We perform a Hamiltonian analysis of the Green-Schwarz sigma model on a supercoset target with Z4m grading. The fundamental Poisson brackets between the spatial component of the flat currents depending on a continuous parameter, which can be thought of as a first step in the complete calculation of the algebra of the transition matrices, are obtained. When m = 1, our results are reduced to the results of the type IIB Green-Schwarz superstring on AdS5×S5 background obtained by Das, Melikyan and Sato.
In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.
In this paper, we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra. We construct a graphical category corresponding to q-boson algebra, q-Fock states correspond to some kind of 1-morphisms, and the graded dimension of the graded vector space of 2-morphisms is exactly the inner product of the corresponding q-Fock states. We also find that this graphical category can be used to categorify q-fermion algebra.
