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Modularity of tadpole Nahm sums in ranks 4 and 5.

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Title: Modularity of tadpole Nahm sums in ranks 4 and 5.
Authors: Shi, Changsong1 (AUTHOR) changsong@whu.edu.cn, Wang, Liuquan1 (AUTHOR) mathlqwang@163.com
Source: Journal of Number Theory. Jul2026, Vol. 284, p214-245. 32p.
Subjects: Modular forms, Theta functions, Mathematical formulas
Abstract: Around 2016, Calinescu, Milas and Penn conjectured that the rank r Nahm sum associated with the r × r tadpole Cartan matrix is modular, and they provided a proof for r = 2. The r = 3 case was recently resolved by Milas and Wang. We prove this conjecture for the next cases r = 4 , 5. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers–Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 191664245
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Label: Title
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  Data: Modularity of tadpole Nahm sums in ranks 4 and 5.
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  Data: <searchLink fieldCode="AR" term="%22Shi%2C+Changsong%22">Shi, Changsong</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> changsong@whu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Liuquan%22">Wang, Liuquan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> mathlqwang@163.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Jul2026, Vol. 284, p214-245. 32p.
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  Data: <searchLink fieldCode="DE" term="%22Modular+forms%22">Modular forms</searchLink><br /><searchLink fieldCode="DE" term="%22Theta+functions%22">Theta functions</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Around 2016, Calinescu, Milas and Penn conjectured that the rank r Nahm sum associated with the r × r tadpole Cartan matrix is modular, and they provided a proof for r = 2. The r = 3 case was recently resolved by Milas and Wang. We prove this conjecture for the next cases r = 4 , 5. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers–Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.jnt.2025.12.010
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 32
        StartPage: 214
    Subjects:
      – SubjectFull: Modular forms
        Type: general
      – SubjectFull: Theta functions
        Type: general
      – SubjectFull: Mathematical formulas
        Type: general
    Titles:
      – TitleFull: Modularity of tadpole Nahm sums in ranks 4 and 5.
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Shi, Changsong
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          Name:
            NameFull: Wang, Liuquan
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          Dates:
            – D: 01
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
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            – Type: issn-print
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              Value: 284
          Titles:
            – TitleFull: Journal of Number Theory
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