Alladi Ramakrishnan Hall

Two dimensional QCD with quarks in large gauge representations

Naveen S. Prabhakar

ICTS, Bangalore

QCD in 1+1 dimensions is a great toy model for understanding the
various intricate phenomena exhibited by gauge theories. In 1974,
SU(N) QCD in 1+1 dimensions coupled to quarks in the fundamental
representation was solved by 't Hooft in the large N limit: an elegant
integral equation was obtained for the meson spectrum by directly
summing the diagrams in the Bethe-Salpeter equation. The integral
equation was shown to have a discrete spectrum, thereby explicitly
demonstrating quark confinement.

The same integral equation was derived by Dhar et. al. in 1994 by
observing that the phase space of the quark can be described as a
particular coadjoint orbit of a $W_\infty$ algebra generated by the
gauge invariant meson operators. The large N limit was interpreted as
a semi-classical limit and the equation of motion for the fluctuations
along the coadjoint orbit was found to be 't Hooft's integral
equation. We work in a different, novel limit of QCD that exists only
in 1+1 dimensions where the representation of the quarks is taken to
be large. We treat the case of SU(2) QCD explicitly, with the quarks
being in the spin J representation of SU(2) with J taken to be
large. In this limit, we use $W_\infty$ algebra techniques and derive a
't Hooft-like integral equation for the meson spectrum.