Equivalence of Conformal Superalgebras to Hamiltonian Superoperators1

Since 1970s, Lie algebras have played more important and extensive roles in nonlinear partial differential equations and theoretical physics than they did before. One of the most interesting examples is the birth of the theory of Hamiltonian operators in middle 1970s, which was a work of Gel’fand, Dikii and Dorfman (cf. [GDi1-2], [GDo]). The existence of certain Hamiltonian operators associated with a nonlinear evolution equation implies its complete integrability. Another interesting example is the theory of vertex operator algebras introduced by Borcherds [Bo] (in initial form) and by Frenkel, Lepowsky and Meurman [FLM] (in revised form) in middle 1980s, in order to solve the problem of the moonshine representation of the Monster group. It is clear now that vertex operator algebras are the fundamental algebraic structures in conformal field theory. Both Hamiltonian operators and vertex operator algebras are essentially algebraic objects with one-variable structure. We observed that there should be a connection between Hamiltonian operators and vertex operator algebras many years ago. Kac [K1] introduced a concept of “conformal superalgebra” which is the local structure of a “super conformal 1991 Mathematical Subject Classification. Primary 17A 30, 17A 60; Secondary 17B 20, 81Q 60 Research supported by Hong Kong RGC Competitive Earmarked Research Grant HKUST709/96P.