All chiral $$ \mathcal{W} $$-algebra extensions of $$ \mathfrak{so}\left(2,3\right) $$

A bstract We show that there are four chiral $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -algebra extensions of $$ \mathfrak{so}\left(2,3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfenced> </mml:math> algebra and construct them explicitly. We do this by a simple identification of each of the inequivalent embeddings of a copy of $$ \mathfrak{sl}\left(2,\mathbb{R}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>sl</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℝ</mml:mi> </mml:mfenced> </mml:math> in the $$ \mathfrak{so}\left(2,3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfenced> </mml:math> algebra and the maximal subalgebra $$ \mathfrak{h} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> that commutes with it. Then using the standard 2d chiral CFT techniques we find the corresponding $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -algebra extensions. Two of the four resultant $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -algebras are new, one of which may be thought of as the conformal $$ {\mathfrak{bms}}_3 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>bms</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> algebra valid for finite values of its central charge.