A Quantum Kirwan Map: Bubbling and Fredholm Theory for ... - KIAS

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46 2. BUBBLING FOR VORTICES OVER THE PLANEWe choose a constant δ as in Theorem 82 in Appendix A.2 (Isoperimetric inequality),corresponding to〈·, ·〉 M := ω(·,J·), K ′ , c := 12 − ε .Shrinking δ we may assume that it satisfies the condition of Proposition 83 inAppendix A.2 (Energy action identity) for K ′ . We choose a constant Ẽ0 > 0 as inLemma 72 in Appendix A.1 (called E 0 there), corresponding to K. We define(2.65) E 0 := min{Ẽ 0 , π 32 r2 0δ 2 0,δ 2128πAssume that r,R,p,w are as in the hypothesis. Without loss of generality, we mayassume that r < R.Consider first the case R < ∞, and assume that w extends to a smooth vortexover the compact annulus of radii r and R. We show that inequality (2.61) holds.We define the function(2.66) E : [0, ∞) → [0, ∞), E(s) := E ( w,A(re s ,Re −s ) ) .(2.67)Claim. For every s ∈ [ log 2,log(R/r)/2 ) we havedE(s) ≤ −(2 − ε)E(s).dsProof of the claim. Using the fact r ≥ r 0 and (2.60,2.65), it follows fromLemma 72 in Appendix A.1 (with r replaced by |z|/2) that{(2.68) e w (z) ≤ min δ0,2 δ 2 }4π 2 |z| 2 , ∀z ∈ A(2r,R/2).We define}.Σ s := ( s + log r, −s + log R ) × S 1 , ∀s ∈ R,ϕ : Σ 0 → C, ϕ(z) := e z , ˜w := (Ã,ũ) := ϕ∗ w.(Here we identify Σ 0∼ = C/ ∼, where z ∼ z + 2πin, for every n ∈ Z.) Let s0 ∈[log(2r),log(R/2)]. Combining (2.68) with the fact |µ ◦ u| ≤√ew and Remark 46,it follows that(2.69) ũ(s 0 ,t) ∈ K ′ = µ −1 ( ¯B δ0 ), ∀t ∈ S 1 , ¯l(Gũ(s0 , ·)) ≤ δ.Hence the hypotheses of Theorem 82 (Appendix A.2) are satisfied with K replacedby K ′ and c := 1/(2 − ε). By the statement of that result the loop ũ(s 0 , ·) isadmissible, and definingwe have(2.70)ι s0 : S 1 → Σ 0 ,ι s0 (t) := (s 0 ,t),∣ A(ι∗s0 ˜w )∣ ∣ ≤12 − ε ‖ι∗ s 0d eA ũ‖ 2 2 + 12m 2 K ′ ∥ ∥µ ◦ ũ ◦ ιs0∥ ∥22 .Here A denotes the invariant symplectic action, as defined in Appendix A.2. Furthermore,the L 2 -norms are with respect to the standard metric on S 1 ∼ = R/(2πZ),the metric ω(·,J·) on M, and the operator norm | · | op : g ∗ → R, induced by 〈·, ·〉 g .By (2.63,2.64) and the fact 2r ≤ e s0 , we have1(2.71)2 − ε |ι∗ s 0d eA ũ| 2 0(t) + 1 ∣ ∣ ∣µ ◦ ũ ◦2m 2 ιs0 2 1 (t) ≤ eK2 − ε e2s0 w (e s0+it ), ∀t ∈ S 1 .′
2.6. SOFT RESCALING 47Here the norm | · | 0 is with respect to the standard metric on S 1 ∼ = R/(2πZ), andwe used the fact that for every ϕ ∈ g ∗ ,|ϕ| op ≤ |ϕ| := √ ϕ(ξ)where ξ ∈ g is determined by 〈ξ, ·〉 g = ϕ. We fix s ∈ [ log 2,log(R/r)/2 ) . Recalling(2.66), we have∫E(s) = e 2s0 e w (e s0+it )dtds 0 .Σ sCombining this with (2.70,2.71), it follows that(2.72) −A ( ι ∗ −s+log R ˜w ) + A ( ι ∗ s+log r ˜w ) ≤ − 1 d2 − ε ds E(s).Using (2.69), the hypotheses of Proposition 83 are satisfied with K replaced by K ′ .Applying that result, we haveE(s) = −A ( ι ∗ −s+log R ˜w ) + A ( ι ∗ s+log r ˜w ) .Combining this with (2.72), inequality (2.67) follows. This proves the claim.By the claim the derivative of the function[log 2, log ( ))Rr∋ s ↦→ E(s)e (2−ε)s2is non-positive, and hence this function is non-increasing. Inequality (2.61) follows.We prove (2.62). Let z ∈ A(4r, √ rR). Using (2.60) and the fact E 0 ≤ Ẽ0, itfollows from Lemma 72 (with r replaced by |z|/2) that(2.73) e w (z) ≤ 32 ( )π|z| 2 E w,B |z| (z) .2We define a := |z|/(2r). Then a ≥ 2 and B |z|/2 (z) is contained in A(ar,a −1 R).Therefore, by (2.61) we haveE ( w,B |z| (z) ) ≤ 16r 2−ε |z| −2+ε E(w).2Combining this with (2.73) and the fact |d A u|(z) ≤ √ 2e w (z), it follows that(2.74) |d A u(z)v| ≤ Cr 1− ε 2 |z|−2+ ε 2√E(w)|v|, ∀z ∈ A(4r,√rR), v ∈ C.where C := 32/ √ π. A similar argument shows that(2.75) |d A u(z)v| ≤ CR −1+ ε 2 |z|− ε 2√E(w)|v|, ∀z ∈ A(√rR,R/4).Let now a ≥ 4 and z,z ′ ∈ A(ar,a −1 R). Assume that ε ≤ 1. (This is no realrestriction.) We define γ : [0,1] → C to be the radial path of constant speed,such that γ(0) = z and |γ(1)| = |z ′ |. Furthermore, we choose an angular pathγ ′ : [0,1] → C of constant speed, such that γ ′ (0) = γ(1), γ ′ (1) = z ′ , and γ ′ hasminimal length among such paths. (See Figure 4.)Consider the “twisted length” of γ ∗ (A,u), given by∫ 10∣ dA u ˙γ(t) ∣ dt.It follows from (2.74,2.75) and the fact ε ≤ 1, that this length is bounded aboveby 4C √ E(w)a −1+ε/2 . Similarly, it follows that the “twisted length” of γ ′∗ (A,u)□
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100 A. AUXILIARY RESULTSpoint x ∞
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102 A. AUXILIARY RESULTSsymplectic
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104 A. AUXILIARY RESULTSconverges t
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106 A. AUXILIARY RESULTSρ : [0,
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110 A. AUXILIARY RESULTSis an isome
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112 A. AUXILIARY RESULTSis a G-bund
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118 A. AUXILIARY RESULTS111. Lemma.
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Bibliography[Ad] D. R. Adams, L. I.
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BIBLIOGRAPHY 129[We] K. Wehrheim, U
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