Kac-Moody and Virasoro Algebras in Relation to Quantum Physics

Solitons, τ-functions and hamiltonian reduction for non-Abelian conformal affine Toda theories

“…, r, are the Kac-labels and the dual Kac-labels ofĜ, respectively [27], and a i = a where |λ i are the highest weight states of the fundamental representations ofĜ, andλ i are the corresponding fundamental weights ofĜ. Namely [41] λ 0 = (0, ψ 2 /2, 0) (B.25) λ a = (λ a , l ψ a ψ 2 /2, 0) (B.26)…”

Toward a vertex operator construction of quantum affine algebras

“…Note that in the classical limit α 0 → 0 (or, equivalently q → −1) the coefficient at the nonlocal term I αβ vanishes and we get the "classical" definition of the third generator [1,2]: E α+β = dz : e i(α+β,ϕ) :. The crucial point in our consideration is the Serre identity (8) for U q (sl 3 ) quantum group.…”

“…Vertex operator algebras [1,2,3] in the free field theories [4,5,6] possess a rich mathematical structure. Their connection with the representation theory of affine Lie algebras has been found in [7] and with the theory of quantum groups in [8].…”

“…We assume that ϕ i take values in some appropriate torus. The root lattice associated with the torus gives rise to a Lie algebra g [2]. We consider vertex operators of the conformal dimension one and investigate their commutation relations.…”

D-branes on group manifolds

“…It is general enough to take g as being made only of the Cartan generators [11]: g = exp(iη · H). Then each generator is transformed under this automorphism as…”