Rigorous passage from the Jones quotients of the Temperley-Lieb algebras to the Virasoro minimal unitary models

When $q$ is a root of unity whose square has order $\ell$ (written $|q^2|=\ell$), the Temperley-Lieb algebra $TL_n(q)$ has a canonical semisimple quotient $Q_n(\ell)=TL_n(q)/R_n(q)$, obtained by dividing out the radical $R_n(q)$ of the Jones trace form (generated by the Jones-Wenzl idempotent $E_\ell$); $Q_n(\ell)$ is precisely the algebra generated by Jones' projections. Iohara, Lehrer and Zhang determined the simple modules and dimensions of these 'Jones quotients' and recorded as their motivation the question of how the Virasoro algebra arises as a limit of the Temperley-Lieb algebras or their semisimple quotients, noting that the central charges $c=1-6/(\ell(\ell-1))$ of the Virasoro minimal unitary series match the small-$\ell$ cases (Ising at $\ell=4$, tricritical Ising at $\ell=5$, 3-state Potts at $\ell=6$). Their own 2019 sequel resolved the fusion-rule component at the Grothendieck-ring level, proving that the stable-limit ring of the $Q_n(\ell)$ is the level-$(\ell-2)$ Verlinde fusion ring of $\widehat{\mathfrak{sl}}_2$, a subring of the minimal-model fusion algebra. The surviving open core is the module-level form of the question: construct an explicit limiting or functorial passage from the $Q_n(\ell)$ as $n\to\infty$ to Virasoro modules of the minimal unitary model $M(\ell,\ell-1)$, identifying highest weights, recovering fusion, and accounting for unitarity. Rigorous constructions of this type are known so far only at the level of rings and numerical invariants, and, for the related anyonic-chain setting, only for the Ising case.

Background

The Temperley-Lieb algebra $TL_n(q)$ is the finite-dimensional associative algebra over $\mathbb{C}$ with generators $f_1,\dots,f_{n-1}$ and relations $f_i^2=-(q+q^{-1})f_i$, $f_if_{i\pm1}f_i=f_i$, and $f_if_j=f_jf_i$ for $|i-j|\ge2$; equivalently it is spanned by non-crossing pairings of $2n$ boundary points, and it arose both in the solution of two-dimensional lattice models in statistical mechanics and in Jones' theory of subfactors of von Neumann algebras, whose index constraints first exhibited the special role of roots of unity. When $q$ is a root of unity with $|q^2|=\ell$, the algebra $TL_n(q)$ is non-semisimple for almost all $n$. Jones showed that the canonical trace (the Jones form) on $TL_n(q)$ then has a radical $R_n(q)$ which, for $n\ge\ell-1$, is generated as an ideal by the Jones-Wenzl idempotent $E_\ell\in TL_{\ell-1}(q)$: the unique idempotent annihilated by multiplication with every generator, which exists precisely below the root-of-unity threshold. The quotient $Q_n(\ell):=TL_n(q)/R_n(q)$ is semisimple and is exactly the algebra generated by Jones' projections; following Iohara, Lehrer and Zhang it is called the Jones quotient or Jones algebra. Its representation theory is accessible because $TL_n(q)$ is a cellular algebra in the sense of Graham and Lehrer, i.e. an algebra equipped with a diagram basis and an anti-involution satisfying axioms that produce 'cell modules' $W_t(n)$, labelled by the number $t$ of through-strings, whose simple heads $L_t(n)$ exhaust the simple modules. Iohara, Lehrer and Zhang (2017) determined the dimensions of the simple $Q_n(\ell)$-modules, generating functions for the cell-module dimensions, and $\dim Q_n(\ell)$ (a formula going back to the monograph of Goodman, de la Harpe and Jones on towers of algebras). Two special cases display the contact with physics: for $\ell=4$ one recovers the classical isomorphism of $Q_n(4)$ with the even part of the Clifford algebra, the operator algebra of the Ising model, and for $\ell=5$ the dimensions of the $Q_n(5)$ are the odd-indexed Fibonacci numbers. Andersen (2019) later gave a complete description of the simple modules of the Jones quotient algebras over arbitrary fields, via the tilting theory of the quantum group $U_q(\mathfrak{sl}_2)$ (the $q$-deformed enveloping algebra of $\mathfrak{sl}_2$).

The motivation recorded in the source, however, was the Virasoro algebra. The Virasoro algebra $\mathcal{L}$ is the infinite-dimensional complex Lie algebra with basis $L_i$, $i\in\mathbb{Z}$, and a central element $C$, with $[L_m,L_n]=(m-n)L_{m+n}+\frac{1}{12}(m^3-m)\delta_{m,-n}C$; it is the symmetry algebra of two-dimensional conformal field theory, and its irreducible highest-weight modules $L(c,h)$ are labelled by the central charge $c$ and the $L_0$-eigenvalue $h$. The unitary minimal models $M(m+1,m)$ form the discrete series with $c=1-6/(m(m+1))$ and weights $h_{r,s}$; taking $m=\ell-1$ gives $c=1-6/(\ell(\ell-1))$, so that $\ell=3$ is degenerate ($c=0$), $\ell=4$ is the Ising model ($c=1/2$), $\ell=5$ the tricritical Ising model ($c=7/10$), and $\ell=6$ the 3-state Potts model ($c=4/5$) - precisely the cases named in the source. The fusion rules of the $(1,s)$-type fields of these models are governed by the truncated tensor product of integrable modules over the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ at level $\ell-2$, whose Grothendieck ring is the Verlinde fusion ring with structure constants given by Verlinde's formula.

What is rigorous today sits at the level of numerical invariants and rings. Koo and Saleur (1994) exhibited discretized Virasoro generators built out of Temperley-Lieb generators on the lattice; this construction underlies most later physics analyses but is not a theorem about the Jones quotients. The authors' own sequel (Mathematical Research Letters, 2019) proved that $Q_n(\ell)\cong\mathrm{End}_{U_q(\mathfrak{sl}_2)}(\Delta_q(1)^{\bar\otimes n})$, the endomorphism algebra of the $n$-fold truncated tensor power of the fundamental tilting module (tilting modules are the modules over $U_q(\mathfrak{sl}_2)$ simultaneously filtered by standard and costandard modules; those of non-zero quantum dimension at roots of unity form a semisimple rigid tensor category $\mathcal{C}_{\mathrm{red}}$ with a truncated, associative tensor product $\bar\otimes$). It also established the fusion rule $[L_s(m)]\circ[L_t(n)]=\sum_{|s-t|\le r\le\min(s+t,\,2(\ell-2)-(s+t))}[L_r(m+n)]$ (the sum running over those $r$ with $r\equiv s+t\pmod 2$) for the Grothendieck groups of the $Q_n(\ell)$, and showed that the stable-limit ring $K(Q_\infty)$ is isomorphic to the level-$(\ell-2)$ Verlinde fusion ring of $\widehat{\mathfrak{sl}}_2$, in turn isomorphic to the subring of the fusion algebra of the Virasoro minimal model with central charge $c_{\ell-1}$ generated by the classes $[L(c_{\ell-1},h_{1,s})]$, $1\le s\le\ell-1$. On the rigorous side, Zini and Wang (2018) formulated a precise conjecture relating Temperley-Lieb generators to finite-stage Virasoro operators for the unitary minimal models and verified it only for the Ising case using Ising anyonic chains (one-dimensional chains of anyon fusion spaces whose local algebra is Temperley-Lieb). No construction in the literature produces an action of the Virasoro algebra, or its highest-weight modules, as a limit of the $Q_n(\ell)$ or their modules; the identification remains a Grothendieck-ring-level coincidence, and unitarity - a property of the limiting modules, invisible on $K_0$ - is unexplained.

Problem Statement

Construct a mathematically rigorous passage from the finite-dimensional Jones quotient algebras $Q_n(\ell)=TL_n(q)/R_n(q)$, $|q^2|=\ell$, to the Virasoro minimal unitary models as $n\to\infty$, and prove its characteristic properties. In the formulation left open by Iohara, Lehrer and Zhang - who stated the goal of understanding how the Virasoro algebra arises as a limit of the Temperley-Lieb algebras or of their semisimple quotients, and observed the match between $Q_n(\ell)$ and the minimal unitary series with $c=1-6/(\ell(\ell-1))$ - a complete answer should, for each $\ell\ge4$ (the case $\ell=3$ being degenerate, since $Q_n(3)\cong\mathbb{C}$ and $c=0$ admits no unitary minimal model, and the source's algebraic development accordingly assumes $\ell\ge4$):

1. define an explicit limit, functor, or comparable construction which starts from the algebras $Q_n(\ell)$, or from the modules or categories they generate (for example the stable limit of their fusion structure), and produces modules for the Virasoro algebra with central charge $c=1-6/(\ell(\ell-1))$ on which the Virasoro generators act;
2. identify the highest weights $h_{r,s}$ of the limiting Virasoro modules in terms of the stable labels $t\in\{0,1,\dots,\ell-2\}$ of the simple $Q_n(\ell)$-modules $L_t(n)$; since there are $\ell-1$ stable labels, the identification runs through the weights $h_{1,s}$, $1\le s\le\ell-1$, whose classes $[L(c_{\ell-1},h_{1,s})]$ generate the subring of the minimal-model fusion algebra matched with the stable-limit ring $K(Q_\infty)$;
3. prove compatibility with fusion: the construction must induce the known product $[L_s]\circ[L_t]=\sum_{|s-t|\le r\le\min(s+t,\,2(\ell-2)-(s+t))}[L_r]$ on the stable-limit Grothendieck group, where the sum runs over those $r$ with $r\equiv s+t\pmod 2$, matching the level-$(\ell-2)$ Verlinde ring of $\widehat{\mathfrak{sl}}_2$ and the corresponding subring of the minimal-model fusion algebra; and
4. account for unitarity: prove that the limiting modules carry a positive-definite form compatible with the constructed passage, for instance induced by the $*$-structure carried by Jones' projections.

References

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