Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915173994856448 |
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| author | Wang, Liuquan Zeng, Wentao |
| author_facet | Wang, Liuquan Zeng, Wentao |
| contents | Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$: \begin{align*}\label{eq-lattice-sum} f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T} \in v+L} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}. \end{align*} If $L$ is a full rank lattice and a proper subset of $\mathbb{Z}^r$, then we call $f_{A,B,C,v+L}(q)$ a rank $r$ partial Nahm sum. When the rank $r=1$, we find eight modular partial Nahm sums using some known identities. When the rank $r=2$ and $L$ is one of the lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$ or $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$, we find 14 types of symmetric matrices $A$ such that there exist vectors $B,v$ and scalars $C$ so that the partial Nahm sum $f_{A,B,C,v+L}(q)$ is modular. We establish Rogers--Ramanujan type identities for the corresponding partial Nahm sums which prove their modularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums Wang, Liuquan Zeng, Wentao Number Theory Combinatorics 11P84, 33D15, 33D60, 11F03 Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$: \begin{align*}\label{eq-lattice-sum} f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T} \in v+L} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}. \end{align*} If $L$ is a full rank lattice and a proper subset of $\mathbb{Z}^r$, then we call $f_{A,B,C,v+L}(q)$ a rank $r$ partial Nahm sum. When the rank $r=1$, we find eight modular partial Nahm sums using some known identities. When the rank $r=2$ and $L$ is one of the lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$ or $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$, we find 14 types of symmetric matrices $A$ such that there exist vectors $B,v$ and scalars $C$ so that the partial Nahm sum $f_{A,B,C,v+L}(q)$ is modular. We establish Rogers--Ramanujan type identities for the corresponding partial Nahm sums which prove their modularity. |
| title | Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums |
| topic | Number Theory Combinatorics 11P84, 33D15, 33D60, 11F03 |
| url | https://arxiv.org/abs/2502.19309 |