Evolution of massive population III stars with rotation and magnetic fields

[Abridged] We present a new grid of massive population III star models including the effects of rotation on the stellar structure and chemical mixing, and magnetic torques for the transport of angular momentum. Based on the grid, we also present a phase diagram for the expected final fates of rotating massive Pop III stars. Our non-rotating models become redder than the previous models in the literature, given the larger overshooting parameter adopted in this study. In particular, convective dredge-up of the helium core material into the hydrogen envelope is observed in our non-rotating very massive star models (>~200 Msun), which is potentially important for the chemical yields. On the other hand, the stars become bluer and more luminous with a higher rotational velocity. With the Spruit-Tayler dynamo, our models with a sufficiently high initial rotational velocity can reach the critical rotation earlier and lose more mass as a result, compared to the previous models without magnetic fields. The most dramatic effect of rotation is found with the so-called chemically homogeneous evolution (CHE), which is observed for a limited mass and rotational velocity range. CHE has several important consequences: 1) Both primary nitrogen and ionizing photons are abundantly produced. 2) Conditions for gamma-ray burst progenitors are fulfilled for an initial mass range of 13 - 84 Msun. 3) Pair instability supernovae of type Ibc are expected for 84 -190 Msun and 4) Both a pulsational pair instability supernova and a GRB may occur from the same progenitor of about 56 - 84 Msun, which might significantly influence the consequent GRB afterglow. We find that CHE does not occur for very massive stars (> 190 Msun), in which case the hydrogen envelope expands to the red-supergiant phase and the final angular momentum is too low to make any explosive event powered by rotation.