scholarly journals LANGLANDS DUALITY IN LIOUVILLE-$H^3_+$ WZNW CORRESPONDENCE

2009 ◽  
Vol 24 (16n17) ◽  
pp. 3137-3170 ◽  
Author(s):  
GASTON GIRIBET ◽  
YU NAKAYAMA ◽  
LORENA NICOLÁS
Keyword(s):  
Correlation Functions ◽  
Free Field ◽  
Liouville Theory ◽  
Hamiltonian Reduction ◽  
Langlands Correspondence ◽  
Conformal Field ◽  
Langlands Duality ◽  
Dual Version ◽  
Free Field Realization ◽  
Wznw Model

We show a physical realization of the Langlands duality in correlation functions of [Formula: see text] WZNW model. We derive a dual version of the Stoyanovky–Riabult–Teschner (SRT) formula that relates the correlation function of the [Formula: see text] WZNW and the dual Liouville theory to investigate the level duality k - 2 → (k - 2)-1 in the WZNW correlation functions. Then, we show that such a dual version of the [Formula: see text]-Liouville relation can be interpreted as a particular case of a biparametric family of nonrational conformal field theories (CFT's) based on the Liouville correlation functions, which was recently proposed by Ribault. We study symmetries of these new nonrational CFT's and compute correlation functions explicitly by using the free field realization to see how a generalized Langlands duality manifests itself in this framework. Finally, we suggest an interpretation of the SRT formula as realizing the Drinfeld–Sokolov Hamiltonian reduction. Again, the Hamiltonian reduction reveals the Langlands duality in the [Formula: see text] WZNW model. Our new identity for the correlation functions of [Formula: see text] WZNW model may yield a first step to understand quantum geometric Langlands correspondence yet to be formulated mathematically.

THE SU(2)0 WZNW MODEL

2003 ◽  
Vol 18 (25) ◽  
pp. 4685-4702
Author(s):  
ALEXANDER NICHOLS
Keyword(s):  
Correlation Functions ◽  
Free Field ◽  
Hamiltonian Reduction ◽  
Close Correspondence ◽  
Quantum Hamiltonian Reduction ◽  
Quantum Hamiltonian ◽  
Wznw Model

We analyse the correlation functions and operator content of the SU(2)0 model using both the free field description and the solutions to the Knizhnik-Zamolodchikov equations. We show that there is a very close correspondence between this model and the well studied c=-2 theory. We also demonstrate that the quantum Hamiltonian reduction of SU(2)0 leads very directly to the correlation functions of the c=-2 model.

scholarly journals THE FREE FIELD REPRESENTATION OF SU(3) CONFORMAL FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 4031-4053
Author(s):  
HOVIK D. TOOMASSIAN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Free Field ◽  
Field Representation ◽  
Conformal Field ◽  
Point Correlation

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

QUANTUM HAMILTONIAN REDUCTION AND WB ALGEBRA

1992 ◽  
Vol 07 (20) ◽  
pp. 4885-4898 ◽  
Author(s):  
KATSUSHI ITO
Keyword(s):  
Lie Algebras ◽  
Free Field ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Hamiltonian Reduction ◽  
Affine Lie Algebras ◽  
Quantum Hamiltonian Reduction ◽  
Free Field Realization ◽  
Coset Construction ◽  
Quantum Hamiltonian

We study the quantum Hamiltonian reduction of affine Lie algebras and the free field realization of the associated W algebra. For the nonsimply laced case this reduction does not agree with the usual coset construction of the W minimal model. In particular, we find that the coset model [Formula: see text] can be obtained through the quantum Hamiltonian reduction of the affine Lie superalgebra B(0, n)(1). To show this we also construct the Feigin-Fuchs representation of affine Lie superalgebras.

scholarly journals LIE SUPERALGEBRA AND EXTENDED TOPOLOGICAL CONFORMAL SYMMETRY IN NON-CRITICAL W3-STRINGS

1994 ◽  
Vol 09 (33) ◽  
pp. 3063-3075 ◽  
Author(s):  
KATSUSHI ITO ◽  
HIROAKI KANNO
Keyword(s):  
String Theory ◽  
Root System ◽  
Simple Root ◽  
Similarity Transformation ◽  
Conformal Symmetry ◽  
Free Field ◽  
Lie Superalgebra ◽  
Hamiltonian Reduction ◽  
Free Field Realization ◽  
Quantum Hamiltonian

We obtain a new free field realization of N = 2 super W3-algebra using the technique of quantum Hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra sl (3|2)(1) associated with a non-standard sl (2) embedding. After twisting and a similarity transformation, this W-algebra can be identified as the extended topological conformal algebra of non-critical W3 string theory.

scholarly journals HAMILTONIAN REDUCTION AND TOPOLOGICAL CONFORMAL ALGEBRA IN c<1 NON-CRITICAL STRINGS

1994 ◽  
Vol 09 (15) ◽  
pp. 1377-1388 ◽  
Author(s):  
KATSUSHI ITO ◽  
HIROAKI KANNO
Keyword(s):  
Topological Algebra ◽  
Free Field ◽  
Lie Superalgebra ◽  
Conformal Algebra ◽  
Hamiltonian Reduction ◽  
Quantum Case ◽  
Operator Formalism ◽  
Free Field Realization ◽  
Brst Cohomology ◽  
Lax Operator

We study the Hamiltonian reduction of affine Lie superalgebra sl(2|1)(1). Based on a scalar Lax operator formalism, we derive the free field realization of the classical topological algebra which appears in the c≤1 non-critical strings. In the quantum case, we analyze the BRST cohomology to get the quantum free field expression of the algebra.

scholarly journals AN EXPLICIT CONSTRUCTION OF WAKIMOTO REALIZATIONS OF CURRENT ALGEBRAS

1996 ◽  
Vol 11 (24) ◽  
pp. 1999-2011 ◽  
Author(s):  
JAN DE BOER ◽  
LÁSZLÓ FEHÉR
Keyword(s):  
Free Field ◽  
Explicit Construction ◽  
Hamiltonian Reduction ◽  
Energy Tensor ◽  
Parabolic Subalgebra ◽  
Standard Case ◽  
Normal Ordering ◽  
Free Field Realization ◽  
Stress Energy ◽  
Wakimoto Realization

It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra [Formula: see text] can be associated with each parabolic subalgebra [Formula: see text] of the Lie algebra [Formula: see text], where in the standard case [Formula: see text] is the Cartan and [Formula: see text] is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltonian reduction of the WZNW model, we first derive a Poisson bracket realization of the [Formula: see text]-valued current in terms of symplectic bosons belonging to [Formula: see text] and a current belonging to [Formula: see text]. We then quantize the formula by determining the correct normal ordering. We also show that the affine-Sugawara stress-energy tensor takes the expected quadratic form in the constituents.

MODULAR INVARIANT ONE-POINT CORRELATION FUNCTIONS FOR SU(2) WESS-ZUMINO MODEL

1992 ◽  
Vol 07 (25) ◽  
pp. 6257-6272 ◽  
Author(s):  
O.D. ANDREEV
Keyword(s):  
Correlation Functions ◽  
Riemann Surfaces ◽  
Free Field ◽  
Necessary Condition ◽  
Modular Invariant ◽  
Field Representation ◽  
Conformal Field ◽  
Higher Genus ◽  
Point Correlation ◽  
Zumino Model

We calculate one-point correlation functions of SU(2) Wess-Zumino model (WZM) on a torus using the Wakimoto free field representation. Their modular invariance is proved. It is a necessary condition of extending the WZ conformal field theory to higher genus Riemann surfaces.

scholarly journals LIOUVILLE THEORY AND LOGARITHMIC SOLUTIONS TO KNIZHNIK–ZAMOLODCHIKOV EQUATION

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4821-4862 ◽  
Author(s):  
GASTÓN GIRIBET ◽  
CLAUDIO SIMEONE
Keyword(s):  
Correlation Functions ◽  
Three Dimensional ◽  
Liouville Theory ◽  
Operator Product ◽  
De Sitter ◽  
Conformal Field ◽  
Point Correlation ◽  
Symmetry Transformations ◽  
Logarithmic Singularities ◽  
Zamolodchikov Equation

We study a class of solutions to the SL (2, ℝ)k Knizhnik–Zamolodchikov equation. First, logarithmic solutions which represent four-point correlation functions describing string scattering processes on three-dimensional anti-de Sitter space are discussed. These solutions satisfy the factorization ansatz and include logarithmic dependence on the SL (2, ℝ)-isospin variables. Different types of logarithmic singularities arising are classified and the interpretation of these is discussed. The logarithms found here fit into the usual pattern of the structure of four-point function of other examples of AdS/CFT correspondence. Composite states arising in the intermediate channels can be identified as the phenomena responsible for the appearance of such singularities in the four-point correlation functions. In addition, logarithmic solutions which are related to nonperturbative (finite k) effects are found. By means of the relation existing between four-point functions in Wess–Zumino–Novikov–Witten model formulated on SL (2, ℝ) and certain five-point functions in Liouville quantum conformal field theory, we show how the reflection symmetry of Liouville theory induces particular ℤ2 symmetry transformations on the WZNW correlators. This observation allows to find relations between different logarithmic solutions. This Liouville description also provides a natural explanation for the appearance of the logarithmic singularities in terms of the operator product expansion between degenerate and puncture fields.

Sign in / Sign up

Export Citation Format

Share Document