Normalized phase shifts #
The universal cover G̃L⁺(2, ℝ) is standardly presented as pairs (T, f)
with T ∈ GL⁺(2, ℝ) and f : ℝ ≃o ℝ satisfying f (φ + 1) = f φ + 1,
subject to T and f inducing the same map on S¹. This file builds the
phase-relabeling factor f as a self-contained mathematical object.
The f component acts on slicings by relabelling phases; the T component
acts on central charges. Splitting them exposes the algebraic and analytic
properties of phase relabelling independently.
An order-isomorphism of ℝ commuting with φ ↦ φ + 1.
The phase-relabelling half of an element of G̃L⁺(2, ℝ). The φ + 1
equivariance is what makes it descend to the circle ℝ / 2ℤ, and it is why
the shift functor [1] interacts with the action the way §8 needs.
On ≃o rather than "increasing". Bridgeland §8 asks only for an
increasing f with f(φ + 1) = f(φ) + 1, and a priori that is weaker:
φ ↦ ⌊φ⌋ + (φ - ⌊φ⌋)/2 is strictly increasing and +1-equivariant but is not
a bijection. The narrowing is harmless because pairing with an invertible T
forces bijectivity — but that forcing happens in GLTilde, not here, so this
structure on its own is strictly narrower than the paper's phrase.
The underlying increasing bijection of
ℝ.Equivariance for the unit shift.
Instances For
A normalized shift is determined by its underlying order-isomorphism.
map_add_one is a Prop, so it carries no data to disagree about.
The inverse of a normalized shift is again +1-equivariant.
Apply f to both sides and use injectivity: this is the only step of the
group structure that is not immediate from composition.
Normalized shifts form a group under composition.
The phase-relabelling factor of G̃L⁺(2, ℝ) is a group in its own right,
before pairing it with GL⁺(2, ℝ).
Equations
- One or more equations did not get rendered due to their size.