Superconductivity in UTe$_2$ from local noncentrosymmetricity

Superconductivity in UTe$_{2}$ has garnered significant attention, as it is widely recognized as a promising candidate for a spin-triplet superconductor. However, the symmetry of superconductivity and the microscopic origin of spin-triplet pairing remain subjects of debate. Nevertheless, various experiments imply an intimate coupling between magnetism and superconductivity. In this paper, we analyze a multi-sublattice periodic Anderson model that incorporates a spin-orbit coupling allowed in locally noncentrosymmetric crystals to discuss magnetic fluctuations and superconductivity in UTe$_2$. Due to the sublattice-dependent spin-orbit coupling, magnetic fluctuations become anisotropic, and the spin degeneracy of superconducting states is lifted. Our calculations reveal anisotropic antiferromagnetic fluctuations along the $b$- and $c$-axes, anisotropic ferromagnetic fluctuations along the $a$-axis, and their coexistence. These can be tuned by the $f$-electron's level. Superconductivity in the $A_u$ representation is predominant for a wide range of parameters, whereas the $B_{2u}$ representation is almost degenerate and can be stabilized. The direction of the $d$-vector changes as we increase the spin-orbit coupling. We discuss the consistency between our results and several experiments.