Analytic Surgery of the zeta-determinant of the Dirac operator

zeta正则化行列式的圈汤表示与等变Symanzik恒等式

Loop soup representation of zeta-regularised determinants and equivariant Symanzik identities

本文推导了流形上向量丛拉普拉斯型算子行列式的随机表示（逆幂为布朗圈汤和乐乘积的期望，适用于2/3维紧致流形），并得出连续性、共形不变性等推论，还构造了最小耦合标量场并证明其满足Symanzik恒等式。

Adiabatic decomposition of the zeta-determinant and the Dirichlet to Neumann operator

We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also describe the adiabatic limit of the zeta-determinant of the Dirichlet to Neumann operator, which plays the essential role of Burghelea-Friedlander-Kappeler's Meyer-Vietoris type formula of the zeta determinant.

度量映射环面的zeta行列式与解析挠率

The zeta-determinants and anlaytic torsion of a metric mapping torus

该研究利用BFK粘合公式计算等距诱导的度量映射环面的zeta行列式与解析挠率，给出克莱因瓶等流形的应用结果，证明相关热迹渐近展开一致性，并重得已有文献结论。

Adiabatic decomposition of the zeta-determinant of the Dirac Laplacian I. The case of an invertible tangential operator. With an appendix by Yoonweon Lee

we discuss the decomposition of the zeta-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold in the case of an invertible tangential operator.

Zeta Functions, Determinants and Torsion for Open Manifolds

On an open manifold, the spaces of metrics or connections of bounded geometry, respectively, split into an uncountable number of components. We show that for a pair of metrics or connections, belonging to the same component, relative $ζ$-functions, determinants, torsion for pairs of generalized Dirac operators are well defined.

The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing form…

Zeta determinant for double sequences of spectral type

We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence. We particularize this technique to the case of sums of sequences of spectral type, and we give two applications: the first concerning some special functions appearing in number the…

Adiabatic decomposition of the zeta-determinant and Scattering theory

We discuss the decomposition of the zeta-determinant of the square of the Dirac operator into contributions coming from the different parts of the manifold. The easy case was worked in the previous paper of authors. Due to the assumptions made on the operators in the previous paper, we were able to avoid the presence of the small eigenvalues which provide the large time contibution to the determinant. In the present work we analyze the general case. We offer a detailed analysis of the contribution made by the small eigenvalues.

关于 Robin 边界条件的 zeta-行列式的 BFK 型粘合公式

The BFK type gluing formula of zeta-determinants for the Robin Boundary condition

本文研究了紧 Riemann 流形上 Robin 边界条件下 Laplacian 的 zeta-行列式的 BFK 型粘合公式，计算了 Robin 与 Dirichlet 边界条件下的行列式差，并推广了圆柱上 Laplacian 的 zeta-行列式计算结果。

The ratio of two zeta-determinants of Dirac Laplacians associated with unitary involutions on a compact manifold with cylindrical end

Given two unitary involutions $σ_{1}$ and $σ_{2}$ satisfying $G σ_{i} = - σ_{i} G$ on $ker B$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with $σ_{1}$ and $σ_{2}$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This…

三维黎曼叶状动力系统 zeta 函数的正则化行列式公式

Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems

本文利用分布动力 Lefschetz 迹公式沟通动力谱 ξ 函数与 zeta 函数，证明了 Deninger 猜想的三维黎曼叶状动力系统 zeta 函数正则化行列式公式。

Regularizing infinite sums of zeta-determinants

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revo…

Relative Zeta Functions, Determinants, Torsion, Index Theorems and Invariants for Open Manifolds

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac operators, we prove for the pair $(D,D')$ relative index theorems, define relative $ζ$-- and $η$--functions, relative determinants and in the case of $D=Δ$ relative analytic torsion. To define relative $ζ$-- and $η$--functions, we assume additionally that the essential spectrum of $…

Zeta Forms and the Local Family Index Theorem

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formu…

Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function

We obtain explicit expressions for the determinants of the Laplacians on zero and one forms for an infinite class of three dimensional lens spaces $L(p,q)$. These expressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As a consequence we obtain an infinite class of formulae for the Riemann zeta function $ζ(3)$. The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces $L(6,1)$, $L(10,3)$ and $L(12,5)$ is also noted. (…

紧黎曼曲面上锥形拉普拉斯算子的Krein公式

Weyl Curves and Zeta Determinants of Conic Laplacians on Riemann Surfaces

通过临界渐近边界数据的有限维辛空间，建立紧黎曼曲面上锥形拉普拉斯算子自伴扩张的Krein公式，并导出迹恒等式用于比较不同Lagrangian边界条件下的正谱zeta行列式。

形式上 Steklov 问题的 Dirichlet-to-Neumann 算子的 zeta-determinant

The zeta-determinant of the Dirichlet-to-Neumann operator of the Steklov Problem on forms

本文将带边界黎曼流形上作用于q-形式的Dirichlet-to-Neumann算子的zeta-行列式对数，表示为绝对与Dirichlet边界条件Laplacian的zeta-行列式对数之差及曲率附加项，显式计算了2维和3维情形，并利用共形缩放方法恢复了2维情形的共形不变性结果。

Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps

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 …

Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

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…

紧李群上的次拉普拉斯算子：热核、距离与泽塔行列式

Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants

该研究针对紧李群闭连通子群确定的次拉普拉斯算子，推导热核积分表示、小时间热迹展开等结果，并应用于$\rm SU(N)$块子群得到经典谱。

Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s=3

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3. The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, gr…

Eta forms and determinant lines

We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.

Dirac operators, heat kernels and microlocal analysis Part II: analytic surgery

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…

Determinant of Laplacian on tori of constant positive curvature with one conical point

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.

Determinant of Laplacians on Heisenberg Manifolds

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the "radius" of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We s…

Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.

Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces

We show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell-Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell-Miller. This generalises author's pre…

Morse-Smale flow, Milnor metric, and dynamical zeta function

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.