Diagonalization of three-dimensional pseudo-Riemannian metrics

We give a short proof of the fact that any Riemannian or Lorentzian real analytic metric in dimension 3 can be locally adapted to the diagonal form. We use the classical Cauchy–Kowalevski Theorem to this purpose.     

On the Fefferman class of metrics associated with a three-dimensional CR space

We derive the explicit forms of Fefferman's metric for a Cauchy-Riemann space admitting a solution of the tangential Cauchy-Riemann equation and of the corresponding Weyl tensor. We show that its Petrov type is 0 in the case of the hyperquadric or N in all other cases, and that the Fefferman class of metrics does not contain any nonflat solution of Einstein's vacuum equations with cosmological constant.Work supported in part by the Polish Ministry of Science and Higher Education, Research Problem CPBP 01.03.     

Deformations of representations

A connection between deformation of Lie group representations and deformations of associated Lie algebra representations is established. Applications are given to the theory of analytic continuation of K-finite quasi-simple representations of semi-simple Lie groups. A construction process of all TCI representations of SL(2,R) is obtained.     

Deformations of super-KMS functionals

We investigate the stability of the super-KMS property under deformations. We show that a family of continuous deformations of the super-derivation in the quantum algebra yields a continuous family of deformed super-KMS functionals. These functionals define a family of cohomologous, entire cocycles.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065Visiting from the Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland     

Deformations of duality-symmetric theories

We prove that a sum of free non-covariant duality-symmetric actions does not allow consistent, continuous and local self-interactions that deform the gauge transformations. For instance, non-abelian deformations are not allowed, even in 4 dimensions where Yang–Mills type interactions of 1-forms are allowed in the non-manifestly duality-symmetric formulation. This suggests that non-abelian duality should require to leave the standard formalism of perturbative local field theories. The analyticity of self-interactions for a single duality-symmetric gauge field in four dimensions is also analyzed.     

Deformations of quantum hyperplanes

We consider the quantum hyperplanex ⁱ x ^j =q _ij x ^j x ⁱ (i,j = 1..n) and define and consider deformations of the formx ⁱ x ^j =q _ij x ^j x ⁱ + _{k k} ^ij x ^k + ^ij , where _k ^ij and ^ij are complex numbers. We prove that for genericq _ij no nontrivial deformations exist forn 3.     

Deformations of Dispersionless KdV Hierarchies

The obstructions to the existence of a hierarchy of hydrodynamic conservation laws are studied for a multicomponent dispersionless KdV system. It is proved that if the lowest order obstruction vanishes then all higher obstructions automatically vanish, if and only the underlying algebra is a Jordan algebra. Deformations of these multicomponent dispersionless KdV-type equations are also studied. It is shown that no new obstructions appear and, hence, that the existence of a fully deformed hierarchy depends only on the existence of a single purely hydrodynamic conservation law.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Deformations of generalized holomorphic structures

A deformation theory of generalized holomorphic structures in the setting of (generalized) principal fibre bundles is developed. It allows the underlying generalized complex structure to vary together with the generalized holomorphic structure. We study the related differential graded Lie algebra, which controls the deformation problem via the Maurer–Cartan equation. As examples, we check the content of the Maurer–Cartan equation in detail in the special cases where the underlying generalized complex structure is symplectic or complex. A deformation theorem, together with some non-obstructed examples, is also included.     

Dragged metrics

We show that the path of any accelerated body in an arbitrary spacetime geometry $g_{\mu \nu }$ can be described as a geodesic in a dragged metric $\hat{q}_{\mu \nu }$ that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kinds of non-gravitational force as modifications of the spacetime metric. This method of effective elimination of the forces by changing the metric of the substratum can be understood as a generalization of the d’Alembert principle applied to all relativistic processes.     

Deformations of Modules of Differential Forms

Abstract

We study non-trivial deformations of the natural action of the Lie algebra Vect(?ⁿ) on the space of differential forms on ?ⁿ. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of [3].     

Deformations in Mathematics and Physics

This text is my introductory talk given at the Workshop “Deformations and Contractions in Mathematics and Physics” in Oberwolfach in January 2006. The author is grateful for the Mathematisches Forschungsinsitut Oberwolfach for being able to organize this Meeting. The work was partially supported bt grants OTKA T043641 and T043034.     

Deformations and renormalisations ofW ∞

Deformations of the infiniteN limit of the ZamolodchikovW _N algebra are discussed. A recent one, due to Pope, Romans and Shen with non-zero central extensions for every conformal spin is shown to be formally renormalisable to one representable in Moyal bracket form. Another deformation is discovered which, like the algebra of Pope et al. possesses automatic closure, but has non-zero central extension only in the Virasoro subalgebra.On Research Leave from the University of Durham UK; research supported in part by the Department of Energy under Grant DE[FG02/88/ER25065, and by a grant from the Alfred P. Sloan Foundation     

Deformations of the canonical commutation relations

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an Nth root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.     

Chiral Forms and Their Deformations

We systematically study deformations of chiral forms with applications to string theory in mind. To first order in the coupling constant, this problem can be translated into the calculation of the local BRST cohomological group at ghost number zero. We completely solve this cohomology and present detailed proofs of results announced in a previous letter. In particular, we show that there is no room for non-abelian, local, deformations of a pure system of chiral p-forms. Received: 17 April 2000 / Accepted: 13 July 2001     

Associative Deformations of Complex Analytic Spaces

A theory of associative deformations is developed for general complex analytic spaces. Deformation–quantization and commutative deformation are particular cases of this concept. Deformation cohomology and obstruction are studied. It is proved that any compact analytic space has a formal versal associative deformation.      

Integrable Deformations of Heisenberg Supermagnetic Model

We construct the integrable deformations of the Heisenberg supermagnet model with the quadratic constraints （i） S2=3S - 2I, for S ∈ USPL（2/1）/S（U（2）×U（1）） and （ii） S2=S, for S ∈ USPL（2/1）/S（L（1/1）×U（1））. Under the gauge transformation, their corresponding gauge equivalent counterparts are derived. They are the Grassman odd and super mixed derivative nonlinear Schrodinger equation, respectively.