$${\mathcal {N}}=(2,2)$$ extended $${\mathfrak {sl}}(3|2)$$ Chern–Simons $$AdS_3$$ supergravity with new boundaries

Abstract

AbstractWe present the first example of $${\mathcal {N}}=(2,2)$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> formulation for the extended higher-spin $$AdS_3$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> supergravity with the most general boundary conditions as an extension of the $${\mathcal {N}}=(1,1)$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> work, discovered recently by us (Özer and Filiz in Eur Phys J C 80(11):1072, 2020). Using the method proposed by Grumiller and Riegler, we restrict a consistent class of the most general boundary conditions to extend it. An important consequence of our method is that, for the loosest set of boundary conditions it ensures that their asymptotic symmetry algebras consist of two copies of the $${\mathfrak {sl}}(3|2)_k$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>sl</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math>. Moreover, we impose some restrictions on the gauge fields for the most general boundary conditions, leading to the supersymmetric extensions of the Brown and Henneaux boundary conditions. Based on these results, we finally find out that the asymptotic symmetry algebras are two copies of the super $${\mathcal {W}}_3$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> algebra for $${\mathcal {N}}=(2,2)$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> extended higher-spin supergravity theory in $$AdS_3$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math>.