Paolo Furlan, Ludmil K. Hadjiivanov, Ivan T. Todorov
Canonical Approach to the Quantum WZNW Model
Preprint series: ESI preprints

MSC:

17B37 Quantum groups and related deformations, See also {16W30, 81R50, 82B23}

58B30 Noncommutative differential geometry and topology, See also {46L30, 46L87, 46L89}

81R50 Quantum groups and related algebraic methods, See Also {16W30, 17B37}

81T40 Two-dimensional field theories, conformal field theories, etc.

Abstract: The canonical approach to the chiral $SU(n)$ WZNW model with a monodromy
independent $r$--matrix is reviewed. Taking the quantum group symmetry of the
model (which reflects its classical Poisson--Lie symmetry) as a guiding
principle, we derive a complete set of exchange relations in the enlarged
chiral
phase space that includes the Borel components $M_\pm$ of the monodromy
matrix.
Regarded as new dynamical variables the elements of $M$ in the left and right
sectors cannot be identified: their Poisson
brackets have opposite signs. This
is a technical reason why the canonical reduction of the pair of
chiral models
to a single 2--dimensional theory that involves the left and right
movers' fields does not respect the quantum group symmetry. A
simple modification of
the Poisson brackets which does lead to an $SL_q(n)$ invariant model is
proven unacceptable as a substitute for the 2D theory.
As a way out we suggest a weak form of the monodromy
and quantum
group invariance of the extended 2D theory (involving mean values
in a physical subspace of the tensor product of chiral state
spaces).


Keywords: chiral WZNW model, quantum group, exchange relations