  We analyze the ac Stark shift of a cesium atom interacting with
far-off-resonance guided light fields in the nanofiber-based two-color optical
dipole trap realized by Vetsch \textit{et al.} [Phys. Rev. Lett. \textbf{104},
203603, (2010)]. Particular emphasis is given to the fictitious magnetic field
produced by the vector polarizability of the atom in conjunction with the
ellipticity of the polarization of the trapping fields. Taking into account the
ac Stark shift, the atomic hyperfine interaction, and a magnetic interaction,
we solve the stationary Schr\"odinger equation at a fixed point in space and
find Zeeman-state-dependent trapping potentials. In analogy to the dynamics in
magnetic traps, a local degeneracy of these state-dependent trapping potentials
can cause spin flips and should thus be avoided. We show that this is possible
using an external magnetic field. Depending on the direction of this external
magnetic field, the resulting trapping configuration can still exhibit
state-dependent displacement of the potential minima. In this case, we find
nonzero Franck-Condon factors between motional states of the potentials for
different hyperfine-structure levels and propose the possibility of microwave
cooling in a nanofiber-based two-color trap.
