{
  "id": "2411.09701",
  "version": "v1",
  "published": "2024-11-14T18:59:36.000Z",
  "updated": "2024-11-14T18:59:36.000Z",
  "title": "Counterexamples to Zagier's Duality Conjecture on Nahm Sums",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "First version",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given any positive integer $r$, Nahm's problem is to determine all rational $r\\times r$ positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums wherein the original Nahm sum is modular while its dual is not modular. This provides counterexamples to Zagier's conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-14T18:59:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P84",
      "33D15",
      "33D60",
      "11F03"
    ],
    "keywords": [
      "zagiers duality conjecture",
      "counterexamples",
      "dual nahm sum",
      "dimensional rational vector",
      "original nahm sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}