Conformal transformations and the SLE partition function martingale

Abstract

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half plane is the SLE hull, we show that the partition function is a famous local martingale known to probabilists, thereby unravelling its CFT origin.

41 pages, 4 figures

Cited by in corpus (26)

• SLE for theoretical physicists
• 2D growth processes: SLE and Loewner chains
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• On Staggered Indecomposable Virasoro Modules
• Dipolar SLEs
• On Multiple Schramm-Loewner Evolutions
• Global and Local Multiple SLEs for $\kappa \le 4$ and Connection Probabilities for Level Lines of GFF
• On the Percolation BCFT and the Crossing Probability of Watts
• Twenty-five years of two-dimensional rational conformal field theory
• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• Virasoro Module Structure of Local Martingales of SLE Variants
• On conformal field theory of SLE(kappa; rho)
• Towards a conformal field theory for Schramm-Loewner evolutions
• Conical Twist Fields and Null Polygonal Wilson Loops
• SLE local martingales in logarithmic representations
• Identification of the stress-energy tensor through conformal restriction in SLE and related processes
• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
• Calculus on manifolds of conformal maps and CFT
• Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution
• Critical interfaces of the Ashkin-Teller model at the parafermionic point
• Random loops and conformal field theory
• Schramm--Loewner-evolution-type growth processes corresponding to Wess--Zumino--Witten theories
• Hypotrochoids in conformal restriction systems and Virasoro descendants
• Algebraic properties of CFT coset construction and Schramm-Loewner evolution
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA