M. Sadowski, A. Szczepanski
The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors of an infinite family of closed flat manifolds. It consits of some flat manifolds M with cyclic holonomy groups.
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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.
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.
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.
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.
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.
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。
Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X-Z, and a nowhere zero section on X-Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by the following requirement: The norm of the spinor must extend across Z as a continuous function vanishing on Z. In particular, the vanishing locus of the norm of the spinor is the complement of the set where the real line bundle is defined, and hence where the spinor is defined. …
We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature $(1,1)$. We will investigate the properties of the solutions of these equations and try to relate the conformal invariant dimension of the space of harmonic and twistor spinors to the natural conformal invariants given by the Lorentzian metric. We will introduce the notion of semi-conformally flat surfaces and establish a complete classification of the possible dimensions for this family.
We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.
We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.
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.
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。
Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let σ: W^+ --> Λ^+ be the natural squaring map, taking self-dual (= positive) spinors to self-dual 2-forms. In this paper, we characterize the self-dual 2-forms that are images of self-dual spinor fields through σ. They are those αfor which (off zeros) c_1(α) = c_1(ω), where c_1(α) is a suitably defined Chern class. We also obtain the form…
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。
In this paper connections between different gauge-theoretical problems in high and low dimensions are established. In particular it is shown that higher dimensional asd equations on total spaces of spinor bundles over low dimensional manifolds can be interpreted as Taubes-Pidstrygach's generalization of the Seiberg-Witten equations. By collapsing each fibre of the spinor bundle to a point, solutions of the Taubes-Pidstrygach equations are related to generalized harmonic spinors. This approach is also generalized for arbitrary fibrations (without singular fibres) compatible with an appropriate …
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。
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.
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。
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.
$\mathbb Z_2$-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。
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。
Let M be a compact manifold with a fixed spin structure χ. The Atiyah-Singer index theorem implies that for any metric g on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and χ. We show that for generic metrics on M this bound is attained.
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…
Let M be a closed spin manifold of dimension congruent to 3 modulo 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.
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…
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.
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…
Let G be a compact, semi-simple Lie group and H a maximal rank reductive subgroup. The irreducible representations of G can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space G/H twisted by bundles associated to the irreducible, possibly projective, representations of H. Here, we give a quick proof of this result, computing the index and kernel of this twisted Dirac operator using a homogeneous version of the Weyl character formula noted by Gross, Kostant, Ramond, and Sternberg, as well as recent work of Kostant regarding an algebraic version…
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein man…
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。
$L^2$-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。
We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.
We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theory---specifically to integral estimates which are used when gluing and ungluing PU(2) monopoles (math.DG/9907107).
We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are closed Riemannian manifolds of dimension $n\ge 3$ together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum of its associated operator is close to the disjoint union of the spectra of the two original operators. As an application, we show that in dimension $n\equiv 3$ mod 4 harmonic spinors for the Dirac operator of a spin, $\Spinc$, or $\Spinh$ manifold are not topologically obstruct…
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。
Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations on 3-Manifolds
针对带有奇异集Z0的Z2-调和旋量，在Z0的管状邻域上构造一族依赖于参数ε的双旋量Seiberg-Witten方程局部解，这些解在ε→0时曲率L2范数发散，且重整化后局部收敛到原Z2-调和旋量。
Motivated by Witten's spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.
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。
Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.
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.
In this paper we study the asymptotic behavior of the spectral flow of a one-parameter family $\{D_s\}$ of Dirac operators acting on the spinor bunldle $S$ twisted by a vector bundle $E$ of rank $k$, with the parameter $s\in [0,r]$ when $r$ gets sufficiently large. Our method uses the variation of eta invariant and local index theory technique. The key is a uniform estimate of the eta invariant $\barη(D_r)$ which is established via local index theory technique and heat kernel estimate.
We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X$ and prove that $η(D)=\frac{1}{πi}\log Z_{Γ,Σ}^{\rm o}(0)$, thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant…
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmoni…
We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $Ω^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $π_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-inva…
The eta invariant on two-step nilmanifolds
本文针对eta不变量仅能在少数紧型局部对称空间中直接谱计算的问题，借助Kirillov理论分解旋量Dirac算子，推导闭流形上eta不变量的通用计算公式，给出一般Heisenberg三维流形的具体计算结果，并发现存在eta不变量恒定的连续几何谱互异流形族。
Eguchi-Hanson harmonic spinors revisited
本文重新探讨了 Eguchi-Hanson 空间上的 Dirac 算子零模问题，利用 spin-$c$ 旋量形式体系简化了计算，无需计算旋量联络即可重现扭曲 Dirac 算子的可归一化零模，并将结果推广至更广泛的 Ricci-flat Kähler 流形。
This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.
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。
Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is noncompact, constitute a prototype for the formation of singularities, the so-called bubbles, in geometric analysis. In theoretical physics, they arise from the nonlinear $σ$-model of quantum field theory. That model possesses a supersymmetric extension, coupling a harmonic map like field with a nonlinear spinor field. In the physical model, that spinor fiel…
We study the Yamabe invariants of cylindrical manifolds and compact orbifolds with a finite number of singularities, by means of conformal geometry and the Atiyah-Patodi-Singer $L^2$-index theory. For an $n$-orbifold $M$ with singularities $Σ_Γ = \{(\check{p}_1, Γ_1), ..., (\check{p}_s, Γ_s)\}$ (where each group $Γ_j<O(n)$ is of finite order), we define and study the \emph{orbifold Yamabe invariant} $Y^{\orb}(M)$. We prove that $Y^{\orb}(M)$ coincides with the corresponding $h$-$\emph{cylindrical Yamabe invariant}$ $Y^{h\textrm{-}\cyl}(M \setminus \{\check{p}_1, ..., \check{p}_s\})$ defined by…
We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on t…
We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|dϕ|^2+(ψ,\Dψ)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.
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。
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.
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds
本文研究闭合黎曼流形上解耦狄拉克-杨-米尔斯方程的解，通过扰动理论和指标定理分析其存在性和结构，提出构造显式解的方法。
We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N$ is a manifold admitting Killing spinors.
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