Super Riemann surfaces: Uniformization and Teichm�ller theory

Primary currents and Riemannian geometry in algebras

“…For sake of truth this problem occurs in the usual conformal models [19], but in the context of W algebras it becomes relevant [20] - [28]. The capital role payed by a W symmetry has been pointed out in many physical fields , such as integrable models [29], [30], string theory [31] and supersymmetry [32] - [35]. Many efforts have been made in that direction and important results have been reached [36] - [46].…”

Noncommutative supertori in two dimensions

“…In a similar way, as the result of the supersymmetric version of the uniformization theorem [17], a supertorus is given by a quotient of the two-dimensional superspace by a subgroup of Osp(N |2) which is the anomaly-free part of the superconformal group [17,16]. The action of this subgroup gives the cycles of the supertorus.…”

“…Although the noncommutative torus is a very well known subject, its supersymmetric version, the noncommutative supertorus, still remains virtually unknown. Commutative supertorus was constructed by Rabin and Freund [16] based on the work of Crane and Rabin [17] on super Riemann surfaces. The supertorus was obtained as the quotient of superplane by a subgroup of Osp(1|2) which acts properly discontinuously on the plane together with the metrizable condition.…”

superconformal symmetries and models

“…In this chapter, we introduce N = 2 superconformal transformations and some related notions [28,29,30,6,14]. To keep supersymmetry manifest, all considerations will be carried out in superspace [31,13,14,8], but the projection of the results to ordinary space will be outlined in the end.…”