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# Advances in Theoretical and Mathematical Physics

## Volume 17 (2013)

### Number 6

### On twisted large $N = 4$ conformal superalgebras

Pages: 1393 – 1415

DOI: http://dx.doi.org/10.4310/ATMP.2013.v17.n6.a6

#### Authors

#### Abstract

We explicitly compute the automorphism group of the large $N = 4$ conformal superalgebra and classify the twisted loop conformal superalgebras based on the large $N = 4$ conformal superalgebra. By considering the corresponding superconformal Lie algebras, we validate the existence of only, two (up to isomorphism) such algebras as described in the physics literature. Our approach is based on viewing the objects to be classified as “étale twisted forms” of objects over the Laurent polynomial ring $\mathbb{C}[t \pm 1]$. This allows methods from non-abelian cohomology (torsors) to enter into the picture. It is worth pointing out that the group of automorphisms of the large $N = 4$ conformal superalgebra is larger than the one described in the physics literature. Remarkably enough, both groups have the same étale cohomology over $\mathbb{C}[t \pm 1]$ which explains the agreement on the classification of the corresponding superconformal Lie algebras).