Gaussian free field and conformal field theory

Abstract

In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory. iii iv CONTENTS 7.5. Singular vectors Appendix 8. Existence of the Virasoro field Appendix 9. Operator algebra formalism 9.1. Construction of (local) operator algebras from holomorphic Fock space fields 9.2. Radial ordering 9.3. Commutation identity and normal ordering Lecture 10. Modifications of the Gaussian free field 10.1. Construction 10.2. Vertex fields 10.3. Level two degeneracy and BPZ equations 10.4. Boundary conditions and insertions Appendix 11. Current primary fields and KZ equations 11.1. Current primary fields 11.2. KZ equations Lecture 12. Multivalued conformal Fock space fields 12.1. Chiral bosonic fields 12.2. Chiral bi-vertex fields 12.3. Rooted vertex fields Appendix 13. CFT and SLE numerology Lecture 14.

Nesting statistics in the $O(n)$ loop model on random planar maps

“…3 ), the Ising model interface (κ = 3 or 16 3 ) [132,30], the GFF contour lines (κ = 4) [124], and the percolation interface (κ = 6) [131]. Critical phenomena in the plane were earlier well known to be related to conformal field theory [12], a discovery anticipated in the so called Coulomb gas approach to critical 2d statistical models (see, e.g., [113,36]), and now including SLE [10,65,84].…”

Conformal Field Theory of Dipolar Sle(4) With Mixed Boundary Condition

“…We denote by F the OPE family of the Gaussian free field Φ with mixed boundary condition, which is the algebra (over C) spanned by 1, ∂ j ∂k Φ and ∂ j ∂k e * αΦ (α ∈ C) under the OPE multiplication. See [5,Section 3.2] for the definition of OPE multiplication and Subsection 3.2 below for the definition of e * αΦ .…”

“…We also explain the differences between the conformal field theory we consider in this paper and the other theories in [5] and [6]. Unlike the chordal and the radial cases, the current field (the derivative of Gaussian free field) is real on the boundary arc with the Neumann boundary condition.…”

“…See Subsection 3.5 for its definition. Also see [2], [8], [5] for the chordal version and [3], [4], [6] for the radial version of this theorem with general κ > 0.…”

“…We apply definitions and theories developed in [5] and [6] to a different conformal setting with a different boundary condition. In the chordal case ( [5]) and the radial case ( [6]), we consider a simply connected domain D with a marked boundary point and a marked interior point, respectively. Both theories are based on the Gaussian free field with the Dirichlet boundary condition.…”