We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary weights appear naturally in the course of study. We show that for the algebra of densities it is possible to establish a one-to-one correspondence between operators and brackets generated by them. Everything is applicable to supermanifolds as well as to usual manifolds. In the super case the problem is closely connected with the geometry of the Batalin--Vilkovisky formalism in quantum field theory, namely the description of the generating operators for an odd bracket. We give a complete answer. This text is a concise outline of the main results. A detailed exposition is in \texttt{arXiv:math.DG/0212311}.
We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. 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 includes earlier results for odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on $M$ is partitioned into orbits by a natural groupoid whose arrows correspond to the solutions of the quantum Batalin--Vilkovisky equations. …
We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential operators acting on densities of all weights simultaneously. The algebra of densities, which we introduce here, has a natural invariant scalar product. Using it, we prove that there is a one-to-one correspondence between second-order operators in this algebra and the corresponding brackets. A bracket on densities incorporates a bracket on functions, an ``uppe…
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite dimensional. Using Jucys-Murphy elements of the Hecke algebra the eigenvalues of the Laplace-Beltrami operator for the Hopf algebra O(SLq(N)) are computed.
For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure $(S^n,[g_0])$, we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The construction can be regarded as a deformation of the Fefferman-Graham ambient metric construction of GJMS 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 Physics at Bialowieza)
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 differential forms on a purely even Lagrangian submanifold. We establish a criterion of ``normality'' of a volume form on an odd symplectic supermanifold in terms of the canonical odd Laplacian acting on semidensities.
The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
We generalize the result of [Matveev-Topalov 2001] to all signatures: we show that in all signatures the Killing tensors constructed by projectively equivalent metrics correspond to commuting differential operators
In the first part of this paper, we introduce the notion of "cyclic stratum" of a Frobenius manifold $M$. This is the set of points of the extended manifold $\mathbb C^*\times M$ at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to $M$ can be reduced to a scalar differential equation, called the "master differential equation" of $M$. In t…
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 transformation for an odd symplectic form is a polynomial in matrix entries and a complete square. This is a simple but fundamental fact, parallel to Liouville's theorem for an even symplectic structure. We draw attention to the fact that the de Rham complex on $M$ naturally admits an action …
We study the cohomology of the complexes of differential, integral and pseudo forms on odd symplectic manifolds taking the wedge product with the symplectic form as differential. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham dif…
In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it is harmonic.
We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature, and derive its algebra structure. Our construction is based on the Riemannian tractor calculus, allowing to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry.
The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator $Δ$ with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operator…
We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat invariants associated with these operators. As another application, we present new explicit formulas for the matrix Korteweg-de Vries hierarchy.
The caloron correspondence is a tool that gives an equivalence between principal $G$-bundles based over the manifold $M \times S^1$ and principal $LG$-bundles on $M$, where $LG$ is the Fréchet Lie group of smooth loops in the Lie group $G$. This thesis uses the caloron correspondence to construct certain differential forms called "string potentials" that play the same role as Chern-Simons forms for loop group bundles. Following their construction, the string potentials are used to define degree 1 differential characteristic classes for $ΩU(n)$-bundles. The notion of an "$Ω$ vector bundle" is i…
Odd $K$-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential $K$-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We characterise the odd Chern character and its transgression form in terms of a connection and Higgs field and discuss some applications. Our model can be seen as the odd counterpart to the Simons-Sullivan construction of even differential $K$-theory. We use this model to prove a conjec…
Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin-Vilkoviski operator.
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.
We study "higher-dimensional" generalizations of differential forms. Just as differential forms can be defined as the universal commutative differential algebra containing C^\infty(M), we can define differential gorms as the universal commutative bidifferential algebra. From a more conceptual point of view, differential forms are functions on the superspace of maps from the odd line to M and the action of Diff(the odd line) on forms is equivalent to deRham differential and to degrees of forms. Gorms are functions on the superspace of maps from the odd plane to M and we study the action of Diff…
Laplace and Dolbeault operators on Sasakian manifolds
Lebanese University（黎巴嫩大学）；Université de Lorraine CNRS, IECL（洛林大学法国国家科学研究中心计算与实验数学中心）；Texas Christian University（得克萨斯基督教大学）；Wydzial Matematyki i Informatyki UJ（雅盖隆大学数学与信息学院）
We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.
we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.
We show that the universal odd Chern form, defined on the stable unitary group $U$, extends to the loop group $LU$ in a way that is closed with respect to an equivariant-type differential. This provides an odd analogue to the Bismut-Chern form. We also describe the associated transgression form, the so-called Bismut-Chern-Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group. As a corollary, we obtain the Chern character homomorphism from odd K-theory to the periodic cohomology of the free loop space, repr…
The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are…
