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Loop Space Formalism and K-Theoretic Quantum Serre Duality

Xiaohan Yan

Abstract

In this paper, we prove the quantum Serre duality for genus-zero K-theoretic permutation-invariant Gromov-Witten theory. The formulation of the theorem relies on an extension to the formalism of loop spaces and big $\mathcal{J}$-functions more intrinsic to quantum K-theory. With the extended formalism, we also arrive at a re-interpretation of the level structures in terms of twisted quantum K-theories. We discuss the torus-equivariant theory in the end, and as an application generalize the K-theoretic quantum Serre duality to non-primitive vector bundles over flag varieties.

Loop Space Formalism and K-Theoretic Quantum Serre Duality

Abstract

In this paper, we prove the quantum Serre duality for genus-zero K-theoretic permutation-invariant Gromov-Witten theory. The formulation of the theorem relies on an extension to the formalism of loop spaces and big -functions more intrinsic to quantum K-theory. With the extended formalism, we also arrive at a re-interpretation of the level structures in terms of twisted quantum K-theories. We discuss the torus-equivariant theory in the end, and as an application generalize the K-theoretic quantum Serre duality to non-primitive vector bundles over flag varieties.
Paper Structure (26 sections, 37 theorems, 169 equations)

This paper contains 26 sections, 37 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Some background
  3. Main theorems
  4. Idea of proof and the rational loop space formalism
  5. Torus-equivariant theory and the assumption on primitivity
  6. Acknowledgement
  7. Preliminaries
  8. Big $\mathcal{J}$-function
  9. Loop space formalism and pseudo-finite-difference (PFD) operators
  10. Invertible twisting
  11. Level structures
  12. Loop Spaces Generalized
  13. Rational functions in $q$ as inputs
  14. Rational functions in both $q$ and parameters as inputs
  15. Power series expansion
  16. ...and 11 more sections

Key Result

Theorem A

For any point with $J^1_d$ independent of $Q_i'\ (i=1,\cdots,n)$, we have

Theorems & Definitions (44)

  • Theorem A: K-Theoretic Quantum Serre Duality
  • Theorem B: K-Theoretic Quantum Serre Duality
  • Corollary A: KqSD for flag varieties
  • Corollary B: KqSD for flag varieties
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 1: Loop space with rational functions in $q$ allowed as input
  • ...and 34 more