I will discuss algebraic properties of periodic $sl(n+1|n)$ spin-chains with Heisenberg-like interaction. These chains are made of alternating tensor products of the fundamental and conjugate $sl(n+1|n)$ representations. The algebra of local Hamiltonian densities in the chain is provided by a representation of the affine or periodic Temperley-Lieb algebra at the primitive 6th root of unity. The more detailed analysis was carried out for periodic $sl(2|1)$ spin chains (with H. Saleur, N. Read and R. Vasseur), which describe statistical properties of boundaries of 2D percolation clusters on a torus. In this case, the continuum limit of the chains was identified with a bulk Logarithmic CFT at $c = 0$, which is a fixed point theory of a non-linear sigma model on the complex projective superspace $CP^{1|1}$ in the strong coupling regime. We deduced the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the explicit structure of the full vacuum module/sector of the LCFT, which exhibits Jordan cells of arbitrary rank for the Hamiltonian or the dilatation operator.

Desy Hambourg