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Papers by Paolo Furlan

Nuovo Cimento a Nucl Part F, 1983

Eprint Arxiv Hep Th 0409213, Sep 21, 2004

The set of two partial differential equations for the Appell hypergeometric function in two varia... more The set of two partial differential equations for the Appell hypergeometric function in two variables F_4(a,b,c,a+b-c+2-h,x,y) is shown to arise as a null vector decoupling relation in a 2h-dimensional generalisation of the Coulomb gas model. It corresponds to a level two singular vector of an intrinsic Virasoro algebra.

Phys Lett B, 1993

The problem of describing the singular vectors of W3 and W3(2) Verma modules is addressed, viewin... more The problem of describing the singular vectors of W3 and W3(2) Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A 2(1). Singular vectors of an A 2(1) Verma module are mapped into W algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.

Phys Lett B, Aug 8, 1993

The problem of describing the singular vectors of cW3\cW_3cW3 and cW3(2)\cW_3^{(2)}cW3(2) Verma modules is addr... more The problem of describing the singular vectors of cW3\cW_3cW3 and cW3(2)\cW_3^{(2)}cW3(2) Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A(1)_2,A^{(1)}_2\,A(1)2,. Singular vectors of an A(1)2,A^{(1)}_2\,A(1)_2, Verma module are mapped into W\WW algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.

Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical c... more Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical concepts leading to the relevant analogues of the (D=4) Doplicher-Haag-Roberts theory of superselection sectors and internal symmetry. We also review a recently uncovered duality between braid and quantum group representations in an extension of the chiral su(2)_k WZNW model for nonnegative integer level k.

Nuovo Cimento a Nucl Part F, 1971

Fortschritte der Physik/Progress of Physics, 1992

ABSTRACT A constructive approach is developed for studying local chiral algebras generated by a p... more ABSTRACT A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z1, g) Ψ(z2, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Zk-parafermions. The case Δ2 = 4 (Δ1 − 1), where Δv = Δk−v(Δ0 = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δv = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z2 [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ1. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.

Lecture Notes in Physics, 1991

Page 1. Tensor Products of qP -- 1 Quantum Groups and WZW Fusion Rules 1 Paolo Furlan 2,Alexander... more Page 1. Tensor Products of qP -- 1 Quantum Groups and WZW Fusion Rules 1 Paolo Furlan 2,Alexander Ganchev a Valentina Petkova a ... Math. 34 (19779) 97. 8. V. Dobrev, preprint IC/85/9. 9. M. Rosso, Comm. Math. Phys. ... 15. J. Frohlich, contribution to this volume. 16. ...

Summary The algebraic treatment of the eigenvalue equations for quantum systems, based on the in... more Summary The algebraic treatment of the eigenvalue equations for quantum systems, based on the introduction of the spectrum-generating algebraSO 2.1, in the sense of a previous work, is generalized by allowing energy-dependent realizations of the algebra. A basic differential equation (3.10) is derived, which expresses in a concise way the conditions that the Hamiltonian of the system must satisfy in order

La Rivista Del Nuovo Cimento Series 3, 1985

... P. FURLAN International School ]or A dcanced Studies (ISAS) - Trieste, italy Istituto di ]~is... more ... P. FURLAN International School ]or A dcanced Studies (ISAS) - Trieste, italy Istituto di ]~isica Teorica dell' U,tdversith - Trieste, Jtalia Istituto Xazionale di Fisica Nucleare - Sezione di Trieste ... o ~ (*) We use three kVt)es of 2-I)oint functions in tim present paper: Minkowski space ...

La Rivista del Nuovo Cimento, 1989

Nuclear Physics B, 1997

We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing... more We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing correlators based on the admissible representations of A (1) 1 . Exploiting a symmetry of the KZ equations we show that the original infinite sums representing the 4-point chiral correlators can be effectively summed up. Using these simplified expressions with proper choices of the contours we determine the duality (braid and fusion) transformations and show that they are consistent with the fusion rules of Awata and Yamada. The requirement of locality leads to a 1-parameter family of monodromy (braid) invariants. These analogs of the "diagonal" 2-dimensional local 4-point functions in the minimal Virasoro theory contain in general nondiagonal terms. They correspond to pairs of fields of identical monodromy, having one and the same counterpart in the limit to the Virasoro minimal correlators. 0 * Permanent address.

Nuclear Physics B, 1990

We perform explicitly a truncation of the tensor product of two regular representations of Uq(g) ... more We perform explicitly a truncation of the tensor product of two regular representations of Uq(g) for q a root of unity and show that it coincides with the fusion rules for integrable representations in a WZW theory based on the affine Lie algebra g(1). We obtain a new formula for the multiplicities which may be useful in practical calculations and

Nuclear Physics B, 1993

In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral n-p... more In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral n-point functions, as well as the equations governing them, of the A (1) 1 WZNW conformal theory and the corresponding Virasoro minimal models. The WZNW correlators are described as solutions of the Knizhnik -Zamolodchikov equations with rational levels and isospins. The technical tool exploited are certain relations in twisted cohomology. The results extend to arbitrary level k + 2 = 0 and isospin values of the type J = j − j ′ (k + 2), 2j, 2j ′ ∈ Z Z + .

Nuclear Physics B, 1994

The BRST quantisation of the Drinfeld -Sokolov reduction applied to the case of A (1) 2 is explor... more The BRST quantisation of the Drinfeld -Sokolov reduction applied to the case of A (1) 2 is explored to construct in an unified and systematic way the general singular vectors in W 3 and W (2) 3 Verma modules. The construction relies on the use of proper quantum analogues of the classical DS gauge fixing transformations. Furthermore the stability groups W (η) of the highest weights of the W -Verma modules play an important role in the proof of the BRST equivalence of the Malikov-Feigin-Fuks singular vectors and the W algebra ones. The resulting singular vectors are essentially classified by the affine Weyl group W modulo W (η) .

Letters in Mathematical Physics, 1991

We write down a complete set of n-point U (/& Ji(I)) invariants (using a polynomial basis for the... more We write down a complete set of n-point U (/& Ji(I)) invariants (using a polynomial basis for the irreducible finite dimensional U-modules) that are regular for all non-zero values of the deformation parameter q.

Letters in Mathematical Physics, 2007

Journal of Physics A: Mathematical and General, 2003

We define the chiral zero modes' phase space of the G = SU (n) Wess-Zumino-Novikov-Witten (WZNW) ... more We define the chiral zero modes' phase space of the G = SU (n) Wess-Zumino-Novikov-Witten (WZNW) model as an (n − 1)(n + 2)-dimensional manifold M q equipped with a symplectic form q involving a Wess-Zumino term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by dρ(M) = 1 3 tr(M −1 dM) 3 . ( * )

Nuovo Cimento a Nucl Part F, 1983

Eprint Arxiv Hep Th 0409213, Sep 21, 2004

The set of two partial differential equations for the Appell hypergeometric function in two varia... more The set of two partial differential equations for the Appell hypergeometric function in two variables F_4(a,b,c,a+b-c+2-h,x,y) is shown to arise as a null vector decoupling relation in a 2h-dimensional generalisation of the Coulomb gas model. It corresponds to a level two singular vector of an intrinsic Virasoro algebra.

Phys Lett B, 1993

The problem of describing the singular vectors of W3 and W3(2) Verma modules is addressed, viewin... more The problem of describing the singular vectors of W3 and W3(2) Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A 2(1). Singular vectors of an A 2(1) Verma module are mapped into W algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.

Phys Lett B, Aug 8, 1993

The problem of describing the singular vectors of cW3\cW_3cW3 and cW3(2)\cW_3^{(2)}cW3(2) Verma modules is addr... more The problem of describing the singular vectors of cW3\cW_3cW3 and cW3(2)\cW_3^{(2)}cW3(2) Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of A(1)_2,A^{(1)}_2\,A(1)2,. Singular vectors of an A(1)2,A^{(1)}_2\,A(1)_2, Verma module are mapped into W\WW algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.

Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical c... more Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical concepts leading to the relevant analogues of the (D=4) Doplicher-Haag-Roberts theory of superselection sectors and internal symmetry. We also review a recently uncovered duality between braid and quantum group representations in an extension of the chiral su(2)_k WZNW model for nonnegative integer level k.

Nuovo Cimento a Nucl Part F, 1971

Fortschritte der Physik/Progress of Physics, 1992

ABSTRACT A constructive approach is developed for studying local chiral algebras generated by a p... more ABSTRACT A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z1, g) Ψ(z2, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Zk-parafermions. The case Δ2 = 4 (Δ1 − 1), where Δv = Δk−v(Δ0 = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δv = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z2 [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ1. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.

Lecture Notes in Physics, 1991

Page 1. Tensor Products of qP -- 1 Quantum Groups and WZW Fusion Rules 1 Paolo Furlan 2,Alexander... more Page 1. Tensor Products of qP -- 1 Quantum Groups and WZW Fusion Rules 1 Paolo Furlan 2,Alexander Ganchev a Valentina Petkova a ... Math. 34 (19779) 97. 8. V. Dobrev, preprint IC/85/9. 9. M. Rosso, Comm. Math. Phys. ... 15. J. Frohlich, contribution to this volume. 16. ...

Summary The algebraic treatment of the eigenvalue equations for quantum systems, based on the in... more Summary The algebraic treatment of the eigenvalue equations for quantum systems, based on the introduction of the spectrum-generating algebraSO 2.1, in the sense of a previous work, is generalized by allowing energy-dependent realizations of the algebra. A basic differential equation (3.10) is derived, which expresses in a concise way the conditions that the Hamiltonian of the system must satisfy in order

La Rivista Del Nuovo Cimento Series 3, 1985

... P. FURLAN International School ]or A dcanced Studies (ISAS) - Trieste, italy Istituto di ]~is... more ... P. FURLAN International School ]or A dcanced Studies (ISAS) - Trieste, italy Istituto di ]~isica Teorica dell' U,tdversith - Trieste, Jtalia Istituto Xazionale di Fisica Nucleare - Sezione di Trieste ... o ~ (*) We use three kVt)es of 2-I)oint functions in tim present paper: Minkowski space ...

La Rivista del Nuovo Cimento, 1989

Nuclear Physics B, 1997

We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing... more We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing correlators based on the admissible representations of A (1) 1 . Exploiting a symmetry of the KZ equations we show that the original infinite sums representing the 4-point chiral correlators can be effectively summed up. Using these simplified expressions with proper choices of the contours we determine the duality (braid and fusion) transformations and show that they are consistent with the fusion rules of Awata and Yamada. The requirement of locality leads to a 1-parameter family of monodromy (braid) invariants. These analogs of the "diagonal" 2-dimensional local 4-point functions in the minimal Virasoro theory contain in general nondiagonal terms. They correspond to pairs of fields of identical monodromy, having one and the same counterpart in the limit to the Virasoro minimal correlators. 0 * Permanent address.

Nuclear Physics B, 1990

We perform explicitly a truncation of the tensor product of two regular representations of Uq(g) ... more We perform explicitly a truncation of the tensor product of two regular representations of Uq(g) for q a root of unity and show that it coincides with the fusion rules for integrable representations in a WZW theory based on the affine Lie algebra g(1). We obtain a new formula for the multiplicities which may be useful in practical calculations and

Nuclear Physics B, 1993

In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral n-p... more In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral n-point functions, as well as the equations governing them, of the A (1) 1 WZNW conformal theory and the corresponding Virasoro minimal models. The WZNW correlators are described as solutions of the Knizhnik -Zamolodchikov equations with rational levels and isospins. The technical tool exploited are certain relations in twisted cohomology. The results extend to arbitrary level k + 2 = 0 and isospin values of the type J = j − j ′ (k + 2), 2j, 2j ′ ∈ Z Z + .

Nuclear Physics B, 1994

The BRST quantisation of the Drinfeld -Sokolov reduction applied to the case of A (1) 2 is explor... more The BRST quantisation of the Drinfeld -Sokolov reduction applied to the case of A (1) 2 is explored to construct in an unified and systematic way the general singular vectors in W 3 and W (2) 3 Verma modules. The construction relies on the use of proper quantum analogues of the classical DS gauge fixing transformations. Furthermore the stability groups W (η) of the highest weights of the W -Verma modules play an important role in the proof of the BRST equivalence of the Malikov-Feigin-Fuks singular vectors and the W algebra ones. The resulting singular vectors are essentially classified by the affine Weyl group W modulo W (η) .

Letters in Mathematical Physics, 1991

We write down a complete set of n-point U (/& Ji(I)) invariants (using a polynomial basis for the... more We write down a complete set of n-point U (/& Ji(I)) invariants (using a polynomial basis for the irreducible finite dimensional U-modules) that are regular for all non-zero values of the deformation parameter q.

Letters in Mathematical Physics, 2007

Journal of Physics A: Mathematical and General, 2003

We define the chiral zero modes' phase space of the G = SU (n) Wess-Zumino-Novikov-Witten (WZNW) ... more We define the chiral zero modes' phase space of the G = SU (n) Wess-Zumino-Novikov-Witten (WZNW) model as an (n − 1)(n + 2)-dimensional manifold M q equipped with a symplectic form q involving a Wess-Zumino term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by dρ(M) = 1 3 tr(M −1 dM) 3 . ( * )