Geometric Quantization and Two Dimensional QCD

Abstract

In this article, we will discuss geometric quantization of 2d QCD with fermionic and bosonic matter fields. We identify the respective large-N c phase spaces as the infinite dimensional Grassmannian and the infinite dimensional Disc. The Hamiltonians are quadratic functions, and the resulting equations of motion for these classical systems are nonlinear. In [33], it was shown that the linearization of the equations of motion for the Grassmannian gave the 't Hooft equation. We will see that the linearization in the bosonic case leads to the scalar analog of the 't Hooft equation found in [36].

Super-Grassmannian and large N limit of quantum field theory with bosons and fermions

“…Using exactly the same methods as in [7,18], we can show that it is closed and nondegenerate. In fact the above form is also a homogeneous two-form invariant under the group action, as can be verified in a simple manner.…”

“…The other possibility is to write the symplectic form in the action and use single valuedness of the path integral as is done in [7]. (We note in passing that there is a factor of 2 missing in the reference [18], due to an error in the conventions, but we scale the Hamiltonian with the same parameter so the large-N c results are the same. The geometric quantization parameter instead should have been 1/N c ).…”

“…However, it is simpler to first look at the linearization where everything decouples (equations for M and N were analyzed in this approximation in previous publications [7,18,6]). We will see that we get the same equations for M, N as in [7,18] in our linearized theory. Let us ignore all the quadratic terms in the equations of motion and all the quadratic terms in the constraints.…”

“…The approach of [7] was extended to the bosonic case in [18] (see also [19] for a similar approach to the problem).…”

Large N limit of SO(N) scalar gauge theory

“…8 Scalar QCD was worked out using the same methods in Ref. 9, where it was shown that the phase space of the theory is an infinite dimensional disk. Originally scalar two-dimensional QCD was worked out by Shei and Tsao in Ref.…”