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<p>We describe a family of indefinite theta functions of signature (1,&#xa0;1) that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell–Lerch sums. We use these relations to write the McKay–Thompson series of umbral moonshine at lambency <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5376_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell =8,12,16\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>8</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>16</mn> </mrow> </math></EquationSource> </InlineEquation> in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.</p>

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Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

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  • 发表日期：

    2025年8月1日

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  • Miranda C. N. Cheng,
  • Gabriele Sgroi

摘要

We describe a family of indefinite theta functions of signature (1, 1) that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell–Lerch sums. We use these relations to write the McKay–Thompson series of umbral moonshine at lambency \(\ell =8,12,16\) ℓ = 8 , 12 , 16 in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.