superalgebra

and

supergeometry

# Contents

## Idea

The formulation of the sigma-model 2d CFTs that are used in (super) string theory had originally been of two types:

1. In the RNS-string forumlation the worldsheet is a supermanifold, while the target space is an ordinary manifold.

The advantage of this formulation is that its quantization is comparatively tractable: it is a 2-dimensional superconformal field theory, coming with a super vertex operator algebra.

The disadvantage is that, as one says, “target space supersymmetry is not manifest”: as described at string theory, every 2d CFT gives rise to an effective background quantum field theory (that quantum field theory whose S-matrix approximates the S-matrix obtained by evaluating the given CFT correlators on all possible surfaces and summing over the conformal moduli space and all genera). One can identify explcitly in the vertex operator algebra of the 2d CFT those operators that correspond to the fields of this effective background field theory (the vertex operators). One finds that these fields are those of a theory of supergravity and super Yang-Mills theory. So in particular there is a super Lie algebra acting on them. But this “supersymmetry” of the effective target space theory appears like an accident if one unwraps all this: there is no structural one-line argument (known) that would guarantee that the effective background quantum field theory of a 2d super CFT is itself supersymmetric.

2. In the GS-string? sigma-model formulation, the worldsheet is an ordinary manifold, but target space is a supermanifold.

This has the immediate advantage that target space supersymmetry is now “manifest”: as described at supergravity it is just the super-diffeomorphism invariance of the theory, and the sigma-models in question all manifestly have this property.

The big disadvantage is, that it is not known how to quantize this system. The reason is that the standard procedure for quantization shows that the GS-type sigma models are constrained? systems with second-class constraints. Little to nothing is known how to deal with that.

Berkovits was motivated by the desire to find a formulation of the superstring sigma model CFT that would combine the advantages of both the RNS formulation and the Green-Schwarz formulation. The goal was to have a CFT that looked like a free field theory (as the RNS string does for a flat Minkowski background, but as the GS string does not) if one just looked at a small part of the range of its fields, but which was globally constrained: the spinorial worldsheet fields here are maps into a space of pure spinor?s that locally lools like a Euclidean space, but globally has a cone geometry (…details…).

Berkovits originally wrote down some more or less ad-hoc expressions. Later it was understood that what described is a sheaf of vertex operator algebras (in general, not a sheaf but a stack, in fact a gerbe) on target space: to each contractible open patch of target space is associate the vertex operator algebra of a free sigma-model CFT whose fields take values just in that patch, such that on overlaps these vertex operator algebras glue in some way.

This at once made the previously mathemtically rather unjustified approach make close contact with the developing theory of chiral deRham complex?, which is one of the most-studied examples of sheaves of vertex operator algebras.

## References

A good reference that explains the sheaf of vertex operator algebra perspective on the Berkovits superstring is

• Nikita Nekrasov, Lectures on curved $\beta$-$\gamma$ systems, pure spinors, and anomalies (arXiv)

The standard reference on the closely related mathematical theory of the chiral deRham complex is