Exact c=1 boundary conformal field theories.

We present a solution of the problem of a free massless scalar field on the half line interacting through a periodic potential on the boundary. For a critical value of the period, this system is a conformal field theory with a non-trivial and explicitly calculable S-matrix for scattering from the boundary. Unlike all other exactly solvable conformal field theories, it is non-rational (i.e. has infinitely many primary fields). It describes the critical behavior of a number of condensed matter systems, including dissipative quantum mechanics and of barriers in “quantum wires”. 11/93 ♦ [email protected] ♣ On leave from Princeton University. ♠ [email protected] 1. Boundary Conformal Field Theory Conformal field theory is usually defined on a two-dimensional manifold without boundaries (the simplest case being the plane). It can also be defined on manifolds with boundaries (like the disk or strip), provided that appropriate boundary conditions are imposed [1]. The Dirichlet and Neumann boundary conditions on scalar worldsheet fields are familiar, if trivial, examples. Non-trivial conformal boundary conditions arise from the interaction of boundary degrees of freedom with worldsheet fields. A wide range of systems, including open string theory [2,3,4], monopole catalysis [5], the Kondo problem [6], dissipative quantum mechanics [7,8] and junctions in quantum wires [9] can be described this way. The technology for dealing with boundary conformal field theory is easily stated [10]: Consider a bulk conformal field theory C confined to a strip of width L with boundary conditions A and B on the two ends. This theory has a partition function Z open = tr(e −TL 0 ) where T is the time interval and L 0 is the open string Hamiltonian. If the boundary conditions are conformal, the partition function will be a sum Z open = ∑ nhχh(e −2πT/l) over Virasoro characters of the open string primary fields (the h are the highest weights and the nh are the integer multiplicities of the characters). The partition function can also be computed as the amplitude for closed string propagation between states |A〉 and |B〉 of the bulk closed string created by the boundary conditions A and B: Z closed = 〈A|e−l(L0+L̃0)|B〉, where L0 and L̃0 are the leftand right-moving closed string Hamiltonians. For the theory as a whole to be conformal, the boundary states must satisfy a reparametrization invariance condition (Ln − L̃−n)|A〉 = 0 [2] which implies that each primary field contributes to |A〉 a piece C h ∑ n |n〉|ñ〉, where the sum is over all the states of the Virasoro module and C h is a coefficient to be determined [11]. This gives a different expansion of the partition function in terms of Virasoro characters: Z closed = ∑ h C A h C B h χh(e −2πl/T ). The dynamical problem is to find the specific primary fields φh appearing in both the open and closed string expansions along with their multiplicities and weights. The two expansions must of course be identical under the “modular transformation” e−2πT/l → e−2πl/T between open and closed string variables. This consistency condition is often enough to explicitly determine the (finitely many) boundary states of a rational conformal field theory such as the WZW theory (which underlies the Kondo model [12]). Non-rational conformal field theories are much harder to deal with since they have an infinite number of primary fields and, presumably, boundary states. In this Letter we will