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Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models

Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce

TL;DR

This work analyzes modular covariant torus partition functions for dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models in their simplest regimes, using four torus boundary conditions and loop fugacities. It introduces transfer-matrix proofs via Markov traces in enlarged Temperley–Lieb algebras, conjectures scaling-limit traces, and expresses the resulting conformal partition functions as Coulomb-gas and Verma-character sesquilinear forms. A central result is the exact equality of the conformal partition functions for the dense and dilute models across $0<\frac{p}{p'}<1$, providing strong evidence for universality between these logarithmic minimal models. At $\alpha=2$, the spectra map onto affine $u(1)$ coset theories and into 6-vertex and Izergin–Korepin 19-vertex models, with modular covariant partition functions written as affine $u(1)$ sesquilinears using Bezout conjugates, linking to the Coulomb and $O(n)$ frameworks and highlighting modular covariance.

Abstract

Yang-Baxter integrable dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions $(h,v)$ is applied in the horizontal and vertical directions with $h,v=0$ and $1$ for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by $α$ and $β$ respectively where $β$ is simply related to the crossing parameter $λ$. At roots of unity, when $λ/π\in\mathbb Q$, these models are the dense ${\cal LM}(p,p')$ and dilute ${\cal DLM}(p,p')$ logarithmic minimal models with $p,p'$ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with $d$ defects and deduce the conformal partition functions ${\cal Z}_{\textrm{dense}}^{(h,v)}(α)$ and ${\cal Z}_{\textrm{dilute}}^{(h,v)}(α)$ using Markov traces. These are expressed in terms of functions ${\cal Z}_{m,m'}(g)$ known from the Coulomb gas arguments of Di Francesco, Saleur and Zuber and subsequently as sesquilinear forms in Verma characters. Crucially, we find that the partition functions are identical for the dense and dilute models. The coincidence of these conformal partition functions provides compelling evidence that, for given $(p,p')$, these dense and dilute theories lie in the same universality class. In root of unity cases with $α=2$, the $(h,v)$ modular covariant partition functions are also expressed as sesquilinear forms in affine $u(1)$ characters involving generalized Bezout conjugates. These also give the modular covariant partition functions for the 6-vertex and Izergin-Korepin 19-vertex models in the corresponding regimes.

Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models

TL;DR

This work analyzes modular covariant torus partition functions for dense and dilute loop models in their simplest regimes, using four torus boundary conditions and loop fugacities. It introduces transfer-matrix proofs via Markov traces in enlarged Temperley–Lieb algebras, conjectures scaling-limit traces, and expresses the resulting conformal partition functions as Coulomb-gas and Verma-character sesquilinear forms. A central result is the exact equality of the conformal partition functions for the dense and dilute models across , providing strong evidence for universality between these logarithmic minimal models. At , the spectra map onto affine coset theories and into 6-vertex and Izergin–Korepin 19-vertex models, with modular covariant partition functions written as affine sesquilinears using Bezout conjugates, linking to the Coulomb and frameworks and highlighting modular covariance.

Abstract

Yang-Baxter integrable dense and dilute loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions is applied in the horizontal and vertical directions with and for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by and respectively where is simply related to the crossing parameter . At roots of unity, when , these models are the dense and dilute logarithmic minimal models with coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with defects and deduce the conformal partition functions and using Markov traces. These are expressed in terms of functions known from the Coulomb gas arguments of Di Francesco, Saleur and Zuber and subsequently as sesquilinear forms in Verma characters. Crucially, we find that the partition functions are identical for the dense and dilute models. The coincidence of these conformal partition functions provides compelling evidence that, for given , these dense and dilute theories lie in the same universality class. In root of unity cases with , the modular covariant partition functions are also expressed as sesquilinear forms in affine characters involving generalized Bezout conjugates. These also give the modular covariant partition functions for the 6-vertex and Izergin-Korepin 19-vertex models in the corresponding regimes.

Paper Structure

This paper contains 19 sections, 10 theorems, 135 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Dense and dilute loop models on the torus
  3. Transfer matrices, Markov traces and partition functions
  4. Scaling limit and conformal partition functions
  5. Partition functions as sesquilinear forms in Verma characters
  6. Partition functions for $\boldsymbol{\alpha = 2}$ as affine $\boldsymbol{u(1)}$ sesquilinear forms
  7. Classical partition functions
  8. Partition functions as sesquilinear forms in affine $\boldsymbol{u(1)}$ characters
  9. Bezout conjugates and sesquilinear forms using affine $\boldsymbol{u(1)}$ indices
  10. Conclusion
  11. Partition functions of the $\boldsymbol{O(n)}$ model
  12. Integer and half-integer Bezout conjugates
  13. Example sesquilinear forms
  14. $\boldsymbol{(p,p')=(1,2)}$ with $\boldsymbol{c = -2}$
  15. $\boldsymbol{(p,p')=(1,3)}$ with $\boldsymbol{c = -7}$
  16. ...and 4 more sections

Key Result

Proposition 5.1

For $d \neq 0$, we have where

Figures (4)

  • Figure 1: Typical configurations of the dense (left panels) and dilute loop model (right panels). In the upper panels, configurations are shown directly on the torus for $(M,N)=(22,52)$. Projections of typical configurations onto a doubly periodic rectangle are shown in the lower panels for $(M,N)=(8,7)$.
  • Figure 2: Example loop configurations for the dilute $\hbox{$A_2^{\textrm{ $(2)$}}$}$ loop model for the four possible $(h,v)$ boundary conditions. Generic horizontal and vertical lines cross loop segments $H$ and $V$ times, respectively, with $h=H \textrm{ mod } 2$ and $v=V \textrm{ mod } 2$. The left/right edges and top/bottom edges are identified to form a torus. Loop configurations for the dense $\hbox{$A_1^{\textrm{ $(1)$}}$}$ loop model are similar but with each square face visited by two loop segments. In this case, $h=N \textrm{ mod } 2$ and $v=M \textrm{ mod } 2$.
  • Figure 3: Kac tables of Bezout conjugates $\{j,\overline j\}|_{h=0}$ and $\{j\!+\!\tfrac{1}{2},\overline{j\!+\!\tfrac{1}{2}}\}|_{h=1}$ for $(p,p')=(3,4)$ in the upper panels and $(p,p')=(3,5)$ in the lower panels. The Bezout conjugators are $\omega_0|_{h=0}=7$ and $\omega_0|_{h=1}=31$ for $(p,p')=(3,4)$, and $\omega_0|_{h=0}=\omega_0|_{h=1}=19$ for $(p,p')=(3,5)$. The periodicity is $P=2n$ in the left panels and $P=4n$ in the right panels. Only Bezout conjugates within the framed box in the lower left contribute to the modular covariant partition functions.
  • Figure 4: Kac tables of Bezout conjugates $\{j,\overline j\}$ for $(p,p')=(4,5)$, for $h=0$ in the left panel and $h=1$ in the right panel, with $\omega_0|_{h=0}=9$, $\omega_0|_{h=1}=29$ and $P=2n$. Only Bezout conjugates within the framed box in the lower left contribute to the modular covariant partition functions.

Theorems & Definitions (11)

  • Conjecture 1: Scaling limits of the traces in the standard modules
  • Proposition 5.1
  • Proposition 6.1
  • Proposition 6.2
  • Proposition 6.3
  • Proposition A.1
  • Proposition A.2
  • Proposition A.3
  • Lemma B.1: Bezout's Lemma
  • Corollary B.2
  • ...and 1 more