The Algebra & Geometry of Modern Physics

Krzysztof Bielas: Some remarks on non-planar Feynman diagrams

Non-planar Feynman diagrams are interesting due to various reasons. The most recent is the new tools for calculating scattering amplitudes in N=4 SYM. Since these methods are applied in general only to planar sector of the theory, the question arises whether it is possible to extend these tools to non-planar diagrams and what the obstacles are. In particular, it is interesting whether categorical structures could allow to give more insight into interplay between planar and non-planar diagrams.

Jerzy Król: Quasimodularity and physics of exotic smooth R4

In the 1980-ties mathematicians established the existence of two infinite families (uncountably many each) of distinct small and large exotic smooth structures on the topological trivial R^4. This phenomenon is possible only in dim. 4 - for other n we have precisely one smooth structure on every R^n. Since then there were rather substantial effort for deriving physical effects driven by these exotic R^4. Small exotic R^4 happened to be connected with the codimension-1 foliations of some 3-manifold and hence, via surgery along a link, of S^3. Connes-Moscovici approach gives the interpretation for the universal Godbillon-Vey class of the foliation as the Eisenstein second series. This is quasimodular object rather than modular one. In the talk I will discuss the quasimodularity issue of exotic small R^4 from the perspective of some susy YM theories on this exotic backgrounds: 1. N=2 SYM and also the effective low energy SW theory, and 2. N=4 topologically twisted (Witten-Vafa) SYM. One indeed finds for the correlation function on small exotic R^4 the expressions which are functions of Eisenstein E_2 series, and for the large case, the quasi-modular 3/2 forms (so called mock theta function of order 7, as proposed originally by Ramanujan). I will discuss possible physical meaning of the results.