The decoration transformation — summing over extra spin variables inserted on the edges of a lattice so that the decorated lattice's partition function equals the parent lattice's up to an explicit multiplicative factor — was recently given new solutions by Catak and Mullahasanoglu in the hyperbolic, lens-hyperbolic, trigonometric, and rational regimes of the gauge/YBE correspondence, which maps supersymmetric gauge-theory dualities to integrable lattice models with continuous spins and special-function Boltzmann weights. The authors explicitly highlight that these solutions are directly applicable to constructing decorated lattices — naming the Kagome lattice, obtainable from the hexagonal lattice via the decoration transformation and the star-triangle relation — but no such construction, explicit weights, or partition-function relation is carried out, and the conclusion leaves open the role of the decoration transformation within the web of integral identities (star-triangle, star-square, star-star) generated by supersymmetric dualities. What remains open is to build the decorated (paradigmatically Kagome) lattice models with their exact partition-function relations in each regime with known decoration solutions, and to determine the structural position of decoration among the gauge/YBE integrability conditions.

Background

The gauge/YBE correspondence connects supersymmetric gauge theories with two-dimensional integrable lattice models: when two supersymmetric gauge theories are dual to each other, the equality of their supersymmetric partition functions (over certain backgrounds) translates into an integral identity satisfied by the Boltzmann weights of an associated spin model, and this identity is precisely the model's integrability condition. The best-known example is the star-triangle relation — an integral (or sum) over a central spin that converts three interaction weights meeting at a point into three weights of a dual interaction graph, generalizing Onsager's Ising-model identity — and the correspondence also produces star-square and star-star (four-leg) analogues. The spin variables are continuous, x∈Rx\in\mathbb{R}, in every regime, and in several regimes they carry an additional discrete component m∈Zm\in\mathbb{Z}, written together as σ=(x,m)∈R×Z\sigma=(x,m)\in\mathbb{R}\times\mathbb{Z} (the hyperbolic model of these papers uses the continuous component alone, while its trigonometric and rational models carry both components), and the weights are built from special functions organized by analytic regime: hyperbolic gamma (Barnes double-sine/quantum-dilogarithm type) functions, elliptic gamma functions, basic hypergeometric functions, and ordinary Euler gamma functions, roughly parallel to the trigonometric/hyperbolic/elliptic classification of Yang–Baxter equations. The decoration transformation (decoration–iteration transformation) is a classical tool of lattice statistical mechanics: one decorates a lattice by inserting additional spin variables on its edges (or faces), lets them interact with the endpoints, and sums over the inserted spins. If the sum factorizes, the decorated lattice's partition function equals the parent lattice's times a computable multiplicative factor, so exact results transfer between the two lattices. Fisher systematized this for Ising models in 1959, and the standard route to solving the Ising model on the Kagome lattice — the two-dimensional lattice of corner-sharing triangles — is precisely a decoration of the honeycomb (hexagonal) lattice combined with a star-triangle transformation; that classical solution, however, is restricted to discrete Ising spins and elementary weights. In 2024 Catak and Mullahasanoglu initiated the decoration of the gauge/YBE correspondence itself: they solved the decoration transformation for the hyperbolic (and lens-hyperbolic) model arising from three-dimensional N=2\mathcal{N}=2 supersymmetric dualities, sketched the corresponding Kagome lattice model in a figure, and stated that the details of the derivation are left to further studies. A 2025 follow-up derived further hyperbolic, trigonometric, and rational solutions, including the previously missing trigonometric and rational cases. Both papers highlight that such solutions are directly applicable to constructing decorated lattices whose partition functions differ from the parent models by specific multiplicative factors, with the Kagome lattice — constructible from the hexagonal lattice using only the decoration transformation and the star-triangle relation — given as the example. What remains is to actually perform this program in the continuous-spin, special-function setting, and to locate the decoration transformation within the full web of gauge/YBE integral identities.

Problem Statement

Using the decoration-transformation solutions of Catak and Mullahasanoglu for the hyperbolic, lens-hyperbolic, trigonometric, and rational models of the gauge/YBE correspondence, carry out the decorated-lattice program that both papers highlight but leave undone, in two linked components. (1) Explicitly construct the decorated lattices these identities make solvable — concretely, the Kagome lattice from the hexagonal (honeycomb) lattice via the decoration transformation combined with the star-triangle relation, and analogously in the other analytic regimes — giving the explicit Boltzmann weights of the decorated model in terms of the parent model's special-function weights, and deriving the exact relation between the decorated lattice's partition function and the parent model's partition function (the multiplicative factor, equivalently the free-energy difference per unit cell). (2) Determine the role of the decoration transformation within the web of integral identities and dualities arising from supersymmetric gauge theories — the star-triangle, star-square, and star-star integrability conditions surveyed in the gauge/YBE literature — for the decoration transformations derived in these works: which of these relations they reduce to or compose with. Component (1) is the concrete application the sources promise; component (2) is the structural question the source's conclusion poses explicitly; together they fix what the decoration transformation contributes to exactly solvable lattice statistical mechanics beyond the classical Ising case.

References

  1. Erdal Catak, Mustafa Mullahasanoglu, "More solutions to the decoration transformation", arXiv:2509.11842 (2025), DOI 10.48550/arXiv.2509.11842, https://arxiv.org/abs/2509.11842 (primary source)
  2. Erdal Catak, Mustafa Mullahasanoglu, "Decorating the gauge/YBE correspondence", European Physical Journal C 84 (2024) 962, DOI 10.1140/epjc/s10052-024-13347-8, arXiv:2403.14485, https://arxiv.org/abs/2403.14485
  3. M. E. Fisher, "Transformations of Ising models", Physical Review 113 (1959) 969, DOI 10.1103/PhysRev.113.969, https://journals.aps.org/pr/abstract/10.1103/PhysRev.113.969
  4. Mustafa Mullahasanoglu, "From dual gauge theories to dual spin models", Bulgarian Journal of Physics 52 (2025) no. s1, 114–119, DOI 10.55318/bgjp.2025.52.s1.114, arXiv:2510.26535, https://arxiv.org/abs/2510.26535
  5. Zehra Akbulut, Ilmar Gahramanov, Anıl Kahraman, Mustafa Mullahasanoglu, Yaren Yıldırım, "Bailey chain approach to 2d N=(0,2)\mathcal{N}=(0,2) dualities", arXiv:2510.19713 (2025), https://arxiv.org/abs/2510.19713
  6. Oğuz Alp Ağırbaş, Anıl Ata, Eren Demirci, Ilmar Gahramanov, Tuğba Hırlı, R. Semih Kanber, Ahmet Berk Kavruk, Mustafa Mullahasanoğlu, Zehra Özcan, Cansu Özdemir, Irmak Özgüç, Sinan Ulaş Öztürk, Uveys Turhan, Ali Mert T. Yetkin, Yunus Emre Yıldırım, Reyhan Yumuşak, "Ising 100: review of solutions", arXiv:2504.09661 (2025), https://arxiv.org/abs/2504.09661
  7. Ilmar Gahramanov, "Integrability from supersymmetric duality: a short review", Analysis and Mathematical Physics 16 (2026) no. 4, 98, DOI 10.1007/s13324-026-01224-4, arXiv:2201.00351, https://arxiv.org/abs/2201.00351
  8. Erdal Catak, Mustafa Mullahasanoglu, "Flipping relation as a reduced star-star relation", European Physical Journal C 85 (2025) 1176, DOI 10.1140/epjc/s10052-025-14931-2, arXiv:2508.19941, https://arxiv.org/abs/2508.19941