The Picard-Lefschetz theory of complexified Morse functions 1 ...

Complexified Morse functions 33
Here,
Sr(T ∗ S 3 ) = {(u, v) ∈ T ∗ S 3 : |v| = r}, D[t,r](T ∗ S 3 ) = {(u, v) ∈ T ∗ S 3 : |v| ∈ [t, r]}.
q has well-defined symplectic parallel transport maps between any two regular fibers,
where the connection is given by the symplectic orthogonal to the fibers,
Tz(q −1 (w)) ω = z C.
(Indeed, parallel transport preserves the level sets of k, so it is well-defined.) For any
embedded path
γ : [0, 1] −→ D 2 with γ(0) = w = 0,γ(0) = 0
the corresponding Lefschetz thimble (or vanishing disk) is
and the vanishing sphere in q −1 (w) is
∆γ = ∪tΣγ(t)
Vγ = Σw ⊂ q −1 (w).
Φ can be described more conceptually as follows.
Lemma 4.1 Consider the map
Φ: C n+1 \ Σ −→ q −1 (0) \ Σ0
given by the radial symplectic parallel transport maps from q −1 (w)\Σw to q −1 (0)\Σ0 .
Then Φ = ρ0 ◦ Φ.
(We omit the proof. One can check this using the rotational symmetry of the radial
transport map as explained in the proof of lemma 1.10 in [S03A].)
For any θ ∈ [0, 2π], 0 < s, let
τθ : q −1 (s) −→ q −1 (se iθ )
be the parallel transport along γ(t) = se iθt , 0 ≤ t ≤ 1. Take the restriction
For w ∈ D 2 , let
τθ = τθ| (q −1 (s)\Σs) : q −1 (s) \ Σs −→ q −1 (se iθ ) \ Σ se iθ.
be the restriction of Φ. Then it turns out
Φw : q −1 (w) \ Σw −→ T ∗ S n \ S n
Φ se iθ ◦ τθ ◦ Φ −1
s : T ∗ S n \ S n −→ T ∗ S n \ S n