Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains

Baran Bayraktaroglu; Konstantin Izyurov; Tuomas Virtanen; Christian Webb

doi:10.1007/s00220-025-05392-5

Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains

Baran Bayraktaroglu
, Konstantin Izyurov
, Tuomas Virtanen
, Christian Webb^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › Scientific › peer-review

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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 Szegö kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.

Original language	English
Article number	222
Number of pages	58
Journal	Communications in Mathematical Physics
Volume	406
Issue number	9
DOIs	https://doi.org/10.1007/s00220-025-05392-5
Publication status	Published - Sept 2025
MoE publication type	A1 Journal article-refereed

Funding

Open Access funding provided by University of Helsinki (including Helsinki University Central Hospital). T.V. and C.W. were supported by the Emil Aaltonen Foundation. C.W. was supported by the Academy of Finland through the grant 348452. T.V. is also grateful for the financial support from the Doctoral Network in Information Technologies and Mathematics at Åbo Akademi University. B. B. and K.I. were supported by Academy of Finland through academy project “Critical phenomena in dimension two”. We are grateful to Mikhail Basok for useful remarks, as well as to two anonymous reviewers, whose careful reading and helpful comments have improved the article.

Keywords

Ising model
Correlation functions
Conformal field theory
Bosonization
Gaussian free field

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10.1007/s00220-025-05392-5Licence: CC BY

Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar DomainsFinal published version, 998 KBLicence: CC BY

https://urn.fi/URN:NBN:fi-fe2025091295676

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title = "Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains",

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{\"o} kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.",

keywords = "Ising model, Correlation functions, Conformal field theory, Bosonization, Gaussian free field",

author = "Baran Bayraktaroglu and Konstantin Izyurov and Tuomas Virtanen and Christian Webb",

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T1 - Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains

AU - Bayraktaroglu, Baran

AU - Izyurov, Konstantin

AU - Virtanen, Tuomas

AU - Webb, Christian

PY - 2025/9

Y1 - 2025/9

N2 - 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 Szegö kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.

AB - 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 Szegö kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.

KW - Ising model

KW - Correlation functions

KW - Conformal field theory

KW - Bosonization

KW - Gaussian free field

UR - https://arxiv.org/abs/2312.02960

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DO - 10.1007/s00220-025-05392-5

M3 - Article

SN - 0010-3616

VL - 406

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 9

M1 - 222

ER -

Bosonization of Primary Fields for the Critical Ising Model on Multiply Connected Planar Domains

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