The preprint constructs an index-two symmetric monoidal subextension of the signature bordism bicategory for three-dimensional oriented bordism. It presents that construction as the half-signature extension and pairs it with a classification of the extension's linear representations. The document is arXiv version 1, dated 26 Aug 2026.
This is a theoretical mathematics result, not a study based on an empirical sample. The paper works with bicategories, presentations and linear representations. A bicategory is the formal structure whose processes can be composed in vertical and horizontal directions; the added extension data are recorded by a bicategorical cocycle, along with coherence functions that keep those composition rules consistent.
A division with limits
The signature in this construction is tracked through gluing. Changes under gluing are calculated with Wall's invariant and specified Lagrangian subspaces. Those calculations supply the composition data the extension needs to work across the bicategory.
The key arithmetic step is a parity correction. A parity function is used so that subtracting its coboundary makes the signature cocycle 2Z-valued, meaning its values are even integers. The adjusted cocycle can then be divided by 2, which yields the half-signature construction. In the relevant extension group, the signature extension class is consequently divisible by 2.
That division has a firm limit in the paper's framework. The signature bordism category has no index-four subextension. The half-signature extension class is also not divisible by any integer n greater than or equal to 2 in the stated extension group. Put plainly, the construction reaches a half-signature, but the resulting class cannot be uniformly divided again within that group.
From construction to representation
The paper checks the construction at two levels. In the componentwise version, the half-signature presentation is equivalent to the componentwise half-signature bordism bicategory. In the global version, the global presentation is equivalent to the half-signature bordism bicategory. In this setting, "equivalent" is the categorical claim that the presentation captures the corresponding bicategory.
Those equivalences are also where the paper's main qualification enters. The document states that the proof of the oriented-bordism presentation theorem used by later results is forthcoming. That disclosure qualifies the presentation equivalences and the representation classifications that rely on them.
For the componentwise half-signature bicategory, linear representations are classified by finite direct sums of modular tensor categories. Here, a modular tensor category is the paper's name for one of the categorical building blocks in that sum: the result concerns how a finite collection of them is combined into a representation. This remains a statement about the specified mathematical structures, not an empirical system.
The global classification adds a matching condition. Its linear representations are finite direct sums of modular tensor categories whose anomalies are equal. In this article, "equal anomalies" means that the anomaly value must match across the summands; the componentwise classification does not state that requirement. That is the central difference between the two representation results.
Taken together, the presentation and representation results describe a chain: the half-signature construction is given a componentwise and a global presentation, and each version receives its own representation classification. The global case carries the extra equal-anomaly condition, while the componentwise case does not.
What remains provisional
Readers should keep the publication status in view. This is an arXiv version 1 preprint dated 26 Aug 2026, and the document says the proof of the oriented-bordism presentation theorem used by later results is forthcoming. The work therefore offers a theoretical construction and stated classifications, while disclosing that a supporting proof is forthcoming.
The acknowledgements report guidance, discussions and comments but no funding statement. Because the paper studies mathematical bicategories, presentations and linear representations rather than an empirical sample, its findings are theoretical statements about the specified constructions.
Paper data and sources
Original title: The half-signature extension of the 3D bordism bicategory and its representations
Authors: Glen Lim
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text