# Instanton Effects in Rank Deformed Superconformal Chern-Simons Theories from Topological Strings

## TL;DR

This paper explores instanton effects in rank deformed superconformal Chern-Simons theories, connecting them to topological string free energies and classifying BPS indices for various deformations.

## Contribution

It introduces a topological string expression for instanton effects in rank deformed theories, extending previous results to new deformation types.

## Key findings

- Classified known diagonal BPS indices by degrees.
- Proposed topological string expressions for combined deformations.
- Connected instanton effects to topological string free energies.

## 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) x U(N+M) x U(N+2M) x U(N+M) and U(N+M) x U(N) x U(N+M) x 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.