19 research outputs found
SUSY vertex algebras and supercurves
This article is a continuation of math.QA/0603633 Given a strongly conformal
SUSY vertex algebra V and a supercurve X we construct a vector bundle V_X on X,
the fiber of which, is isomorphic to V. Moreover, the state-field
correspondence of V canonically gives rise to (local) sections of these vector
bundles. We also define chiral algebras on any supercurve X, and show that the
vector bundle V_X, corresponding to a SUSY vertex algebra, carries the
structure of a chiral algebra.Comment: 50 page
All Stable Characteristic Classes of Homological Vector Fields
An odd vector field on a supermanifold is called homological, if
. The operator of Lie derivative makes the algebra of smooth
tensor fields on into a differential tensor algebra. In this paper, we give
a complete classification of certain invariants of homological vector fields
called characteristic classes. These take values in the cohomology of the
operator and are represented by -invariant tensors made up of the
homological vector field and a symmetric connection on by means of tensor
operations.Comment: 17 pages, references and comments adde
Variational Lie algebroids and homological evolutionary vector fields
We define Lie algebroids over infinite jet spaces and establish their
equivalent representation through homological evolutionary vector fields.Comment: Int. Workshop "Nonlinear Physics: Theory and Experiment VI"
(Gallipoli, Italy; June-July 2010). Published v3 = v2 minus typos, to appear
in: Theoret. and Mathem. Phys. (2011) Vol.167:3 (168:1), 18 page
Minkowski superspaces and superstrings as almost real-complex supermanifolds
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that
mathematicians study are real or (almost) complex ones, while Minkowski
superspaces are completely different objects. They are what we call almost
real-complex supermanifolds, i.e., real supermanifolds with a non-integrable
distribution, the collection of subspaces of the tangent space, and in every
subspace a complex structure is given.
An almost complex structure on a real supermanifold can be given by an even
or odd operator; it is complex (without "always") if the suitable superization
of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we
define the circumcised analog of the Nijenhuis tensor. We compute it for the
Minkowski superspaces and superstrings. The space of values of the circumcised
Nijenhuis tensor splits into (indecomposable, generally) components whose
irreducible constituents are similar to those of Riemann or Penrose tensors.
The Nijenhuis tensor vanishes identically only on superstrings of
superdimension 1|1 and, besides, the superstring is endowed with a contact
structure. We also prove that all real forms of complex Grassmann algebras are
isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to
related recent work by Witten is adde
Gepner-like models and Landau-Ginzburg/sigma-model correspondence
The Gepner-like models of -type is considered. When is multiple
of the elliptic genus and the Euler characteristic is calculated. Using
free-field representation we relate these models with -models on
hypersurfaces in the total space of anticanonical bundle over the projective
space
Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models
We review the construction of homological evolutionary vector fields on
infinite jet spaces and partial differential equations. We describe the
applications of this concept in three tightly inter-related domains: the
variational Poisson formalism (e.g., for equations of Korteweg-de Vries type),
geometry of Liouville-type hyperbolic systems (including the 2D Toda chains),
and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or
the Poisson sigma-models). Also, we formulate several open problems.Comment: Proc. 7th International Workshop "Quantum Theory and Symmetries-7"
(August 7-13, 2011; CVUT Prague, Czech Republic), 20 page