We prove a collection of q-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_k(\mathfrak {osp}_{1|2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="fraktur">osp</mi> <mrow> <mn>1</mn> <mo stretchy="false">|</mo> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into the principal subsuperspace of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_k({\mathfrak {s}}{\mathfrak {l}}_{1|2n+1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">s</mi> <msub> <mi mathvariant="fraktur">l</mi> <mrow> <mn>1</mn> <mo stretchy="false">|</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers N, k the character of the principal subspace of type <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> at level k can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.

A conjecture of Warnaar-Zudilin from deformations of lie superalgebras

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2026年4月2日

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Thomas Creutzig,
Niklas Garner

摘要

We prove a collection of q-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k(\mathfrak {osp}_{1|2n})$ $L_{k} ({osp}_{1 | 2 n})$ into the principal subsuperspace of $L_k({\mathfrak {s}}{\mathfrak {l}}_{1|2n+1})$ $L_{k} (s l_{1 | 2 n + 1})$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers N, k the character of the principal subspace of type $A_N$ $A_{N}$ at level k can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.