In this John Preskill et al paper, a statement is made in page 1:
We will refer to a gauge theory with fermions transforming as a complex representation of the gauge group as a chiral gauge theory, because the gauged symmetry is a chiral symmetry, rather than a vector-like symmetry (such as QCD).
But my question is: why does a Complex Representation of gauge group imply a Chiral Gauge Theory?
If fundamental representation of SU(3) is a complex representation (with complex conjugate anti-fundamental Rep), then isn't QCD with fundamental representation of SU(3) a perfect counter example where the gauge symmetry is vector-like, instead of chiral???
ps. See this page, or learn that from Wiki:
In physics, a complex representation is a group representation of a group (or Lie algebra) on a complex vector space that is neither real nor pseudoreal. In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real.
For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation.