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.

Linear functionals on hypervector spaces

The study of linear functionals, as an important special case of linear transformations, is one of the key topics in linear algebra and plays a significant role in analysis. In this paper we generalize the crucial results from the classical theory and study main properties of linear functionals on hypervector spaces. In this way, we obtain the dual basis of a given basis for a finite-dimensional hypervector space. Moreover, we investigate the relation between linear functionals and subhyperspaces and conclude the dimension of the vector space of all linear functionals over a hypervector space, the dimension of sum of two subhyperspaces and the dimension of the annihilator of a subhyperspace, under special conditions. Also, we show that every superhyperspace is the kernel of a linear functional. Finally, we check out whether every basis for the vector space of all linear functionals over a hypervector space V is the dual of some basis for V.