Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

In the so-called (2, 2) theory, which is the U(N)4 circular quiver superconformal Chern-Simons theory with levels (k, 0, −k, 0), it was known that the instanton effects are described by the free energy of topological strings whose Gopakumar-Vafa invariants coincide with those of the local D5 del Pezzo geometry. By considering two types of one-parameter rank deformations U(N)×U(N + M)×U(N + 2M)×U(N + M) and U(N + M)×U(N)×U(N + M)×U(N), we classify the known diagonal BPS indices by degrees. Together with other two types of one-parameter deformations, we further propose the topological string expression when both of the above two deformations are turned on.