S. Goette
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.
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We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.
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.
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 establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As…
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…
On the eta invariant of (2,3,5) distributions
本文研究闭五流形上一般秩二分布相关的Rumin复形中Heisenberg阶二自伴算子的eta不变量，证明(2,3,5)幂零流形上eta不变量为零但eta函数非平凡，并给出其用基本函数表达的公式。
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…
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 study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.
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 defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant. Using adiabatic limit technique, we give such an intrinsic interpretation.
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.
We give a survey on eta invariants including methods of computation and applications in differential topology.
We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the eta-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the probl…
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.
Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178
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 extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we improve a recent result of Han-Zhang on the modularity of a certain power series of eta invariants.
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 compute the eta function $η(s)$ and its corresponding $η$-invariant for the Atiyah-Patodi-Singer operator $\mathcal{D}$ acting on an orientable compact flat manifold of dimension $n =4h-1$, $h\ge 1$, and holonomy group $F\simeq \mathbb{Z}_{2^r}$, $r\in \mathbb{N}$. We show that $η(s)$ is a simple entire function times $L(s,χ_4)$, the $L$-function associated to the primitive Dirichlet character modulo 4. The $η$-invariant is 0 or equals $\pm 2^k$ for some $k\ge 0$ depending on $r$ and $n$. Furthermore, we construct an infinite family $\mathcal{F}$ of orientable $\mathbb{Z}_{2^r}$-manifolds w…
In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (higher dimensional) conformally flat structures in a conformally invariant way. The eta invariant gives rise to a homomorphism from this group to the circle group, which can be highly nontrivial. It remains an interesting question of how to compute this group.
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.
We study the dependence of the eta invariant $η_D$ on the spin structure, where $D$ is a twisted Dirac operator on a (4k+3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a cohomolgy class which is the reduction of a $H^1(M,\Za)$-class is shown to be a half integer. As an application of the technique of proof the generalized Rokhlin invariant is shown to be equal modulo 8 for two spin structures related in this way.
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.
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。
A microlocal pathway to spectral asymmetry: curl and the eta invariant
本文通过旋度算子的分析，提出一种不依赖解析延拓的eta不变量计算方法，为研究非半有界伪微分系统的谱不对称性提供了新途径。
We present an alternate proof of the Bismut-Zhang localization formula for $η$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.
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。
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 first apply the method and results in the previous paper to give a new proof of a result (hold in $ {\bf C}/{\bf Z}$) of Gilkey on the variation of h-invariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain h-invariant appearing in recent papers of Braverman-Kappeler on what they call refined analytic torsion, and propose an alternate formulation of their definition of the refined analytic torsion. A refinement in ${\bf C}$ of the above variation formula is also proposed.
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…
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…
The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P. Gilkey. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented. This result answers the question of P. Gilkey (1989) concerning the existence of even-order operators on odd-dimensional manifolds with nontrivial fractional pa…
Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$
研究Seifert有理同调3球面的Floer同调类型及eta不变量，证明其为L空间，且Floer同调类型为S^0的悬垂，同时计算Ozsváth-Szabó d-不变量及Seiberg-Witten δ-不变量。
We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $ρ$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes.
We express the real connective $K$ theory groups of the quaternion QL group of order $2^j\ge8$ in terms of the representation theory of by showing $ko_{4k-1}(BQL)=KSp(S^{4k+3}/τQL)$ where $tau$ is any fixed point free representation of QL in U(2k+2)
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.
On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a Kähler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the Kähler metric admits a special Kähler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the Kähler metric is part of an ambitoric structure, in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toric one. The formulas are derived using the Duisterm…
We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents.
We present a new proof, as well as a ${\bf C/Q}$ extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of $η$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott analytic torsion form can be derived naturally from the transgression of the $η$-forms appearing in the adiabatic limit computations.
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.
Eta-Invariants and Determinant Lines
研究了奇维有边流形上的η-不变量。对边界条件的依赖性最好通过将（取幂）η-不变量视为边界的（逆）行列式线的元素来总结。我们证明了这个不变量的胶合定律和变分公式。这产生了一个新的，更简单的证明的完整公式的行列式线丛的一个家庭的狄拉克运营商，也被称为“全球异常”的公式。本文使用AMSTeX 2.1编写，可以通过FTP从美国数学学会获得（包括说明）。以uuencoded焦油压缩格式单独提交了一份附有数字的附言文件。
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford r…
The purpose of this paper is to give a new proof of results of Moscovici and Stanton on the orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals using the hypoelliptic Laplacian, with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct the proper It{ô} calculus associated with certain hypoelliptic diffusions.
Jumps of the eta invariant
本文研究了由Atiyah-Patodi-Singer定义的η-不变量，它是一个有向奇维黎曼流形的实值不变量，该流形具有其基本群的酉表示.当表示解析地变化时，对应的η-不变量可能具有积分跳跃，也称为谱流。本文的主要结果建立了一个公式，这种频谱跳跃的签名的一些同调形式，自然定义的道路表示。这些签名也可以通过由变形数据定义的厄米特形式的谱序列来计算。本文的谱跳跃定理推广到任意自伴椭圆算子的解析族。作为一个应用，我们考虑的问题同伦不变性的rho不变。我们给出了一个内在的同伦理论的定义的rho不变，不确定性的形式的一个局部常数函数的空间的酉表示。在S.Weinberger所写的附录中，证明了（使用本文的结果）同伦等价流形的rho不变量的差总是有理的。
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。
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…
Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of the fractional part of the eta-invariant in this case. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces. It also utilizes K-theory with coefficients Z_n. In particular, it is shown that the group K(T^*M,Z_n) is realized by elliptic operators (symbols) acting in appropriate subspaces.
There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of ${\rm{SO}}(4)$ acting freely on $S^3$ is given. An application of this is a non-existence result for Ricci-flat ALE metrics on certain spaces. (ii) A formula for the orbifold correction term that arises in the index of the self-dual deformation complex is proved for all finite subgroups of ${\rm{SO}}(4)$ which act freely on $S^3$. Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed. (iii) Two infinite …
We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. In particular we prove that contact versions of the relative eta-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.
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.
Let $Γ$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a $Γ$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $Γ\to \hat{M}\to \hat{M}/Γ$ is a Galois covering of a compact manifold with boundary. Let $(D^+ (θ))_{θ\in T}$ be a $Γ$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r Γ)$. If, in addition, $Γ$ is of polynomial growth, we define higher indeces by …
I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of elliptic boundary problems. The theme of the 90s is the eta invariant. The paper with Douglas was fundamental for establishing spectral invariants for manifolds with boundary and for the investigation of the behavior of spectral invariants under analytic surgery. This was so influ…
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 $Γ$…
We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliat…
Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8
本文提出通过指标不变量和Eta不变量来确定微分KO理论在8的倍数次数中的情况，并发展了两个家庭指标定理。
Using adiabatic limits of Eta invariants, Rho invariants of the total space of a fiber bundle are investigated. One concern is to formulate the aspects of local index theory for families of Dirac operator in terms of the odd signature operator, and place known results in a context which permits the treatment of Rho invariants. The main focus is, however, to use this to compute Rho invariants for explicit classes of fibered 3-manifolds. More precisely, we consider $\U(1)$-Rho invariants of principal $S^1$-bundles over closed, oriented surfaces as well as mapping tori with torus fiber. Hyperboli…
We study elliptic theory on manifolds with boundary represented as a covering space. Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. We show that elliptic elements define Fredholm operators and prove an index formula. For the identity covering, our formula reduces to the Atiyah-Bott index formula for classical boundary value problems. Secondly, we consider Atiyah-Patodi-Singer boundary value problems for operators adapted to the covering. We show that the corresponding symbols have a natural h…
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.
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