Supermultiplets and Koszul Duality

Abstract

There is a deep and general correspondence connecting constrained objects, superalgebras and supermultiplets. The most natural way of displaying this correspondence is using partition functions and BRST formalism. The partition functions of the constrained objects can be found to be dual to that of a Lie superalgebra and in some cases L1 algebras. The duality reveals itself through the partition functions being each others inverses. We find the pure spinor partition functions in D = 10 to contain the supermultiplet for D = 10 linearised super-Yang-Mills. We find the dual algebra to be an extension of D5 with an odd null root, defining an infinite dimensional graded Lie superalgebra called a Borcherds superalgebra. The algebra is proven to be freely generated by the super-Yang-Mills multiplet from order 3. Further investigation concerns the case of D = 11 supergravity. The dual algebra is no longer just a Lie superalgebra. In addition to the Lie bracket structure there is also, at least, a 3- and a 4-bracket structure. It is conjectured that this algebra, from order 4, is freely generated, under the Lie bracket, by the D = 11 supergravity multiplet.