Permutation-equivariant quantum K-theory of Fermat singularities

TL;DR

This paper computes the genus-0 permutation-equivariant quantum K-theory for Fermat singularities, extending Givental-Tonita's formalism and revealing new insights into the Landau-Ginzburg/Calabi-Yau correspondence in K-theory.

Contribution

It extends the formalism of adelic Lagrangian cones to singularity theory and provides explicit I-functions satisfying specific q-difference equations.

Findings

Explicit I-functions for Fermat singularities are derived.

The I-functions satisfy the same q-difference equations as those of associated hypersurfaces.

A discrepancy between Landau-Ginzburg and Calabi-Yau sides emerges in K-theory.

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.