Using the D=2 (1/k,1/k) superalgebra (i.e. the two-dimensional generalized supersymmetry generated by spin s=1/k and -1/k; , charge operators Q and satisfying among other conditions and where P and are the light components of the energy--momentum operator), we build a superspace representation of this exotic symmetry. This realization generalizing the usual D=2 (1/2,1/2) supersymmetric one is based on the use of parafermionic variables and of spin s=-1/k and 1/k obeying as well as generalized commutation relation rules. Two-dimensional (1/3,0) and (1/3,1/3) invariant scalar superfield models extending the well-known D=2 (1/2,0) and (1/2,1/2) supersymmetric models are given. The link between this exotic symmetry and the periodic representation of with is worked out. Other features are also discussed.

We study mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties. We mainly discuss the case where the linear sigma A-model contains as many fermionic fields as there are U (1) factors in the gauge group. In the mirror super-Landau-Ginzburg B-model, focus is on the bosonic structure obtained after integrating out all the fermions. Our key observation is that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model, and that the degree of the associated superpotential in the B-model is given in terms of the determinant of the fermion charge matrix of the A-model.

We use local mirror symmetry to study a class of local Calabi-Yau supermanifolds with bosonic sub-variety V b having a vanishing first Chern class. Solving the usual super-CY condition, requiring the equality of the total U (1) gauge charges of bosons Φ b and the ghost like fields Ψ f one b q b = f Q f , as b q b = 0 and f Q f = 0, several examples are studied and explicit results are given for local A r super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where q b = 0, ∀b and f Q f = 0 is also made.

Methods developed for the analysis of integrable systems are used to study the problem of hyper-Kähler metrics building as formulated in D=2, N=4 supersymmetric harmonic superspace. We show in particular that the constraint equation [Formula: see text] and its Toda-like generalizations are integrable. Explicit solutions together with the conserved currents generating the symmetry responsible for the integrability of these equations are given. Other features are also discussed.

Denoting by D = 2(1/3, 1/3) superalgebra the off critical symmetry of the φ 5/7,5/7 perturbation of the c = 6/7 conformal theory, we build a new superspace solution of the (1/3, 1/3) − subalgebra generated by spin ±1/3 charge operators extending the usual (1/2, 1/2) supersymmetry generated by spin ±1/2 charges (Saidi et al). This solution is based on the use of two Grassmann variables instead of one parafermionic variable θ ±1/3 satisfying the cubic nilpotency condition (θ ±1/3 ) 3 = 0. Known results on the c = 6/7 tricritical Potts model are recovered as special features. The relation with N = 2 Landau-Ginzburg models is also discussed.