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

The free fermion system is the simplest quantum field theory which has the symmetry of the Ding-Iohara-Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism, which defines the transformation of generators. We introduce the second set of the fermionic basis (S-dual basis) which implements 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 is proportional to the Hopf link invariant. We also rewrite the general topological vertex in the form of an Awata-Feigin-Shiraishi intertwiner and show that it becomes more symmetric for the duality transformation.