Root Of Unity Limit Research Articles

Q-Virasoro algebra at root of unity limit and 2d-4d connection

We propose a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space ℝ4/ℤr. This proceeding is based on arXiv:1308.2068

2d–4d connection between q-Virasoro/W block at root of unity limit and instanton partition function on ALE space

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five-dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five-dimensional instanton partition function onto that on the ALE space R4/Zr. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q→−1, t→−1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q,t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n,r) that accomplishes the automatic projection through the limit.

Spinless Basis for Spin-Singlet FQH States

We investigate an alternative description of the SU(M)-singlet FQH state by using the spinless basis. The SU(M)-singlet Halperin state is obtained via the q-deformation of the Laughlin state and its root of unity limit, by applying the Yangian Gelfand-Zetlin basis for the spin Calogero-Sutherland model. The squeezing rule for the SU(M) state is also investigated in terms of the spinless basis.

Quantum superalgebras at roots of unity and non-Abelian symmetries of integrable models

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang–Baxter equation. As is well known, the latter can be generically constructed from quantum affine superalgebras Uq(ĝ). These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed. If q is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-Abelian subalgebras which are translation invariant and commute with the transfer matrix and therefore with all charges of the model. In the line of argument, we introduce the restricted quantum superalgebra Uresq(ĝ) and investigate its root of unity limit. We prove several new formulae involving supercommutators of arbitrary powers of the Chevalley–Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.