Instanton slices and their superpolynomials

@misc{cherednik2026instanton,
  title = {{Instanton slices and their superpolynomials}},
  author = {Ivan Cherednik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.25666},
  eprint = {2607.25666},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.25666},
}

TY  - UNPB
TI  - Instanton slices and their superpolynomials
AU  - Ivan Cherednik
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2607.25666
UR  - https://arxiv.org/abs/2607.25666
AB  - The key theorem is a connection between motivic superpolynomials of plane curve singularities in any ranks with superpolynomials of the corresponding instanton slices, Nekrasov-type instanton sums with conductors. In this case, instanton slices are related to Compactified Jacobians, but they form a much wider class and, generally, have no connection to plane curve singularities. Thus, we extend significantly theory of affine Springer fibers in type A. For instance, we obtain a motivic interpretation (counting 𝔽_q-points of some stacks) of superpolynomials for hyperbolic knots K12n242, K12n725, among many others. New formulas for Nekrasov's instanton sums as limits of superpolynomials of families of plane curve singularities are another application, with an impressive connection to nonsymmetric Macdonald polynomials. The main conjecture states that instanton superpolynomials for arbitrary Young diagrams coincide with the corresponding (new) DAHA superpolynomials if and only if the latter are positive or (equivalently) the former are superdual. Only very few such diagrams come from plane curve singularities. Our DAHA superpolynomials are parallel to the EHA ones, but we used the nonsymmetric approach. The connection with Khovanov-Rozansky polynomials of the corresponding Coxeter knots is expected. Among applications: the Volume Conjecture reformulated in terms of instanton slices, topological invariance of unibranch motivic superpolynomials for semigroups with up to 4 generators, a mixed-characteristic conjecture, upgraded Weak Riemann Hypothesis, and superpolynomials for Gorenstein curve singularities and multibranch ones.
ER  -