The star–square relation and the generalized star–triangle relation from 3d supersymmetric dualities I

Eur. Phys. J. Plus (2024) 139:643 https://doi.org/10.1140/epjp/s13360-024-05444-0 Regular Article The star–square relation and the generalized star–triangle relation from 3d supersymmetric dualities I Mustafa Mullahasanoglua Department of Physics, Bogazici University, Bebek, 34342 Istanbul, Turkey Received: 23 February 2024 / Accepted: 11 July 2024 © The Author(s) 2024 Abstract We study duality transformations of the star–square relation and the generalized star–triangle relation for Ising-like lattice spin models. The lattice spin models are obtained via gauge/YBE correspondence which connects the supersymmetric gauge theories and lattice spin models of statistical mechanics. By the use of integral identities coming from the duality of threedimensional supersymmetric gauge theories, we construct hyperbolic, lens hyperbolic, trigonometric, and rational solutions to the duality transformations. These duality transformations allow us to construct spin lattice models with four-spin (the star–square relation) and three-spin (the generalized star–triangle relation) interactions. 1 Introduction The gauge/YBE correspondence [1, 2] is one of the studies focusing on the links between supersymmetric gauge theories and integrable lattice spin models of statistical mechanics. The correspondence, see exhaustive reviews e.g. [3, 4], provides solutions to integrability conditions of lattice spin models the star–triangle relation [5] and the star–star relation [6] via the integral identities coming from the equality of partition functions of supersymmetric gauge theories. Along with various solutions to the integrability equations, the most general solution is obtained in terms of the lens elliptic gamma function [7, 8] by using the duality of four-dimensional N  1 supersymmetric gauge theories on Sb3 /Zr × S 1 . This lens elliptic model can be reduced to the lens hyperbolic model by the dimensional reduction and the star–triangle relation of this integrable model corresponds to the duality of three-dimensional N  2 supersymmetric gauge theories on Sb3 /Zr [9]. Besides the dimensional reduction, the gauge symmetry-breaking method from the gauge theory perspective also provides different integrable models. As an example, the generalized Faddeev-Volkov model [10, 11] is acquired by breaking gauge symmetry SU(2) to the U(1) gauge group for the dualities of three-dimensional N  2 supersymmetric gauge theories on Sb3 /Zr . The other known integrable models in this context are also obtained from the lens elliptic model [12, 13]. The Ising-like models have spins sitting on sites and nearest-neighbor spins interact with each other through edges. However, one can also acquire the IRF-type (interaction round a face) models [7, 11, 14, 15], and vertex-type (spins interacting at a vertex) [16–18] integrable lattice spin models by using integrability conditions of edge interacting models and constructing Bailey pairs of them, respectively. In this study, we investigate the non-planar lattice spin models consisting of higher-spin interactions by using the duality of the three-dimensional supersymmetric gauge theories. To achieve this investigation, we obtain solutions to the star–square relation [19, 20] and the generalized star–triangle relations [21, 22] with the help of the gauge/YBE correspondence. This is the first time that higher-spin interactions for Ising-like models are presented in the aspect of the gauge/YBE correspondence and Boltzmann weights of the dual higher-spin interacting models are studied in terms of hyperbolic, lens hyperbolic, trigonometric, and rational functions. The duality transformations—the star–square relation and the generalized star–triangle relation- equate partition functions of two different spin models up to some coefficient. In this duality of the lattice spin models, one model possesses only nearest-neighbor interactions and the dual model has various kinds of spin interactions such as higher spin1 (triple or quadruple) interactions. From the supersymmetric gauge theory perspective, the corresponding solutions to the duality transformations are the integral identities resulting from the equality of the partition functions of the dual three-dimensional N  2 supersymmetric gauge theories on Sb3 , Sb3 /Zr and S 2 × S 1 . Therefore, Boltzmann weights of the models are written in terms of hyperbolic gamma function [23, 24], lens hyperbolic gamma function [9–11], trigonometric hypergeometric function [25–27], Euler’s gamma function [28, 29]. 1 The term ’higher spin’ is employed to keep the consistency with terminology in statistical mechanics and it should not be mistaken for its modern usage. a e-mail: (corresponding author) 0123456789().: V,-vol 123 643 Page 2 of 11 Eur. Phys. J. Plus (2024) 139:643 In the future work [30], we will apply the duality transformations for the integrable lattice spin models studied by the gauge/YBE correspondence, that is, the star–square relation and the generalized star–triangle relation will be solved in terms of Boltzmann weights belonging to the integrable models. The paper is organized as follows. In Sect. 2, we briefly introduce all special functions used throughout the work. In the remaining two sections, we study the solutions to the star–square relation and the generalized star–triangle relation, respectively. 2 Notations Let us start with introducing the q-Pochhammer symbol (z; q)∞  ∞  (1 − zq i ) , (2.1) i0 and the shorthand notation for the product of two q-Pochhammer symbols (z, x; q)∞  (z; q)∞ (x; q)∞ . (2.2) The hyperbolic gamma function which is a variant of Faddeev’s non-compact quantum dilogarithm [31, 32] is another key special function that we utilize πi γ (2) (z; ω1 , ω2 )  e 2 B2,2 (z;ω1 ,ω2 ) (e −2πi ωz (e 2 −2πi ωz q̃; q̃)∞ 1 , (2.3) ; q)∞ where q̃  e2πiω1 /ω2 and q  e−2πiω2 /ω1 with the complex variables ω1 , ω2 and with the second Bernoulli polynomial B2,2 (z; ω1 , ω2 )  z2 z z ω1 ω2 1 − − + + + . ω1 ω2 ω1 ω2 6ω2 6ω1 2 There are several integral representations for the hyperbolic gamma function, see, e.g. [33, 34]. We give here one of them   ∞   d x sinh x(2z − ω1 − ω2 ) 2z − ω1 − ω2 γ (2) (z; ω1 , ω2 )  exp − − , x 2 sinh (xω1 ) sinh (xω2 ) 2xω1 ω2 0 where Re(ω1 ), Re(ω2 ) > 0 and Re(ω1 + ω2 ) > Re(z) > 0. The hyperbolic gamma function reduces to the Euler gamma function by the asymptotic property z 1  ω (z/ω1 ) ω2 − 2 (2) 2 . lim γ (z; ω1 , ω2 )  √ ω2 →∞ 2πω1 2π (2.4) (2.5) (2.6) The hyperbolic gamma function has the reflection property γ (2) (ω1 + ω2 − z; ω1 , ω2 )γ (2) (z; ω1 , ω2 )  1 , (2.7) γ (2) (±z; ω1 , ω2 )  γ (2) (+z; ω1 , ω2 )γ (2) (−z; ω1 , ω2 ) . (2.8) and the shorthand notation is The reflection property is helpful while reducing the number of flavors in the supersymmetric gauge theories and the integral identities studied as the star–square relation will be reduced to obtain the generalized star–triangle relation. Lens hyperbo (...truncated)