Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAbout 30 Years of Integrable Chiral Potts Model, Quantum Groups at Roots of Unity and Cyclic Hypergeometric Functions
56
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang--Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts model leads to special identities for basic hypergeometric series in the $q$ a root-of-unity limit. Many of the well-known summation formulae for basic hypergeometric series do not work in this case. However, if we require the summand to be periodic, then there are many summable series. For example, the integrability condition, namely, the star-triangle equation, is a summation formula for a well-balanced ${}_4Phi_3$ series. We finish with a few remarks about the relation with quantum groups at roots of unity.
rate research
Read More
At roots of unity the $N$-state integrable chiral Potts model and the six-vertex model descend from each other with the $tau_2$ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different $q$-Pochhammer symbols, one construction only working well for $N$ odd, the other equally well for all $N$. We also address the generalization based on the sl$(m,n)$ vertex model.
In the first part of this paper I shall discuss the round-about way of how the integrable chiral Potts model was discovered about 30 years ago. As there should be more higher-genus models to be discovered, this might be of interest. In the second part I shall discuss some quantum group aspects, especially issues of odd versus even $N$ related to the Serre relations conjecture in our quantum loop subalgebra paper of 5 years ago and how we can make good use of coproducts, also borrowing ideas of Drinfeld, Jimbo, Deguchi, Fabricius, McCoy and Nishino.
We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to emph{stable patterns} and emph{signed-patterns}, we give general results which allow us to find emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representation
We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2F1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2F1 hypergeometric functions and modular forms.
We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type A_1, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1,p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum sl_2, in conjunction with a natural PSL_2-action on quantum sl_2 provided by our de-equivariantization construction, in order to deduce linear equivalences between extended quantum groups, the singlet vertex operator algebra, and the (1,p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type A_1, which we intend to eliminate in a subsequent paper.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
