Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Adams operations in quasimap K-theory

Shaoyun Bai, Jae Hee Lee

TL;DR

This paper defines quantum Adams operators $Q\psi_{\mathcal{F}}^k$ in equivariant $K$-theory via $\,\mu_k$-equivariant quasimap counts to Higgs branches, establishing them as quantum deformations of classical Adams operations and linking them to PSZ quantum $K$-theory. It proves a central result that the $p$-curvature of the Kahler $q$-difference connection coincides with descendant quantum Adams operators, thereby interpreting $p$-curvature as quantum power operation and connecting to quantum Steenrod theory through cohomological degeneration. The work develops two flavors of quantum cyclic powers and uses $\,\mu_k$-localization to compute explicit structure constants, exemplified for $X=T^*\mathbb{P}^n$ and in hypertoric/abelian cases, illustrating how root-of-unity specializations control the algebraic structure. A major contribution is a K-theoretic quantum Hikita conjecture at roots of unity, including an arithmetic refinement involving Lonergan’s Frobenius center, with concrete verification for abelian gauge theories. Altogether, the paper provides a geometric, moduli-space-based framework for quantum power operations in $K$-theory, connects them to $q$-difference modules, and suggests rich interactions with 3D mirror symmetry and arithmetic geometry.

Abstract

We define quantum deformations of Adams operations in $K$-theory, in the framework of quasimap quantum $K$-theory. They provide $K$-theoretic analogs of the quantum Steenrod operations from equivariant symplectic Gromov--Witten theory. We verify the compatibility of these operations with the Kahler and equivariant $q$-difference module structures, provide sample computations via $\mathbb{Z}/k$-equivariant localization, and identify them with $p$-curvature operators of the Kahler $q$-difference connections as studied in Koroteev-Smirnov. We also formulate and verify a $K$-theoretic quantum Hikita conjecture at roots of unity, and propose an indirect algebro-geometric definition of quantum Steenrod operations

Quantum Adams operations in quasimap K-theory

TL;DR

This paper defines quantum Adams operators in equivariant -theory via -equivariant quasimap counts to Higgs branches, establishing them as quantum deformations of classical Adams operations and linking them to PSZ quantum -theory. It proves a central result that the -curvature of the Kahler -difference connection coincides with descendant quantum Adams operators, thereby interpreting -curvature as quantum power operation and connecting to quantum Steenrod theory through cohomological degeneration. The work develops two flavors of quantum cyclic powers and uses -localization to compute explicit structure constants, exemplified for and in hypertoric/abelian cases, illustrating how root-of-unity specializations control the algebraic structure. A major contribution is a K-theoretic quantum Hikita conjecture at roots of unity, including an arithmetic refinement involving Lonergan’s Frobenius center, with concrete verification for abelian gauge theories. Altogether, the paper provides a geometric, moduli-space-based framework for quantum power operations in -theory, connects them to -difference modules, and suggests rich interactions with 3D mirror symmetry and arithmetic geometry.

Abstract

We define quantum deformations of Adams operations in -theory, in the framework of quasimap quantum -theory. They provide -theoretic analogs of the quantum Steenrod operations from equivariant symplectic Gromov--Witten theory. We verify the compatibility of these operations with the Kahler and equivariant -difference module structures, provide sample computations via -equivariant localization, and identify them with -curvature operators of the Kahler -difference connections as studied in Koroteev-Smirnov. We also formulate and verify a -theoretic quantum Hikita conjecture at roots of unity, and propose an indirect algebro-geometric definition of quantum Steenrod operations

Paper Structure

This paper contains 42 sections, 38 theorems, 188 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum deformation of cohomological operations
  4. $p$-curvature in $q$-difference modules
  5. Large center at roots of unity
  6. Statement of results
  7. Further directions
  8. Organization
  9. Acknowledgements
  10. Background on quasimaps
  11. Stable quasimaps and Higgs branches
  12. Quantum $K$-theory
  13. Quantum Adams operations
  14. Adams operations
  15. Quantum cyclic powers
  16. ...and 27 more sections

Key Result

Theorem 1.4

$Q\psi_{\mathcal{F}}^k$ is a $z$-linear endomorphism of $K_{\mathbf{T}}(X)[\![ z^{\mathrm{eff}} ]\!]$, where $z^{\mathrm{eff}}$ denotes the Kähler variable from the cone of effective curve classes, such that the following holds.

Figures (2)

  • Figure 1: A degeneration argument
  • Figure 2: Branched covers of the reducible curves $\overline{\mathcal{C}}_{0,4}|_{D_1}$ and $\overline{\mathcal{C}}_{0,4}|_{D_0}$

Theorems & Definitions (109)

  • Theorem 1.4: See Proposition \ref{['prop:adams-property']} and Lemma \ref{['lem:qadams-additivity']}
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: = Theorem \ref{['thm:qadams=pcurvature']}
  • Conjecture 1.8: = \ref{['conj:arithmetic-k-hikita']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3: Nakajima quiver varieties Nak94
  • Example 2.4: Hypertoric varieties hyper-toric
  • Definition 2.5: Gluing operator
  • ...and 99 more