Infinite-dimensional algebras and superalgebras

My present motivation stems from the wish to understand how a well-known conformally invariant theory described by a Wess-Zumino-Witten-Novikov action based on the coset sl(2/1;R)/sl(2/1;R) can provide an insight into the proper treatment of the coupling of two-dimensional gravity to strings.

This has led me, together with my long-term collaborators Peter Bowcock (Durham), Boris Feigin (Landau Institute, Moscow) and Alexei Semikhatov (Lebedev Institute, Moscow) , through the meandres of Representation Theory of Affine Lie Superalgebras.  Most recently, we presented a realisation of the affine Lie superalgebra sl(2/1;C) and of the exceptional affine superalgebra D(2/1;alpha) as vertex operator extensions of two affine sl(2,C) algebras at dual levels k and k', where the duality relation between the levels is given by (k+1)(k'+1)=1. This is a particular instance of the general problem of extending infinite-dimensional Lie algebras by vertex operators, a famous example of which is the extension by vertex operators of a sum of two Virasoro algebras with appropriate central charges (which leads to matter coupled to gravity in the conformal gauge)  (hep-th/9907171).

At the moment, together with Alexei Semikhatov (Lebedev Institute, Moscow) and two of my graduate students Jafar Sadeghi (Durham) and Mehrdad Ghominejad (Durham) ,  we analyse the modular properties  of characters for certain classes of affine sl(2/1;C) fractional level representations. The difficulty here is that the characters do not enjoy periodicity properties which would allow them to be written in terms of elliptic theta functions,  in which case modular properties would be trivial. Very much the same happens in non-unitary N=2 minimal models, and the techniques developed here will hopefully be helpful when revisiting the representation theory of two-dimensional N=4 superconformal algebras which play an important role in String Theory.