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Recently there has been some controversy in the literature concerning theexistence of a fermion sign problem in the $\mathcal{N}=(2,2)$ supersymmetricYang--Mills (SYM) theories on the lattice. In this work, we address this issueby conducting Monte Carlo simulations not only for $\mathcal{N}=(2,2)$ but alsofor $\mathcal{N}=(8,8)$ SYM in two dimensions for the U(N) theories with N=2,using the new ideas derived from topological twisting followed by geometricdiscretization. Our results from simulations provide the evidence that thesetheories do {\it not} suffer from a sign problem as the continuum limit isapproached. These results thus boost confidence that these new latticeformulations can be used successfully to explore the nonperturbative aspects ofthe four-dimensional $\mathcal{N}=4$ SYM theory.

Investigating the sign problem for two-dimensional $\mathcal{N}=(2,2)$ and $\mathcal{N}=(8,8)$ lattice super Yang--Mills theories