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<p>Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1194_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma _0(N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1194_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <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>. In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade–Russell mod 9 conjecture and Capparelli’s identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews–Gordon identities and Bressoud’s identities.</p>

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Modularity of some Nahm sums as vector-valued functions

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Liuquan Wang,
  • Huohong Zhang

摘要

Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup \(\Gamma _0(N)\) Γ 0 ( N ) with \(N=1,2,3,4\) N = 1 , 2 , 3 , 4 . In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade–Russell mod 9 conjecture and Capparelli’s identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews–Gordon identities and Bressoud’s identities.