The Complete structure of the nonlinear W 4 and W 5 algebras from quantum Miura transformation

Starting from the well-known quantum Miura transformation for the Lie algebra An, we compute explicitly the OPEs for n = 3 and 4. The primary fields with spin 3, 4 and 5 are found (for general n). By using these primary fields and the OPEs from quantum Miura transformation, we derive the complete structure of the nonlinear W4 and W5 algebras. It is known that the quantum Miura transformation for the Lie algebra An ≃ sl(n + 1) gives a quadratic nonlinear algebra [1]. This algebra is believed to be identical with the nonlinear extended conformal algebra Wn+1, generated by fields Wk’s with the integer k ranging from 2 to n + 1. For n = 1 and n = 2, this gives the Virasoro algebra and the well-known Zamolodchikov’s nonlinear W3 algebra [2]. For the general case such identification is not established explicitly. The problem with this identification comes from the fact that the basis fields in the quantum Miura transformation are not primary fields and the higher spin fields in Wn are all primary fields (by definition). It is still an important open problem to find a primary basis in the quantum Miura transformation. Given the difficulty of this problem, in this paper we will establish such identification for W4 and W5 (commonly known as W4 and W5 algebras) by explicitly computing the operator product expansions (OPEs). In another word we will derive the complete structure of the nonlinear W4 and W5 algebras directly from the quantum Miura transformation. In fact the structure of the W4 algebra is known in literature [3,4]. So our derivation serves as a non-trivial check to their results. The method we used was then applied to derive the more complicated W5 algebra. As a first remark we note that most of our computations are done by computer symbolic calculation Mathematica [5]. There exists also a Mathematica package for computing and simplifying OPEs [6] 1) , but I didn’t make use of it in this paper. 1. The Quantum Miura Transformation Let {~εi, i = 1, 2, · · · , n + 1} be a set of vectors in an n-dimensional space. They are normalized as ~εi · ~εj = δij − 1 n + 1 , (1) and satisfy the constraint n+1