Torsion Invariants for Families

Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds

We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $λ$ on $L$. Much of…

Absolute torsion and eta-invariant

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.

具有对合的不可约全纯辛四重形的解析挠率，I：一个不变量的构造

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant

本文利用等变解析挠率，为具有反辛对合的 $K3^{[2]}$ 型不可约全纯辛流形构造了一个新不变量，并给出了该不变量对数的复 Hessian 公式。

Invariant torsion and G_2-metrics

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G_2 that arises from SO(3)-…

Refined Analytic Torsion

For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase i…

Einstein connections with skew-torsion on Berger spheres

The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to $\mathbb{S}^3$. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, …

Removing Metric Anomalies from Ray Singer Torsion

Ray Singer torsion is a numerical invariant associated with a compact Riemannian manifold equipped with a flat bundle and a Hermitian structure on this bundle. In this note we show how one can remove the dependence on the Riemannian metric and on the Hermitian structure with the help of a base point and of an Euler structure, in order to obtain a topological invariant. A numerical invariant for an Euler structure and additional data is also constructed.

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 $…

Hyperkähler torsion structures invariant by nilpotent Lie groups

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying no…

Complex valued Ray-Singer torsion

In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations.

Analytic torsion for Calabi-Yau threefolds

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa.

A remark on the higher torsion invariants for flat vector bundles with finite holonomy

We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.

Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

Let X be a compact oriented Riemannian manifold and let $ϕ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $ϕ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $ϕ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $ϕ$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $ϕ$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs)…

Torsion type invariants of singularities

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on ${\mathbb C}^n$ by a quasi-homogeneous polynomial $f$. Under some mild assumption on $f$, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time $t$. Then we prove a local index formula which expresses the Milnor number of $f$ by a Gaussian type integral. Furthermore, the heat kernel expansion provides spectral invariants of $f$. Especially, we define torsion type invaria…

微局指数定理与解析扭量不变量在偏微分方程几何理论中的应用

Microlocal index theorems and analytic torsion invariants in the geometric theory of partial differential equations

本文提出微局和导出几何框架，统一非线性PDE、扭量不变量和模理论，揭示BCOV不变量与de Rham系统的关联，并拓展至配置空间应用量子场论。

Analytic torsion for log-Enriques surfaces and Borcherds product

We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac{1}{4}(1,1)$. The moduli space of such log-Enriques surfaces with $k$ singular points is a modular variety of orthogonal type %of dimension $10-k$ associated with a unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed as a function on the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$. As a consequence, th…

Twisted Analytic Torsion

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle an…

Determinant lines, von Neumann algebras and $L^2$ torsion

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced $L^2$ homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dime…

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M)>10

We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.

Geometry of growth: Approximation theorems for L^2 invariants

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homolog…

Analytic Torsion of a Bounded Generalized Cone

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted product of zeta-regularized determinants of Laplace operators on differential forms. The celebrated Cheeger-Mueller Theorem, established independently by J. Cheeger and W. Mueller, proved equality between the analytic …

A generalization of analytic torsion via differential forms on spaces of metrics

We introduce multi-torsion, a spectral invariant generalizing Ray-Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of multi-torsion is inspired by an interpretation of each of analytic torsion and the eta invariant as a regularized integral of a closed differential form on a space of metrics on a vector bundle or on a space of elliptic operators. We generalize the Stokes' theorem argument expla…

Combinatorial invariants computing the Ray-Singer analytic torsion

It is shown that for any piecewise-linear closed orientable manifold of odd dimension there exists an invariantly defined metric on the determinant line of cohomology with coefficients in an arbitrary flat bundle E over the manifold (E is not required to be unimodular). The construction of this metric (called Poincare - Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister type construction with Poincare duality. The main result of the paper states that the Poincare-Reidemeister metric computes combinatorially the Ray-Singer metric. It is shown also that the Ray-S…

On invariants of almost symplectic connections

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.

Reidemeister torsion in generalized Morse theory

In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological invariant. It is hoped that this will provide a model for possible generalizations to Floer theory.

Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, I

This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II). The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance--the Reidemeister torsion (τ_F) of the Floer-Novikov complex of (possibly non-hamiltonian) symplectomorphisms. (τ_F) turns out NOT to be invariant under hamiltonian isotopies, but this failure may be fixed by introducing certain ``correction term'': We define…

Discrete Curvature and Torsion from Cross-Ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and …

Torsion, TQFT, and Seiberg-Witten invariants of 3-manifolds

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg-Witten invariants of 3-manifolds by making use of a `topological quantum field theory,' which makes the calculation completely explicit. We also realize a version of the Seiberg-Witten invariant of X as the intersection number of a pair of totally real submanifolds of a pr…

Torsion Discriminance for Stability of Linear Time-Invariant Systems

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $τ(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a measurable set $E_1$ with positive Lebesgue measure, such that $r(0)\in E_1$ implies that $\lim\limits_{t\to+\infty}τ(t)\neq0$ or $\lim\limits_{t\to+\infty}τ(t)$ does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set $E_2$ with positive Lebesgue measure, such that $r(0)\in E_2$ implies that $\lim\limits_{t\to+\infty}τ(…

带挠率流形上非极小算子的热不变量 E_2

Heat Invariant E_2 for Nonminimal Operator on Manifolds with Torsion

本文利用计算机代数方法，计算了带挠率和规范场的弯曲流形上非极小算子热核展开的第二系数 E_2，给出了任意维数 n 及 n=2 情形的完整表达式。