Full-Plane Kadanoff-Ceva Fermions

• Full-plane Kadanoff-Ceva fermions are fundamental entities derived from spin and disorder operators that capture scaling limits of local and nonlocal observables in planar Ising models.
• They are constructed through s-holomorphic propagation and converge to spinorial solutions of a conjugate Beltrami equation, linking to quasiconformal mappings and emergent conformal structures.
• This framework rigorously recovers energy-energy correlations and generalizes classical conformal invariance to settings with variable geometric structures.

Full-plane Kadanoff-Ceva fermions provide the fundamental object for capturing the scaling limits of local and nonlocal observables in planar Ising models on generic, non-degenerate $s$-embeddings. The scaling limits of these fermionic observables, constructed from spin and disorder insertions, can be expressed in terms of solutions to conjugate Beltrami equations with prescribed singularities. Their rigorous analysis not only recovers the energy-energy correlations in the Ising model but also uncovers a connection to quasiconformal mappings and emergent non-Euclidean conformal structures, especially for (near-)critical regimes. This framework generalizes conformal invariance to a setting involving Lipschitz, possibly highly variable, conformal structures.

1. $s$-Embeddings and Geometric Structures

An $s$-embedding is a map $S: \Lambda(G)\cup \diamondsuit(G)\to\mathbb{C}$ for a planar graph $G$ (not necessarily finite), based on a unique solution $X$ to the $3$-term propagation (s-holomorphicity) relation: $$X(c_{pq}) = \cos\theta_z\, X(c_{p,1-q}) + \sin\theta_z\, X(c_{1-p,q})$$ where $(c_{pq})$ enumerate the four corners of every quad $z\in\diamondsuit(G)$. Here, planar Ising weights $x(e)\in(0,1)$ on edges $e$ are encoded via angles $\theta_z=2\arctan x(e)\in (0,\pi/2)$. This construction determines discrete edge lengths $\delta_c=|S(v^\bullet(c))-S(v^\circ(c))|=X(c)^2$ and quad inradii $r_z$ that realize tangential quadrilaterals. The origami map $Q: \Lambda(G)\to\mathbb{R}$, defined by $Q(v^\bullet(c))-Q(v^\circ(c)) = \delta_c$, encodes a further geometric structure that, under non-degeneracy conditions ($\mathrm{Lip}(\kappa,\delta)$ and $\mathrm{ExpFat}$), produces a controlled discrete conformal structure (Mahfouf, 23 Dec 2025).

2. Construction of Full-Plane Kadanoff-Ceva Fermions

Given two points $a$, $c$ in $G$ and associated corners $a, c \in\Upsilon(G)$, the Kadanoff-Ceva fermion is $$\chi_c\,\chi_a = \mu_{v^\bullet(c)}\,\sigma_{u^\circ(c)}\,\mu_{v^\bullet(a)}\,\sigma_{u^\circ(a)}$$, where $\sigma$ denotes the spin and $\mu$ the disorder operator. In a finite subgraph $\Lambda_R$ with wired boundary conditions, the correlator

$$X_R^{(a)}(c) := \left\langle \mu_{v^\bullet(c)}\,\sigma_{u^\circ(c)}\,\mu_{v^\bullet(a)}\,\sigma_{u^\circ(a)} \right\rangle^{(w)}_{\Lambda_R}$$

is a spinor on the double cover branching at $u^\circ(a), v^\bullet(a)$ and satisfies the same $3$-term s-holomorphic propagation. Taking $R\to\infty$ (full-plane limit), one obtains a unique, full-plane 3-term-harmonic spinor $X_S^{(a)}$ with normalization $X_S^{(a)}(a^\pm)=\pm1$ (Mahfouf, 23 Dec 2025).

3. Scaling Limits and the Conjugate Beltrami Equation

For a sequence of mesh-refinements $S^\delta$ ($\delta\to 0$) satisfying the embedding assumptions, the discrete correlators

$$F^\delta_{(a^\delta)}(z) = (\delta_{a^\delta}\,\delta_z)^{-1/2}\left\langle \chi_z\,\chi_{a^\delta} \right\rangle_{S^\delta}$$

converge to continuous full-plane Kadanoff-Ceva fermions $F^{[\eta]}_{\vartheta}(z,a)$. In the conformal parametrization $$\zeta \mapsto (z(\zeta), \vartheta(\zeta)) \subset \mathbb{R}^{2,1}$$, the primitive

$$g(\zeta) = \int \left( \overline\varsigma\,F(z(\zeta))\,dz + \varsigma\,\overline{F(z(\zeta))}\,d\vartheta \right)$$

solves the conjugate Beltrami equation: $$\partial_{\bar\zeta}g(\zeta) = i\,\overline{\nu(\zeta)}\,\overline{\partial_\zeta g(\zeta)}, \qquad |\nu(\zeta)|<1,$$ where $\nu(\zeta) = -\,\vartheta_\zeta / z_\zeta$ and equivalently $\zeta_{\bar z} = \mu(z)\, \zeta_z$, with $\mu(z) = -\vartheta_z^2/(1-2|\vartheta_z|^2)$ (Mahfouf, 23 Dec 2025). This result identifies the scaling limits as spinorial solutions of a quasiconformal structure set by the (possibly varying) geometry of the s-embedding.

4. Prescribed Singularities and Fermionic Boundary Conditions

The continuous Kadanoff-Ceva fermion $F^{[\eta]}_{\vartheta}(z,a)$ exhibits a half-integer spinor singularity at the insertion point $a$: $$F^{[\eta]}_{\vartheta}(z,a) = \overline{\eta}\,\frac{1}{2\pi (z-a)} + O(1)\quad(z\to a),$$ or equivalently, $f(z)\simeq (z-a)^{-1/2}\times[\text{holomorphic at$a$}]$. These singularities enforce the spinor boundary conditions of Kadanoff-Ceva type and determine unique normalization up to sign on the double cover, reflected in the behavior of the primitive $g$ (with $2\pi i\overline{\eta}$ jump around $a$) (Mahfouf, 23 Dec 2025).

5. Local Scaling Factors and Geometric Normalizations

Normalization of observables relies exclusively on local geometric data from the embedding: $$\delta_c = |S(v^\bullet(c)) - S(v^\circ(c))|,\quad r_z = \text{inradius of quad }z,\quad \cos\theta_z = \frac{x(e)}{\sqrt{1+x(e)^2}}.$$ For energy observables at an edge $e$, one rescales via

$$\widetilde\varepsilon_e = r_e^{-1}\cos\theta_e \left(\varepsilon_e - \mathbb{E}[\varepsilon_e]\right),$$

ensuring consistency with the continuous, conformal-covariant fields in the scaling limit (Mahfouf, 23 Dec 2025).

6. Scaling Limits of Energy-Energy Correlations

Let $e_1, e_2$ be edges approximating $a_1\neq a_2$. The rescaled, centered energy correlation functions

$$\frac{\cos\theta_{e_1}\cos\theta_{e_2}}{r_{e_1} r_{e_2}}\, \mathbb{E}\left[\widetilde\varepsilon_{e_1}\,\widetilde\varepsilon_{e_2}\right] \xrightarrow[\delta\to 0]{} \frac{1}{\pi^2} \left(|F^\star_\vartheta(a_1, a_2)|^2 - |F_\vartheta(a_1, a_2)|^2\right)$$

are conformally covariant under the generalized conformal structure on $(z,\vartheta(z))\subset\mathbb{R}^{2,1}$ and agree with the behavior of fields of half-integer conformal spin (Mahfouf, 23 Dec 2025).

7. Emergent Conformal and Quasiconformal Structures

The limiting structure of Kadanoff-Ceva fermions depends on the behavior of the function $\vartheta$ from the embedding:

• Critical Euclidean case: $\vartheta\equiv0$, $\nu\equiv 0$, reducing the Beltrami equation to $\bar\partial g=0$ and recovering classical holomorphicity on the plane.
• Near-critical “massive” regime: With $\vartheta$ of regularity $C^2$, the limiting field solves a massive Dirac equation.
• Generic full-plane case: For Lipschitz $\vartheta$, the limit is governed by quasiconformal structures, with the underlying surface $\{(z,\vartheta(z))\}\subset\mathbb{R}^{2,1}$ inheriting a non-trivial conformal geometry.
• Maximal surfaces ($\varrho\equiv0$): Here, $\vartheta$ is harmonic, $m=0$, and the conformal structure is realized on a Lorentzian surface in $\mathbb{R}^{2,1}$, restoring full conformal invariance modulo the embedding (Mahfouf, 23 Dec 2025).

This analysis confirms Chelkak’s conjecture: the scaling limits of (near-)critical planar Ising models generically reside on quasiconformal models and, in special regimes, on Lorentzian maximal surfaces, thus extending the classical scope of Euclidean conformal invariance.