Quantum Hamiltonian Reduction Research Articles

Quasi-superconformal algebras in two dimensions and hamiltonian reduction

In the standard quantum hamiltonian reduction, constraining the SL(3.ℝ) WZNW model leads to a model of Zamolodchikov's W 3-symmetry. In recent work, Polyakov and Bershadsky have considered an alternative reduction which leads to a new algebra. W 3− 2 a nonlinear extension of the Virasoro algebra by a spin-1 current and two bosonic spin-2/3 currents. Motivated by this result, we display two new infinite series of nonlinear extended conformal algebras, containing 2 N bosonic spin-3/2 and spin-1 Kac-Moody currents for either U( N)or Sp(2 N); the W 3 2 algebra appears as the N = 1 member of the U( N) series. We discuss the relationship between these algebras and the Knizhnik-Bershadsky superconformal algebras, and provide realisations in terms of free fields coupled to Kac-Moody currents. We propose a specific procedure for obtaining the algebras for general N through hamiltonian reduction.

Conformal field theories via Hamiltonian reduction

Constraining theSL(3) WZW-model we construct a reduced theory which is invariant with respect to the new chiral algebraW 3 2 . This symmetry is generated by the stress-energy tensor, two bosonic currents with spins 3/2 and theU(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also discuss the quantum Hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras.

Quantum hamiltonian reduction and N = 2 coset models

We study the quantum hamiltonian reduction of the affine Lie superalgebra A( n, n−1) (1) = sl( n + 1, n) (1) ( n ⩾1), whose central charge is zero. After a BRST gauge fixing the model has a W algebra structure with N = 2 superconformal symmetry. We show that this model is the N = 2 coset model C P n = SU (n + 1)/ SU (n) × U (1) constructed by Kazama and Suzuki. We also discuss a topological field theoretical aspect of the SL ( n + 1, n) Wess-Zumino-Novikov-Witten model.

REMARKS ON THE BRST QUANTIZED GAUGED WZNW MODELS AND THE TODA FIELD THEORIES

It is shown that the quantum Hamiltonian reduction proposed by Bershadsky and Ooguri enables us to connect the gauged WZNW models with fractional levels to the quantum Toda field theories, and the coupling constants of the Toda field theories with the fractional levels. The BRST framework is applied to the SL (n, ℝ)-WZNW models.

HeiddenSL(n) symmetry in conformal field theories

We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of theSL(2, ℛ) current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism. Thus there is aSL(2, ℛ) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrainedSL(2, ℛ) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of theSL(2, ℛ) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of theSL(2, ℛ) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of theSL(2, ℛ) current algebra and its relation to theW n -algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for theW n -algebra despite its non-Lie-algebraic nature.