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arXiv:1211.1067

A q-Virasoro Algebra at roots of unity, Free Fermions and Temperley Lieb Hamiltonians

Alessandro Nigro

Published Nov 5, 2012• 1 citation
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Abstract

In this work we introduce a novel q-deformation of the Virasoro algebra expressed in terms of free fermions, we then realize that this algebra, when the deformation parameter is a root of unity can be realized exactly on the lattice. We then study the relations existing between this lattice deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.

DOI
10.48550/arxiv.1211.1067

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In this work we introduce a novel q-deformation of the Virasoro algebra expressed in terms of free fermions, we then realize that this algebra, when the deformation parameter is a root of unity can be realized exactly on the lattice. We then study the relations existing between this lattice deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.
In this work, we consider the q-deformation of the Virasoro algebra [M. Chaichian and P. Presnajder, Phys. Lett. B 277, 109 (1992)] expressed in terms of free fermions, and we then realize this algebra, when the deformation parameter is a root of unity, on the lattice in a truncated form in terms of the Clifford algebra of Γ matrices. For this finite size truncation, the commutation relations of the Deformed algebra hold exactly albeit without central extension term. We then study the relations existing between this lattice truncation of the deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.
We investigate in details how the Virasoro algebra appears in the scaling limit of the simplest lattice models of XXZ or RSOS type. Our approach is straightforward but to our knowledge had never been tried so far. We simply formulate a conjecture for the lattice stress-energy tensor motivated by the exact derivation of lattice global Ward identities. We then check that the proper algebraic relations are obeyed in the scaling limit. The latter is under reasonable control thanks to the Bethe-ansatz solution. The results, which are mostly numerical for technical reasons, are remarkably precise. They are also corroborated by exact pieces of information from various sources, in particular Temperley-Lieb algebra representation theory. Most features of the Virasoro algebra (like central term, null vectors, metric properties...) can thus be observed using the lattice models. This seems of general interest for lattice field theory, and also more specifically for finding relations between conformal invariance and lattice integrability, since basis for the irreducible representations of the Virasoro algebra should now follow (at least in principle) from Bethe-ansatz computations.
All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.
qqq-Deformed Classical Lie Algebras and their Anyonic Realization
All classical Lie algebras can be realized \\`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their qqq-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter qqq is a phase related to the anyonic statistical parameter. A crucial r\\^ole in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case qqq can also be real.
Abstract We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of W0,q±2 {W}_{0,{\mathfrak{q}}^{\pm 2}} W0,q±2​ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in q±2 {\mathfrak{q}}^{\pm 2} q±2 q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.
This paper is part of an effort to gain further understanding of 2D logarithmic conformal field theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product ), the best known (although maybe not the most relevant physically) LCFTs in the continuum are characterized by a W-algebra symmetry, whose presence is powerful, but whose role as a ‘symmetry’ remains mysterious. We explore here the origin of this symmetry in the underlying lattice models. We consider XXZ spin chains for a root of unity, and argue that the centralizer of the ‘small’ quantum group goes over the W-algebra in the continuum limit. We justify this identification by representation theoretic arguments, and give, in particular, lattice versions of the W-algebra generators. In the case , which corresponds to symplectic fermions at central charge , we provide a full analysis of the scaling limit of the lattice Virasoro and W generators, and show in details how the corresponding continuum Virasoro and W-algebras are obtained. Striking similarities between the lattice W algebra and the Onsager algebra are observed in this case.