Signature Operator Research Articles

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

L‐theory of C∗$C^*$‐algebras

AbstractWe establish a formula for the L‐theory spectrum of real ‐algebras from which we deduce a presentation of the L‐groups in terms of the topological K‐groups, extending all previously known results of this kind. Along the way, we extend the integral comparison map obtained in previous work by the first two authors to real ‐algebras and interpret it using topological Grothendieck–Witt theory. Finally, we use our results to give an integral comparison between the Baum–Connes conjecture and the L‐theoretic Farrell–Jones conjecture, and discuss our comparison map in terms of the signature operator on oriented manifolds.

A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data

For the given signature operator H=Ir⊕−In−r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈⋅,⋅〉H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n×n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊n2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.

A note on traces of conformal twisted signature operators

In this paper, the traces of conformal twisted signature operators and noncommutative residue for compact manifolds with boundary are considered, the classic gravity representation for manifolds with boundary associated with conformal twisted signature operators and Kastler–Kalau–Walze-type theorems is explored.

Rough paths and SPDE

After a brief survey on rough paths theory, from the seminal paper of T. Lyons to its recent developments, this proceeding provides details on C. Bellingeri, Y. Bruned and A. Fermanian’s talks during the Journées MAS 2020: a new formulation and a generalization of the signature operator, an extension of branched rough paths called Volterra branched rough paths, and recurrent neural networks investigated as a kernel method thanks to the signature operator and neural ODEs.

Uniform homotopy invariance of Roe Index of the signature operator

In this paper we study the Roe index of the signature operator of manifolds of bounded geometry. Our main result is the proof of the uniform homotopy invariance of this index. In other words we show that, given an orientation-preserving uniform homotopy equivalence \(f:(M,g) \longrightarrow (N,h)\) between two oriented manifolds of bounded geometry, \(f_\star (Ind_{Roe}D_M) = Ind_{Roe}(D_N)\). Moreover we also show that the same result holds if a group \(\Gamma \) acts on M and N by isometries and f is \(\Gamma \)-equivariant. The only assumption on the action of \(\Gamma \) is that the quotients are again manifolds of bounded geometry.

A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes

The bosonic signature operator is defined for Klein–Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein–Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein–Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.

Reflection positivity via Krein space analysis

We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M+) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U,J,M+) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J, and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U,J,M+), with U assumed unitary and J-self-adjoint, and M+ reflecting the choices of the associated J-positive and U-invariant subspaces. We further consider the wider setting when U is only assumed J-self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M+ will be realized as spaces G(X), graph of a second linear operator X, where the possibilities for such X, as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis.

Stratified surgery and K-theory invariants of the signature operator

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-theory groups. In recent years the (higher) signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we show that secondary invariants, such as the rho-class, also extend from closed manifolds to Cheeger spaces. We revisit a surgery exact sequence for stratified spaces originally introduced by Browder-Quinn and obtain a natural transformation analogous to that of Higson-Roe. We also discuss geometric applications.

A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces

The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator H=diag(1,1,…,1,−1), the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator H. In this paper we solve this problem by applying Sylvester’s identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known Lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective.

How to choose the signature operator such that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable?

Using a discrete version of Floquet theory in a space with indefinite metric we reconstruct periodic pseudo-Jacobi matrices from given spectral data. We use two methods to characterize the signature operator H so that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable. As a by-product of the obtained results, we precisely polish some non-rigorous assumptions in the existing literature.

On localized signature and higher rho invariant of fibered manifolds

The higher index of the signature operator is a far-reaching generalization of the signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator, called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for the higher index and the higher rho invariant of the signature operator on a fibered manifold. Our result implies the classical product formula for the numerical signature of a fibered manifold obtained by Chern, Hirzebruch, and Serre (1957). We also give a new proof of the product formula for the higher rho invariant of the signature operator on a product manifold, which is parallel to the product formula for the higher rho invariant of Dirac operator on a product manifold obtained by Xie and Yu (2014) and Zeidler (2016).

$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

The main result of this paper is the $G$-homotopy invariance of the $G$-index of signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$ manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^{*}$-algebra of the group $G$. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required, so this is a generalization of the classical case of closed manifolds. Using this result we can deduce the equivariant version of Novikov conjecture for proper co-compact $G$-manifolds from the Strong Novikov conjecture for $G$.

A mod k index theorem for signature operators

A mod k index theorem for signature operators

The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

From the signature theorem to anomaly cancellation

We survey the Hirzebruch signature theorem as a special case of the Atiyah–Singer index theorem. The family version of the Atiyah–Singer index theorem in the form of the Riemann–Roch–Grothendieck–Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

The fermionic signature operator in de Sitter spacetime

The fermionic projector state is a distinguished quasi-free state for the algebra of Dirac fields in a globally hyperbolic spacetime. We construct and analyze it in the four-dimensional de Sitter spacetime, both in the closed and in the flat slicing. In the latter case we show that the mass oscillation properties do not hold due to boundary effects. This is taken into account in a so-called mass decomposition. The involved fermionic signature operator defines a fermionic projector state. In the case of a closed slicing, we construct the fermionic signature operator and show that the ensuing state is maximally symmetric and of Hadamard form, thus coinciding with the counterpart for spinors of the Bunch-Davies state.

A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Phi or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.

THE FINITE GROUP ACTION AND THE EQUIVARIANT DETERMINANT OF ELLIPTIC OPERATORS III

Let $M$ be an even-dimensional closed oriented manifold and $g$ a periodic automorphism of $M$ of order $p$. In this paper, under the assumption that the fixed points of $g^k\;(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D:{\Bbb Z}_p\to{\Bbb R}/{\Bbb Z}$ defined in [6] for equivariant twisted signature operators $D$ over $M$. The formula gives a new method to study the periodic automorphisms of oriented manifolds. As examples of the application of the formula, results about the existence of the cyclic group action for 2,4,6-dimensional closed oriented manifolds are obtained.