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The \(gl(1|1)\) super-current algebra: the rôle of twist and logarithmic fields. (English) Zbl 1172.81016

Summary: A free field representation of the \(\text{gl}(1|1)_k\) current algebra at arbitrary level \(k\) is given in terms of two scalar fields and a symplectic fermion. The primary fields for all representations are explicitly constructed using the twist and logarithmic fields in the symplectic fermion sector. A closed operator algebra is described at integer level \(k\). Using a new super spin charge separation involving \(\text{gl}(1|1)_N\) and \(\text{su}(N)_0\), we describe how the \(\text{gl}(1|1)_N\) current algebra can describe a non-trivial critical point of disordered Dirac fermions. Local \(\text{gl}(1|1)\) invariant lagrangians are defined which generalize the Liouville and sine-Gordon theories. We apply these new tools to the spin quantum Hall transition and show that it can be described as a logarithmic perturbation of the \(\text{osp}(2|2)_k\) current algebra at \(k=-2\).

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
22E70 Applications of Lie groups to the sciences; explicit representations