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

2.3. AN EXAMPLE: THE GINZBURG-LANDAU SETTING 27R 2 \ {0}. 13 The one-form α := r22 γ is a primitive of ω 0, and therefore, by Stokes’theorem,∫ ∫(2.20)v ∗ ω 0 = v ∗ α.B RBy elementary arguments, we have∫(v ∗ γ = 2π deg SR 1 ∋ z ↦→ v(z) )SR1 |v(z)| ∈ S1 = 2π ∑deg u (z) = 2π deg(W).z∈B R(Here in the last equality we used the assumption that v(z) ≠ 0 if |z| ≥ R.) Itfollows that∫(2.21) π deg(W) min |v(z)| 2 ≤z∈SR1 v ∗ α ≤ π deg(W) max |v(z)| 2 .z∈SR1S 1 ROn the other hand, using the estimates E(w) < ∞ and |µ ◦u| ≤ √ e w , Lemma 72 inAppendix A.1 implies that |µ ◦ v(Rz)| = 1 2 (1 − |v(Rz)|2 ) converges to 0, uniformlyin z ∈ S 1 , as R → ∞. Combining this with (2.21,2.20,A.1), equality (2.19) follows.This proves Proposition 24.□This result has the following consequence.25. Corollary. Let w := (P,A,u) be a smooth vortex over C with positiveand finite energy. Then the image of u contains the open unit ball B 1 ⊆ C.S 1 RProof of Corollary 25. Consider the setX := { |u(p)| ∣ ∣ p ∈ P}.This set is connected, and hence an interval. Since E(w) < ∞, for every r < 1there exists a point p ∈ P such that |u(p)| ≥ r. On the other hand, positivity ofthe energy and Proposition 24 imply that u vanishes somewhere. It follows that Xcontains the interval [0,1). Since the image of u is invariant under the S 1 -action,it follows that it contains the ball B 1 . This proves Corollary 25.□Proposition 23 and Corollary 25 imply that the image of u equals B 1 , forevery smooth vortex over C with positive and finite energy. Fix now d ∈ N 0 . Wedenote by Sym d (C) the d-fold symmetric product. By definition this is the quotienttopological space for the action of the symmetric group S d on C d given byσ · (z 1 ,...,z d ) := ( z σ −1 (1),...,z σ −1 (d)).We identify Sym d (C) with the set ˜Sym d (C) of all maps m : C → N 0 such thatm(z) ≠ 0 for only finitely many points z ∈ C, and∑(2.22)m(z) = d,z∈Cby assigning to z := [z 1 ,...,z d ] ∈ Sym d (C) the multiplicity map m z : C → N 0 ,given bym z (z) := # { i ∈ {1,...,d} |z i = z } .We can now characterize vortex classes with energy dπ as follows.13 By our convention this form integrates to 2π over any circle centered at the origin.