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