Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models

Abstract

We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results. arXiv:1806.02790v3 [hep-th] 15 Aug 2018 1.0.3 The singularitiesThe 4-point conformal block on the sphere is a function of six parameters: the Virasoro central charge, the conformal dimension of the Virasoro representation that flows in the internal channel, and the conformal dimensions of the four external fields. The solution of the elliptic recursion relation displays a rich structure of poles. These poles are physical in the sense that they correspond to the propagation of states for suitable choices of the central charge and conformal dimensions. In a numerical 2D bootstrap based on Virasoro conformal blocks that are computed using the elliptic recursion relation, one must deal with these poles when exploring the space of possible crossing-symmetric CFT solutions. This happens, for example, in studies of percolation in the 2D Ising model [9]. When the central charge is such that one deals with minimal-model conformal blocks, additional poles appear. These additional poles are non-physical and appear due to resonances of conformal dimensions (see equation (11)) at rational values of the central charge. This complication requires a careful study of the pole structure of the elliptic recursion relation in the case of Virasoro minimal models, which is the aim of the present work. The present work.We study the cancellation of the non-physical poles in computations of minimal-model conformal blocks using Zamolodchikov's elliptic recursion relation for the 4-point conformal block on the sphere. But the 4-point conformal block on the sphere is not the only or the simplest conformal block that can be computed using a recursion relation. In 2009, Poghossian [10], and independently Fateev and Litvinov [3] proposed recursion relations to compute Liouville 1-point conformal blocks on the torus. These recursion relations are equivalent [6], and we use the Fateev-Litvinov version to study minimalmodel 1-point functions on the torus, and their 0-point limits (when the vertex operator insertion is the identity) which are Virasoro minimal-model characters, as the simplest examples of solutions of a Zamolodchikov-type elliptic recursion relation. Outline of contentsIn section 2, we recall basic facts related to the Virasoro algebra, its representations, and conformal blocks. In section 3, we consider the 4-point conformal blocks on the sphere as solutions of the recursion relation, study their singularities and their behaviour in the context of the Virasoro generalized minimal models and minimal models. In section 4, we consider the 1-point conformal block on the torus as solutions of the Fateev-Litvi...

On 2d CFTs that interpolate between minimal models

“…It would be interesting to find a recursive representation of blocks for rational central charges that would be manifestly finite, and that would explicitly exhibit these higher-order poles. This may involve the combinatorial structures that appear when recovering minimal model characters from the recursive representation of torus blocks [9]. It may also be useful to notice that the double pole term of H q(p+1) ∆ (3.28) has the same expression (up to a factor 2) as the double pole term of H pq+qs−pr ∆ (3.18), if we set (r, s) = (0, 1).…”

Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance

“…(B.3) vanishes as well. The use of the recursive formula for calculating Virasoro conformal blocks for rational values of the central charge c < 1 has been discussed also in [71,72]. Since F c ρ (η) = F(η, c, ρ, {h, h, h, h}), we used Eqs.…”

Crossing-symmetric twist field correlators and entanglement negativity in minimal CFTs

“…Even if there are closed expressions for the null vectors, the construction of such a basis becomes cumbersome at higher levels, due to the resonances mentioned above. We refer to [52] where this particular issue is discussed in more detail. Here we will follow an alternative path, inspired by the AGT approach to the minimal models [53][54][55], see also [56].…”