THE FULL STRUCTURE OF QUANTUM N=2 SUPER-$W_3^{\left( 2 \right)} $ ALGEBRA

Abstract

We present the complete structure of the nonlinear N = 2 super extension of PolyakovBershadsky, W algebras. For c → ∞ limit, the algebra reduces to the classical one, which has been studied previously. The 'hybrid' field realization of this algebra is also discussed. *

QUANTUM N = 2${\rm SUPER}-W_3^{\left( 2 \right)} $ ALGEBRA IN SUPERSPACE

“…In the present article we show that the quantum N = 2 super W (2) 3 algebra of ref. [6] also admits a similar superfield description with respect to the modified N = 2 SCA. We give the relevant SOPEs in the explicit form and compare them with the classical expressions of ref.…”

“…This extension comprises four additional fermionic currents with non-canonical integer spins (1,1,2,2), besides N = 2 SCA and W (2) 3 algebra as two different subalgebras. For the quantum case, it was further studied in [6]. A new feature of the quantum case is the necessity to include, in the r.h.s.…”

“…The reason is that OPEs between spin 1/2 currents do not contain central terms the presence of which is crucial for these currents to be decoupled [9]. By the same reason one cannot decouple the spin 1/2 supercurrents in the superfield form of N = 2 super W Let us also assume that all the SOPEs which in the classical case involve no nonlinearities retain their structure in the quantum case too (an analogous assumption in the component approach proved to be true [6]). Then the remaining four supercurrents have the abovementioned spins with respect to this N = 2 SCA…”

“…After inclusion of all possible composite currents with undetermined structure constants we are left with more than 200 terms, in contrast to the classical consideration. In the component approach [6], we have experienced that the quantum Jacobi identities are not satisfied if one specializes only to those algebraic structures which are present at the classical level. Nonetheless, we will see that only three extra composite currents finally survive.…”

“…This difficulty can be got round by the following argument. Using the correspondence (11) and resorting to the component results of [6] we are already guaranteed that the Jacobi identities…”