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arxiv: 2312.02960 · v1 · submitted 2023-12-05 · 🧮 math-ph · math.CV· math.MP· math.PR

Bosonization of primary fields for the critical Ising model on multiply connected planar domains

Baran Bayraktaroglu , Konstantin Izyurov , Tuomas Virtanen , Christian Webb This is my paper

Pith reviewed 2026-05-24 05:21 UTC · model grok-4.3

classification 🧮 math-ph math.CVmath.MPmath.PR
keywords bosonizationcritical Ising modelscaling limitsGaussian free fieldmultiply connected domainsHejhal-Fay identitySzegő kernelSchottky double
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The pith

The scaling limits of critical Ising correlations in finitely connected planar domains are given by correlations of the compactified Gaussian free field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes bosonization identities that relate the scaling limits of critical Ising model correlations in multiply connected planar domains to those of a compactified Gaussian free field. These identities provide explicit expressions for the Ising correlations using geometric quantities like the period matrix, Green's function, and harmonic measures of the domain. A reader would care because this connects the fermionic Ising model to bosonic field theory, enabling new ways to compute correlations in non-simply connected geometries. The approach uses a limiting form of the Hejhal-Fay identity and operator product expansions for both models.

Core claim

We prove bosonization identities for the scaling limits of the critical Ising correlations in finitely-connected planar domains, expressing those in terms of correlations of the compactified Gaussian free field. This, in particular, yields explicit expressions for the Ising correlations in terms of domain's period matrix, Green's function, harmonic measures of boundary components and arcs, or alternatively, Abelian differentials on the Schottky double. Our proof is based on a limiting version of a classical identity due to D. Hejhal and J. Fay relating Szegő kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the b

What carries the argument

The bosonization identities relating Ising correlations to compactified Gaussian free field correlations, established via a limiting Hejhal-Fay identity and operator product expansions.

If this is right

  • Ising correlations admit expressions in terms of the domain's period matrix and Green's function.
  • Alternative expressions are available using Abelian differentials on the Schottky double.
  • The identities apply to finitely connected planar domains.
  • The proof combines the limiting Hejhal-Fay identity with operator product expansions for both Ising and bosonic fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identities may allow computation of Ising correlations in complex domains using bosonic simulation techniques.
  • Similar bosonization approaches could be explored for other conformal field theories on multiply connected domains.
  • This work bridges discrete lattice models and continuous field theories in planar domains with holes.

Load-bearing premise

The existence of the scaling limits of the Ising correlations together with the validity of a limiting version of the Hejhal-Fay identity on the relevant Riemann surfaces.

What would settle it

A numerical or analytical computation of Ising spin correlations in a doubly-connected domain, such as an annulus, that deviates from the predicted compactified Gaussian free field correlations.

Figures

Figures reproduced from arXiv: 2312.02960 by Baran Bayraktaroglu, Christian Webb, Konstantin Izyurov, Tuomas Virtanen.

Figure 1.1
Figure 1.1. Figure 1.1: An illustration of a triply connected planar domain with boundary conditions. The dashed part of the boundary forms the set {free} and solid parts form the set {wired}. The points where the boundary con￾ditions changes from fixed to free are marked as dots. Because of (1.2), we may restrict our attention to domains whose boundary consists of disjoint analytic Jordan curves, or even circular domains whose… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A genus 2 Riemann surface with a canonical homology basis {Aj , Bj} 2 j=1. Definition 3.2. A function f from a Riemann surface M to the Riemann sphere Cˆ is called meromorphic if f ◦ z −1 α is meromorphic for any coordinate map α. A 1-form ω is meromorphic (holomorphic) if in any coordinate chart zα : Uα → U, it is represented as fα ◦ zα dzα, where fα : U → Cˆ is meromorphic (holomorphic). Meromorphic 1-… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: A construction of the surface Ωbε in the case g = 2, n = 2, k = 1. Two copies Ωε, Ω ⋆ ε with small cuts near v1,2 and v ⋆ 1,2 are glued together, forming small “holes”. The Torelli marking is drawn with A￾loops in red and B-loops in blue. In the final stage, small cuts near b1, b2 are glued together, so that the curve B5 becomes a closed loop. • For each marked point v1, . . . , vn ∈ Ω, make straight cut… view at source ↗
read the original abstract

We prove bosonization identities for the scaling limits of the critical Ising correlations in finitely-connected planar domains, expressing those in terms of correlations of the compactified Gaussian free field. This, in particular, yields explicit expressions for the Ising correlations in terms of domain's period matrix, Green's function, harmonic measures of boundary components and arcs, or alternatively, Abelian differentials on the Schottky double. Our proof is based on a limiting version of a classical identity due to D.~Hejhal and J.~Fay relating Szeg\H{o} kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove bosonization identities for the scaling limits of critical Ising correlations on finitely-connected planar domains. These identities express the Ising correlations in terms of correlations of the compactified Gaussian free field, yielding explicit formulas involving the domain's period matrix, Green's function, harmonic measures of boundary components and arcs, or Abelian differentials on the Schottky double. The proof is based on a limiting version of the classical Hejhal-Fay identity relating Szegő kernels and Abelian differentials, together with systematic use of operator product expansions for both the Ising and bosonic fields.

Significance. If the result holds, the work provides explicit, computable expressions for Ising correlations in multiply connected domains, extending bosonization techniques beyond simply connected cases and linking discrete statistical mechanics to continuum CFT objects such as the period matrix and Schottky doubles. This could enable new calculations in conformal field theory on Riemann surfaces with multiple boundaries.

major comments (1)
  1. [Abstract] Abstract (proof strategy paragraph): the central claim rests on the existence of scaling limits of Ising correlations in finitely-connected domains together with the validity of a limiting Hejhal-Fay identity; the manuscript invokes both without deriving the limiting identity from first principles or citing a prior rigorous convergence result specific to multiply-connected planar domains, which is load-bearing for all subsequent OPE arguments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and the constructive feedback provided. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (proof strategy paragraph): the central claim rests on the existence of scaling limits of Ising correlations in finitely-connected domains together with the validity of a limiting Hejhal-Fay identity; the manuscript invokes both without deriving the limiting identity from first principles or citing a prior rigorous convergence result specific to multiply-connected planar domains, which is load-bearing for all subsequent OPE arguments.

    Authors: The referee correctly notes that the proof relies on these two elements. The existence of scaling limits for the critical Ising model in finitely connected planar domains follows from established results in the theory of discrete complex analysis and convergence of Ising observables to their continuum counterparts, as developed in works on multiply connected domains. For the limiting Hejhal-Fay identity, the manuscript applies the classical identity on the Schottky double and takes the appropriate limit corresponding to the scaling limit of the discrete model. While this is implicit, we agree that an explicit reference or sketch would improve clarity. We will revise the manuscript by adding a paragraph in the introduction that cites the relevant convergence theorems for the Szegő kernel and Abelian differentials in this setting, and briefly explains the limiting procedure. This addresses the load-bearing nature of these steps for the OPE arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external classical identity

full rationale

The paper states its proof is based on a limiting version of the classical Hejhal-Fay identity (external authors) together with standard OPE techniques for Ising and bosonic correlations. No self-citation load-bearing steps, self-definitional reductions, fitted inputs renamed as predictions, or ansatz smuggling via own prior work appear in the provided abstract or described strategy. The target bosonization identities are derived from these external inputs rather than presupposed by them. The existence of scaling limits is treated as a standing assumption, not constructed from the result itself. This is a normal non-circular case relying on independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of scaling limits for Ising correlations and on standard results from Riemann surface theory; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of scaling limits of critical Ising correlations in finitely-connected domains
    Required for the bosonization identities to hold in the scaling limit.
  • standard math Validity of the limiting version of the Hejhal-Fay identity relating Szegő kernels and Abelian differentials
    Invoked as the key classical input to the proof.

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Forward citations

Cited by 1 Pith paper

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