Decomposition of Level-1 Representations of D_4^(1) With Respect to its Subalgebra G_2^(1) in the Spinor Construction

TL;DR

This paper analyzes how level-1 representations of the affine algebra D_4^(1) decompose into G_2^(1) representations using spinor and vertex operator constructions, coset Virasoro algebras, and explicit decompositions.

Contribution

It provides an explicit construction of the G_2^(1) decomposition of D_4^(1) representations via coset Virasoro modules and spinor methods, extending previous theoretical frameworks.

Findings

Explicit construction of coset Virasoro operators with central charges 1/2 and 7/10.

Decomposition of D_4^(1) representations into G_2^(1) modules using intermediate subalgebras.

Spinor constructions of Virasoro modules with specific central charges.

Abstract

In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertex operator para-algebra (abelian intertwining algebra) V = V^0 \oplus V^1 \oplus V^2 \oplus V^3, whose summands are four level-1 irreducible representations of the affine Kac-Moody algebra D_4^(1). The triality group S_3 = < \sigma,\tau | \sigma^3 = 1 = \tau^2, \tau\sigma\tau = \sigma^{-1} > in Aut(V) was constructed, preserving V^0 and permuting the V^i, for i=1,2,3. V is (1/2)Z-graded where V^i_n denotes the n-graded subspace of V^i. Vertex operators Y(v,z) for v in V^0_1 represent D_4^(1) on V, while those for which \sigma(v) = v represent G_2^(1). We investigate branching rules, how V decomposes into a direct sum of irreducible G_2^(1) representations. We use a two-step process, first decomposing with respect to the intermediate subalgebra B_3^(1), represented by Y(v,z) for \tau(v) = v. There are three vertex operators, Y(\omega_{D_4},z), Y(\omega_{B_3},z), and Y(\omega_{G_2},z) each representing the Virasoro algebra given by the Sugawara constructions from the three algebras. The Goddard-Kent-Olive coset construction [GKO] gives two mutually commuting coset Virasoro representations, provided by the vertex operators Y(\omega_{D_4}-\omega_{B_3},z) and Y(\omega_{B_3}-\omega_{G_2},z), with central charges 1/2 and 7/10, respectively. The first one commutes with B_3^(1), and the second one commutes with G_2^(1). This gives the space of highest weight vectors for G_2^(1) in V as tensor products of irreducible Virasoro modules L(1/2,h^{1/2}) \otimes L(7/10,h^{7/10}). This dissertation explicitly constructs these coset Virasoro operators, and uses them to describe the decomposition of V with respect to G_2^(1). This work also provides spinor constructions of the 7/10 Virasoro modules, and of the two level-1 representations of G_2^(1) inside V.