Inverse Reduction for Hook-Type W-Algebras

Abstract Originating in the work of A.M. Semikhatov and D. Adamović, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine $$\mathfrak {sl}_{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> vertex operator algebra and the minimal $$\mathfrak {sl}_{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> W-algebra exists. This generalises the realisations for $$n=1,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> in Adamović (Commun Math Phys 366:1025–1067, 2019), Adamović (Math Ann 1–44, 2023). A similar argument is then used to show that inverse reduction embeddings exists between all hook-type $$\mathfrak {sl}_{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> W-algebras, which includes the principal/regular, subregular, minimal $$\mathfrak {sl}_{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> W-algebras, and the affine $$\mathfrak {sl}_{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> vertex operator algebra. This generalises the regular-to-subregular inverse reduction of Fehily (Commun Contemp Math 2250049, 2022), and similarly uses free-field realisations and their associated screening operators.