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Mathematics · Document 7

Orbifolds

T³/Z₂, fixed points, and the origin of chirality

1. What is an Orbifold?

An orbifold is a space obtained by quotienting a manifold by a group action. Unlike a smooth manifold, orbifolds can have singularities at fixed points.

Orbifold = M / G (manifold M quotiented by group G)
Definition
Manifold:
Smooth everywhere (like a sphere)
Orbifold:
Singularities at fixed points (like a cone tip)
The Z² framework uses the orbifold T³/Z₂ — the 3-torus with opposite points identified. This creates 8 fixed points and introduces chirality!

2. The Z₂ Action on the Torus

The Z₂ group has one non-trivial element σ that acts by reflection through the origin:

σ: (x, y, z) → (-x, -y, -z) [mod 2π]
🎮 Interactive: Z₂ Action on T² (2D slice of T³)
fixedfixedfixedfixedP02π2π
Z₂ action: σ: (x, y) → (2π - x, 2π - y)
Red points are fixed: σ(P) = P. These create orbifold singularities!

Properties of the Z₂ Action

• Involution: σ² = identity (applying twice returns to original)

• Free action: σ has no fixed points (NOT true here!)

• Fixed points: Points where σ(P) = P (orbifold singularities)

3. The 8 Fixed Points

Points fixed by the Z₂ action satisfy (x, y, z) = (-x, -y, -z) mod 2π. This means each coordinate is either 0 or π.

🎮 Interactive: Fixed Points on T³/Z₂
12345678xyz
8 Fixed Points
= 2³ (two choices per dimension)
Why 8? The Z₂ action sends x → -x (mod 2π). Fixed points satisfy x = -x, so x = 0 or π. With 3 coordinates, that's 2³ = 8 fixed points.
2 × 2 × 2 = 8 fixed points
Fixed point count
Z²
8 fixed points
Each fixed point creates a singularity where twisted sector strings can live

4. Twisted Sectors

In string theory on orbifolds, there are two types of string states:

🎮 Interactive: Untwisted vs Twisted Sectors
String stuck at fixed point(Boundary conditions: σ(X) = X)
Untwisted Sector
  • • String moves freely on orbifold
  • • States must be Z₂-invariant
  • • Gives gauge bosons
Twisted Sector
  • • String stuck at fixed point
  • • One sector per fixed point (8 total)
  • • Gives chiral fermions!

Physical Significance

SectorBoundary ConditionPhysics
UntwistedX(σ+2π) = X(σ)Gauge bosons, gravity
TwistedX(σ+2π) = σ(X(σ))Chiral fermions!
Chirality from orbifolds: The twisted sector states are localized at fixed points and naturally produce chiral (left-handed or right-handed) particles. This is why the weak force only affects left-handed particles!

5. Orbifold Geometry

Near each fixed point, the orbifold looks like ℝ³/Z₂ — a cone with a singular tip.

○
Away from fixed points
Locally looks like T³
△
At fixed points
Conical singularity
◆
Blowing up
Replace singularity with CP¹

Resolution of Singularities

The conical singularities can be "resolved" by replacing each fixed point with a small 2-sphere (blowup). This is important for understanding the geometry!

6. Connection to Z² Framework

Z²
T³/Z₂
The fundamental compactification space of the Z² framework
Z²
8 fixed points
Create 8 twisted sectors, contributing to the particle spectrum
Z²
Chirality
Z₂ projection removes half the states, leaving only left-handed weak interactions

Why Orbifolds Matter

  • • T³/Z₂: Provides the compact extra dimensions
  • • 8 fixed points: Give twisted sector contributions
  • • Chirality: Z₂ action creates matter-antimatter asymmetry
  • • D-branes: Wrap cycles of T³/Z₂, giving gauge groups

Exercises

  1. Verify that σ(x) = -x (mod 2π) satisfies σ² = identity.
  2. List all 8 fixed points of the Z₂ action on T³. Show they are (kπ, lπ, mπ) for k,l,m ∈ {0,1}.
  3. Why does T²/Z₂ have 4 fixed points while T³/Z₂ has 8?
  4. The orbifold ℝ²/Z₂ is a cone. What is the deficit angle?
  5. Explain why the Z₂ projection removes half the states from the spectrum.