On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

Abstract

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra g \mathfrak {g} . The main ingredients in the proof are: Garland’s result on the Lie algebra cohomology of u ^ := g ⊗ t C [ t ] \hat {\mathfrak {u}} := \mathfrak {g}\otimes t\mathbb {C}[t] ; Kostant’s result on the ‘diagonal’ cohomolgy of u ^ \hat {\mathfrak {u}} and its connection with abelian ideals in a Borel subalgebra of g \mathfrak {g} ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.

Lie algebra fermions

“…It will turn out that the Hamiltonian evaluated on an eigenstate equals the level of the state minus the quadratic Casimir 15 . This property was first noted in [39] 16 and a detailed published proof and generalisation can be found in [40]. The proof of [40] is clear.…”

“…15 The level of a state is given by the sum of the lower indices on the fermion operators that create the state. 16 The calculation was omitted though the guiding remark 'The collapsing is considerable' was offered.…”

“…Moreover, we describe an interesting relation of the supersymmetric quantum mechanics to N = 1 supersymmetric conformal field theories in two dimensions. Lastly, a subalgebra of the exterior algebra plays a crucial role in the conjectured description of the chiral ring of pure N = 1 supersymmetric Yang-Mills theories in four dimensions [12][13][14][15][16][17]. More generally, the mathematics that we render more accessible applies to a degree to all quantum mechanical and field theoretical models with Lie algebra valued fermions.…”

Lie Algebra Fermions

“…For all cases, it is known that the glueball operator S = TrW α W α = 2W a 1 W 2a generates the ring. Moreover, there are conjectures that in turn imply the conjecture, which chop up the general problem into potentially more tractable ones [15,16]. However, there is no proof of the conjecture for general Lie algebra g, let alone a uniform proof.…”

“…The zero modes behave as the fermions in the model (2.7) 15. The level of a state is given by the sum of the lower indices on the fermion operators that create the state 16. The calculation was omitted though the guiding remark "The collapsing is considerable."…”