Fusion rules in N=1 superconformal minimal models

Communications in Mathematical Physics, 1990

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that the possible choices of S matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to the Aff level one algebra.

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.

Journal of Physics A: Mathematical and General, 2002

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field φ of dimension 2 . It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of φ is shown to coincide with a simpler one, generated by a bilocal scalar field V (x 1 , x 2 ) of dimension (1, 1) . The modes of V together with the unit operator span an infinite dimensional Lie algebra L V whose vacuum (i.e. zero energy lowest weight) representations only depend on the central charge c . Wightman positivity (i.e. unitarity of the representations of L V ) is proven to be equivalent to c ∈ N .

Physics Letters B, 1997

The conjecture that the states of the ferminonic quasiparticles in minimal conformal field theories are eigenstates of the integrals of motion to certain eigenvalues is checked and shown to be correct only for the Ising model.

1990

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that the possible choices of S matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to the Aff level one algebra.

1990

Abstract The problem of deriving linear differential equations for correlation functions of Rational Conformal Field Theories is considered. Techniques from the theory of fuchsian differential equations are used to show that knowledge of the central charge, dimensions of primary fields and fusion rules are enough to fix differential equations for one-and two-point functions on the tours. Any other correlation function can be calculated along similar lines. The results settle the issue of “exact solution” of rational conformal field theories.

The space of two dimensional quantum field theories: deformations of conformal models, 2022

In this thesis we study some deformations of conformal minimal models. In particular we study the imaginary magnetic deformation of the tricritical Ising model. The results and methodologies of this thesis are to be considered linked to what was known and discovered by the authors at the time of writing the thesis. There are errors (in the last chapter), then corrected in the paper: 2211.01123

International Journal of Modern Physics A, 2004

In this work, we exploit the operator content of the (D4, A6) conformal algebra. By constructing a Z2-invariants fusion rules of a chosen subalgebra and by resolving the bootstrap equations consistent with these rules, we determine the structure constants of the subalgebra.

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.

Nuclear Physics B, 1999

For each lattice one can define a free boson theory propagating on the corresponding torus. We give an alternative definition where one employs any automorphism of the group M * /M. This gives a wealth of conformal data, which we realize as some bosonic theory, in all the 'regular' cases. We discuss the generalization to affine theories. As a byproduct, we compute the gauss sum for any lattice and any diagonal automorphism. Free bosons systems in two dimensions have been a source of a simple environment in which both string theories and condensed matter systems could be studied. Ideas such as Narain compactification [1] and Orbifolds [2] have shown that although simple, these systems offer a vast richness. In ref. [3] one of this authors described the notion of a "pseudo" bosonic system. This is a conformal field theory in which one formally replaces the level in the modular matrix S with a fractional number. We denote this system as M 1/q where M is the lattice defining the boundary condition of the boson and q is some number strange to |M * /M |, where M * is the dual lattice of M. The primary fields are elements of M * /M and the modular matrix is, S λ,µ = |M * /M | − 1 2 exp(−2πiqλµ). (1) These bosonic systems are important in other constructions described in ref. [3] of new RCFT, as well as being mathematically intriguing. Actually, the definition of M 1/q can be generalized vastly. Consider the group G ≈ M * /M. Then for each automorphism of G, denoted by h, we can define a set of conformal data. The S matrix is given by, S λ,µ = |M * /M | − 1 2 exp(−2πih(λ) • µ). (2) This S matrix is symmetric, it has the same fusion rules for all h and it generalizes M 1/q when one takes h(λ) = qλ mod M. For cyclic groups, all automorphisms are of this form. Otherwise, M h is more general. Note, in passing that it is not important how we define the scalar product. Take g(λ, µ) to be a real valued, non degenerate, bi-linear, symmetric function on G × G. Then it is easy to show that g can be written as g(λ, µ) = h (λ) • µ where h is some automorphism of G, and vice versa. Thus, we can adopt any definition for the scalar product.