论文标题
Zagier排名NAHM问题的三个示例的明确表格和证明
Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem
论文作者
论文摘要
令$ r \ geq 1 $为正整数，$ a $ a $ a真正的正面半定义对称$ r \ times r $ r $理性矩阵，$ b $是长度$ r $的理性向量，以及$ c $ a a a a a $ c $ a是理性标量。 NAHM的问题是要找到所有三元组$（a，b，c）$，以便$ r $ -fold $ q $ q $ -hyperemotem $$ f_ f_ {a，b，c}（q）：= \ sum_ {n =（n_1，\ dots，\ dots，n_r dots，n_r） 0}）^r} \ frac {q^{\ frac {1} {2} n^\ mathrm {t} an+n^\ mathrm {t} b+c}}}}} {（q; q; q; q; q; q; q; q） $（a，b，c）$ a模块化三倍。当等级$ r = 3 $，经过大量的计算机搜索后，Zagier提供了十二套猜想的模块化三元组，并证明了其中三个。我们证明了许多涉及三重总和的Rogers-Ramanujan类型身份。这些身份给出了模块化形式表示形式，从而验证了Zagier的所有等级三个示例。特别是，我们证明了Zagier的猜想身份。
Let $r\geq 1$ be a positive integer, $A$ a real positive semi-definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the $r$-fold $q$-hypergeometric series $$f_{A,B,C}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T}\in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}$$ becomes a modular form, and we call such $(A,B,C)$ a modular triple. When the rank $r=3$, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier.
