Integrability and quantum symmetries

Nuclear Physics B, 1991

Communications in Mathematical Physics, 1990

Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid-and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.

1992

Braid invariant quantum group extended su(2 )k current algebra models and minimal conformal models are constructed using a pair of regular bases of n-point invariants in the space of conformal blocks and in the quantum group space.

Nuclear Physics B, 1991

The quantum group symmetry of the c < 1 Rational Conformal Field Theory. in its Coulomb gas version, is formulated in terms of a new type of screened vertex operators, which define the representation spaces of a quantum group Q. The conformal properties of these operators show a deep interplay between the quantum group Q and the Virasoro algebra.

Nuclear Physics B - Proceedings Supplements, 1997

Two approaches to the construction of symmetry generators for the quantum group U q (l(2)) in conformal field theory are presented, in the concrete context of 2d gravity. The first works with an extension of the physical phase space and has been successfully applied already to WZW theory. We show that the result can be used also for Liouville theory and related models by employing Hamiltonian reduction. The second is based on a completely new idea and realizes the quantum group symmetry intrinsically, on the physical phase space alone.

Communications in Mathematical Physics, 2004

We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.

Communications in Mathematical Physics, 1990

Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant) r-matrices and this geometrical approach.

Annals of Physics, 1969

Communications in Mathematical Physics, 1993

Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space M , and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworldM , i.e. the universal covering of the Dirac-Weyl compactification of M . As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincaré representation is unique in this case. * Supported in part by Ministero della Ricerca Scientifica and CNR-GNAFA. • Supported in part by INFN, sez. Napoli.

Communications in Mathematical Physics, 1989

Using the Feigin-Fuchs representation of minimal conformal models in a form introduced recently by one of us, the braid group representation matrices, describing the analytic continuation properties of conformal blocks, are computed. In a suitable normalization, their matrix elements are shown to essentially factorize into pairs of Boltzmann weights of critical RSOS models in a certain limit of the spectral parameter. These Boltzmann weights are related to quantum group K-matrices by the vertex-SOS transformation. We show that the crossing symmetry of the four-point function in left-right symmetric models follows from a quantum group relation, also called crossing symmetry. This observation gives a simple way to evaluate the structure constants.