Ordinary modules for vertex algebras of 𝔬𝔰𝔭1|2𝑛

We show that the affine vertex superalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">o</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo fence="false">|</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Vᵏ(osp1|2n) at generic level 𝑘 embeds in the equivariant 𝒲-algebra of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> sp₂ₙ times <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> 4n free fermions. This has two corollaries: (1) it provides a new proof that, for generic 𝑘, the coset <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Com</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">o</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo fence="false">|</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Com(Vᵏ(sp₂ₙ),Vᵏ(osp1|2n)) is isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="script">W</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> W^{}(sp₂ₙ) for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">ℓ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> =-(n+1)+(k+n+1)/(2k+2n+1) , and (2) we obtain the decomposition of ordinary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">o</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo fence="false">|</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Vᵏ(osp1|2n) -modules into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo rspace="0.055em" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.222em">⊗</m:mo> <m:msup> <m:mi mathvariant="script">W</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msup> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Vᵏ(sp₂ₙ)^{}(sp₂ₙ) -modules. Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> sp₂ₙ , we show that the simple algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">o</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo fence="false">|</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Lₖ(osp1|2n) is an extension of the simple subalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo rspace="0.055em" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.222em">⊗</m:mo> <m:msub> <m:mi mathvariant="script">W</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Lₖ(sp₂ₙ)⊗{W}_{}(sp₂ₙ) . Using the theory of vertex superalgebra extensions, we prove that the category of ordinary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">o</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo fence="false">|</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Lₖ(osp1|2n) -modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">W</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> W_{}(sp₂ₙ) -modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">s</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="fraktur">p</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Lₖ(sp₂ₙ) -modules are rigid.