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

TL;DR

This paper provides an explicit description of the TMF-module state space for closed 3-manifolds using torsion linking pairings, offering new invariants and models that connect topology, algebra, and TMF theory.

Contribution

It introduces a canonical, computable package of invariants for torsion linking pairings and constructs explicit models for the TMF state spaces of 3-manifolds.

Findings

Explicit model for GKMP state space in terms of a rank-one TMF-module

Identification of values on $CP^2$ and $S^2\times S^2$ with Hopf elements

Establishment of a rank-one time-reversal duality for all integers

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 $\pm\nu$ and $\eta$, respectively. Finally, we establish a rank-one time-reversal duality $L_{(-n)}\simeq L_{(n)}^\vee[1]$ for all integers $n$.