The patching of critical points using quantum group
  Following our recent conjecture to model the phenomenona of antiferromagnetism and superconductivity by quantum symmetry groups, we discuss in the present note how to construct a workable scenario using this symmetry. In particular we propose to patch the relevant critical points. This means we identify fixed points, corresponding to various $k$ or $q$ [since the two are related] and make expansion around these points, to control these expansion we can impose gauge structure and we thus arrive at quantum group based gauge theory or collection of classical gauge theories which represent the condensed matter system such as cuprates. This is different than ordinary gauge theories in several ways, for in ordinary field theory one has well-defined critical point here the critical points are not unique or simple. In short the real transition than in condensed matter system is represented by collection or {\it ensemble average} of the several chosen critical points which come from ordinary field theory [gauge theory].   This idea may reveal the connection between Hubbard model and gauge theory and string theory.   In short it can lead to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians. 
Sher Alam (ETL), M. O. Rahman (GUAS & Kek), M. Ando (GUAS & Kek), S.   B. Mohamed(ETL) and T. Yanagisawa(ETL)