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Canonical torsion linking pairings and explicit TMF state spaces of closed 3-manifolds

Ruiliang Li

TL;DR

The paper provides a complete, computable framework for the TMF-valued GKMP state spaces of closed 3-manifolds by introducing Canon(A), a canonical, Kirby-stable invariant of the torsion linking form. It builds a realization dictionary 𝓡 from canonical tokens to TMF-modules, yielding Z_TMF(Y) ≃ 𝓡(Canon(A)) and revealing a prime-local, assembly-like structure of state spaces. The authors identify the Hopf elements ν and η in the TMF-stem as invariants arising from CP² and S²×S², and establish a rank-one time-reversal duality L(−n) ≃ L(n)ᵛ[1], giving a deep duality picture. Together, these results provide a computable, canonical, and structurally rich description of GKMP TMF-state spaces with clear connections to classical invariants and dualities, enabling practical calculations and conceptual insight into TMF TQFTs.

Abstract

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 $n$.

Canonical torsion linking pairings and explicit TMF state spaces of closed 3-manifolds

TL;DR

The paper provides a complete, computable framework for the TMF-valued GKMP state spaces of closed 3-manifolds by introducing Canon(A), a canonical, Kirby-stable invariant of the torsion linking form. It builds a realization dictionary 𝓡 from canonical tokens to TMF-modules, yielding Z_TMF(Y) ≃ 𝓡(Canon(A)) and revealing a prime-local, assembly-like structure of state spaces. The authors identify the Hopf elements ν and η in the TMF-stem as invariants arising from CP² and S²×S², and establish a rank-one time-reversal duality L(−n) ≃ L(n)ᵛ[1], giving a deep duality picture. Together, these results provide a computable, canonical, and structurally rich description of GKMP TMF-state spaces with clear connections to classical invariants and dualities, enabling practical calculations and conceptual insight into TMF TQFTs.

Abstract

We study the TMF-valued -dimensional TQFT of Gukov--Krushkal--Meier--Pei and give an explicit description of the TMF-module state space assigned to a closed -manifold. Our starting point is the torsion linking pairing on , viewed as a discriminant form. We construct a canonical, computable package of invariants for torsion linking pairings (uniformly for odd and -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 with a canonical degree shift determined by signature data. As applications we identify the values on and, conditional on a natural functoriality/duality statement in GKMP, on with the Hopf elements and , respectively. Finally, we establish a rank-one time-reversal duality for all integers .
Paper Structure (81 sections, 82 theorems, 290 equations, 1 table)

This paper contains 81 sections, 82 theorems, 290 equations, 1 table.

Table of Contents

  1. Introduction
  2. Surgery matrices, linking forms, and the canonicalization problem
  3. Main results
  4. Proof strategy and organization
  5. Background and conventions
  6. Surgery matrices, $2$--handlebodies, and Kirby moves
  7. Kirby matrices.
  8. Kirby moves in matrix form.
  9. Nondegenerate versus singular matrices.
  10. From a matrix to a discriminant group and linking pairing
  11. The discriminant group.
  12. The matrix linking form.
  13. Topological interpretation.
  14. Kirby invariance of the isometry class.
  15. Orientation reversal.
  16. ...and 66 more sections

Key Result

Lemma 2.2

If $\det(A)\neq 0$, then $\lambda_A$ is well-defined and is a nonsingular symmetric bilinear pairing.

Theorems & Definitions (204)

  • Remark 1.1: Not a classification of $3$--manifolds
  • Remark 1.2: Equivalences versus canonical equivalences
  • Remark 2.1: Orientation convention
  • Lemma 2.2
  • proof
  • Proposition 2.3: Homology and the torsion linking pairing
  • proof
  • Proposition 2.4: Congruence and stabilization invariance
  • proof
  • Remark 2.5: The torsion linking form for a singular surgery matrix
  • ...and 194 more