On the Cachazo-douglas-seiberg-witten Conjecture for Simple Lie Algebras

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. We prove the conjecture for type G2, and also verify a consequence of the conjecture in the general case. 1. The CDSW conjecture Let g be a simple finite dimensional Lie algebra over C. We fix an invariant form on g and do not distinguish g and g∗. Let g be the dual Coxeter number of g. Consider the associative algebra R = ∧(g ⊕ g). This algebra is naturally Z+bigraded (with the two copies of g sitting in degrees (1,0) and (0,1), respectively). The degree (2,0),(1,1), and (0,2) components of R are ∧g, g⊗ g,∧g, respectively, hence each of them canonically contains a copy of g. Let I be the ideal in R generated by these three copies of g, and A = R/I. The associative algebra A may be interpreted as follows. Let Πg denote g regarded as an odd vector space. Then R may be thought of as the algebra of regular functions on Πg×Πg. We have the supercommutator map {, } : Πg×Πg → g given by the formula {X,Y } = XY +Y X , where the products are taken in the universal enveloping algebra (this is a morphism of supermanifolds). The ideal I in R is then defined by the relations {X,X} = 0, {X,Y } = 0, {Y, Y } = 0, so A is the algebra of functions on the superscheme defined by these equations. In [CDSW,W], the following conjecture is proposed, and proved for classical g: Conjecture 1.1. (i) The algebra A of g-invariants in the algebra A is generated by the unique invariant element S of A of degree (1,1) (namely, S = Tr|V (XY ), where V is a non-trivial irreducible finite dimensional representation of g). (ii) S = 0. (iii) S 6= 0. Here we prove the conjecture for g of type G2. We believe that the method of the proof should be relevant in the general case. We also prove in general the following result, which is a consequence of the conjecture. Proposition 1.2. Any homogeneous element of A is of degree (d, d) for some d; therefore, the algebra A is purely even, and the natural action of sl(2) on it (by linear transformations of X,Y ) is trivial. Remark. Conjecture 1.1 has the following cohomological interpretation. Let g[x, y] denote the Lie algebra of polynomials of x, y with values in g. Coniser the algebra of relative cohomology H∗(g[x, y], g,C). It is graded by the cohomological