Notes on su(1,2)⊕u$(1)$ Chern-Simons theory and torsional Newton-Cartan gravity

In this study, we investigate three-dimensional torsional Newton-Cartan (TNC) gravity by gauging the su(1,2)⊕u$(1)$ algebra and construct its action using the Chern-Simons theory. This TNC exhibits novel features, including the fact that the gauge fields associated with both dilatation and rotation symmetries transform non-trivially under Galilean boosts. This theory also reproduces the Schr"odinger gravity acquired by gauging the extended $z=2$ Schr"odinger algebra via a large speed of light ($1/c$)-expansion. In particular, we explain that the $z=2$ Lifshitz vacuum solution appearing in Schr"odinger gravity is related to the null reduction of 4d Ω-background up to a conformal factor. Based on these results, we revisit the identification between the extended Schr"odinger algebra and bosonic analogue of super BMS algebra. We interpret that this relation originates from the W₃⁽²⁾ algebra which acts as the bosonic analogue of N=2 superconformal algebra.