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

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30 2. BUBBLING FOR VORTICES OVER THE PLANEthe set of all stable maps of (combinatorial) type T. G T acts on M(T) as follows.For every (f,(ϕ α )) ∈ G T and (W,z) ∈ M(T) we defineW ′ α := ϕ ∗ f(α) W f(α), ∀α ∈ T 1 , ū ′ α := ū α ◦ ϕ f(α) , ∀α ∈ T ∞ ,z ′ αβ:= ϕ−1f(α) (z f(α)f(β)),∀αEβ,α i ′ := f(α i), z i ′ := ϕ−1α(z ′ α ′i i), ∀i = 0,...,k.(Here we set W α := ū α if α ∈ T ∞ . Furthermore, for ϕ ∈ T C we set ϕ(∞) := ∞.)30. Definition. We define)(f,(ϕ α )) ∗ (W,z) :=(T 0 ,T 1 ,T ∞ ,E,(W α) ′ α∈T1 ,(ū ′ α) α∈T∞ ,(z αβ) ′ αEβ ,(α i,z ′ i) ′ i=0,...,k .This defines an action of G T on M(T). Let now (M,J) be an almost complexmanifold. Recall that a J-holomorphic map u : S 2 → M is called multiply coverediff there exists a holomorphic map ϕ : S 2 → S 2 of degree at least two, and aJ-holomorphic map v : S 2 → M, such that u = v ◦ϕ. Otherwise, u is called simple.We call a stable map (W,z) simple iff the following conditions hold: For everyα ∈ T ∞ the ¯J-holomorphic map ū α is constant or simple. Furthermore, if α,β ∈ T 1are such that α ≠ β and E(W α ) ≠ 0, and ϕ ∈ T C , then ϕ ∗ W α ≠ W β . Moreover, ifα,β ∈ T ∞ are such that α ≠ β and ū α is nonconstant, and if ϕ ∈ PSL(2, C), thenū α ◦ ϕ ≠ ū β . We denote byM ∗ (T) := M ∗( T 0 ,T 1 ,T ∞ ,E ) ⊆ M(T)the subset of all simple stable maps. The action of G T on M(T) leaves M ∗ (T)invariant.31. Proposition. The action of G T on M ∗ (T) is free.The proof of this result uses the following lemma.32. Lemma. The action of T C on M
2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 31Note also that the action of Isom + (C) on the set of vortex classes of positiveenergy is not always free, as the next example shows. ✷34. Example. Consider the action of G := S 1 ⊆ C on M := C by multiplication.Let d ∈ N 0 be an integer. By Proposition 26 there exists a unique finiteenergy vortex class W over C such that{ d, if z = 0,deg W (z) =0, otherwise.For every rotation R ∈ SO(2) we havedeg R ∗ W = R ∗ deg W = deg W ,where SO(2) acts in a natural way on Sym d (C). Thus the action of Isom + (C) onthe set of vortex classes of positive energy is not free. ✷2.5. Compactness modulo bubbling and gauge for rescaled vorticesIn this section we formulate and prove a crucial ingredient (Proposition 37) ofthe proof of Theorem 3, which states the following. Consider a sequence of rescaledvortices over C with image in a fixed compact subset of M and uniformly boundedenergies. We assume that (M,ω) is aspherical. Then there exists a subsequencethat away from finitely many bubbling points and up to regauging, converges toa rescaled vortex over C. If the rescalings converge to ∞, then the limit objectcorresponds to a ¯J-holomorphic sphere in M.The proof of this result is based on compactness for rescaled vortices over thepunctured plane with uniformly bounded energy densities (Proposition 38 below).It also uses the fact that at each bubbling point at least the energy E min > 0 islost, which is the minimal energy of a vortex over C or pseudo-holomorphic spherein M. This is the content of Proposition 40 below, which is proved by a hardrescaling argument, using Proposition 38 and Hofer’s lemma. Another ingredientof the proof of Proposition 37 is Lemma 42 below, which says that the energydensities of a convergent sequence of rescaled vortices converge to the density ofthe limit.In order to explain the main result of this section, let M,ω,G,g, 〈·, ·〉 g ,µ,J,Σ,j, and ω Σ be as in Chapter 1. We fix a triple w = (P,A,u) ∈ ˜B Σ (defined as in(1.7)). Recall the definition (1.11) of the energy density ew ωΣ,j = e w .35. Remark. This density has the following transformation property: Let Σ ′be another surface, and ϕ : Σ ′ → Σ a smooth immersion. Consider the pullbackϕ ∗ w := ( ϕ ∗ P,Φ ∗ A,u ◦ Φ ) ,where the bundle isomorphism Φ : ϕ ∗ P → P is defined by Φ(z,p) := p. A straightforwardcalculation shows that(2.26) e ϕ∗ (ω Σ,j)ϕ ∗ w= e ωΣ,jw ◦ ϕ.Note also that w is a vortex with respect to (ω Σ ,j) if and only if ϕ ∗ w is a vortexwith respect to ϕ ∗ (ω Σ ,j). ✷36. Remark. If w is a vortex (with respect to (ω Σ ,j)) then(2.27) e ωΣ,jw = |∂ J,A u| 2 + |µ ◦ u| 2 ,
Page 1: A Quantum Kirwan Map: Bubbling and
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92 A. AUXILIARY RESULTSwhere the no
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96 A. AUXILIARY RESULTSA.2. The inv
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100 A. AUXILIARY RESULTSpoint x ∞
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102 A. AUXILIARY RESULTSsymplectic
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106 A. AUXILIARY RESULTSρ : [0,
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108 A. AUXILIARY RESULTSProof of Cl
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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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A.7. FURTHER AUXILIARY RESULTS 121T
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A.7. FURTHER AUXILIARY RESULTS 125a
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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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