The $r^\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds
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| Format: | Preprint |
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2026
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| _version_ | 1866911680939687936 |
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| author | Pigazzini, Alexander Toda, Magdalena |
| author_facet | Pigazzini, Alexander Toda, Magdalena |
| contents | 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 $M = Σ_g \times M_2$ equipped with a product metric, when $r^\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_β\subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_β$ (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 irreducible 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $r^\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds Pigazzini, Alexander Toda, Magdalena Differential Geometry 53C29, 53C05, 58A14, 57R19, 81T30 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 $M = Σ_g \times M_2$ equipped with a product metric, when $r^\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_β\subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_β$ (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 irreducible 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. |
| title | The $r^\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds |
| topic | Differential Geometry 53C29, 53C05, 58A14, 57R19, 81T30 |
| url | https://arxiv.org/abs/2605.13603 |