The integrable structure of nonrational conformal field theory

Abstract

Using the example of Liouville theory, we show how the separation into left-and right-moving degrees of freedom in a nonrational conformal field theory can be made explicit in terms of its integrable structure. The key observation is that there exist separate Baxter Q-operators for leftand right-moving degrees of freedom. Combining a study of the analytic properties of the Q-operators with Sklyanin's Separation of Variables Method leads to a complete characterization of the spectrum. Taking the continuum limit allows us in particular to rederive the Liouville reflection amplitude using only the integrable structure.

The DOZZ formula from the path integral

“…5 We set φ = c + ϕ where we separate the constant zero mode c and let ϕ be orthogonal to 2 As long as µ > 0; the case µ = 0 gives a different theory, the Gaussian Free Field theory. 3 Our proof is based on [13] where we established the BPZ equations for the degenerate field insertions and on [14], where we used them along with a probabilistic identification of the reflection principle to prove the DOZZ formula. 4 In the literature on LCFT two conventions for the vertex operators are used and in [5,13,14] we used the one where the interaction term is 4πµe γφ and vertex operators are denoted by e αφ(z) .…”

Baxter Operator and Baxter Equation for q-Toda and Toda2 Chains

“…An alternative proof of a similar identity using only the pentagonal identity can be found in [18]. 4 The Q operator and the Baxter equation…”

“…Thus, the two limiting cases seem to be very special in respect to their analytic structure. The pattern of poles can be argued by following the reasoning developed by Bytsko-Teschner [4]. Let z → Ψ q (z; p) be an Eigenfunction of Q(λ; p) associated with the Eigenvalue q(λ) and let ϕ be a test function.…”

Lax matrix solution of c = 1 conformal field theory

“…We have chosen the solutions Ψ (1) and Ψ (2) so that the associated W 1 functions (4.12) behave simply near y = z 1 and y = z 2 respectively. From our definition of Ψ (1) , we have…”

“…Integrable structures have been investigated in particular in the cases of minimal models [1] and Liouville theory [2]. In both cases, the approach was to associated an integrable model to a conformal field theory.…”