We show the strong graded locality of all unitary minimal W-algebras, so that they give rise to irreducible graded-local conformal nets. Among these unitary vertex superalgebras, up to taking tensor products with free fermion vertex superalgebras, there are the unitary Virasoro vertex algebras (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and the unitary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N=1,2,3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> super-Virasoro vertex superalgebras. Accordingly, we have a uniform construction that gives, besides the already known <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N=0,1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> super-Virasoro nets, also the new <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N=3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> super-Virasoro nets. All strongly rational unitary minimal W-algebras give rise to previously known completely rational graded-local conformal nets and we conjecture that the converse is also true. We prove this conjecture for all unitary W-algebras corresponding to the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N=0,1,2,3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> super-Virasoro vertex superalgebras.

Conformal Nets from Minimal W-Algebras

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2025年10月30日

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Sebastiano Carpi,
Tiziano Gaudio

摘要

We show the strong graded locality of all unitary minimal W-algebras, so that they give rise to irreducible graded-local conformal nets. Among these unitary vertex superalgebras, up to taking tensor products with free fermion vertex superalgebras, there are the unitary Virasoro vertex algebras ( $N=0$ $N = 0$ ) and the unitary $N=1,2,3,4$ $N = 1, 2, 3, 4$ super-Virasoro vertex superalgebras. Accordingly, we have a uniform construction that gives, besides the already known $N=0,1,2$ $N = 0, 1, 2$ super-Virasoro nets, also the new $N=3,4$ $N = 3, 4$ super-Virasoro nets. All strongly rational unitary minimal W-algebras give rise to previously known completely rational graded-local conformal nets and we conjecture that the converse is also true. We prove this conjecture for all unitary W-algebras corresponding to the $N=0,1,2,3,4$ $N = 0, 1, 2, 3, 4$ super-Virasoro vertex superalgebras.