Anja Wittmann
We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.
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In this study, we obtain a spinorial Gauss formula for a lightlike hypersurface in Lorentzian manifold with 4-dimension. Then, we take into account the changes caused by degenerate metric on hypersurface and investigate Dirac operator for lightlike hypersurface. Later, we establish the relation between Dirac operators and Riemannian curvatures of manifold and hypersurface.
In this paper we give a new proof of Bismut-Freed's result on the local regularity of the eta invariant of a Dirac operator in odd dimension.
Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $Σ\subset M$. Building on past and recent works of Bär and Strohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operator on compact Lorentzian spaces to the case, where $M$ is diffeomorphic to a product of $Σ$ with a compact time intervall and the hypersurface is a Galois covering with respect to a group $Γ$. We follow the first approach of both authors in this extended setting, where a well-posedness result of the Cauchy problem for the Dirac operator on non-compact manifolds is needed in p…
We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present p…
Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds
本文研究非紧致流形上Dirac-Schrödinger算子的相对eta不变量，推导非紧致流形上的谱流公式，并推广Atiyah-Patodi-Singer索引定理至非紧边界情况。
We extend to the eigenvalues of the hypersurface Spin$^c$ Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spin$^c$ Killing spinor on homogeneous 3-dimensional manifolds $\mathbb E^*(κ, τ)$ with 4-dimensional isometry group and isometric immersions of $\mathbb E^*(κ, τ)$ into the complex space form $\mathbb M^4(c)$ of constant holomorphic sectional curvature $4c$, for some $c\in \mathbb R^*$. As applications, we show the non-existence of totally umbilic surfaces in $\mathbb E^*(κ, τ)$ a…
Let $\mathcal{A}_0$ and $\mathcal{A}_1$ be two self-adjoint Fredholm Dirac-type operators defined on two non-compact manifolds. If they coincide at infinity so that the relative heat operator is trace-class, one can define their relative eta function as in the compact case. The regular value of this function at the zero point, which we call the relative eta invariant of $\mathcal{A}_0$ and $\mathcal{A}_1$, is a generalization of the eta invariant to non-compact situation. We study its variation formula and gluing law. In particular, under certain conditions, we show that this relative eta inva…
We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrödinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.
New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.
We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $Ł^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infi…
The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-invariants for the family of Berger metrics on the odd dimensional spheres.
Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderon projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott-Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and…
Given a fiber bundle with closed connected fibers, and a family of separating hypersurfaces, we study the behavior of the Bismut-Lott analytic torsion form, and the eta form for a duality bundle, under analytic surgery in the sense of Hassell, Mazzeo and Melrose. We find that under the surgery limit, the rescaled heat kernel is non-singular, while both the Bismut-Lott analytic torsion form and eta form can be written as the sum of a logarithmic term, which satisfies the Igusa additivity property, the b- Bismut-Lott analytic torsion form (respectively the b- eta form), and an error term coming …
On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire.
We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.
We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space.
We describe a new family of examples of hypersurfaces in the sphere satisfying the limiting-case in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator.
The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to…
Note on a numerical equality regarding the eta invariant on Berger spheres
本文通过数值计算发现Berger球面上的Dirac APS eta不变量与圆球面上的Dirac共形反常一致，并用广义Bernoulli多项式验证了已知猜想，同时得到了Weingart生成函数。
We show that the R/Z part of the analytically defined eta invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, we discuss the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and prove an R refinement in the case where the Dirac operator is replaced by the Signature operator. We also extend the above discussion to the case of eta invariants associated to Hermitian vector bu…
Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self…
Eta Form and Spectral Sequence for the Composition of Fibrations
本文受纤维复合符号算子谱序列启发，定义纤维wise Dirac算子的谱序列，并用热核方法与解析局部化技术证明eta不变量绝热极限公式的等变族版本，推广了Dai和Bunke-Ma的余项构造。
Cauchy Problem for the Dirac operator on spatially non-compact spacetimes
本文证明了在具有完备类空Cauchy超曲面的全局双曲流形上Dirac算子Cauchy问题的适定性，为未来针对非紧致Cauchy超曲面证明Fredholm性结果奠定基础。
We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.
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。
We derive a formula for the eta invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres.
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.
We consider families of Dirac operators on the unit interval which depend on parameters via boundary conditions. We study the associated eta forms and Maslov cocyles. With this simple example we show how previous results of Lesch/Woiciechowski and the first author on the eta invariant of cylinders generalize to the family case.
An index formula for families of end-periodic Dirac operators
本文为端周期流形上的Dirac算子族建立了指标公式，通过转降公式和新的端周期eta形式，推广了已有的eta不变量。
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detailed analysis of the unipotent terms give the relation between the eta invariant and the Selberg zeta function of odd type. This allows us to show the vanishing of the unipotent contribution. We show that the eta invariant and the Selberg zeta function of odd type satisfy certain …
We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly c…
The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher genera…
In this paper, using the equivariant version of the Dai-Zhang higher spectral flow, we generalize the variation formula, embedding formula and the adiabatic limit formula for the Atiyah-Patodi-Singer eta invariants to the equivariant Bismut-Cheeger eta forms for a fiberwise compact Lie group action when the kernels of corresponding fiberwise Dirac operators form equivariant vector bundles.
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…
Starting from an even definite lattice, we construct a principal circle bundle covered by a certain three-step nilpotent Lie group G. On the base space, which is again a nilmanifold, we then study the Dirac operator twisted by the associated complex line bundles. Noting that the whole situation fibers over the circle, we are able to determine the reduced eta-invariant of these Dirac operators in the adiabatic limit. As an application, we consider the total space of the circle bundle, equipped with a parallelism induced by G, as an element in the stable homotopy groups of the sphere and use the…
In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah-Patodi-Singer, which is for odd dimensional manifolds. It is associated to $K^1$ representatives on even dimensional manifolds and is closely related to the so called WZW theory in physics. In fact, it is an intrinsic interpretation of the Wess-Zumino term without passing to the bounding 3-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral int…
We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces $\M^3(κ)\times\R$, in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results for surfaces by T. Friedrich, B. Morel and the two authors. The main argument is the interpretation of the energy-momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an …
The eta invariant of a circle bundle on a Fano manifold
研究正线丛上圆包的spin-c狄拉克算子，计算其eta不变量并扩展了adiabatic参数的范围。
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.
Atiyah-Patodi-Singer index and domain-wall eta invariants
本文提出一个无需边界 Dirac 算子可逆性假设的公式，将广义 Atiyah-Patodi-Singer 指标用畴壁大质量 Dirac 算子的 eta 不变量表示，避免了全局谱投影边界条件。
We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schrödinger operator D_f = D - f/2.
For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\widetilde M$ is a $Γ$-regular covering space of $M$. Let $\langle α\rangle$ be the conjugacy class of a non-identity element $α\in Γ$. Suppose $\{Γ_i\}$ is a sequence of finite-index normal subgroups of $Γ$…
A partitioned manifold index theorem for noncompact hypersurfaces
该研究将Roe的分区流形指标定理从紧超曲面情形推广到非紧超曲面情形，在嵌入假设下得到N的Roe代数K-理论中两类的等式，补充了相关领域的研究成果。
Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formulæfor several geometric and spectral invariants associated to a Dirac-type operator ð_X on X are presented. Considered in detail are: the index of ð_X, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as…
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.
Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit version of this at the level of the determinant bundle. Their result, that the eta invariant of the interior family trivializes the determinant bundle of the boundary family, is extended here to the wider context of pseudodifferential families of cusp type.
We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial data of its Milnor algebra. This is done by representing this eta-invariant as the eta-invariant of the odd signature operator on a certain closed fibrewise double of the original bundle and expressing the latter as the mapping cylinder of a specific fibrewise isometry. In this si…
On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with invertible Dirac operators is disconnected in dimensions $n \equiv 0,1,3,7 \mod 8$, $n \geq 5$. As a corollary follows results on the existence of metrics with harmonic spinors by Hitchin and Bär. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics…
We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the classic Atiyah-Patodi-Singer index formula for $(X,\F)$. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relativ…
We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.
A Reilly-Type Inequality for CR Dirac Operators on CR Manifolds and Applications
本文推导CR流形上CR Dirac算子的Reilly型不等式，并应用于p-质量下界估计及有界域上CR Dirac方程的求解。
Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the $η$-series in terms of values of Riemann-Hurwitz zeta functions, and the $η$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\Z_2^…
We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered by applying the theorem to Dirac operators twisted by the Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental group.
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the…
We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.
We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant introduced by Braverman and Shi. This extends recent results of Bär and Strohmaier, who studied the index of a hyperbolic Dirac operator on a spatially compact globally hyperbolic manifold.
We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuations combined with the Birkhoff normal form and local index theory.
In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this estimate, we obtain an asymptotic expansion for the first eigenvalue of the Dirac operator on large spheres in three dimensional asymptotically flat manifolds. We also study this expansion for small geodesic spheres in a three dimensional Riemannian manifold. We finally discuss how this method can be adapted to yield similar results in the hyperbolic space.
We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included)…
An extended variational formula for the Bismut-Cheeger eta form and its applications
本文扩展了Bismut-Cheeger eta形式的变分公式，允许spin$^c$ Dirac算子被同构向量丛扭曲，并建立了其$\mathbb{Z}_2$-分次可加性，进而给出了微分$K$-理论中解析指标良定义性和Riemann-Roch-Grothendieck定理的替代证明。
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