We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and their corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

We try to determine the partition function of N = 4 ADE gauge theory on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies gap condition.

Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal sixmanifold M with static SU(2) structure and mirror M̂ , it is argued that the product M×M̂ is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and M̂ . The mirror map is then realized by T-dualizing the fibers. Mirror-symmetric properties of the fluxes, both geometric an...

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-proje...

Two distinguished types of semidualizing complexes are the shifts underlying rings and dualizing complexes. Let C be a complex for an analytically irreducible local ring R set n:=supC d:=dimRC. We show that is quasi-isomorphic to shift if only nth Betti number one. Also, we dth Bass

The geometry of the Lubin-Tate space of deformations of a formal group is studied via anétale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory.