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.