Conformal field theory and hyperbolic geometry.

We examine the correspondence between the conformal field theory of boundary operators and two-dimensional hyperbolic geometry. by consideration of domain boundaries in two-dimensional critical systems, and the invariance of the hyperbolic length, we motivate a reformulation of the basic equation of conformal covariance. The scale factors gain a new, physical interpretation. We exhibit a fully factored form for the three-point function. A doubly-infinite discrete series of central charges with limit c = −2 is discovered. A correspondence between the anomalous dimension and the angle of certain hyperbolic figures emerges. * Present and permanent address. e-mail: [email protected] In this letter, we establish several connections between the conformal field theory of boundary operators and two-dimensional hyperbolic geometry. First, by consideration of domain boundaries in two-dimensional critical systems, and the invriance of the hyperbolic length, we motivate a reformulation of the basic equation of conformal covariance. The scale factors gain a new, physical interpretation. They operate to keep the distance from the end of the domain boundary to the boundary of the system fixed. We also point out that for any geometry conformally equivalent to the half-plane, domain boundaries in twodimensional critical systems follow hyperbolic geodesics. Their energy per unit hyperbolic length is finite. Motivated by these results, we next exhibit a completely factored form for the three-point function (and the prefactor of the four-point function). Here, a connection between the anomalous dimension of a primary operator and the angle of a hyperbolic figure appears. Finally, we impose the condition that the Schwarz function defined by a four-point function of opertors degenerate at level two correspond to a hyperbolic tiling, or tessellation. this leads to a new, doubly-infinite discrete set of minimal models. The angle-dimension correspondence is again encountered. To begin, we establish a connection between conformal field theory and hyperbolic geometry in the language of the theory of phase transitions. However, it should be emphasized that our results are generally valid,a nd not dependent on this particular realization of the theory. As demonstrated elsehwere [1], a domain boundary in the upper half-plane is created by boundary operators φ(x) [1–5] located at its endpoints on the real axis. These operators act to change the boundary condition along the edge of the system [3], the real axis. Boundary operators may also be defined by letting bulk operator in the system with a boundary approach the boundary, and making use of the bulk-boundary oeprtor product expansion [1,4]. Although a domain boundary at a critical point exhibits large fluctuations, and has an energy that is not proportional to its length, it is a well-defined object. Conformal invariance implies universality, which allows one to study it in general. The (extra free) energy of such a boundary is F = −ln 〈φ(x1)φ(x2)〉 , (1) as described in [1]. For completeness, we note tht Equation (1) ignores both universal [3,6] and non-universal constants independent of x1, x2. The former are associated with the boundary states on the real axis, while the latter arise in computing the free energy of the boundary of any real system of statistical mechanical model. Evaluating the correlation funciton, we find [1] F = 2∆ln|x1 − x2|, (2) where ∆ is the critical dimension of φ. Now, Equation (1) also gives the domain boundary free energy in any geometry conformally equivlaent to the half-plane, if we evaluate the correlation function in the new geometry. This is done by making use of the basic equation of conformal covariance of correlation functions [7], as applied to boundary operators, 〈φ1(x1)φ2(x2) . . .〉 = |w(x1)| ∆1 |w(x2)| ∆2 〈φ1(w1)φ2(w2) . . .〉 . (3)