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Defining chiral gauge theories on a 2-d lattice

Defining chiral gauge theory on a finite lattice has been a longstanding difficult problem. In the Ginsparg-Wilson formalism, the very existence of such theories are intriguingly tied to the presence or absence of gauge anomalies, and the connection arises from a topological consideration. It has been fully understood that abelian chiral gauge theories can be defined in this formalism if and only if the fermion content is anomaly-free. But, for nonabelian gauge theories, the complete understanding is still missing. In 2-d, significant simplifications allow one to go a bit further and treat the nonabelian theories much more easily. In this talk, I will review the challenge for defining chiral lattice gauge theories in the Ginsparg-Wilson formalism, discuss the topology of the gauge field configuration space in 2-d, and explain how the gauge anomaly-free condition arises as a topological requirement for the theory to be self-consistent when the gauge group is an arbitrary compact Lie group.