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The weakly interacting tenfold way

Lucas C. P. A. M. Müssnich, Renato Vasconcellos Vieira

Abstract

We present implementations of the topological K-theory spectra $KU$ and $KO$ in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We also introduce a geometric definition of weakly interacting time evolution operators, and show how associated spectra $KU^{wi}$ and $KO^{wi}$ deformation retract to $KU$ and $KO$. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.

The weakly interacting tenfold way

Abstract

We present implementations of the topological K-theory spectra and in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We also introduce a geometric definition of weakly interacting time evolution operators, and show how associated spectra and deformation retract to and . We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.
Paper Structure (18 sections, 3 theorems, 119 equations, 2 tables)

This paper contains 18 sections, 3 theorems, 119 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Notation and terminology
  4. Acknowledgements
  5. Classification of free fermion systems with symmetries
  6. Nambu space model for fermions
  7. Nambu space symmetries
  8. Reduction to grotesque fermion systems
  9. Convenient choice of basis
  10. Spaces of equivariant free Hamiltonian
  11. Classification by symmetric spaces of compact type
  12. K-theoretical classification
  13. Cohomology theories and representing spectra
  14. Construction of $KU$ and $KO$ in terms of time-evolution operators
  15. Weakly interacting systems
  16. ...and 3 more sections

Key Result

Proposition 3.2

Let $G$ be a compact Lie group equipped with a Cartan involution $\tau$. Let $s$ be a geodesic segment on the symmetric space $\frac{G}{\text{Fix}_\tau}$ from the coset $\text{Fix}_\tau$ to a coset $g\text{Fix}_\tau$, with $g$ in the normalizer of $\text{Fix}_\tau$. Let $K_s$ be the centralizer of $ If $s$ contains no conjugate point of $e$ in its interior, then the induced homomorphism is surjec

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 10 more