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We address some issues relating to a supersymmetric (SUSY) Ward-Takahashi(WT) identity in Sugino's lattice formulation of two-dimensional (2D)$\mathcal{N}=(2,2)$ $SU(k)$ supersymmetric Yang-Mills theory (SYM). Aperturbative argument shows that the SUSY WT identity in the continuum theoryis reproduced in the continuum limit without any operatorrenormalization/mixing and tuning of lattice parameters. As application of thelattice SUSY WT identity, we show that a prescription for the hamiltoniandensity in this lattice formulation, proposed by Kanamori, Sugino and Suzuki,is justified also from a perspective of an operator algebra amongcorrectly-normalized supercurrents. We explicitly confirm the SUSY WT identityin the continuum limit to the first nontrivial order in a semi-perturbativeexpansion.

SUSY WT identity in a lattice formulation of 2D $\mathcal{N}=(2,2)$ SYM