This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the $$ \mathfrak{su}{(3)}_k $$ affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum matrix of the $$ {U}_q\left(\mathfrak{sl}(3)\right) $$ quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of matrix elements.

Israel

Vladimir Belavin

Doron Gepner

J. Cabezas

Boris Runov

On different approaches to IRF lattice models. Part II

We consider two explicit constructions of states in the orbifolds of a product of Minimal N = (2, 2) models which are based on twisting by spectral flow, mutual locality and operator algebra requirement. It is shown that these two constructions lead to the Berglund-Hubsh-Krawitz dual orbifold groups which define mirror pairs of isomorphic models. Then we generalize our construction for the orbifolds of Gepner models of superstring compactification and explicitly build IIA/IIB mirror map of the space of states of the superstrings using light-cone gauge.

Russia

Sergej Parkhomenko

Conformal bootstrap and mirror symmetry of states in Gepner models

In Gepner's pioneering work, the requirement that leads to a model having the desired <math><mi>N</mi><mo>=</mo><mn>1</mn></math> Spacetime supersymmetry and <math><mi>E</mi><mo>(</mo><mn>8</mn><mo>)</mo><mo>×</mo><mi>E</mi><mo>(</mo><mn>6</mn><mo>)</mo></math> Gauge symmetry was the requirement that the spacetime symmetry is compatible with modular invariance. In this work we show that the requirement for the simultaneous fulfillment of mutual locality of the left-moving vertices of physical states with the space-time symmetry generators and of right-moving vertices with generators of <math><mi>E</mi><mo>(</mo><mn>8</mn><mo>)</mo><mo>×</mo><mi>E</mi><mo>(</mo><mn>6</mn><mo>)</mo></math>-gauge symmetry, which arises after some special reduction together with the requirement of mutual locality of complete (left-right) vertices of physical states among themselves leads to the same Gepner models.

Alexander Belavin

Conformal bootstrap and Heterotic string Gepner models

The Gepner model (2)4 describes the sigma model of the Fermat quartic K3 surface. Moving through the nearby moduli space using conformal perturbation theory, we investigate how the conformal weights of its fields change at first and second order and find approximate minima. This serves as a toy model for a mechanism that could produce new chiral fields and possibly new nearby rational CFTs.

United States

Christoph Keller

Conformal perturbation theory on K3: the quartic Gepner point

Motivated by the principles of the conformal bootstrap, primarily the principle of Locality, simultaneously with the requirement of space-time supersymmetry, we reconsider constructions of compactified superstring models. Starting from requirements of space-time supersymmetry and mutual locality, we construct a complete set of physical fields of orbifolds of Gepner models. To technically implement this, we use spectral flow generators to construct all physical fields from the chiral primary fields. The set of these spectral flow operators forms a so-called admissible group <math altimg='si1.svg'><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>m</mi></mrow></msub></math>, which defines a given orbifold. The action of these operators produces a collection of physical fields consistent with the action of supersymmetry generators. The selection of mutually local fields from this collection is carried out using the mirror group <math altimg='si2.svg'><msubsup><mrow><mi>G</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>m</mi></mrow><mrow><mo>&#8270;</mo></mrow></msubsup></math>. The permutation of <math altimg='si1.svg'><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>m</mi></mrow></msub></math> and <math altimg='si2.svg'><msubsup><mrow><mi>G</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>m</mi></mrow><mrow><mo>&#8270;</mo></mrow></msubsup></math> replaces the original orbifold with a mirror one that satisfies the same conditions as the original one. This also implies that the resulting model is modular invariant.

Sergey Parkhomenko

Mirror symmetry and new approach to constructing orbifolds of Gepner models

We conjecture the inversion relations for thermalized solvable interaction round the face (IRF) two dimensional lattice models. We base ourselves on an ansatz for the Baxterization described in the 90's. We solve these inversion relations in the four main regimes of the models, to give the free energy of the models, in these regimes. We use the method of Baxter in the calculation of the free energy of the hard hexagon model. We believe these results to be quite general, shared by most of the known IRF models. Our results apply equally well to solvable vertex models. Using the expression for the free energy we calculate the critical exponent α, and from it the dimension of the perturbing (thermal) operator in the fixed point conformal field theory (CFT). We show that it matches either the coset <math><mi>O</mi><mo>/</mo><mi>G</mi></math> or <math><mi>G</mi><mo>/</mo><mi>O</mi></math>, where <math><mi>O</mi></math> is the original CFT used to define the model and <math><mi>G</mi></math> is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.

On the free energy of solvable lattice models

Solvable vertex models in two dimensions are of importance in conformal field theory, phase transitions and integrable models. We consider here the <math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math> spin vertex models, for n which is odd. The models involve also the anti–spinor representation. We describe here the Boltzmann weights for these representations using crossing symmetry from the previously known spinor representations. For calculation reasons we limit ourself to the <math><mi>n</mi><mo>=</mo><mn>3</mn></math> and <math><mi>n</mi><mo>=</mo><mn>5</mn></math> cases, which are described explicitly.

On the Spin Vertex Models for odd

Abstract We study a natural generalization of the results given in K. Aoyama and Y. Sugawara, Prog. Theor. Exp. Phys. 2020, 103B01 (2020) to heterotic strings. Namely, starting from the generic Gepner models for Calabi –Yau three-folds, we construct non-SUSY heterotic string vacua with vanishing cosmological constant at the one-loop level. We especially focus on asymmetric orbifolding based on some discrete subgroup of the chiral  action which acts on both the Gepner model and the  or  sector. We present a classification of the relevant orbifold models leading to the string vacua with the properties mentioned above. In some cases, the desired vacua can be constructed in a manner quite similar to those given in the previous paper for the type II string, in which the orbifold groups contain two generators with discrete torsions. On the other hand, we also have simpler models that are just realized as asymmetric orbifolds of cyclic groups with only one generator.

Japan

Koji Aoyama

Yuji Sugawara

Non-SUSY heterotic string vacua of Gepner models with vanishing cosmological constant

Abstract In this article we discuss a construction of non-SUSY type II string vacua with the vanishing cosmological constant at the one-loop level based on the generic Gepner models for Calabi–Yau 3-folds. We make an orbifolding of the Gepner models by , which asymmetrically acts with some discrete torsions incorporated. We demonstrate that the obtained type II string vacua indeed lead to the vanishing cosmological constant at the one-loop level, whereas any space-time supercharges cannot be constructed as long as we assume chiral forms such as “”. We further discuss possible generalizations of the models described above.

Non-SUSY Gepner models with vanishing cosmological constant

We describe the Boltzmann weights of the <math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub></math> algebra spin vertex models. Thus, we find the <math><mi>S</mi><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> spin vertex models, for any N, completing the <math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub></math> case found earlier. We further check that the real (self–dual) SO(N) models obey quantum algebras, which are the Birman–Murakami–Wenzl (BMW) algebra for three blocks, and certain generalizations, which include the BMW algebra as a sub–algebra, for four and five blocks. In the case of five blocks, the <math><msub><mrow><mi>B</mi></mrow><mrow><mn>4</mn></mrow></msub></math> model is shown to satisfy additional twenty new relations, which are given. The <math><msub><mrow><mi>D</mi></mrow><mrow><mn>6</mn></mrow></msub></math> model is shown to obey two additional relations.