# A note on S-dual basis in free fermion system

## TL;DR

This paper introduces a new S-dual fermionic basis in free fermion systems, linking duality transformations to the Hopf link invariant and enhancing symmetry in topological vertex formulations.

## Contribution

It constructs the S-dual basis implementing S-duality in free fermion systems, connecting it to Hopf link invariants and reformulating the topological vertex symmetrically.

## Key findings

- Inner product between bases relates to Hopf link invariant
- S-dual basis acts as Fourier dual to standard basis
- Topological vertex becomes more symmetric under duality

## Abstract

Free fermion system is the simplest quantum field theory which has the symmetry of Ding-Iohara-Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism which defines the transformation of generators. In this note, we introduce the second set of the fermionic basis (S-dual basis) which implement the duality transformation. It may be interpreted as the Fourier dual of the standard basis, and the inner product between the standard and the S-dual ones is proportional to the Hopf link invariant. We also rewrite the general topological vertex in the form of Awata-Feigin-Shiraishi intertwiner and show that it becomes more symmetric for the duality transformation.