28,497 research outputs found
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled
by two parameters and . The aim of the present work is to investigate
the lowest weight representations of CGA with for any integer value of
. First we focus on the reducibility of the Verma modules. We give a
formula for the Shapovalov determinant and it follows that the Verma module is
irreducible if and the lowest weight is nonvanishing. We prove that
the Verma modules contain many singular vectors, i.e., they are reducible when
. Using the singular vectors, hierarchies of partial differential
equations defined on the group manifold are derived. The differential equations
are invariant under the kinematical transformation generated by CGA. Finally we
construct irreducible lowest weight modules obtained from the reducible Verma
modules
The form factors in the finite volume
The form factors of integrable models in finite volume are studied. We
construct the explicite representations for the form factors in terms of
determinants.Comment: 16 pages, Latex, no figure
Cluster Complexes via Semi-Invariants
We define and study virtual representation spaces having both positive and
negative dimensions at the vertices of a quiver without oriented cycles. We
consider the natural semi-invariants on these spaces which we call virtual
semi-invariants and prove that they satisfy the three basic theorems: the First
Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition
Theorem. In the special case of Dynkin quivers with n vertices this gives the
fundamental interrelationship between supports of the semi-invariants and the
Tilting Triangulation of the (n-1)-sphere.Comment: 34 page
Kac and New Determinants for Fractional Superconformal Algebras
We derive the Kac and new determinant formulae for an arbitrary (integer)
level fractional superconformal algebra using the BRST cohomology
techniques developed in conformal field theory. In particular, we reproduce the
Kac determinants for the Virasoro () and superconformal () algebras.
For there always exist modules where the Kac determinant factorizes
into a product of more fundamental new determinants. Using our results for
general , we sketch the non-unitarity proof for the minimal series;
as expected, the only unitary models are those already known from the coset
construction. We apply the Kac determinant formulae for the spin-4/3
parafermion current algebra ({\em i.e.}, the fractional superconformal
algebra) to the recently constructed three-dimensional flat Minkowski
space-time representation of the spin-4/3 fractional superstring. We prove the
no-ghost theorem for the space-time bosonic sector of this theory; that is, its
physical spectrum is free of negative-norm states.Comment: 33 pages, Revtex 3.0, Cornell preprint CLNS 93/124
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