Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

TL;DR

This paper computes the exact spectra of conformal weights in certain two-dimensional supersymmetric nonlinear sigma models, revealing their non-rational conformal field theories and connections to statistical mechanics and quantum Hall systems.

Contribution

It provides the complete spectra of conformal weights for these models and introduces a novel method involving lattice, loop, and Coulomb gas models to analyze strong-coupling regimes.

Findings

Exact conformal weights are rational but the theories are non-rational.

Most theories lack affine superalgebra structures in their chiral algebras.

Results apply to models like percolation, polymers, and quantum Hall transitions.

Abstract

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[U(1)\times U(n+m-1|n)] \cong CP^{n+m-1|n} (projective superspaces) and OSp(2n+m|2n)/OSp(2n+m-1|2n) \cong S^{2n+m-1|2n} (superspheres), n, m integers, -2\leq m\leq 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau-Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder.