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Modularity of tadpole Nahm sums in ranks 4 and 5

Changsong Shi, Liuquan Wang

TL;DR

The paper proves the modularity conjecture for tadpole Nahm sums in ranks $r=4,5$ by developing rank-reduction formulas that decompose higher-rank Nahm sums into mixtures of lower-rank Nahm-type sums and theta functions. Using these formulas, it derives explicit Rogers–Ramanujan type identities and constructs modular expressions for rank-four and rank-five cases, including five modular rank-four instances and nine modular rank-five instances, each accompanied by precise $q$-product/theta representations. The approach connects to Zagier’s modular triples via dual transformations and validates modularity through theta-function identity checks (Frye–Garvan). The results provide uniform machinery for extending Nahm-sum modularity to higher ranks and illustrate the intricate interplay between rank reduction, Rogers–Ramanujan identities, and theta-function theory.

Abstract

Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ 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.

Modularity of tadpole Nahm sums in ranks 4 and 5

TL;DR

The paper proves the modularity conjecture for tadpole Nahm sums in ranks by developing rank-reduction formulas that decompose higher-rank Nahm sums into mixtures of lower-rank Nahm-type sums and theta functions. Using these formulas, it derives explicit Rogers–Ramanujan type identities and constructs modular expressions for rank-four and rank-five cases, including five modular rank-four instances and nine modular rank-five instances, each accompanied by precise -product/theta representations. The approach connects to Zagier’s modular triples via dual transformations and validates modularity through theta-function identity checks (Frye–Garvan). The results provide uniform machinery for extending Nahm-sum modularity to higher ranks and illustrate the intricate interplay between rank reduction, Rogers–Ramanujan identities, and theta-function theory.

Abstract

Around 2016, Calinescu, Milas and Penn conjectured that the rank Nahm sum associated with the tadpole Cartan matrix is modular, and they provided a proof for . The case was recently resolved by Milas and Wang. We prove this conjecture for the next cases . 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.

Paper Structure

This paper contains 8 sections, 10 theorems, 108 equations, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of the rank reduction formulas
  4. The rank four tadpole Nahm sums
  5. Proof of Theorem \ref{['thm-dual-Zagier']}
  6. Proof of Theorem \ref{['thm-rank4']}
  7. The rank five tadpole Nahm sums
  8. Concluding remarks

Key Result

Theorem 1.2

If $x_{2i}x_{2i+1}=1$ for $i=0,1,\dots,r-1$, then we have If $x_{2i-1}x_{2i}=1$ for $i=1,\dots,r$, then we have where $n_0=0$.

Theorems & Definitions (22)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 12 more