Prepotential formulation of SU(3) lattice gauge theory

TL;DR

This paper reformulates SU(3) lattice gauge theory using prepotential harmonic oscillators, simplifying the Mandelstam constraints and enabling a local, gauge-invariant description of the Hilbert space and loop states.

Contribution

It introduces a novel prepotential formulation of SU(3) lattice gauge theory that simplifies constraints and generalizes to SU(N).

Findings

Mandelstam constraints become local and simple.

Hilbert space is equivalent to prepotential formulation with Sp(2,R) constraints.

Construction of linearly independent SU(3) loop states is achieved.

Abstract

The SU(3) lattice gauge theory is reformulated in terms of SU(3) prepotential harmonic oscillators. This reformulation has enlarged $SU(3)\otimes U(1) \otimes U(1)$ gauge invariance under which the prepotential operators transform like matter fields. The Hilbert space of SU(3) lattice gauge theory is shown to be equivalent to the Hilbert space of the prepotential formulation satisfying certain color invariant Sp(2,R) constraints. The SU(3) irreducible prepotential operators which solve these Sp(2,R) constraints are used to construct SU(3) gauge invariant Hilbert spaces at every lattice site in terms of SU(3) gauge invariant vertex operators. The electric fields and the link operators are reconstructed in terms of these SU(3) irreducible prepotential operators. We show that all the SU(3) Mandelstam constraints become local and take very simple form within this approach. We also discuss the construction of all possible linearly independent SU(3) loop states which solve the Mandelstam constraints. The techniques can be easily generalized to SU(N).