Results on formally dual sets in finite abelian groups of size 64 obtained from a graph search algorithm

We shortly present two small results regarding the study formal duality in finite abelian groups as introduced by Cohn, Kumar, Reiher and Sch\"urmann. In particular, we give a new example of a formally self dual set in $\mathbb{Z}_2^2\times\mathbb{Z}_4^2$ and computed nonexistence of primitive formally dual sets of size $8$ in $\mathbb{Z}_8^2$

Dieudonn\'e modules and pp-divisible groups associated with Morava KK-theory of Eilenberg-Mac Lane spaces

We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonn\'e correspondence between bicommutative Hopf algebras and Dieudonn\'e modules. We extend P. Goerss's results on the bilinear products of such Hopf algebras and corresponding Dieudonn\'e modules.Comment: 23 page

S-duality in Abelian gauge theory revisited

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.Comment: LaTeX; 22 pages, no figure