Noncommutative coordinates on Riemann surfaces

Abstract

We show that the definition of a projective coordinate frame within a LaguerreForsyth scheme, leads to the extension of the factorized diffeomorphism algebra. The quantum improvement of this symmetry can be performed only if these coordinates switch, at the quantum level, into a non-commutative regime.

Large chiral diffeomorphisms on Riemann surfaces and W-algebras

“…The general ansatz (6.17) given in [15] and exemplified in (6.22) and (6.33), (6.34) for s = 3, 4 respectively, can be put into relation with some previous pioneer work [6,5,8,10]. Indeed, [10] will be of particular interest.…”

“…(6.14) Equation (6.14) can be disassembled into: 15) where the first summand looks like the variation coming from a symplectic approach [11] to W-algebra.…”

“…In order to recover the familiar expressions [12,23] nontrivial redefinitions of the tensorial ghosts involving derivative terms must be performed. The ansatz (6.11) relating the jet ghosts to the tensorial ones is recast into the form [15]: …”

“…The inhomogeneous coordinates have to satisfy a s + 1-th order equations. Accordingly, our point of view proposed in [15] traced back in the literature, and found out an unexpected origin in old past [14] related to some algebraic type differential equations, once covariantly formulated over a generic Riemann surface. Indeed the definition of general well-defined differential operators over a (two dimensional) Riemann Surface requires some care [16], and puts forward a deep insight on the links between covariance required by physical considerations and projective geometry.…”

“…In [15] the notion of Laguerre-Forsyth frames were promoted. In order to avoid a possible confusion with the named Laguerre-Forsyth form of a conformally covariant differential equation [8], in the present paper, we adopt the name Forsyth frames.…”