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Complexity of quantum cohomology

Xiaobo Liu, Chongyu Wang

Abstract

Circuit complexity for two-dimensional topological quantum field theories (2D TQFT) was defined by Couch, Fan, and Shashi in [12]. In this paper, we study complexity for the 2D TQFT given by quantum cohomology of compact symplectic manifolds. We will estimate the number of states with finite approximate complexity of arbitrarily small tolerance for Fano complete intersections and (co)minuscule homogeneous varieties. We will give an upper bound for the dimension of the space spanned by states with finite complexity. In the case of Gr(2, n), this bound is sharp and we also obtain a precise description for this subspace. For (co)minuscule homogeneous varieties, we prove a positivity result for the eigenvalues of quantum multiplication by the handle element (also called the quantum Euler class) divided by the class of a point.

Complexity of quantum cohomology

Abstract

Circuit complexity for two-dimensional topological quantum field theories (2D TQFT) was defined by Couch, Fan, and Shashi in [12]. In this paper, we study complexity for the 2D TQFT given by quantum cohomology of compact symplectic manifolds. We will estimate the number of states with finite approximate complexity of arbitrarily small tolerance for Fano complete intersections and (co)minuscule homogeneous varieties. We will give an upper bound for the dimension of the space spanned by states with finite complexity. In the case of Gr(2, n), this bound is sharp and we also obtain a precise description for this subspace. For (co)minuscule homogeneous varieties, we prove a positivity result for the eigenvalues of quantum multiplication by the handle element (also called the quantum Euler class) divided by the class of a point.
Paper Structure (24 sections, 39 theorems, 198 equations)

This paper contains 24 sections, 39 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complexity in 2D TQFT
  4. Quantum cohomology
  5. Projective spaces and quadrics
  6. Projective spaces
  7. Quadrics $\mathbb{Q}^{r}$
  8. (Co)minuscule homogeneous varieties
  9. Quantum cohomology for (co)minuscule homogeneous varieties
  10. Action of handle operator
  11. Estimate the size of $\mathfrak{S}_\infty$
  12. Estimate dimension of $\mathfrak{F}$
  13. Grassmannians
  14. Quantum cohomology of Grassmannians
  15. Computing $\Delta$ for Grassmannians
  16. ...and 9 more sections

Key Result

Theorem 1.1

For the small quantum cohomology of all (co)minuscule homogeneous varieties and Fano complete intersections of complex dimension bigger than $2$, $\mathfrak{S}_\infty$ is a finite set for arbitrary reference state. Moreover, the number of points in $\mathfrak{S}_\infty$ is less than or equal to the

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • Lemma 4.4
  • ...and 67 more