COMPLEX K-THEORY SPECTRA

The type represents -twisted -equivariant complex K-theory group (generalized K-theory, complex K-theory spectra), a topological invariant used to classify vector bundles or operator algebras over a topological space up to stable isomorphisms, twisted by a groupoid. It is a cornerstone of algebraic topology and mathematical physics, with applications in quantum field theory, string theory, and index theory.

group naturally classifies the ground states of su(2)-anyons like Majorana or Fibonacci anyons. These anyons are defects in the material, and their wavefunctions are modeled by elements in these K-theory groups. The configuration space represents the positions of these defects, the twist and group action account for the topological and symmetry properties of the system. A term of type represents the specific quantum state, and a transport operation along a braid path represents the application of a quantum gate, preserving topological protection.

Formation

(-Formation). The generalized K-theory type is formed over a term , a groupoid , and a twist , yielding a type in the universe .

def KU_G^τ (X : Type) (G : Group) (τ : X → BPU)
  : Spectra := KU_G^τ(X;G;τ)

Introduction

(-Introduction). A term of type is introduced by constructing a generalized K-theory class, representing a stable equivalence class of vector bundles or operators over , twisted by and .

def intro-ku (X : Type) (G : Group) (τ : X → BPU)
    (f : Maps X (Fred^0 H))
  : KU(X;G;τ) := ku-intro f

Elimination

(-Elimination). The eliminator for allows reasoning about generalized K-theory classes by mapping them to properties or types dependent on , typically by analyzing the underlying bundle or operator structure over .

def elim-ku (X : Type) (G : Group) (τ : X → BPU) (k : KU_G^τ(X;G;τ))
  : Maps X (Fred^0 H) := ku-elim k

Computation

def ku-β (X : Type) (G : Group) (τ : X → BPU) (f : Maps X (Fred^0 H))
  : Path (Maps X (Fred^0 H)) (elim-ku X G τ (intro-ku X G τ f)) f
 := refl _ f

Theorems

(K-Theory Stability). The type is stable under suspension, meaning it is invariant under the suspension operation in the spectrum, reflecting its role in stable homotopy theory.

def stability (X : Type) (G : Group) (τ : X → BPU)
  : Path Spectra (KU_G^τ(X; G; τ)) (KU_G^τ(susp X; G; τ)))
 := refl Spectra (KU_G^τ(X; G; τ))

(Refinement to Differential K-Theory). The type can be refined to differential K-theory by incorporating a connection, as provided by .

def refinement (X : Type) (G : Group) (τ : X -> BPU) (conn : Ω^1(X))
  : KU(X; G; τ) -> KU_♭(X; G; τ; conn)
 := \ (x : KU(X; G; τ)), intro-ku_♭(x, conn)