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Symplectic fermions in general domains

David Adame-Carrillo

Abstract

We review the basic features of a logarithmic conformal field theory that arise in the context of the scaling limit of lattice models. The theory of interest is the symplectic fermions, whose central charge is $-2$. We provide an explicit construction of its space of fields as a logarithmic Fock space, and discuss its logarithmic structure as a representation of the Virasoro algebra. The construction of the correlation functions is presented following the ideas of the bootstrap approach. The text aims to be accessible to readers with little or no expertise in conformal field theory.

Symplectic fermions in general domains

Abstract

We review the basic features of a logarithmic conformal field theory that arise in the context of the scaling limit of lattice models. The theory of interest is the symplectic fermions, whose central charge is . We provide an explicit construction of its space of fields as a logarithmic Fock space, and discuss its logarithmic structure as a representation of the Virasoro algebra. The construction of the correlation functions is presented following the ideas of the bootstrap approach. The text aims to be accessible to readers with little or no expertise in conformal field theory.
Paper Structure (18 sections, 2 theorems, 102 equations, 4 figures)

This paper contains 18 sections, 2 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statistical mechanics and conformal field theory.
  3. Logarithmic CFT.
  4. Dimers, spanning trees, dense polymers and sandpiles.
  5. Organization of the paper.
  6. The space of fields
  7. The chiral module
  8. Ground states and currents
  9. Logarithmic Virasoro action
  10. Interlude: highest-weight representations at $c=-2$
  11. Staggered (sub)module of $\mathcal{F}^{\textnormal{(chi)}}$
  12. The full non-chiral representation
  13. A glimpse of correlation functions
  14. Correlation functions
  15. Conformal covariance
  16. ...and 3 more sections

Key Result

Proposition 2.1

We have the isomorphism as Virasoro representations.

Figures (4)

  • Figure 2.1: The ground states, the currents, and their relation via the action of the current modes.
  • Figure 2.2: The ground states, the currents, and their relation via the Virasoro modes.
  • Figure 2.3: The ground states, the currents, and their relation through the action of the current modes.
  • Figure 2.4: The ground states, the currents, and their relation through the action of the Virasoro modes.

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.4: KR-staggered
  • Proposition 2.1
  • ...and 13 more