摘要 : We work out the factorization of the 3d superconformal index for \( \mathcal{N}=2 \) U(N c ) gauge theory with one adjoint chiral ... 展开 We work out the factorization of the 3d superconformal index for \( \mathcal{N}=2 \) U(N c ) gauge theory with one adjoint chiral multiplet as well as N f fundamental, N a anti-fundamental chiral multiplets. Using the factorization, one can prove the Seiberg-like duality for \( \mathcal{N}=4 \) U(N c ) theory with N f hypermultiplets at the index level. We explicitly show that monopole operators violating unitarity bound in a bad theory are mapped to free hypermultiplets in the dual side. For \( \mathcal{N}=2 \) U(N c ) theory with one adjoint matter X, N f fundamental, N a anti-fundamental chiral multiplets with superpotential W = trX n+1, we work out Seiberg-like duality for this theory. The index computation provides combinatorial identities for a dual pair, which we carry out intensive numerical checks 收起