Canonical torsion linking pairings and explicit TMF state spaces of closed 3-manifolds
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915796583710720 |
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| author | Li, Ruiliang |
| author_facet | Li, Ruiliang |
| contents | We study the TMF-valued $(3+1)$-dimensional TQFT of Gukov--Krushkal--Meier--Pei and give an explicit description of the TMF-module state space assigned to a closed $3$-manifold. Our starting point is the torsion linking pairing on $H_1$, viewed as a discriminant form. We construct a canonical, computable package of invariants for torsion linking pairings (uniformly for odd and $2$-primary parts), and from it a canonical tokenization together with an explicit symmetric integral matrix representative realizing the same stable class. This yields an explicit model for the GKMP state space in terms of a rank-one TMF-module $L_b$ with a canonical degree shift determined by signature data. As applications we identify the values on $CP^2$ and, conditional on a natural functoriality/duality statement in GKMP, on $S^2\times S^2$ with the Hopf elements $\pmν$ and $η$, respectively. Finally, we establish a rank-one time-reversal duality $L_{(-n)}\simeq L_{(n)}^\vee[1]$ for all integers $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12795 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Canonical torsion linking pairings and explicit TMF state spaces of closed 3-manifolds Li, Ruiliang Algebraic Topology Geometric Topology Quantum Algebra 57R56 (Primary), 55N34, 57M27 (Secondary) We study the TMF-valued $(3+1)$-dimensional TQFT of Gukov--Krushkal--Meier--Pei and give an explicit description of the TMF-module state space assigned to a closed $3$-manifold. Our starting point is the torsion linking pairing on $H_1$, viewed as a discriminant form. We construct a canonical, computable package of invariants for torsion linking pairings (uniformly for odd and $2$-primary parts), and from it a canonical tokenization together with an explicit symmetric integral matrix representative realizing the same stable class. This yields an explicit model for the GKMP state space in terms of a rank-one TMF-module $L_b$ with a canonical degree shift determined by signature data. As applications we identify the values on $CP^2$ and, conditional on a natural functoriality/duality statement in GKMP, on $S^2\times S^2$ with the Hopf elements $\pmν$ and $η$, respectively. Finally, we establish a rank-one time-reversal duality $L_{(-n)}\simeq L_{(n)}^\vee[1]$ for all integers $n$. |
| title | Canonical torsion linking pairings and explicit TMF state spaces of closed 3-manifolds |
| topic | Algebraic Topology Geometric Topology Quantum Algebra 57R56 (Primary), 55N34, 57M27 (Secondary) |
| url | https://arxiv.org/abs/2602.12795 |