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

Abstract

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 D 5 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.

Hanany-Witten transition in quantum curves

“…which played an important role in computing the partition function of the super Chern-Simons matrix model in [20].…”

“…Although there are multiple ranks, as we have mentioned around (2.1) and (2.2) for the corresponding brane configuration, the overall number of D3-branes decouples from the other relative numbers in the Hanany-Witten transition and we naturally identify this overall rank as the particle number to be dualized. Also, as noted in [20], for the correspondence to the topological string theory, we need to fix the power of the fugacity to be one of the ranks, which we identify as ¶ The phase factor is a natural generalization from those of the ABJM matrix model. The sign function is defined by sign(k a ) = (+1, 0, −1) for k a = (+k, 0, −k) respectively.…”

“…The overall phase was not investigated in [20]. Hence, strictly speaking, to discuss the correspondence to topological strings, we need to take the absolute value for the partition function here and later for example in (5.21) and (5.22).…”

“…To understand the relation in more details, rank deformations of the (2, 2) model were studied in [20]. Combined with the results obtained from rank deformations of the (1, 1, 1, 1) model [21] through the Hanany-Witten transition [22], it was found that the parameter spaces of both models are connected smoothly.…”

“…Combined with the results obtained from rank deformations of the (1, 1, 1, 1) model [21] through the Hanany-Witten transition [22], it was found that the parameter spaces of both models are connected smoothly. Among others it was pointed out that, though in rank deformations we have several ranks appearing, for the relation to topological strings to work correctly, we have to fix the power of the fugacity so that it matches to one of the ranks in defining the grand canonical partition function [20]. Then, the grand canonical partition function with rank deformations is described by the free energy of topological strings if we assume that the BPS indices are split suitably.…”