We consider antiPoisson superalgebras realized on the smooth Grassmann-valued functions with compact supports in R^n and with the grading inverse to Grassmanian parity. The lower cohomologies of these superalgebras are found.
Let $ 0rightarrow mathfrak{a} rightarrow mathfrak{e} rightarrow mathfrak{g} rightarrow 0$ be an abelian extension of the Lie superalgebra $mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $mathfrak{a}$ and lifting
endomorphisms of $mathfrak{g}$ to certain endomorphisms of $mathfrak{e}$. We connect these problems to the cohomology of $mathfrak{g}$ with coefficients in $mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.
Poisson superalgebras realized on the smooth Grassmann valued functions with compact support in R^n have the central extensions. The deformations of these central extensions are found.
We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominato
r identities for the chiral partition functions of the Conway module $V^{s atural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.
Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on R^2 are investigated under suitable continuity restrictions on cochains. The zeroth, first, and second cohomology spaces in the adjoin
t representation of the Poisson superalgebra are found for the case of a nondegenerate constant Poisson superbracket.
We compute the quantum cohomology of symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb b
ranch due to complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds derived by other means. We also explain why a (0,2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.