<p>Vafa and Warner observed that the Landau–Ginzburg model associated to the potential <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$E_6$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation>) is a product of two other models, associated to the potentials <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ A_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A_2 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ A_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>). We translate this along the Landau–Ginzburg/conformal field theory correspondence to a conjecture about the unitary minimal quotients <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$M_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> of the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$N=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> superconformal algebra of central charge <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$c_d=3-\frac{6}{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>d</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>-</mo> <mfrac> <mn>6</mn> <mi>d</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>: there should be a conformal embedding <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$M_{12}\hookrightarrow M_{3} \otimes M_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mn>12</mn> </msub> <mo stretchy="false">↪</mo> <msub> <mi>M</mi> <mn>3</mn> </msub> <mo>⊗</mo> <msub> <mi>M</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$M_{30}\hookrightarrow M_{3} \otimes M_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mn>30</mn> </msub> <mo stretchy="false">↪</mo> <msub> <mi>M</mi> <mn>3</mn> </msub> <mo>⊗</mo> <msub> <mi>M</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) that exhibits the product as Ostrik’s <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$E_6$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$E_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation>) algebra in the <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\textbf{Rep}(su(2)_{10})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Rep</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>u</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>10</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\textbf{Rep}(su(2)_{28})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Rep</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>u</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>28</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) factor of the NS sector of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\textbf{Rep}(M_{12})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Rep</mi> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mn>12</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (resp. <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\textbf{Rep}(M_{30})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Rep</mi> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mn>30</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>). We motivate, formulate, and prove this conjecture.</p>

Exceptional embeddings of $N=2$ minimal models

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2026年7月2日

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Ana Ros Camacho,
Thomas A. Wasserman

摘要

Vafa and Warner observed that the Landau–Ginzburg model associated to the potential $E_6$ $E_{6}$ (resp. $E_8$ $E_{8}$ ) is a product of two other models, associated to the potentials $A_2$ $A_{2}$ and $ A_3$ $A_{3}$ (resp. $A_2 $ $A_{2}$ and $ A_4$ $A_{4}$ ). We translate this along the Landau–Ginzburg/conformal field theory correspondence to a conjecture about the unitary minimal quotients $M_d$ $M_{d}$ of the $N=2$ $N = 2$ superconformal algebra of central charge $c_d=3-\frac{6}{d}$ $c_{d} = 3 - \frac{6}{d}$ : there should be a conformal embedding $M_{12}\hookrightarrow M_{3} \otimes M_4$ $M_{12} ↪ M_{3} \otimes M_{4}$ (resp. $M_{30}\hookrightarrow M_{3} \otimes M_5$ $M_{30} ↪ M_{3} \otimes M_{5}$ ) that exhibits the product as Ostrik’s $E_6$ $E_{6}$ (resp. $E_8$ $E_{8}$ ) algebra in the $\textbf{Rep}(su(2)_{10})$ $Rep (s u {(2)}_{10})$ (resp. $\textbf{Rep}(su(2)_{28})$ $Rep (s u {(2)}_{28})$ ) factor of the NS sector of $\textbf{Rep}(M_{12})$ $Rep (M_{12})$ (resp. $\textbf{Rep}(M_{30})$ $Rep (M_{30})$ ). We motivate, formulate, and prove this conjecture.

Exceptional embeddings of \(N=2\) minimal models

摘要