Abstract
This is a joint work with I.~Shchepochkina.
The list of simple Lie superalgebras of vector fields over C with polynomial or formal coefficients was obtained by the authors by 1997. There are 35 series and 15 exceptional superalgebras which if we ignore gradings or filtrations can be united into 12-14 series and 5 exceptional families. None of their real forms have a unitary algebra as the linear part and this is, perhaps, the reason why physicists' community ignores Broadhurst-Kac's recent interpretations of the elements of some of the simple exceptional Lie superalgebras in terms of The Standard Model.
Contrariwise, the real forms of stringy or "superconformal" superalgebras (namesakes of the above examples with Laurent polynomials rather than polynomials as coefficients) do have unitary linear parts. Serganova classified all the real forms of stringy superalgebras but published the announcement in a mathematical journal which, though famous, physicists obviously seldom read.
I will show that to classify simple Lie superalgebras of vector fields over R is not "just take the real forms of those over C" and even if it were, it is not as simple as one might think.
We also list differential operators acting in tensor fields over supercircle and invariant with respect to centerless stringy superalgebras
Time permitting, I will review critical dimensions and Shapovalov determinant for modules over stringy Lie superalgebras (work with Grozman)