v 1 3 F eb 1 99 7 A TRINOMIAL ANALOGUE OF BAILEY ’ S LEMMA AND N = 2 SUPERCONFORMAL INVARIANCE

We propose and prove a trinomial version of the celebrated Bailey’s lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N = 2 superconformal field theory (SCFT). We also establish interesting relations between N = 1 and N = 2 models of SCFT with central charges 3 2 ( 1− 2(2−4ν) 2(4ν) ) and 3 ( 1− 2 4ν ) . A number of new mock theta function identities are derived. 1. Brief review of Bailey’s method and its generalizations. It may come as a surprise that Manchester, England was an ideal setting for pure mathematics during the height of World War II. However, a variety of historical coincidences conspired to make this the case. In particular, mathematics that would later prove extremely valuable in the development of statistical mechanics and conformal field theory (CFT) flourished there. Essentially, Bailey, extending the original ideas of Rogers, came up with a new method [1, 2] of deriving Rogers-Ramanujan type identities during the winter 1943– 44. Hardy who was then editor for the Journal of London Mathematical Society sent a referee report with Dyson’s name on it back to Bailey. Bailey’s reply was immediate. A charming account of Dyson-Bailey collaboration appears in Dyson’s article, Ramanujan Garden [3]. A few years later, Slater, in a study building on Bailey’s work, systematically derived 130 identities of Rogers-Ramanujan type [4,5]. In the last decade, Bailey’s 1)e-mail: [email protected] 2)e-mail: berkov [email protected] Partially supported by National Science Foundation Grant: DMS-9501101.