[Paper Review] SLE martingales in coset conformal field theory

Anton Nazarov|arXiv (Cornell University)|May 28, 2012

Advanced Algebra and Geometry2 references3 citations

TL;DR

This paper establishes a connection between Schramm-Loewner evolution (SLE) and coset conformal field theory (CFT) by identifying correlation functions in coset CFT that are martingales under SLE with additional Brownian motion on the coset space G/A. The authors derive explicit relations between SLE parameters (κ, τ) and CFT data (weights, Casimir eigenvalues), confirming consistency with parafermionic and minimal models, and propose a framework for studying off-critical SLE via massive perturbations of coset CFTs.

ABSTRACT

Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT we consider CFT correlation functions which are martingales with respect to SLE. We establish connection between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset conformal field theory. Then we check the consistency of our approach with the behaviour of parafermionic and minimal models. Coset models are connected with off-critical massive field theories and we discuss implications for SLE.

Motivation & Objective

• To bridge SLE and coset CFT by identifying observables that are martingales under SLE with Brownian motion on G/A.
• To derive explicit relations between SLE parameters (κ, τ) and algebraic data of coset CFT, such as weights and Casimir eigenvalues.
• To verify consistency of the framework with known results in parafermionic and minimal models.
• To explore implications for off-critical SLE by connecting massive perturbations of coset CFT to drift terms in SLE.

Proposed method

• Formulate SLE on the coset space G/A using a stochastic differential equation driven by Brownian motion with a drift parameter τ.
• Define boundary condition-changing operators in coset CFT labeled by weights (μ, ν) of G and A, respectively.
• Use conformal mapping to express correlation functions on the slit domain H_t in terms of full-plane correlation functions.
• Apply Ito calculus to derive necessary martingale conditions, leading to two equations (12) and (13) relating κ, τ to CFT data.
• Use the quadratic Casimir eigenvalues C_μ and C_ν for the G and A Lie algebras to express the martingale conditions algebraically.
• Verify consistency with known models: Z_N parafermions (N=2,3) and minimal unitary models via the coset construction.

Experimental results

Research questions

• RQ1 Which correlation functions in coset CFT are martingales with respect to SLE on the coset space G/A?
• RQ2 How are the SLE parameters κ and τ related to the conformal weights and Casimir eigenvalues of the coset CFT?
• RQ3 Does the proposed framework reproduce known SLE parameters for parafermionic and minimal models?
• RQ4 Can massive perturbations of coset CFT, realized as affine Toda field theories, be linked to off-critical SLE with drift terms?
• RQ5 Do the τ terms in the SLE dynamics correspond to the effects of perturbing primary fields in the correlation functions?

Key findings

• For the Z_2 parafermion model (Ising CFT), the martingale condition yields κ=3 and τ=0 for the h=1/2 field, and κ=16/3, τ=0 for the h=1/16 field, matching known SLE values.
• For the Z_3 parafermion model, the method yields three distinct (κ, τ) pairs: (208/25, 242/225), (10/3, 0), and (80/19, 14/171), consistent with field content and symmetry.
• The field with h=2/5 in the Z_3 model corresponds to a Z_3 singlet, leading to τ=0, confirming the absence of additional Brownian motion.
• The system of equations (12) and (13) is consistently satisfied for minimal unitary models realized as cosets, with τ=0 and standard κ(c) relation.
• The framework suggests a correspondence between massive perturbations of G/A coset CFTs (via affine Toda theories) and off-critical SLE with drift, opening a path to study domain walls away from criticality.
• The derived relations between κ, τ and CFT data provide a systematic method to compute SLE parameters directly from coset representation theory.