χ的拓扑量从下有界。我们表明，通用度量M上达到这个界限。
Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field
利用Clifford代数的极小左理想作为旋量，给出了一般曲面浸入四维复空间的广义Weierstrass公式的表示.讨论了mVN族曲面的可积变形与曲面上旋量场的可积变形之间的关系。
Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs $(Σ, ψ)$ where $Σ$ is a $C^1$-embedding simple closed curve in $M$, $ψ$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $Σ$, and $\|ψ\|_{L^2_1}\neq 0$. We prove that when $Σ$ is $C^2$, a neighborhood of $(Σ, ψ)$ in the moduli space can be parametrized by the space of Riemannian metrics on $M$ locally as the kernel of a Fredholm operator.
Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators that transform gauged twistor spinors to gauged harmonic spinors are found. Symmetry operators of gauged harmonic spinors in terms of conformal Killing-Yano forms are obtained. Algebraic conditions to obtain solutions of the Seiberg-Witten equations are discussed.
In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.
Eigenvalue Estimates of the ${\rm spin}^c$ Dirac Operator and Harmonic Forms on Kähler-Einstein Manifolds
我们建立了定义在正数量曲率的紧致Kähler-Einstein流形上的Dirac算子的特征值的一个下界，并赋予了特殊的${\rm spin}^c$结构.极限情形的特征是在旋量丛的某个子丛中存在Kählerian Killing ${\rm spin}^c$旋量。此外，我们证明了一个有效的调和形式和一个Kählerian Killing ${\rm spin}^c$旋量场之间的Clifford乘法为零。这将A的结果推广到${\rm spin}^c$的情况。Moroianu指出，在一个紧凑的凯勒-爱因斯坦流形的复杂尺寸$4\ell+3$携带一个复杂的切触结构，克利福德乘法之间的有效谐波形式和凯勒的Killing旋量是零。
$L^2$-harmonic forms and spinors on stable minimal hypersurfaces
本文研究了稳定极小超曲面上的$L^2$-harmonic形式和spinors的消失定理，并证明了在特定条件下这些超曲面不存在非平凡的$L^2$-harmonic spinors。
Flat manifolds, harmonic spinors, and eta invariants
本文的目的是计算无穷族闭平坦流形的调和旋量空间的eta不变量和维数。它由一些具有循环和乐群的平坦流形M构成。
Homogeneous $\mathbb Z/2$-Harmonic Forms and Spinors on $\R^4$ from Regular 4-Polytopes
研究四维空间中齐次Z/2调和1形式、自对偶2形式和旋量的局部奇异性模型，基于正则四维正多面体的1骨架锥结构。
Gauge theory, calibrated geometry and harmonic spinors
本文建立了高维和低维的不同规范理论问题之间的联系。特别是它表明，高维asd方程的全空间的旋量丛低维流形上可以解释为Taubes-Pidstrygach的推广的Seiberg-Witten方程。通过将旋量丛的每个纤维坍缩到一个点，Taubes-Pidstrygach方程的解与广义调和旋量有关。这种方法也适用于任意纤维化（没有奇异纤维）与适当的校准兼容。
Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds
本文通过构造一个参数族的双旋量Seiberg-Witten单极，研究Z₂-调和旋量在规范化的极限下的收敛性，采用交替方法克服无限维障碍束的困难。
Riemannian Metrics and Harmonic Sections of Spinor Bundles
研究度量变化下 Dirac 算子最低非负特征值的聚集现象，并证明维数大于等于四的闭 Spin 流形存在允许非平凡调和旋量的 Riemannian 度量。
This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.
Harmonic spinors on axisymmetric 3-manifolds with Melvin ends
证明了具有非负数量曲率的轴对称黎曼三维流形中调和旋量场的存在性，并证明了调和旋量场渐近于Melvin磁宇宙的常时超曲面。这样的旋量可以用于磁化史瓦西解的唯一性的证明。
On deformations of metrics and their associated spinor structures
考虑四维时空流形中Minkowski型度量的光滑变形。与此度量的基本自旋张量场的变形计算和它们的应用程序计算的大质量自旋1/2粒子的能量动量张量。
We study Kohn-Dirac operators $D_θ$ on strictly pseudoconvex CR manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb Z}$. Certain components of $D_θ$ are CR invariants. We also derive CR invariant twistor operators of weight $\ell$. Harmonic spinors correspond to cohomology classes of some twisted Kohn-Rossi complex. Applying a Schrödinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.
A remark on the space of metrics having non-trivial harmonic spinors
设M是维数全等于3模4的闭自旋流形.本文给出了一个简单的证明，证明了具有可逆Dirac算子的M上的度量空间要么是空的，要么有无穷多个路径分支。
Vortex Harmonic Spinors on the Nappi-Witten Space
建立了平坦黎曼曲面上的涡旋方程与Nappi-Witten空间（欧几里得群SE(2)的中心扩张的群流形）上调和旋量之间的对应关系，通过涡旋构型构造了扭曲Dirac方程的显式解，并利用Nappi-Witten度量的共形平坦性在四维Minkowski空间上诱导出调和旋量，从而从涡旋数据几何构造了平坦Minkowski时空上的阿贝尔磁零模。
The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and $P^2(C)$ with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of $C^2$ and extend to $P^2(C)$ as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Pat…
Harmonic spinors on homogeneous spaces
设G是紧的单李群，H是极大秩约化子群. G的不可约表示可以构造为关于齐性空间G/H上的狄拉克算子的调和旋量空间，该空间由与H的不可约（可能是射影）表示相关的丛扭曲。在这里，我们给出了一个快速证明这一结果，计算指数和内核的扭曲狄拉克算子使用齐次版本的外尔字符公式注意到格罗斯，Kostant，拉蒙德，和斯滕伯格，以及最近的工作Kostant关于代数版本的狄拉克算子。
We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to rescaling of $\mathbb{R}^3$ with their zero locus consisting of four or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example; and they point from those of a centered octahedron and a centered icosahedron in two others.
On the nu-invariant of two-step nilmanifolds with closed G2-structure
研究了具有闭G2结构的二步nilmanifolds上的nu不变量，通过计算调和旋量和η不变量，确定了其存在性和奇偶性，并推导出在不变调和旋量上nu的消失。
A characterization of Dirac morphisms
通过对全空间上和水平共形浸入基流形上的Dirac算子的研究，刻画了Dirac态射，即把（局部）调和旋量场拉回到（局部）调和旋量场上的映射.
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
An index theorem for Z/2-harmonic spinors branching along a graph
本文提出关于沿图分支的Z/2调和旋量的指标定理，通过不同证明方法扩展到更一般情形，并应用于更广泛的扭曲Dirac算子及Z/2调和1形式。
This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exoti…
We study the deformation theory of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds. There is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$ structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homoge…
This paper studies the space of $L ^2 $ harmonic forms and $L ^2 $ harmonic spinors on Taub-bolt, a Ricci-flat Riemannian 4-manifold of ALF type. We prove that the space of harmonic square-integrable 2-forms on Taub-bolt is 2-dimensional and construct a basis. We explicitly find a 2-parameter family of $L ^2 $ zero modes of the Dirac operator twisted by an arbitrary $L ^2 $ harmonic connection. We also show that the number of zero modes found is equal to the index of the Dirac operator. We compare our results with those known in the case of Taub-NUT and Euclidean Schwarzschild as these manifol…
On the space of connections having non-trivial twisted harmonic spinors
本文考虑奇维紧自旋流形上的Dirac算子，该流形被一个积丛扭曲。我们证明了如果非扭曲Dirac算子是可逆的，并且扭曲丛的维数足够大，那么扭曲丛上产生可逆算子的联络空间有无穷多个连通分支.
The Calderón problem for harmonic spinors
本文证明具有MIT边界条件的Dirac算子的边界共轭映射唯一确定紧致旋量流形的共形类，并推广到扭曲Dirac算子及实解析酉联络的局部规范等价性。
A Kato-Yau inequality for harmonic spinors and decay estimate for eigenspinors
我们证明了调和旋量服从著名的逐点Kato不等式的一个加强版本的向量丛部分的连接。然后，我们证明了衰减估计本征旋量使用此加藤-丘估计和由此产生的微分不等式。我们简要地描述了规范理论的一些应用-特别是当粘合和解开PU（2）单极时使用的积分估计（数学DG/9907107）。
We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.
Harmonic Spinors for Twisted Dirac Operators
我们证明了，对于一个合适的类的“Dirac-like”的运营商有一个胶合定理的连接和。更精确地说，如果$M_1$和$M_2$是维数为$n\ge 3$的闭黎曼流形，那么连通和$M_1 # M_2$可以被赋予一个黎曼度量，使得其相关算子的谱接近于两个原始算子的谱的不交并。作为一个应用，我们证明了在维数$n\equiv 3$ mod 4的Dirac算子的自旋，$\Spinc$，或$\Spinh$流形的调和旋量不被拓扑阻塞.
Deformations of harmonic function germs
在右等价下研究了二元调和首项函数芽奇点的分类问题。我们观察到拉普拉斯算子的多重作用出现在这类函数芽的分类中。
Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds
本文研究了3-流形上 $\mathbb Z_2$-调和旋量通用模空间的局部结构，利用 Nash-Moser 隐函数定理克服了无限维阻碍和正则性丧失问题，证明其投射为参数空间中的余维数1子流形。
Witten spinors on nonspin manifolds
受Witten关于正质量定理的旋量证明的启发，我们研究了完备渐近平坦非旋流形上具有非负数量曲率的渐近常数调和旋量。
In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic maps is smooth and has dimension two. We also show that the points that correspond to minimal tori (conformal harmonic maps) are either smooth points of dimension two or singular.
$\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds
通过粘合论证，在连通和与环面和上构造$\mathbb Z_2$-调和旋量与1-形式，并证明每个闭三维流形存在无穷多个具有不同奇异链的$\mathbb Z_2$-调和旋量。
A note on coupled Dirac-harmonic maps
本文提出一种从 $S^1\ imes M$ 型流形构造耦合狄拉克-调和映射的方法，其中 $M$ 为带满足范数条件的调和旋量对的闭自旋流形。
Metric perturbations and deformations of k-nondegenerate Z/2-harmonic 1-forms
研究退化Z/2调和1形式的度量扰动与变形理论，证明通过适当扰动度量可将形式变形为附近的非退化Z/2调和1形式，结合局部展开分析与量化Nash-Moser隐函数定理。
Nowhere-zero harmonic spinors and their associated self-dual 2-forms
设M是一个闭的定向四维流形，具有黎曼度量g，由一个殆复结构ω诱导出一个自旋^C结构.行列式线丛上的每个联络A诱导唯一的联络\nabla^A，旋量场上的狄拉克算子\D^A。设σ：W^+ --> Λ^+为自然平方映射，取自对偶（=正）旋量为自对偶2-形式。本文通过σ刻画了自对偶旋量场的自对偶2-形式。它们是那些α满足（非零点）c_1（α）= c_1（ω）的情形，其中c_1（α）是一个适当定义的Chern类。我们还得到公式：|| ϕ|| ^2 D^A ^= i（2 d^* σ（σ）+ < \nabla^A σ，i σ>）* σ.利用这些，我们建立了{Kahler形式α与g的度量标量倍相容，且c_1（α）= c_1（ω）}和{对的规范类（λ，A），且\nabla^A λ = 0}之间的双射对应，以及以下两个之间的双射对应：{辛形式α与共形于g的度量相容，且c_1（α）= c_1（ω）}和{规范对类（λ，A），其中D^ A λ = 0，且< \nabla^A λ，i λ> = 0，和无源零}。
Exploring certain geometric and harmonic properties of the Berger-type metric conformal deformation on the Para-Kähler-Norden manifold
该研究在Para-Kähler-Norden流形上构造了由Berger型度量共形形变得到的新度量类，分析了其Levi-Civita联络与各类曲率，并研究了相关的一类调和映射。
We study Rarita-Schwinger fields on 6-dimensional compact strict nearly Kähler manifolds. In order to investigate them, we clarify the relationship between some differential operators for the Hermitian connection and the Levi-Civita connection. As a result, we show that the space of the Rarita-Schwinger fields coincides with the space of the harmonic 3-forms. Applying the same technique to a deformation theory, we also find that the space of the infinitesimal deformations of Killing spinors coincides with the direct sum of a certain eigenspace of the Laplace operator and the space of the Killi…
Deformation theory of nearly Kähler manifolds
几乎凯勒流形是黎曼6-流形承认真正的杀戮旋量。等价地，几乎Kähler流形上的黎曼锥具有包含在G2中的完整性。本文研究了近似Kähler流形的形变理论，证明了它一般是阻塞的。更精确地说，我们证明了旗流形上齐次近似Kähler结构的无穷小变形都被阻塞到二阶。
A note on the spectral deformation of harmonic maps from the two-sphere into the unitary group
在[5]中，与J. C. Wood等，作者给出了从$2$ -球面到酉群$U(n)$的所有调和映射在$S^2$上的任意亚纯函数的完全显式公式.最简单的调和映射是迷向映射。Burstall和Guest [1]利用莫尔斯理论证明了调和映射是以各向同性映射为标记的簇。在这项工作中，使用上述调和映射的公式，我们明确地描述了这一过程，显示如何所有的调和映射可以建立从各向同性的。[1]F.~ E. Burstall和M.~ A. Guest，\textit{Harmonic two-spheres in compact symmetric spaces, revisited}，Math.Ann.309（1997）541- 572. [5]M.~ J. Ferreira，B.~ A Similes和J.~ C. Wood \emph{All harmonic $2$-spheres in the unitary group, completely explicitly}，Math. Z. { \bf 266}（2010），953- 978.
A class of generalised Killing spinors determined by the Ricci tensor and the metric
本文引入仿射Killing旋量（AKS），在调和曲率和局部共形平坦条件下分类允许AKS的黎曼自旋流形，刻画带平行1-形式的流形，并证明三维曲率齐次流形的局部齐次性及Bianchi度量的完整分类。
The Energy-Momentum tensor on $Spin^c$ manifolds
在$Spin^c$流形上，我们研究了与旋量场相关的能量动量张量。首先，我们给出了$Spin^c$流形上定向超曲面的旋量Gauss型公式。利用广义柱的概念，导出了Dirac算子在度量变形下的变分公式，指出能量动量张量自然地表现为等距浸入的第二基本型.最后，我们证明了Codazzi能量动量张量的广义$Spin^c$ Killing旋量是平行旋量的限制。
Geometric Rigidity via Almost-Harmonic Twisted Spinors
研究Gromov精确提升二形式方法在标量曲率几何中的应用，通过扭曲L^2指标构造调和旋量，证明锐利双曲标量曲率比较，并分析等式情形得到原度量是Einstein的。
We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.
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