On Cachazo–Douglas–Seiberg–Witten Conjecture for Simple Lie Algebras
نویسنده
چکیده
Let g be a finite dimensional simple Lie algebra over the complex numbers C. Consider the exterior algebra R := ∧ (g⊕ g) on two copies of g. Then, the algebra R is bigraded under R = ∧p(g)⊗∧q(g). The diagonal adjoint action of g gives rise to a g-algebra structure on R compatible with the bigrading. There are three ‘standard’ copies of the adjoint representation g in R. Let J be the (bigraded) ideal of R generated by these three copies of g (in R) and define the bigraded g-algebra A := R/J. The Killing form gives rise to a g-invariant S ∈ A. Motivated by supersymmetric gauge theory, Cachazo–Douglas–Seiberg–Witten made the following conjecture. Conjecture. (i) The subalgebra A of g-invariants in A is generated, as an algebra, by the element S. (ii) S = 0. (iii) Sh−1 6= 0, where h is the dual Coxeter number of g. The aim of this talk is to give a uniform proof of the above conjecture part (i). This theorem is proved by using Garland’s result on the Lie algebra cohomology of û := g ⊗ tC[t]; Kostant’s result on the ‘diagonal’ cohomology of û and its connection with abelian ideals in b; and a certain deformation of the singular cohomology of Y introduced by Belkale–Kumar. 3:00 3:50 pm HA 335 Faculty and Students are invited to attend.
منابع مشابه
On the Cachazo-douglas-seiberg-witten Conjecture for Simple Lie Algebras, Ii
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تاریخ انتشار 2006