A Remark on Deformations of Hurwitz Frobenius Manifolds

In this note, we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux–Egoroff system. As an application, we explain how Shramchenko’s deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental–van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.     

Optico-Mechanical Analogy: an Axiomatic Approach

An axiomatic characterization of a ‘two-level Hamiltonian structure’ is proposed, which expresses the optico-mechanical analogy by representing optics and mechanics as (disjoint) classes of models satisfying the axioms. There is the ‘Hamilton–Jacobi level’, which involves a differential manifold on which the characteristic function satisfying the Hamilton–Jacobi equation is defined; and the ‘symplectic level’, involving the Hamiltonian, defined on the cotangent bundle of the manifold. The two levels, with the (analogous) structures on them, concern both optics and mechanics.     

Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.     

On Ruelle–Perron–Frobenius Operators.¶I. Ruelle Theorem

We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be presented in a forthcoming paper. Received: 31 May 2000 / Accepted: 1 June 2001     

Modular Classes of Poisson–Nijenhuis Lie Algebroids

The modular vector field of a Poisson–Nijenhuis Lie algebroid A is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian A-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson–Nijenhuis structure.      

Analysis of novel optical properties of subwavelength double-layers metallic grating

We present a theoretical Fabry–Pèrot-Like model for the novel optical response of the subwavlength double-layers metal grating. The Fabry–Pèrot-Like resonance phenomenon has been found in this structure, and the complex cavities model is carried out for the interpretation of the resonant frequency shift due to the SPPs’ effect for the TM mode. The SPPs exist or not for TE and TM mode lead to different resonance frequency. We conclude that the reflection coefficient minimum corresponds to the condition satisfying the Fabry–Pèrot-Like resonance situation no matter for TE or TM mode. PACS 42.79.Dj; 71.36.+c; 73.20.Mf; 78.66.Bz     

Proof of a conjecture on the genus two free energy associated to the An singularity

In a recent paper Dubrovin et al. (1998), it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two $G$-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two $G$-functions vanish. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.     

On the genus two free energies for semisimple Frobenius manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as the sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called ??genus two G-function.?? Conjecturally, the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities, as well as for ?¹-orbifolds with positive Euler characteristics. We explain the reasons for the conjecture and prove it in particular cases.     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Algebras from Multisymplectic Geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L _∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism.     

Quantum Group of Isometries in Classical and Noncommutative Geometry

We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. The idea of ‘quantum families’ (due to Woronowicz and Soltan) are relevant to our construction. A number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed. Supported in part by the Indian National Academy of Sciences.     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999     

Semiclassical Almost Isometry

Let (M , ω , J) be a compact and connected polarized Hodge manifold, an isodrastic leaf of half-weighted Bohr–Sommerfeld Lagrangian submanifolds. We study the relation between the Weinstein symplectic structure of and the asymptotics of the the pull-back of the Fubini–Study form under the projectivization of the so-called BPU maps on .     

On a Banks-Casher relation in MQCD

We discuss the meaning of a Banks–Casher relation for the Dirac operator eigenvalues in MQCD. It is argued that the eigenvalue can be identified with a coordinate involved in the string compactification manifold. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 1, 3–7 (10 January 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Knot Theory Based on the Minimal Braid in Lorenz System

By means of symbolic dynamics in Lorenz map, after studying spatial topological structure of dynamical knot constructed by the minimal braid assumption, we pry into the spatial structure of three-dimensional manifold from low-dimensional space. Lorenz dynamical knot provides a scheme about suspension. So, we are able to understand partly dynamical behaviors’ topological properties of high-dimensional differential manifold by studying dynamical knot’s properties. We hope to afford an approach and understand the nature of physical reality, especially in the study of DNA sequences, 20 amino acids symbolic sequences of proteid structure, and time series that can be symbolic in finance market et al.     

Noncommutative gravity, a ‘no strings attached’ quantum-classical duality, and the cosmological constant puzzle

There ought to exist a reformulation of quantum mechanics which does not refer to an external classical spacetime manifold. Such a reformulation can be achieved using the language of noncommutative differential geometry. A consequence which follows is that the ‘weakly quantum, strongly gravitational’ dynamics of a relativistic particle whose mass is much greater than Planck mass is dual to the ‘strongly quantum, weakly gravitational’ dynamics of another particle whose mass is much less than Planck mass. The masses of the two particles are inversely related to each other, and the product of their masses is equal to the square of Planck mass. This duality explains the observed value of the cosmological constant, and also why this value is nonzero but extremely small in Planck units. Second Award in the 2008 Essay Competition of the Gravity Research Foundation.     

Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations

We show that given a finite-dimensional real Lie algebra acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations. Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche Scientifique PROTARS III.     

Cartan–Weyl Dirac and Laplacian Operators,Brownian Motions: The Quantum Potential and Scalar Curvature,Maxwell’s and Dirac-Hestenes Equations,and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q     

Explicit Formula for the Natural and Projectively Equivariant Quantization

In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecome conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence using the framework of Thomas–Whitehead connections. In [Lett Math Phys 72(3):183–196, 2005], we gave a new proof of the same theorem thanks to the Cartan connections. After these works, there was no explicit formula for the quantization. In this paper, we give this formula using the formula in terms of Cartan connections given in [Lett Math Phys 72(3):183–196, 2005]. This explicit formula constitutes the generalization to any order of the formulae at second and third orders soon published by Bouarroudj in [Lett Math Phys 51(4):265–274, 2000] and [C R Acad Sci Paris Sér I Math 333(4):343–346, 2001].     

Orderings and Non-formal Deformation Quantization

We propose suitable ideas for non-formal deformation quantization of Fréchet Poisson algebras. To deal with the convergence problem of deformation quantization, we employ Fréchet algebras originally given by Gel’fand–Shilov. Ideas from deformation quantization are applied to expressions of elements of abstract algebras, which leads to a notion of “independence of ordering principle”. This principle is useful for the understanding of the star exponential functions and for the transcendental calculus in non-formal deformation quantization. Akira Yoshioka was partially supported by Grant-in-Aid for Scientific Research (#19540103.), Ministry of Education, Science and Culture, Japan.