We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…
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48 results for “odd Laplacians”
Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.
problem Understanding cohomology classes on odd symplectic manifolds and their relation to Lagrangian submanifolds.
method Investigates complexes of differential, integral, and pseudo forms, introduces new operators, and proves cohomology isomorphisms.
result Proves isomorphism between de Rham cohomology and BV Laplacian cohomology on odd symplectic manifolds.
We observe that an anti-symplectic manifold locally always admits a parity structure. The parity structure can be viewed as a complex-like structure on the manifold. This induces an odd metric and its Levi-Civita connection, and thereby a new notion of an odd Kaehler geometry. Oversimplified, just to capture the idea, …
Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\primeΛand det L as the metric on M varies, where Δdenotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals de…
Analyze second order self-adjoint operators on supermanifolds, focusing on their potential and modular classes.
problem Geometrical meaning and modular classes of odd Laplacians on supermanifolds.
method Analyse the potential as a compensating field, relate to symplectic structures and modular classes.
result Modular classes of odd Poisson manifolds via Δ-operators, find non-trivial modular classes.
Study connects supermanifold Laplacian to harmonic superfunctions via sigma model.
problem Understanding harmonic superfunctions on supermanifolds.
method Calculus of variations applied to a supersymmetric sigma model.
result Relates Laplacian to harmonic superfunctions on supermanifolds.
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction. (Based on a talk at XXII Workshop on Geometric Methods in Physi…
On odd Laplace operatorsmath.DG
We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold . We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This…
New CR manifolds found with same Kohn Laplacian spectra.
problem Spectrum of Kohn Laplacian doesn't determine CR manifold equivalence.
method Constructed pairs of odd-dimensional elliptic manifolds.
result Found manifolds with same spectrum but different CR structures.
We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…
We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a cor…
In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsi…
The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditionsmath.DG
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 co…
A numerical expression in the form of an integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a curious sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. A few graphs are drawn.
Study heat kernel and spectrum on manifolds with wedge singularities.
problem Analyzing the spectrum and heat kernel on manifolds with wedge singularities.
method Uniform constructions of the resolvent and heat kernel on manifolds with corners, using spectral gap and analytic torsion.
result Obtain a Cheeger-Müller theorem relating analytic torsion and Reidemeister torsion.
For a Riemannian closed spin manifold and under some topological assumption (non-zero -genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the o…
The paper equates the index of a vector bundle to the manifold's index and introduces an analytic torsion.
problem Equating the index of a vector bundle to the manifold's index.
method Adiabatic limit and Witten Laplacian.
result The Quillen metric of the analytic torsion on the vector bundle coincides with the manifold's metric.
Study cohomology of GL₂n(Z) and graph complexes using Pfaffian forms.
problem Cohomology of GL₂n(Z) and related graph complexes.
method Use Pfaffian forms on symmetric spaces and dual Laplacians on graphs.
result First cocycle gives a non-trivial class in H⁻⁶(GC₃).
Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cuspsmath.DG
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 detai…
Eigenfunctions on spheres and tori with complex nodal sets.
problem Finding high-energy eigenfunctions with specific nodal sets.
method Constructing eigenfunctions on spheres and tori with given nodal sets.
result Eigenfunctions with complex nodal sets exist for sufficiently large eigenvalues.
The paper analyzes heat kernel asymptotics for real powers of Laplacians on manifolds.
problem Analyzing the small-time behavior of heat kernels for real powers of Laplacians.
method Analyzes asymptotics on the diagonal and away from it, proving non-triviality and non-locality of coefficients.
result Logarithmic terms appear only if the manifold dimension is odd and the power is rational with even denominator.
Study on the distribution of nodal sets on random superpositions of eigenfunctions.
problem Understanding the distribution of nodal sets on random superpositions of eigenfunctions.
method Analysis of the conormal cycle and integration current on nodal sets.
result Equidistribution of the conormal cycle on fibers of cotangent bundle in odd dimensions.
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental inte…
We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction …
The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of…
This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This giv…
We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, for instance, pairs of flat manifolds o…
In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pen…
We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bi…
The paper defines odd symplectic structures and compares them to Riemannian structures.
problem Defining and comparing odd symplectic structures on supermanifolds.
method Analyzing the Cartan prolongation of Lie algebras to describe differences between odd Riemannian and odd symplectic structures.
result Formulated an analogue of the Levi-Civita theorem for odd symplectic supermanifolds.
In this article we study the sub-Riemannian geometry of the spheres and , arising from the principal bundle structure defined by the Hopf map and the principal bundle structure given by the quaternionic Hopf map respectively. The action leads to the classical contact geometry of $…
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
Transforms odd-dimensional hyperbolic spaces using parallel formulas.
problem Developing formulas for odd-dimensional hyperbolic spaces.
method Parallel formulas to odd-dimensional spheres.
result Isometry and inversion formulas for Segal--Bargmann transform.
Cobordisms of graphs. A sliceness criterion for stably odd knots and related results on cobordismsmath.GT
The paper provides a sliceness criterion for stably odd knots.
problem Sliceness of stably odd knots
method Analyzes cobordisms and free knots to derive a sliceness criterion
result A sliceness criterion for stably odd knots
Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…
Proves conditions for odd pretzel knots to be doubly slice.
problem Identifying conditions for odd pretzel knots to be doubly slice.
method Analyzes knots with specific twist parameters and odd integers.
result Identifies all doubly slice odd pretzel knots.
On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is even dimensional. We then build a conformal invariant functional for the maps betw…
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM manifolds of non-negative curvature.
This paper extends NCFI to odd codimension and computes examples.
problem Extending NCFI to foliations of odd codimension.
method Computing NCFI for various foliated manifolds in both even and odd codimensions.
result NCFI is an invariant of foliations in odd codimension, requiring an odd \(K_1\)-class.
The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hype…
The paper solves sliceness for odd free knots.
problem Determining if odd free knots have a spanning disc.
method Using the pairing of knot diagram chords.
result An explicit criterion for sliceness of odd free knots.
We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd -theory. Moreover we discover that in odd dimen…
Adapts scanning algorithm for odd Khovanov homology.
problem Computing odd Khovanov homology efficiently.
method Uses mapping cone construction instead of tensor product.
result Determines odd Khovanov homology of 3-strand torus links.
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and o…
Extends Landweber-Stong result to torus actions on manifolds of generalized odd type.
problem Signature vanishing for manifolds with torus actions of generalized odd type.
method Extends Landweber-Stong result to torus actions.
result Signature vanishes for manifolds with torus actions of generalized odd type.
ODD Metricsmath.DG
New ODD metrics defined on manifolds with degeneracy conditions.
problem Defining metrics on manifolds with degeneracy conditions.
method Introducing ODD metrics that degenerate on submanifolds while maintaining compatibility.
result ODD metrics satisfy basic properties and induce metric space structures.
Study proposes new methods to convert betting odds into accurate probabilities for sports forecasting.
problem Convert betting odds to accurate outcome probabilities for sports forecasting and market efficiency analysis.
method Proposes two methods: Odds-Only-Equal-Profitability-Confidence (OO-EPC) and Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM).
result Proposed methods outperform existing methods in empirical tests and real-world applications.