Conformally covariant boundary correlation functions with a quantum group

Abstract

Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm-Loewner evolutions (SLE). In this article, we introduce a correspondence and establish its fundamental properties, which are used in the companion articles [JJK16, KP16] for explicitly solving two such problems. The correspondence associates Coulomb gas type integrals to vectors in a tensor product representation of a quantum group, a q-deformation of the Lie algebra sl 2 . We show that desired properties of the functions are guaranteed by natural representation theoretical properties of the vectors. denote the vector obtained by identifying v as a vector in an (n − 1)-fold tensor product representation. Then, as |x j+1 − x j | → 0, the function F[v] has the asymptotic behaviorwhere the constant B = B dj ,dj+1 d and the exponent ∆ = ∆ dj ,dj+1 d are explicit. The analogous statement holds also for F (x0) [v].In practice, this theorem is applied as follows. In specific problems, we are looking for particular solutions to systems of PDEs of conformal field theory. Typically, the sought solution has specific Möbius covariance properties and specific boundary conditions as the distance of some of its arguments tend to zero. By our correspondence, the task of finding a function with these properties is translated to the problem of finding a corresponding vector in the tensor product representation. The different parts (PDE), (COV), and (ASY) of the theorem state that a careful choice of the vector would ensure the desired properties of the function -even the most delicate boundary conditions for the function can be guaranteed by (ASY) if the vector has appropriate projections to certain subrepresentations. All of the requirements are explicit linear conditions on the vector living in a finite dimensional vector space, and we moreover have a variety of representation theoretical tools at our disposal to solve for such a vector. By outlining a few case studies in Sections 1.4 and 1.5, we exemplify how the correspondence thus allows us to translate the original, possibly rather complicated problem to an explicitly solvable one, and to eventually express the function of interest explicitly as a linear combination of integral form functions.Before example applications, we make a few further observations about the interpretation of the constructed correspondence, and comparisons to related research.

Toward a conformal field theory for Schramm-Loewner evolutions

“…However, PDE system (3.9) does have other solutions too -for instance, solutions satisfying other Möbius covariance properties (where infinity is a special point). This kind of solutions are also discussed in [KP19,Pel19], and all of these solutions satisfy the conclusions in Theorem 4.1. The real problem in Question 4.7 is therefore whether there exist solutions to PDEs (3.9) other than the Coulomb gas integral functions studied in [FK15a, FK15b, FK15c, FK15d, KP19, Pel19].…”

“…The rough idea is to write the function Z ω as a sum of terms, each of which has a prescribed asymptotics as claimed in the assertion (4.16). This is established using the quantum group symmetry developed in [KP19,Pel19]. The terms with m ≥ j, j+1 (ω) are already immediate from (4.15).…”

“…The functions are constructed in integral form (as so-called Coulomb gas integrals). The key tool for this construction is a quantum group symmetry on the solution space (3.11) of the second order PDE system (3.9) [KP19]. This symmetry is very useful also for analyzing the solutions -indeed, we used it, e.g., to establish Proposition 4.9 in Section 4 D.…”

“…The key in the construction of Z α is a judicious choice of the integration contours Γ(α), certain closed N-surfaces designed in such a way that the functions Z α do satisfy the asymptotics properties in (3.14). We refer the interested reader to [KP16,KP19,Pel19].Proposition D.1. Let κ ∈ (0, 8) \ Q.The functions appearing in Theorem 3.6 can be written in the formZ α (x 1 , .…”

Asymptotic Analysis of Dotsenko–Fateev Integrals

“…[7]) shows that (1.1) can be viewed as an analytic continuation of Appell's function. We also mention [5,9,10] which discuss related questions in the context of the Schramm-Loewner Evolution (SLE).…”