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arXiv · 2410.17730

Permutation-equivariant quantum K-theory of Fermat singularities

Maxime Cazaux
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Abstract

We compute the genus-0 permutation-equivariant quantum K-theory of Fermat singularities, in parallel with the Givental-Lee theory for projective varieties. We extend Givental-Tonita's formalism of adelic Lagrangian cones to the singularity theory, and we obtain explicit $I$-functions for the invariants, which satisfy the same $q$-difference equation as Givental's $I$-function of the associated hypersurface. This can be regarded as an extension of the Landau-Ginzburg/Calabi-Yau correspondence, although a discrepancy between the two sides sides emerges in K-theory. In the case of the quintic threefold, both generating functions satisfy a $q$-difference equation of degree $25$; the hypersurface $I$-function only spans a $5$-dimensional subspace of solutions, while the singularity $I$-function spans the full space of solutions.

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Maxime Cazaux. 2024-10-23. Permutation-equivariant quantum K-theory of Fermat singularities. https://arxiv.org/abs/2410.17730

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