Topology in 2D non-Abelian Lattice Gauge Theories

In two dimensions, $U(N_c)$ gauge theories exhibit a non-trivial topological structure, while $SU(N_c)$ theories are topologically trivial. Hence, for $G=U(N_c)$ the phase space is divided into topological sectors, characterized by a topological index (a.k.a. topological charge''). These sectors are separated by action barriers, which diverge if the lattice spacing is taken small, resulting in an algorithmic problem known as topological freezing’'. We study these theories in various box sizes and at various couplings. With the help of gradient flow we derive instanton-like solutions for 2D $U(N_c)$ theory with a specific focus on the case of $N_c=2$.