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Mathematics · Document 8

Index Theory

The Atiyah-Singer index theorem, eta invariants, and counting zero modes

1. What is Index Theory?

Index theory connects analysis (solutions of differential equations) to topology (global shape of spaces). The central result is that the number of solutions is determined by topological invariants!

ind(D) = dim(ker D) - dim(ker D†)
The Index
ker D:
Solutions to Dψ = 0 (zero modes)
ker D†:
Solutions to D†ψ = 0 (adjoint zero modes)
The index is topological! It doesn't change under continuous deformations of the operator D. This is why we can compute it from topology alone.

2. The Dirac Index

For the Dirac operator on spinors, the index counts the difference between left-handed and right-handed zero modes:

🎮 Interactive: Index = Left - Right
Left-handed
+
+
+
+
+
−
Right-handed
−
−
=
Index
3
ind(D) = n₊ - n₋ = 5 - 2 = 3
The index counts the net number of chiral zero modes
Key insight: The index is a topological invariant — it doesn't change under continuous deformations of the operator D!
ind(D) = n₊ - n₋ = ∫_M Â(R) ch(F)
Dirac index on manifold M

Physical Interpretation

• n₊: Number of left-handed fermion zero modes

• n₋: Number of right-handed fermion zero modes

• ind(D): Net chirality — the "excess" of one handedness

Z²
n₊ - n₋ = 3
The index on T³/Z₂ gives 3 generations of chiral fermions

3. The Atiyah-Singer Index Theorem

The Atiyah-Singer theorem (1963) says that the analytical index equals a topological integral:

ind(D) = ∫_M Â(M) ∧ ch(E)
Atiyah-Singer Theorem

Ingredients

TermNameMeaning
Â(M)A-hat genusBuilt from Riemann curvature
ch(E)Chern characterBuilt from gauge field strength
∫_MIntegrationOver the manifold M
Analysis = Topology: The number of solutions to a differential equation (left side) equals a topological integral (right side)!

4. The APS Theorem (with Boundary)

When the manifold M has a boundary ∂M, there's an additional correction term: the eta invariant.

🎮 Interactive: Atiyah-Patodi-Singer Theorem
M (manifold)Dirac operator D∂Mη(∂M)ind(D) = ∫_M â - η(∂M)/2
ind(D)
Analytical index
∫_M â
A-hat genus (curvature)
η(∂M)
Eta invariant (boundary)
ind(D) = ∫_M Â(R) - η(∂M)/2
APS Theorem

The Eta Invariant

The eta invariant η measures the spectral asymmetry of the Dirac operator on the boundary:

🎮 Interactive: Eta Invariant: Spectral Asymmetry
0-1λ=-2.5-1λ=-1.2-1λ=-0.3+1λ=0.5+1λ=1.8+1λ=3.1−+
Click eigenvalues to include/exclude them
η = Σ sign(λ) = 0
(Sum of signs of non-zero eigenvalues)
Eta invariant: η = Σ_{λ≠0} sign(λ)|λ|^{-s}|_{s=0} (regularized sum)
Measures the asymmetry of the spectrum around zero.

5. Index Theory for Orbifolds

On orbifolds like T³/Z₂, the index theorem gets contributions from fixed points:

ind(D) = ∫_{T³/Z₂} Â + Σ_{fixed} η_p / 2
Orbifold Index
Bulk contribution
∫ Â over the smooth part
Fixed point contributions
Eta invariants at the 8 singularities
Z²
η contributions
Each of the 8 fixed points on T³/Z₂ contributes to the index

6. Connection to Z² Framework

Z²
ind(D) = 3
Index theorem on T³/Z₂ gives exactly 3 generations of fermions
Z²
η invariant
Boundary corrections important for D-brane physics
Z²
 genus
Gravitational contribution to anomalies

Why Index Theory Matters

  • • Generation counting: Index = number of fermion generations
  • • Anomaly cancellation: Related to consistency of the theory
  • • Topological protection: Index is stable under perturbations
  • • D-brane charges: Counted by K-theory and indices

Exercises

  1. If a Dirac operator has 5 left-handed and 2 right-handed zero modes, what is the index?
  2. The  genus of a 4-torus T⁴ is 0. What does this tell us about the Dirac index?
  3. Why is the index an integer? (Hint: dimensions of vector spaces)
  4. For the sphere S², show that χ(S²) = 2 using V - E + F for a triangulation.
  5. The eta invariant is regularized. Why is the naive sum Σ sign(λ) problematic?