The fermionic DGFF and its scaling limit logCFT

TL;DR

The paper proves that the fermionic discrete Gaussian free field (fDGFF) on a lattice has a scaling limit described by the logarithmic conformal field theory of symplectic fermions with central charge $c=-2$, by establishing a one-to-one correspondence between fDGFF local observables and local fields of the SF CFT. It gives a rigorous construction of the local field space $ ext{𝔉}$ and shows that it is isomorphic to the symplectic-fermions Fock space, with current modes providing a twofold Virasoro action via a discrete Sugawara construction. The main results (Theorems 1 and 2) show that renormalised discrete correlation functions converge to the corresponding SF CFT correlators, up to a scale ambiguity reflecting logCFT self-isomorphisms; this yields explicit scaling limits for observables in the uniform spanning tree and Abelian sandpile models. The work thus provides a rigorous bridge from lattice models to a logCFT description, enabling precise computation of scaling limits for UST and ASM observables and enriching the understanding of logarithmic structures in two-dimensional critical systems.

Abstract

In this paper, we identify the scaling limit of the fermionic discrete Gaussian free field (fDGFF) as a logarithmic conformal field theory (CFT) in two dimensions. We first establish a one-to-one correspondence between the space of local observables of the fDGFF and the space of local fields of the symplectic fermions CFT, a logarithmic CFT with central charge $c = - 2$. This correspondence is meaningful in the sense that, when appropriately renormalised, the fDGFF correlation functions converge to corresponding CFT correlation functions in the scaling limit. As an application to these results, we interpret (the scaling limit of) certain local observables in the uniform spanning tree and the Abelian sandpile model as local fields of the symplectic fermions.

Figures (7)

• Figure 2.1: Lattices involved in our tools of discrete complex analysis.
• Figure 2.2: A corner contour and, zoomed in, an integration step $(c_{j-1},c_j)$ with its closest medial vertex $\mathbf{u}_j^\textnormal{m} \in {\mathbb Z}^2_\textnormal{m}$ and diamond vertex $\mathbf{u}_j^\diamond \in {\mathbb Z}^2_\diamond$.
• Figure 4.1: On the left, a discrete (Jordan) domain $\Omega^{\delta}$ and its boundary $\partial\Omega^{\delta}$ (black) in the infinite square grid ${\mathbb Z}^2$ (gray). On the right, its nearest-neighbour graph with wired boundary $(V_{\Omega^{\delta}}, E_{\Omega^{\delta}})$; the thick line along the boundary vertices indicates they all correspond to the same vertex in $V_{\Omega^{\delta}}$.
• Figure 4.2: An example of a spanning tree with wired boundary conditions.
• Figure 4.3: A configuration of the Abelian sandpile model with wired boundary conditions in a discrete domain. This configuration corresponds to the spanning tree in Figure \ref{['fig: ust']} via Dhar's burning algorithm.

Theorems & Definitions (63)

• Remark 2.1
• Lemma 2.2
• Theorem 2.3: AdaCar-symplectic_fermions
• Remark 2.4
• Example 2.5
• Remark 2.6
• Remark 2.7
• Proposition 2.8: HKV
• Remark 3.1
• Remark 3.2