HOPF ALGEBRA STRUCTURE OF (2+1)-DIMENSIONAL QUANTUM SUPERSPACE, ITS DUAL AND ITS DIFFERENTIAL CALCULUS

We consider a (2+1)-dimensional quantum superspace which has noncommuting coordinates in Manin sense and it was shown that this space has a Hopf algebra structure, i.e. the quantum supergroup, when it is extended by the inverse of the bosonic variable. Differential structures on this space were given by constructing the differential calculus in the sense of Woronowicz. Thus, we deduce that the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is non-cocommutative q-deformed. Finally, dual Hopf superalgebra was given.

Vacuum Expectation Value of the Higgs Field and Dyon Charge Quantisation from Spacetime Dependent Lagrangians

The spacetime dependent Lagrangian formalism of Refs. 1 and 2 is used to obtain a classical solution of Yang–Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field viz. ϕa = A/e, where A is a constant and e is the Yang–Mills coupling (related to the usual electric charge). The solution can also accommodate noncommuting coordinates on the boundary of the theory which may be used to construct D-brane actions. The formalism is also used to obtain the Deser–Gomberoff–Henneaux–Teitelboim results10 for dyon charge quantisation in Abelian p-form theories in dimensions D = 2(p+1) for both even and odd p.

SPACE–TIME DEPENDENT LAGRANGIANS AND WEAK–STRONG DUALITY: SINE-GORDON AND MASSIVE THIRRING MODELS

The formalism of space–time dependent Lagrangians developed in Ref. 1 is applied to the sine-Gordon and massive Thirring models. It is shown that the well-known equivalence of these models (in the context of weak–strong duality) can be understood in this approach from the same considerations as described in Ref. 1 for electromagnetic duality. A further new result is that all these can naturally be linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of noncommuting coordinates residing on the boundaries.