For an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-framed modular operad P, we introduce its “Feynman compactification" denoted by FP which is a modular operad. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{\mathbb {M}^\textsf{fr}(g,n)\}_{(g,n)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msup> <mrow> <mi mathvariant="double-struck">M</mi> </mrow> <mi mathvariant="sans-serif">fr</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> be the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F\mathbb {M}^\textsf{fr}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <msup> <mrow> <mi mathvariant="double-struck">M</mi> </mrow> <mi mathvariant="sans-serif">fr</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H_*(\overline{M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mover> <mi>M</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$[\overline{M}_{g,n}/S_n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mover> <mi>M</mi> <mo>¯</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> in terms of Sen-Zwiebach’s string vertices. As an immediate application, we prove Costello’s categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.

Categorical enumerative invariants of the ground field

期刊论文
发表日期：

2026年7月18日

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Junwu Tu

摘要

For an $S^1$ $S^{1}$ -framed modular operad P, we introduce its “Feynman compactification" denoted by FP which is a modular operad. Let $\{\mathbb {M}^\textsf{fr}(g,n)\}_{(g,n)}$ ${M^{fr} (g, n)}_{(g, n)}$ be the $S^1$ $S^{1}$ -framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of $F\mathbb {M}^\textsf{fr}$ $F M^{fr}$ is isomorphic to $H_*(\overline{M})$ $H_{*} (\bar{M})$ , the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of $[\overline{M}_{g,n}/S_n]$ $[{\bar{M}}_{g, n} / S_{n}]$ in terms of Sen-Zwiebach’s string vertices. As an immediate application, we prove Costello’s categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.