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Discrete symplectic fermions on double dimers and their Virasoro representation

David Adame-Carrillo

TL;DR

This work constructs a rigorous lattice realization of the logarithmic CFT of symplectic fermions with central charge $c=-2$ on the square lattice via the double-dimer model. It develops a discrete holomorphic fermion framework, connects correlation functions to Grassmann/Wick formalisms, and shows that the space of local fields on $oldsymbol{Z}^2$ carries a Virasoro representation obtained through a Sugawara construction on current modes. The analysis spans finite temperleyan domains and their thermodynamic limit, identifying the two-point function with the derivative of the full-plane Green’s function and establishing multipoint correlators by Wick’s theorem. The main result is a fully discrete Virasoro action at the lattice level, providing a concrete bridge between lattice dimer models and logarithmic CFT structures with $c=-2$.

Abstract

A discrete version of the Conformal Field Theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $-2$.

Discrete symplectic fermions on double dimers and their Virasoro representation

TL;DR

This work constructs a rigorous lattice realization of the logarithmic CFT of symplectic fermions with central charge on the square lattice via the double-dimer model. It develops a discrete holomorphic fermion framework, connects correlation functions to Grassmann/Wick formalisms, and shows that the space of local fields on carries a Virasoro representation obtained through a Sugawara construction on current modes. The analysis spans finite temperleyan domains and their thermodynamic limit, identifying the two-point function with the derivative of the full-plane Green’s function and establishing multipoint correlators by Wick’s theorem. The main result is a fully discrete Virasoro action at the lattice level, providing a concrete bridge between lattice dimer models and logarithmic CFT structures with .

Abstract

A discrete version of the Conformal Field Theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge .
Paper Structure (19 sections, 24 theorems, 85 equations, 13 figures)

This paper contains 19 sections, 24 theorems, 85 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Discrete symplectic fermions in finite domains
  3. Simple paths and fermions.
  4. Discrete holomorphicity of fermions.
  5. Grassmann formalism and Wick's theorem.
  6. Discrete symplectic fermions and double dimers.
  7. Symplectic fermions on $\mathbb{Z}^2$
  8. Temperleyan domains.
  9. Green's functions and $2$-point function.
  10. Thermodynamic limit.
  11. Proof of Theorem \ref{['the_convergence']}.
  12. Local fields and current modes
  13. Preliminaries: Discrete integration and discrete monomials.
  14. Local fields and null fields.
  15. Fermionic current modes.
  16. ...and 4 more sections

Key Result

Proposition 2.1

For $w_1,\ldots,w_n\in\mathcal{V}_{\circ}$ and $b_1,\ldots,b_n\in\mathcal{V}_\bullet$, where the second sum runs over the set of $n$ simple paths from $w_i$ to $b_{\sigma(i)}$ that do not intersect each other.

Figures (13)

  • Figure 2.1: A double-dimer cover $(\omega,\overline{\omega})$ on a subgraph of the square lattice and an odd simple path $\lambda: w \rightsquigarrow b$ adapted to $(\omega,\overline{\omega})$.
  • Figure 2.2: Involution $\iota$ in the proof of Proposition \ref{['disjoint']}.
  • Figure 2.3: The bijection $\phi$ in the proof of Theorem \ref{['discrete_holomorphicity']}.
  • Figure 2.4: The involution $\iota$ in the proof of Theorem \ref{['discrete_holomorphicity']}.
  • Figure 3.1: A temperleyan domain $\mathcal{V}=\mathcal{V}_{\bullet{\mathbf{1}}}\sqcup\mathcal{V}_{\bullet{\mathbf{0}}}\sqcup\mathcal{V}_\circ$ with a distinguished boundary point (red) and its associated dimerable graph $\mathcal{G}_{}$.
  • ...and 8 more figures

Theorems & Definitions (56)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Corollary 2.1.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 46 more