Mori theory and symplectic reduction for polygons and quivers
arXiv:2608.20930
2026
Geometry
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper identifies weighted configurations on \(\mathbb{P}^{1}\), quiver moduli, polygon spaces, and symplectic reductions as different presentations of the same quotient geometry. The transferable asset is not the particular moduli space, but the combination of weighted balancing constraints and invariance under a global group action: configurations are meaningful only modulo simultaneous \(PGL_{2}\) or \(SU(2)\) transformations. This suggests an attention or token-interaction module whose values lie on fixed-radius \(SU(2)\)-coadjoint spheres and whose aggregate is forced to satisfy an additive closure constraint. A practical first version is a gauge-invariant regularizer for attention heads, with the constraint gradually activated so that it can be tested against ordinary attention for stability and parameter efficiency.
Ideas from this paper
Unverified
2026
Represent each token value in an attention head as a bounded three-dimensional Lie-algebra vector and impose a weighted polygon-closure condition on the aggregate value vectors. The module is invariant to a common \(SU(2)\simeq SO(3)\) rotation of all token vectors, preventing the head from spending capacity on an arbitrary global orientation. A soft closure penalty gives a drop-in experiment, while projection onto the zero-sum manifold provides a harder constrained variant.
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