The $r^\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds

We show that the cohomological invariant $r^\sharp$, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion $3$-form under both dimensional reduction and Buscher T-duality.

On a product $M = \Sigma_g \times M_2$ equipped with a product metric, when $r^\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_\beta \subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_\beta$ (the parallel regime); when $r^\sharp = 1$, no such circle factor exists, and the $H$-flux survives T-duality along every flat circle factor as $H$-flux in the dual background (the transversely non-reducible regime).

When $M_2 = N \times T^k$ contains a torus factor, we prove that the Bouwknegt--Evslin--Mathai obstruction to successive T-dualities vanishes automatically for $H$-flux of pure bidegree $(2,1)$, that the resulting dualities are non-interfering and order-independent, and that $r^\sharp$ detects the \emph{irreducible kernel} of the $H$-flux: the component that survives T-duality along every flat circle factor and cannot be converted into geometric or non-geometric flux in any duality frame.

This provides a metric refinement of topological T-duality: while the latter disregards the Riemannian metric entirely, $r^\sharp$ detects whether the cohomological coupling is aligned with the flat sub-factors identified by the Levi-Civita parallel-form strata.