Completing exact bosonization of the critical Ising model in bounded domains: flat-boundary dimer observables, the half-integer electric endpoint, and lattice-to-continuum constants

Dubédat's exact bosonization identities (arXiv:1112.4399, 2011) express products of dual critical Ising correlators exactly as expectations of local observables of a bipartite dimer height field, and yield full-plane asymptotics of multipoint spin-disorder correlations - but leave open the finite-domain theory (deferred to a sequel that never appeared), the delicate half-integer-charge endpoint $\alpha=1/2$ of the electric-observable asymptotics (the known analysis is sharp on $\alpha\in(-1/2,1/2)$), and the closed-form identification of the lattice-to-continuum constants $c_{m,n}$. Later work resolved the continuum targets by other methods: fermionic observables give all primary-field correlations in finitely connected domains (Chelkak-Hongler-Izyurov), and the Hejhal-Fay route proves the continuum bosonization identities there (Bayraktaroglu-Izyurov-Virtanen-Webb). What remains open - and is explicitly flagged as a viable unfinished route by that later work - is the completion of the lattice-level program: convergence of the flat-boundary-height dimer observables (including the $\alpha=1/2$ spin/disorder vertex operators and monomer-type operators) in bounded domains with Ising wired/free boundary conditions, together with explicit constants, so that the exact lattice identities yield the continuum correlators and their compactified-free-field representation by the combinatorial route.

Background

The Ising model is the canonical lattice model of ferromagnetism: spins $\sigma_v=\pm 1$ sit on vertices of a planar graph $\Gamma$ and interact along edges. In two dimensions it has a critical temperature at which correlations decay polynomially and the scaling limit is conformally invariant. Besides the spin (order) variables, a central role is played by disorder variables (Kadanoff-Ceva): a disorder insertion at a dual vertex flips the sign of all couplings along a path ending there, and represents the dual degree of freedom of Kramers-Wannier duality (the self-duality exchanging the high- and low-temperature expansions of the model). Field-theoretic bosonization (Di Francesco-Saleur-Zuber, 1987) predicts that squares of critical Ising correlators - equivalently, correlators of two independent Ising copies - equal correlation functions of a compactified free boson (a free field with circle-valued target, whose correlation functions carry theta-function sectors): spin and disorder insertions become vertex operators of charge $\pm 1/2$.

A planar Ising configuration also maps to a dimer configuration - a perfect matching of the vertices - on a decorated graph. Dimers on bipartite graphs carry a height function: an integer-valued function on faces whose local increments record which edges are occupied; at criticality the height field converges to the Gaussian free field (GFF), a random generalized function with logarithmic covariance, which is the massless boson of the bosonization dictionary (Kenyon, 2001). 'Electric' observables of this field are exponentials $\exp(i\alpha\phi)$ of the height (vertex operators of charge $\alpha$), while 'magnetic' insertions correspond to monomers (unmatched vertices), whose two-point decay is the classical Fisher-Stephenson problem. Dimer statistics are computed by the Kasteleyn operator, a signed adjacency matrix whose determinant counts configurations.

Dubédat's exact bosonization (arXiv:1112.4399) makes the dictionary exact at the lattice level: the product of a spin correlator of an Ising model on $\Gamma$ and a disorder correlator of the dual Ising model on $\Gamma^\dagger$ equals, up to explicit local factors, the expectation of a product of local operators in a bipartite dimer model on a decorated graph $\mathcal{C}$; spin and disorder insertions become the half-integer-charge observables $\cos(\phi/2)$, $\sin(\phi/2)$ and the monomer-type operators $O_{\pm 1}$. Combined with the asymptotic analysis of dimer height observables via families of Cauchy-Riemann operators and their determinants (preprint 'Dimers and analytic torsion I'; published as 'Dimers and families of Cauchy-Riemann operators I', J. Amer. Math. Soc. 2015), which evaluates electric correlators $\langle\exp(i\alpha(\phi(y)-\phi(x)))\rangle$ for charges $\alpha\in(-1/2,1/2)$ - a range noted there to be sharp, at least for a single pair - and settles the Fisher-Stephenson monomer asymptotics, this yields the full-plane asymptotics of multipoint critical Ising spin-disorder correlators (Theorem 4 of the paper), with two limitations: the multiplicative lattice-to-continuum constant $c_{m,n}$ is asserted only to exist and be nonzero, and the analysis is planar. The endpoint $\alpha=1/2$ - exactly the charge carried by the spin/disorder observables - is described as 'rather delicate'; on the square lattice it can be reached indirectly, through an exact identity between half-integer electric and magnetic (monomer) correlators. Finite domains add a difficulty of principle: wired (respectively free) Ising boundary conditions - spins conditioned to agree along a boundary arc, respectively left unconstrained - translate into flat, i.e. constant, boundary height for the dimer model, rather than the Temperleyan boundary conditions (a natural dimer boundary condition under which the boundary height tracks the winding of the boundary) for which most dimer convergence theorems are proved; boundary corners moreover produce oblique-reflection singularities in the random-walk representation of the inverse Kasteleyn operator. The paper defers the finite-domain theory to forthcoming work by Chelkak-Hongler-Izyurov and to a planned sequel by the author that never appeared.

What later work established by other routes: Chelkak-Hongler-Izyurov (Ann. of Math. 2015 for plane spin correlations; 2021 preprint for the general case) proved convergence of renormalized correlations of all primary fields (the basic local fields: spin, disorder, fermions, energy density) in arbitrary finitely connected domains with plus/free/mixed boundary conditions, with conformal covariance and descriptions via Riemann boundary value problems - using fermionic observables based on s-holomorphicity (a discrete holomorphicity property of Ising fermionic observables), not bosonization; Bayraktaroglu-Izyurov (2025) added the torus. Bayraktaroglu-Izyurov-Virtanen-Webb (Comm. Math. Phys. 2025) then proved the continuum bosonization identities in finitely connected domains - squares of Ising primary-field correlations equal compactified-GFF correlations, with explicit theta-functional formulae built from the domain's period matrix, Green's function, and harmonic measures - via the Hejhal-Fay identity relating the Szegő kernels (canonical bi-differentials encoding fermionic two-point functions) and Abelian differentials on the doubled surface, again not via the lattice identities. On the dimer side, Basok (2023) proved convergence of dimer height functions to the compactified free field on Riemann surfaces in the Temperleyan setting; Chelkak-Deiman (2026) and Basok (2025) extended the height identification to doubly connected and arbitrary bordered surfaces, with an additional discrete-Gaussian sector. Park-Virtanen-Webb (2025) extended continuum bosonization to the near-critical Ising model, related there to the sine-Gordon model at the free fermion point. None of these covers the flat-boundary-height dimer graphs induced by Ising wired/free boundaries, or the $\alpha=1/2$ vertex observables with explicit constants; Bayraktaroglu-Izyurov-Virtanen-Webb state explicitly that carrying out the lattice exact-bosonization program in generality 'could also lead to a proof' of their theorem, and that passing from height-field convergence to convergence of the required observables 'involves a lot of work'.

Problem Statement

Complete Dubédat's exact bosonization program for the critical Ising model at the lattice level in bounded domains. Concretely, for mesh-$\delta$ discretizations of bounded planar domains (simply connected at minimum, finitely connected in full generality) with wired/free Ising boundary conditions - equivalently, for the associated bipartite dimer graphs $\mathcal{C}$ with flat (constant) boundary height - establish:

1. Observable convergence with boundary control: scaling-limit asymptotics of the dimer observables appearing in the exact bosonization identities - the height field with its winding/compactification sectors, the electric (vertex) observables $\exp(i\alpha\phi)$, and the monomer-type magnetic operators $O_{\pm 1}$ - under the flat-boundary-height regime and the corner oblique-reflection singularities specific to Ising wired/free boundaries.

2. The half-integer endpoint: an asymptotic treatment of the endpoint $\alpha=1/2$, i.e. of the observables $\cos(\phi/2)$ and $\sin(\phi/2)$ that carry the Ising spin and disorder fields, in bounded domains, beyond the known charge range $\alpha\in(-1/2,1/2)$ (sharp for the existing analysis) and beyond the existing full-plane, square-lattice setting of the exact electric/monomer identity; extending that identity to bounded-domain dimer graphs is a legitimate route to this endpoint.

3. Constants: closed-form identification of the lattice-to-continuum multiplicative constants - the $c_{m,n}$ of Theorem 4 of the source paper and their finite-domain counterparts, including the Ising/dimer normalization prefactors and their dependence on microscopic lattice conventions. The identification must be derived within the lattice bosonization framework (e.g. from the exact finite-size determinant asymptotics described in the Background), not reverse-defined as a ratio of the known continuum correlators to the lattice asymptotics.

The common objective is that items 1-3, combined with the exact lattice identities, yield a complete lattice-to-continuum derivation of the multipoint critical Ising correlators in bounded domains and of their identification with compactified free-field (theta-functional) correlators - i.e. an independent proof, by the combinatorial bosonization route, of the continuum identities established by Bayraktaroglu-Izyurov-Virtanen-Webb. The derivation is complete when the continuum bosonization identities are recovered from the lattice identities together with the convergence and constant-identification results for bounded finitely connected planar domains with arbitrary wired/free decompositions of the boundary (the generality of the Bayraktaroglu-Izyurov-Virtanen-Webb theorem); a result restricted to simply connected domains counts as partial progress toward this target.

References

1. Julien Dubédat, 'Exact bosonization of the Ising model', preprint arXiv:1112.4399 (2011), DOI 10.48550/arXiv.1112.4399, https://arxiv.org/abs/1112.4399
2. Julien Dubédat, 'Dimers and analytic torsion I', preprint arXiv:1110.2808 (2011, v2 2014), https://arxiv.org/abs/1110.2808
3. Julien Dubédat, 'Dimers and families of Cauchy-Riemann operators I', Journal of the American Mathematical Society 28 (2015) 1063-1167, DOI 10.1090/jams/824, https://doi.org/10.1090/jams/824
4. Dmitry Chelkak, Clément Hongler, Konstantin Izyurov, 'Correlations of primary fields in the critical Ising model', preprint arXiv:2103.10263 (2021), https://arxiv.org/abs/2103.10263
5. Dmitry Chelkak, Clément Hongler, Konstantin Izyurov, 'Conformal invariance of spin correlations in the planar Ising model', Annals of Mathematics (2) 181 (2015) 1087-1138, DOI 10.4007/annals.2015.181.3.5, https://doi.org/10.4007/annals.2015.181.3.5
6. Baran Bayraktaroglu, Konstantin Izyurov, Tuomas Virtanen, Christian Webb, 'Bosonization of primary fields for the critical Ising model on multiply connected planar domains', Communications in Mathematical Physics 406 (2025), article 222, DOI 10.1007/s00220-025-05392-5 (preprint arXiv:2312.02960), https://arxiv.org/abs/2312.02960
7. S. C. Park, Tuomas Virtanen, Christian Webb, 'Near-critical Ising, sine-Gordon at the free fermion point, and bosonization', preprint arXiv:2512.11304 (2025, v2 2026), https://arxiv.org/abs/2512.11304
8. Mikhail Basok, 'Dimers on Riemann surfaces and compactified free field', preprint arXiv:2309.14522 (2023), https://arxiv.org/abs/2309.14522
9. Mikhail Basok, 'Kenyon's identities for the height function and compactified free field in the dimer model', preprint arXiv:2511.03804 (2025), https://arxiv.org/abs/2511.03804
10. Baran Bayraktaroglu, Konstantin Izyurov, 'Critical Ising correlations on a torus', preprint arXiv:2506.11324 (2025), https://arxiv.org/abs/2506.11324
11. Béatrice de Tilière, 'Bipartite dimer representation of squared 2d-Ising correlations', preprint arXiv:1407.6274 (2014), https://arxiv.org/abs/1407.6274
12. Janne Junnila, Eero Saksman, Christian Webb, 'Imaginary multiplicative chaos: moments, regularity and connections to the Ising model', preprint arXiv:1806.02118 (2018), https://arxiv.org/abs/1806.02118
13. R. Kenyon, 'Dominos and the Gaussian free field', Annals of Probability 29 (2001) 1128-1137, DOI 10.1214/aop/1015345599, https://doi.org/10.1214/aop/1015345599
14. M. E. Fisher, J. Stephenson, 'Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers', Physical Review 132 (1963) 1411-1431, DOI 10.1103/PhysRev.132.1411, https://doi.org/10.1103/PhysRev.132.1411
15. P. Di Francesco, H. Saleur, J.-B. Zuber, 'Critical Ising correlation functions in the plane and on the torus', Nuclear Physics B 290 (1987) 527-581, DOI 10.1016/0550-3213(87)90202-1, https://doi.org/10.1016/0550-3213(87)90202-1
16. L. P. Kadanoff, H. Ceva, 'Determination of an operator algebra for the two-dimensional Ising model', Physical Review B 3 (1971) 3918-3939, DOI 10.1103/PhysRevB.3.3918, https://doi.org/10.1103/PhysRevB.3.3918
17. Julien Dubédat, Reza Gheissari, 'Asymptotics of height change on toroidal Temperleyan dimer models', Journal of Statistical Physics 159 (2015) 75-100, DOI 10.1007/s10955-014-1181-x (preprint arXiv:1407.6227, 2014), https://arxiv.org/abs/1407.6227
18. Dmitry Chelkak, Zachary Deiman, 'Domino tilings of black-and-white Temperleyan cylinders', preprint arXiv:2601.13332 (2026), https://arxiv.org/abs/2601.13332
19. Cédric Boutillier, Béatrice de Tilière, 'Height representation of XOR-Ising loops via bipartite dimers', preprint arXiv:1211.4825 (2012), https://arxiv.org/abs/1211.4825