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K-theoretic Heisenberg algebras and permutation-equivariant Gromov--Witten theory

Todor Milanov

Abstract

We found an interesting application of the K-theoretic Heisenberg algebras of Weiqiang Wang to the foundations of permutation equivariant K-theoretic Gromov--Witten theory. We also found an explicit formula for the genus 0 correlators in the permutation equivariant Gromov--Witten theory of the point. In the non-equivariant limit our formula reduces to a well known formula due to Y.P. Lee.

K-theoretic Heisenberg algebras and permutation-equivariant Gromov--Witten theory

Abstract

We found an interesting application of the K-theoretic Heisenberg algebras of Weiqiang Wang to the foundations of permutation equivariant K-theoretic Gromov--Witten theory. We also found an explicit formula for the genus 0 correlators in the permutation equivariant Gromov--Witten theory of the point. In the non-equivariant limit our formula reduces to a well known formula due to Y.P. Lee.
Paper Structure (23 sections, 20 theorems, 177 equations)

This paper contains 23 sections, 20 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. K-theoretic Fock spaces
  3. Genus-0 integrable hierarchies
  4. Permutation-equivariant invariants of the point in genus 0
  5. Acknowledgements
  6. K-theoretic Fock space
  7. Double cosets of the Young subgroups
  8. Induced vector bundles
  9. Anihilation operators
  10. Permutation-equivariant K-theoretic Gromov--Witten theory
  11. Translation symmetry
  12. Givental's permutation-equivariant correlators
  13. Descendants at the permutable entries
  14. String equation
  15. Dilaton equation
  16. ...and 8 more sections

Key Result

Proposition 1.1.1

The Fock space $\mathcal{F}(X)=\mathbb{C}[\nu_{n,\alpha}(1\leq \alpha\leq N, n\geq 0)]$, that is, $\nu_{n,\alpha}$ generate freely $\mathcal{F}(X)$ as a commutative algebra.

Theorems & Definitions (24)

  • Proposition 1.1.1: Wang
  • Theorem 1.2.1
  • Theorem 1.3.1
  • Lemma 2.1.1
  • Lemma 2.1.2
  • proof
  • Lemma 2.1.3
  • proof
  • Proposition 2.1.1
  • Proposition 2.2.1: Frobenius reciprocity
  • ...and 14 more