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

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 $CP<sup>2$ and, conditional on a natural functoriality/duality statement in GKMP, on $S^2\times S<sup>2$ with the Hopf elements $\pmν$ and $η$, respectively. Finally, we establish a rank-one time-reversal duality $L_{(-n)}\simeq L_{(n)}<sup>\vee[1]$ for all integers $n$.

• The paper constructs a complete Kirby-invariant token package for a 3-manifold’s torsion linking pairing, including free rank, prime-local forms, and refined 2-primary data.
• It realizes each package as an explicit shifted tmf-module assembled from canonical plumbing and local matrix blocks, proving the resulting state space agrees with the GKMP TQFT assignment.
• It calibrates the theory with the values ±ν for CP² and conditionally η for S²×S², while establishing rank-one time-reversal duality and identifying remaining canonicity limitations.

This paper by Ruiliang Li (2602.12795) provides an explicit, canonical description of the state spaces of the $\mathrm{tmf}$-valued $(3+1)$-dimensional TQFT constructed by Gukov–Krushkal–Meier–Pei (GKMP) (Gukov et al., 15 Sep 2025). The central contribution is a two-step procedure: an algebraic "Step A" that extracts a complete canonical invariant of the torsion linking pairing of a closed oriented 3-manifold from any integral surgery presentation, and a "Step B" realization functor that converts this invariant into an explicit object in the homotopy category of $\mathrm{tmf}$-modules, identified with the GKMP state space.

Surgery matrices and the canonicalization problem

For a framed link $L\subset S^3$ with symmetric linking matrix $A$, surgery produces $Y_A=\partial W(A)$, and $H_1(Y_A;\mathbb{Z})$ is canonically identified with the discriminant group $G(A)=\operatorname{coker}(A)$ via Poincaré–Lefschetz duality on the 2-handlebody. When $\det A\neq 0$, the pairing

$$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$

is well-defined, symmetric, and nonsingular; when $(3+1)$0, the same construction applies to $(3+1)$1 after splitting off the free summand, with $(3+1)$2 recorded separately. The isometry class of $(3+1)$3 is invariant under unimodular congruence (handle slides) and $(3+1)$4 stabilization (blow-up/down), hence under Kirby moves [Kirby78].

The classical classifications of Wall [Wall63] and Kawauchi–Kojima [KawauchiKojima80] give complete isometry invariants prime by prime, but do not supply canonical representatives compatible with the stabilization operations that appear in the GKMP theory. This gap is what Step A addresses.

Canonical token packages

The paper defines an invariant $(3+1)$5 — equivalently a "token package" $(3+1)$6, which is merely a flattened encoding — recording:

• the free rank $(3+1)$7;
• the torsion order and Smith invariant factors;
• for each odd prime $(3+1)$8 and each exponent layer $(3+1)$9: the dimension $\mathrm{tmf}$0 of the layer form $\mathrm{tmf}$1 on $\mathrm{tmf}$2 together with the Legendre symbol $\mathrm{tmf}$3;
• for $\mathrm{tmf}$4: the Type A / Type E dichotomy (nonalternating versus alternating layer form, detected by the characteristic element $\mathrm{tmf}$5), plus a determinant square class $\mathrm{tmf}$6 ($\mathrm{tmf}$7 for $\mathrm{tmf}$8, $\mathrm{tmf}$9 for $L\subset S^3$0) in Type A, or a normalized Gauss-sum phase $L\subset S^3$1 extracted from the quadratic refinement $L\subset S^3$2 in Type E.

The main algebraic theorem (Theorem A) states that $L\subset S^3$3 depends only on $L\subset S^3$4 up to isometry, is Kirby-invariant, and is complete: equality holds if and only if the torsion linking pairings are isometric and the free ranks agree. Completeness at odd primes follows from Wall's homogeneous splitting plus classification over $L\subset S^3$5 by dimension and determinant class; at $L\subset S^3$6 it rests on Kawauchi–Kojima's system, repackaged through the characteristic element, determinant refinements, and Gauss sums (with Miranda's normal forms [Miranda84] supplying concrete representatives). The paper emphasizes that this is a rigidification of known classification results rather than new invariants.

Stable classification and the realization functor

The bridge to the TQFT is a stable classification statement (Proposition 6.4): two symmetric integral matrices are Kirby stably congruent (related by unimodular congruence and $L\subset S^3$7 stabilizations) if and only if they have equal $L\subset S^3$8 and isometric torsion linking pairings. The nonsingular case is attributed to Murakami–Ohtsuki–Okada [MurakamiOhtsukiOkada92].

Given a token package $L\subset S^3$9, the paper assembles a canonical matrix $A$0 using explicit generators: Hirzebruch–Jung continued-fraction plumbing matrices $A$1 realizing cyclic pairings $A$2, Wall generators at odd primes, and Miranda's diagonal and hyperbolic/F-block matrices at $A$3. The realization functor is then

$A$4

where $A$5 denotes the GKMP module attached to an integral bilinear form and the shift is the GKMP normalization. The principal theorem (Theorem B) identifies

$A$6

using stable congruence between $A$7 and $A$8 together with GKMP's stabilization behavior ($A$9, $Y_A=\partial W(A)$0, exactly cancelled by the shift). A companion assembly formula gives a symmetric monoidal decomposition

$Y_A=\partial W(A)$1

so computation reduces to finitely many local building blocks.

Two caveats are stated plainly. First, the result does not classify 3-manifolds: many non-diffeomorphic manifolds share the same $Y_A=\partial W(A)$2 and hence receive equivalent state spaces. Second, while $Y_A=\partial W(A)$3 and $Y_A=\partial W(A)$4 are canonical representatives of the equivalence class, the comparison equivalence itself is not claimed to be canonical; GKMP show the state space is well-defined only up to non-canonical equivalence.

Calibration: Hopf elements from closed 4-manifolds

The paper computes two closed-manifold values that fix residual normalizations. For $Y_A=\partial W(A)$5, the GKMP value arises from the reduced Becker–Gottlieb $Y_A=\partial W(A)$6-transfer of the universal bundle; restricting to $Y_A=\partial W(A)$7 yields a stable map $Y_A=\partial W(A)$8 whose suspension $Y_A=\partial W(A)$9 is identified as follows. The cofiber $H_1(Y_A;\mathbb{Z})$0 is a two-cell spectrum on which the Thom identities give $H_1(Y_A;\mathbb{Z})$1 mod 2 and $H_1(Y_A;\mathbb{Z})$2 mod 3, so $H_1(Y_A;\mathbb{Z})$3 is Adams-filtration-one detected at both primes by $H_1(Y_A;\mathbb{Z})$4 and $H_1(Y_A;\mathbb{Z})$5 respectively; since $H_1(Y_A;\mathbb{Z})$6, this forces $H_1(Y_A;\mathbb{Z})$7 with $H_1(Y_A;\mathbb{Z})$8. Computing the complex Adams $H_1(Y_A;\mathbb{Z})$9-invariant via the Bernoulli expansion of $G(A)=\operatorname{coker}(A)$0 gives $G(A)=\operatorname{coker}(A)$1, pinning down $G(A)=\operatorname{coker}(A)$2. Hence

$G(A)=\operatorname{coker}(A)$3

with the sign removable by convention. For $G(A)=\operatorname{coker}(A)$4, decomposing along $G(A)=\operatorname{coker}(A)$5 into two 2-handle cobordisms and composing the explicit matrices for the unit-section map $G(A)=\operatorname{coker}(A)$6 and the handle-slide self-equivalence (using $G(A)=\operatorname{coker}(A)$7) yields

$G(A)=\operatorname{coker}(A)$8

conditional on the naturality statement for reversing simply connected cobordisms posed as Question 7.14 in (Gukov et al., 15 Sep 2025). This dependence on an unproven input should be noted: the $G(A)=\operatorname{coker}(A)$9 identification is not unconditional.

Rank-one time-reversal duality

The final structural result establishes, for every integer $\det A\neq 0$0, a canonical equivalence

$\det A\neq 0$1

where $\det A\neq 0$2 is the normalized rank-one module built from the universal spectral elliptic curve. The proof combines the compatibility of internal Hom with suspension, preservation of duals by derived global sections (via 0-affineness of $\det A\neq 0$3, following Mathew–Meier), and spectral Grothendieck duality for the proper smooth morphism $\det A\neq 0$4 of relative dimension one, which gives $\det A\neq 0$5; the shifts then cancel to produce the single suspension. The equivalence is canonical only up to multiplication by $\det A\neq 0$6, an ambiguity traced to the self-duality $\det A\neq 0$7 of the $\det A\neq 0$8-cone; fixing a single test case (the $\det A\neq 0$9 model $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$0) removes it globally. The paper notes this sign must be tracked carefully for strict on-the-nose functoriality.

Limitations and open questions

Several restrictions bound the scope of the results. The identification $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$1 relies on the unresolved naturality question [(Gukov et al., 15 Sep 2025), Question 7.14]; without it, only the $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$2 value is unconditional. The comparison equivalence in the main theorem is not upgraded to a canonical one, so cobordism maps are not computed on the nose by the dictionary. The state space factors through the coarse datum $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$3, which is far from a 3-manifold invariant, so the formula cannot distinguish manifolds with isometric linking forms. Finally, the sign ambiguity in the rank-one duality persists unless a global normalization is fixed, and the accompanying computational implementation serves as verification only, playing no role in the proofs.

Conclusion

The paper converts the GKMP $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$4-valued $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$5-TQFT from a construction depending on auxiliary choices of bounding 4-manifolds into an effectively computable one: given any surgery matrix, the state space is the explicit $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$6-module $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$7, assembled tensorially from finitely many prime-local building blocks determined by a complete, Kirby-invariant, canonical packaging of the torsion linking pairing. The calibration computations identify the structure constants $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$8 and $$\lambda_A(\bar x,\bar y)=x^{\mathsf T}A^{-1}y \bmod \mathbb{Z}\in\mathbb{Q}/\mathbb{Z}$$9 governing the rank-one sector, and the Grothendieck-duality argument establishes time-reversal duality for all rank-one blocks. The remaining gaps — canonicity of the comparison maps, the assumed naturality axiom, and strict functoriality modulo signs — delineate precisely what would be needed for a fully rigid version of the theory.