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

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2 1. MOTIVATION AND MAIN RESULTSThe space QH ∗ G(M,ω) is naturally a module over the equivariant Novikov ringΛ µ ω. 3 We denote by QH ∗ (M,ω) the quantum cohomology of (M,ω) with coefficientsin this ring. A map(1.1) ϕ : H G 2 (M, Z) → Hom Q(H∗G (M),H ∗ (M) )satisfying the equivariant Novikov condition 4 , induces a Λ µ ω-module homomorphism(1.2) ϕ ∗ : QH ∗ G(M,ω) → QH ∗ (M,ω), (ϕ ∗ α)(B) := ∑ ϕ(B 1 )α(B 2 ),where the sum is over all pairs B 1 ,B 2 ∈ H2 G (M, Z) satisfying B 1 + B 2 = B. Wedenote by c G 1 (M,ω) ∈ HG 2 (M, Z) the first G-equivariant Chern class of (TM,ω),and by(1.3) N := inf ({〈 c G 1 (M,ω),B 〉 ∣ ∣ B ∈ HG2 (M, Z) : spherical } ∩ N ) ∈ N ∪ {∞} 5the minimal equivariant Chern number. We call (M,ω,µ) semipositive iff thereexists a constant c ∈ R such that〈[ω − µ],B〉= c〈cG1 (M,ω),B 〉 ,for every spherical class B ∈ H G 2 (M, Z), and if c < 0 then N ≥ 1 2 dimM.1. Conjecture (Quantum Kirwan map, semipositive case). Assume that (H)holds and that (M,ω,µ) is convex at ∞ 6 and semipositive. Then there existsa map ϕ as in (1.1), satisfying the equivariant Novikov condition, such that theinduced map ϕ ∗ as in (1.2) is a surjective ring homomorphism, and(1.4)ϕ(0) = κ,〈〉(1.5) [ω − µ],B ≤ 0, B ≠ 0 =⇒ ϕ(B) = 0.Once proven, this conjecture will give rise to a recursion formula for QH ∗ (M,ω)in terms of QH ∗ G(M,ω) and ϕ. 7 As noticed in [NWZ], without the semipositivitycondition, the conjecture likely needs to be modified as follows:2. Conjecture (Quantum Kirwan map, general situation). Assume that (H)holds, and that (M,ω,µ) is convex at ∞. Then there exists a morphism of cohomologicalfield theories (CohFT’s) from the equivariant Gromov-Witten theory of(M,ω) to the Gromov-Witten theory of (M,ω).For the notion of a morphism between two CohFT’s V and W see [NWZ].Such a morphism consists of a sequence of S n -invariant multilinear mapsψ n : V ×n × H ∗ (M n,1 (A)) → W, n ∈ N 0 := N ∪ {0},3 This ring consists of all maps λ : H G2 (M, Z) → Q satisfying an equivariant version of theNovikov condition, analogous to the one above. The product is given by convolution.4 This condition is analogous to the one above.5 In this memoir N := {1, 2, . . .} does not include 0.6 This means that there exists an ω-compatible and G-invariant almost complex structure Jon M, such that the quadruple `M, ω, µ, J´ is convex at ∞ in the sense explained before Theorem3 below.7 The recursion is over the set(*+ )kX[ω − µ], B i k ∈ N ∪ {0}, B˛˛˛˛˛ i ∈ H2 G (M, Z) : ϕ(B i ) ≠ 0 or GW G B i ≠ 0, i = 1, . . . , k ,i=1where GW G B idenotes the 3-point genus 0 equivariant Gromov-Witten invariant of (M, ω) in theclass B i .
1.2. SYMPLECTIC VORTICES 3satisfying relations involving the composition maps of V and W. Here M n,1 (A)denotes the moduli space of stable n-marked scaled lines. Furthermore, the actionof the symmetric group S n is by permutations of the first n arguments of ψ n . Themap ψ 1 plays the role of ϕ ∗ as in Conjecture 1. The map ψ 0 measures how muchψ 1 fails to be a ring homomorphism. Once proven, Conjecture 2 will give rise toa recursion formula for the Gromov-Witten invariants of (M,ω) in terms of theequivariant Gromov-Witten invariants of (M,ω) and the morphism (ψ n ) n .The present memoir is part of a project whose goal is to prove Conjectures 1and 2. 8 The approach pursued here was suggested by D. A. Salamon. 9 The idea isto construct the maps of the conjectures by counting symplectic vortices over thecomplex plane C. In a first step, we will consider the (symplectically) asphericalcase. This means that∫(1.6)u ∗ ω = 0, ∀u ∈ C ∞ (S 2 ,M).S 2In this case the equivariant quantum cup product is induced by the ordinary cupproduct on HG ∗ (M).1.2. Symplectic vortices, idea of the proof of existence of a quantumKirwan mapTo explain the idea of the proofs of Conjectures 1 and 2, we recall the symplecticvortex equations: Let J be an ω-compatible and G-invariant almost complexstructure on M, 〈·, ·〉 g an invariant inner product on g, (Σ,j) a Riemann surface,and ω Σ a compatible area form on Σ. 10 For every smooth (principal) G-bundle Pover Σ we denote by A(P) the affine space of smooth connection one-forms on P,and by CG ∞ (P,M) the set of smooth equivariant maps from P to M. Consider theclass˜B := ˜B Σ := { w := (P,A,u) ∣ ∣ P smooth G-bundle over Σ,(1.7)A ∈ A(P), u ∈ C ∞ G (P,M) } .The symplectic vortex equations are the equations¯∂ J,A (u) = 0,(1.8)(1.9)F A + (µ ◦ u)ω Σ = 0for a triple (P,A,u) ∈ ˜B. To explain these conditions, note that the pullback bundleu ∗ TM → P descends to a complex vector bundle (u ∗ TM)/G → Σ. 11 For everyx ∈ M we denote by L x : g → T x M the infinitesimal action, corresponding tothe action of G on M. With this notation, ¯∂ J,A (u) means the complex antilinearpart of d A u := du + L u A, which we think of as a one-form on Σ with values in(u ∗ TM)/G → Σ. In (1.9) we view the curvature F A of A as a two-form on Σ withvalues in the adjoint bundle g P := (P × g)/G → Σ 12 . Furthermore, identifying g ∗with g via 〈·, ·〉 g , we view µ ◦ u as a section of g P . The vortex equations (1.8,1.9)were discovered by K. Cieliebak, A. R. Gaio, and D. A. Salamon [CGS], and8 Further relevant results will appear elsewhere, including [Zi4, Zi5].9 Private communication.10 This means that j and ωΣ determine the same orientation of Σ.11 The complex structure on this bundle is induced by the almost complex structure J.12 Here G acts on g in the adjoint way.
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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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