Chiral Fermions on Quantum Four-spheres

We construct wave functions and Dirac operator of spin 1/2 fermions on quantum four-spheres. The construction can be achieved by the q-deformed differential calculus which is manifestly SO(5)q covariant. We evaluate the engenvalue of the Dirac operator on wave functions of the spinors and show that we can define the chiral fermions in such a way that the massless Dirac operator anti-commutes with γ5. e-mail address: [email protected] e-mail address: [email protected] The structure of quantum groups[1][2] has been appeared in many aspect of the field theories. As geometrical counterparts of quantum groups, we have non-commutative geometry[3][4]. The quantum manifolds have been investigated by physician as possible deformations of the space-time[5][6]. However, up to now there is no priori reason why we have to consider quantum manifolds. It seems of no use to consider quantum manifolds instead of classical manifolds. As an attempt to find a reasonable solution, the use of quantum spheres as a regularization of field theories was proposed [8]. Actually, it is shown that the number of wave functions of the scalar fields on quantum spheres can be finite when q is a root of unity by using periodic representations. The parameter is determined by the radius of the spheres and cutoff scale. At low energy, the observable reduces to the ones on classical spheres. We can thus regard the quantum manifold as an effective prescription of the strong quantum fluctuation of space-time at microscopic level. Of course, the investigation of quantum field theories on quantum spheres was far from complete. The calculation of the amplitudes or full second quantization are importaint problem which we should solve for the understanding of the field thoeries on quantum spheres. Before analyzing these questions, we should first construct the fields with internal spin on quantum spheres. Our aim of this paper is to construct spin 1/2 fermions and dirac operators on quantum four-spheres. Since we cannot change variables freely on quantum spheres, it is necessary to use a formulation which is manifestly covariant under the quantum groups. For classical four-spheres, there exist such a formulation. The manifest covariant formulation of the spinors and gauge fields was formulated long ago by Adler [17]. The instanton-(ant-instanton) solution was shown to be expressed elegantly in ref.[18]. We therefore consider the quantum version of the formulation. In the case of quantum twospheres, we have already found the dirac operators[20]. The advantage of this formulation is that we can easily consider the higher dimensional extension, which we are going to pursue.