55 research outputs found
Non-Local Matrix Generalizations of W-Algebras
There is a standard way to define two symplectic (hamiltonian) structures,
the first and second Gelfand-Dikii brackets, on the space of ordinary linear
differential operators of order , . In this paper, I consider in detail the case where the are
-matrix-valued functions, with particular emphasis on the (more
interesting) second Gelfand-Dikii bracket. Of particular interest is the
reduction to the symplectic submanifold . This reduction gives rise to
matrix generalizations of (the classical version of) the {\it non-linear}
-algebras, called -algebras. The non-commutativity of the
matrices leads to {\it non-local} terms in these -algebras. I show
that these algebras contain a conformal Virasoro subalgebra and that
combinations of the can be formed that are -matrices of
conformally primary fields of spin , in analogy with the scalar case .
In general however, the -algebras have a much richer structure than
the -algebras as can be seen on the examples of the {\it non-linear} and
{\it non-local} Poisson brackets of any two matrix elements of or
which I work out explicitly for all and . A matrix Miura transformation
is derived, mapping these complicated second Gelfand-Dikii brackets of the
to a set of much simpler Poisson brackets, providing the analogue of the
free-field realization of the -algebras.Comment: 43 pages, a reference and a remark on the conformal properties for
adde
Robin conditions on the Euclidean ball
Techniques are presented for calculating directly the scalar functional
determinant on the Euclidean d-ball. General formulae are given for Dirichlet
and Robin boundary conditions. The method involves a large mass asymptotic
limit which is carried out in detail for d=2 and d=4 incidentally producing
some specific summations and identities. Extensive use is made of the
Watson-Kober summation formula.Comment: 36p,JyTex, misprints corrected and a section on the massive case
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Influence of plastic deformation on the mechanical-corrosion strength of 60Kh3G8N8V austenitic steel
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