Minimal Models Of Conformal Field Theory Research Articles

Spectral curve duality beyond the two-matrix model

We describe a simple algebraic approach to several spectral duality results for integrable systems and illustrate the method for two types of examples: the Bertola–Eynard–Harnad spectral duality of the two-matrix model and the various dual descriptions of minimal model conformal field theories coupled to gravity.

Inverse bootstrapping conformal field theories

We propose a novel approach to study conformal field theories (CFTs) in general dimensions. In the conformal bootstrap program, one usually searches for consistent CFT data that satisfy crossing symmetry. In the new method, we reverse the logic and interpret manifestly crossing-symmetric functions as generating functions of conformal data. Physical CFTs can be obtained by scanning the space of crossing-symmetric functions. By truncating the fusion rules, we are able to concentrate on the low-lying operators and derive some approximate relations for their conformal data. It turns out that the free scalar theory, the 2d minimal model CFTs, the ϕ4 Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule ϕ1 × ϕ1 = I + ϕ2 + T , where ϕ1, ϕ2 are scalar operators, I is the identity operator and T is the stress tensor.

Entanglement in Nonunitary Quantum Critical Spin Chains.

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well-understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.

A Solution Space for a System of Null-State Partial Differential Equations: Part 4

This article is the last of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Loewner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the first two articles, we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building on these results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $\mathcal{S}_N$. First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $8/\kappa$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of $\mathcal{S}_N$, which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE$_\kappa$ process join in a particular connectivity. Finally, we propose a reason for why the "exceptional speeds" (certain $\kappa$ values that appeared in the analysis of the Coulomb gas solutions in the third article) and the minimal models of CFT are connected.

The black hole interior and a curious sum rule

We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics.

Entanglement entropy of two disjoint intervals from fusion algebra of twist fields

We study the entanglement and Rényi entropies of two disjoint intervals in minimal modelsof conformal field theory. We use the conformal block expansion and fusion rules of twistfields to define a systematic expansion in the elliptic parameter of the trace of thenth power of the reduced density matrix. Keeping only the first few terms we obtain anapproximate expression that is easily analytically continued to , leading to an approximate formula for the entanglement entropy. These predictions arechecked against some known exact results as well as against existing numericaldata.

A non-rational CFT with c = 1 as a limit of minimal models

We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT.

A plasma analogy and Berry matrices for non-abelian quantum Hall states

We present an approach to the computation of the non-abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases, which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlation functions. The non-abelian statistics of multi-quasihole states follows from an additional assumption.