Let T_r=A_2r/Z_2 be the tadpole Dynkin diagram, and let mathbfT_r be its Cartan matrix. For mathbfx_r=(x_1,…,x_r), define the tadpole Nahm sum T_r(mathbfx_r;q)=∑_mathbfn_r=(n_1,…,n_r)∈Z^rfracq^frac12mathbfn_r mathbfT_r mathbfn_r^mathsfTx_1^n_1… x_r^n_r(q)_n_1…(q)_n_r. The sum is principal when x_1=…=x_r=1. Calinescu–Milas–Penn's conjecture. The principal tadpole Nahm sum T_r(1,…,1;q), shifted by a rational power of q, is modular. This conjecture generalizes the confirmed modularity results in ranks two through five; its general status is not resolved in the supplied context.

Calinescu–Milas–Penn conjecture on modularity of principal tadpole Nahm sums

About 1 year old · traced to 2025

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$ , define the tadpole Nahm sum

\mathcal{T}_r(\mathbf{x}_r;q)=\sum_{\mathbf{n}_r=(n_1,\ldots,n_r)\in\mathbb{Z}^r}\frac{q^{\frac{1}{2}\mathbf{n}_r\mathbf{T}_r\mathbf{n}_r^\mathsf{T}}x_1^{n_1}\cdots x_r^{n_r}}{(q)_{n_1}\cdots(q)_{n_r}}.

The sum is principal when $x_1=\cdots=x_r=1$ . Calinescu–Milas–Penn's conjecture. The principal tadpole Nahm sum $\mathcal{T}_r(1,\ldots,1;q)$ , shifted by a rational power of $q$ , is modular. This conjecture generalizes the confirmed modularity results in ranks two through five; its general status is not resolved in the supplied context.

References

Primary source

Shane Chern, Chanh Tran and Tanay Wakhare, “Tadpole Nahm sum as a Wronskian”, arXiv:2607.22904 (2026).

Additional references

2 papers in this index state this conjecture (2025–2026). The statement above is taken from the most recent of them; the others are arXiv:2504.17737.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A July 2026 preprint claims to settle the conjecture in every rank, but the claimed proof has not yet been independently checked.

The conjecture predicts that the principal tadpole sums become modular after multiplying by a suitable rational power of $q$ . It is confirmed in ranks $r=2,3,4,5$ , while the all-rank assertion was previously open.

Known results

$r=2$ : proved by Calinescu, Milas, and Penn.
$r=3$ : confirmed by Milas and Wang, 2023, using Rogers–Ramanujan-type identities.
$r=4,5$ : proved using rank reduction and Rogers–Ramanujan-type identities, 2025.

July 2026 claimed general resolution

The preprint Tadpole Nahm sum as a Wronskian claims an explicit Wronskian formula proving modularity of the principal sum for every $r$ , and also treats the twisted specialization with $q^{1/2}$ . The claim is unverified: no independent check, referee report, or correction is supplied. Its authors attribute discovery of a key connection to the named model, while describing the subsequent argument as human-completed.

Current status (as of August 2026): Ranks $r=2,3,4,5$ are settled, while the claimed all-rank resolution remains unverified.

Sources

No solutions have been posted yet.