Notes on su <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⊕</mml:mo> </mml:mrow> </mml:math> u(1) Chern–Simons theory and to

Abstract In this study, we investigate three-dimensional torsional Newton–Cartan (TNC) gravity by gauging the su <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊕</mml:mo> </mml:mrow> </mml:math> 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 Schrödinger gravity acquired by gauging the extended z = 2 Schrödinger algebra via a large speed of light ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> )-expansion. In particular, we explain that the z = 2 Lifshitz vacuum solution appearing in Schrödinger 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 Schrödinger algebra and bosonic analogue of super BMS algebra. We interpret that this relation originates from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">W</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> algebra which acts as the bosonic analogue of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">N</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> superconformal algebra.