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Index for a mutation loop

  • Tsukasa Ishibashi
  • Yuji Terashima
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Abstract

Motivated by the 3d-3d correspondence, we introduce an index ℐgϕ for a mutation loop ϕ of a quiver and an integer g ≥ 0. We prove its integrality ℐgϕ∈ ℤ under certain conditions, which extends the integrality conjecture of Gang--Kim--Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-g partition function of a (1+1)-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera g as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

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Cite

BibTeXRIS
BibTeX
@misc{ishibashi2026index,
  title = {{Index for a mutation loop}},
  author = {Tsukasa Ishibashi and Yuji Terashima},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18940},
  eprint = {2610.18940},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18940},
}
RIS
TY  - UNPB
TI  - Index for a mutation loop
AU  - Tsukasa Ishibashi
AU  - Yuji Terashima
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18940
UR  - https://arxiv.org/abs/2610.18940
AB  - Motivated by the 3d-3d correspondence, we introduce an index ℐ_g^ϕ for a mutation loop ϕ of a quiver and an integer g ≥ 0. We prove its integrality ℐ_g^ϕ∈ ℤ under certain conditions, which extends the integrality conjecture of Gang--Kim--Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-g partition function of a (1+1)-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera g as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.
ER  -