We investigate a one-point restriction of conformal blocks on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mathbb {P}^1,\infty ,1,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> <mo>,</mo> <mi>∞</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> associated with modules over a vertex operator algebra V. By restricting the module attached to the point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation> to its bottom degree, we obtain a new formula for computing fusion rules in terms of a new left A(V)-module <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M^1 \odot M^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mn>1</mn> </msup> <mo>⊙</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> over the Zhu algebra A(V), constructed from two V-modules <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. As a consequence, for strongly rational vertex operator algebras, the construction of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M^1 \odot M^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mn>1</mn> </msup> <mo>⊙</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> induces the fusion tensor product on the module category <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsf{Mod}(A(V))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">Mod</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

One-point restricted conformal blocks and the fusion tensor product

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2026年2月27日

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Jianqi Liu

摘要

We investigate a one-point restriction of conformal blocks on $(\mathbb {P}^1,\infty ,1,0)$ $(P^{1}, \infty, 1, 0)$ associated with modules over a vertex operator algebra V. By restricting the module attached to the point $\infty $ $\infty$ to its bottom degree, we obtain a new formula for computing fusion rules in terms of a new left A(V)-module $M^1 \odot M^2$ $M^{1} ⊙ M^{2}$ over the Zhu algebra A(V), constructed from two V-modules $M^1$ $M^{1}$ and $M^2$ $M^{2}$ . As a consequence, for strongly rational vertex operator algebras, the construction of $M^1 \odot M^2$ $M^{1} ⊙ M^{2}$ induces the fusion tensor product on the module category $\textsf{Mod}(A(V))$ $Mod (A (V))$ .