We are solving for the case of flat superspace some homological problems that were formulated by Berkovits and Howe. (Our considerations can be applied also to the case of supertorus.) These problems arise in the attempt to construct integrals invari
ant with respect to supersymmetry. They appear also in other situations, in particular, in the pure spinor formalism in supergravity.
We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $leq 6$. In particula
r, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.
In the study of the Type II superstring, it is useful to consider the BRST complex associated to the sum of two pure spinors. The cohomology of this complex is an infinite-dimensional vector space. It is also a finite-dimensional algebra over the alg
ebra of functions of a single pure spinor. In this paper we study the multiplicative structure.
We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring.
We study the multiplication and comultiplication in equivariant cohomology of Sato Grassmannian
We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of curves that are defined in terms of Weierstrass points.
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare algebra. We discuss in detail the calculation in dimensions D=10 and D=6. Our methods can be applied to extended supersymmetry algebra and to other dimensions.
