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1 Definition
2 BRST cohomology as a relative cohomology

We will now extend the superconformal algebra to some infinite-dimensional algebra, just like was extended to SYM.

    1 Definition

    2 BRST cohomology as a relative cohomology

1 Definition

We will take two copies and of the SYM algebra (7), add a copy of , and glue them all together:






In other words:


The consistency of this definition can be verified using the general theory of quadratic-linear algebras, the PBW theorem by Polishchuk-Positselski-Braverman-Gaitsgory.

Since the associated graded algebra is Koszul, it is enough to verify the Jacobi identity for three elementary generators. But for any three elementary generators, the commutation relations are the same as in the twisted loop algebra .

2 BRST cohomology as a relative cohomology

The analogue of (17) is:


It involves relative Lie algebra cohomology. The cochain complex consists of -invariant linear functions:

Therefore an element of is a function of Lie-algebra valued variables with taking values in :

The actual map between and goes as follows. Given a representative cocycle , the representative of the corresponding element in is obtained by the substitution of in place of :