Ideas for the Simons Wall Project

Claude LeBrun Department of Mathematics

Stony Brook

March 5, 2010

Uniformization of a surface of genus 2:

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Index theorem for coupled Dirac operator:

∂̸ : Γ(𝕊_{+} ⊗ E) → Γ(𝕊_{−} ⊗ E)

dim ker(∂̸) − dim ker(∂̸^{∗}) = ∫_{M^{4k}} Â(M) ⌣ ch(E)

Generalized Gauss-Bonnnet theorem:

χ(M^{2n}) = \frac{1}{(8π)^{n}n!} ∫_{M} \underbrace{R_{ab}^{ij} ⋯ R_{cd}^{kℓ}}_{n} ε^{ab···cd}ε_{ij···kℓ} dμ

Classical Gauss-Bonnnet theorem:

χ(M^{2}) = \frac{1}{2π} ∫_{M} K dA

χ(M) = −4  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Einstein’s gravitational field equations:

R_{μν} − \frac{R}{2}g_{μν} = \frac{8πG}{c^{4}}T_{μν}

Hodge theorem:

H^{p}(M) = {φ ∈ Ω^{p}(M) | dφ = 0, d ⋆ φ = 0}

Dirac Equation:

(−iℏγ^{μ}∇_{μ} + mc) ψ = 0
γ^{μ}γ^{ν} + γ^{ν}γ^{μ} = g^{μν}

Seiberg-Witten Equations:

∂̸_{A}ψ = 0
F_{A}^{+} = −\frac{1}{2}ψ ⊙ ψ̄

Elliptic curve, and uniformizing elliptic integral:

× ×  × 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F (z) = ∫_{z_{0}}^{z} \frac{dζ}{\sqrt{ζ(ζ − 1)(ζ − λ)}}