Beyond the Fock Vacuum
in Finite-N Bilocal Algebras of Vector Models
We discuss the finite-N Hilbert space of bilocal operators that underlies bilocal holography of vector models, beyond the sector built on the Fock vacuum. If the microscopic oscillator Fock space carries a multiplicity-free dual action of a compact color group G and a bilocal algebra, Howe duality implies that projection to the derived subgroup [G,G] selects precisely the characters of the abelianization Gab=G/[G,G] that occur in the Fock space. Thus O(N)/SO(N)=Z2 gives two sectors, U(N)/SU(N)=U(1) gives integer determinant-charge sectors, while Sp(N) is perfect and gives no additional sector; Bose/Fermi statistics and the number of modes determine which sectors are actually realized. We construct the epsilon and determinant backgrounds that generate these sectors, together with their finite-N constraints and Casimirs, and find that the two bosonic O(N) sectors are not distinguished by any Casimir. Using group-integral projectors, we derive explicit sector-resolved characters for the O(N) and U(N) theories. Around each new background we solve the constraints with auxiliary oscillators, which form a Fermi sea for bosonic matter. In the free vector model on R×S2 the new sectors consist of baryonic primaries; we determine their energies, degeneracies and spin content exactly at finite N, together with their lowest energy at fixed angular momentum.
