By Fabian Ziltener

Think of a Hamiltonian motion of a compact hooked up Lie team on a symplectic manifold M ,w. Conjecturally, below appropriate assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten thought of M , w to the Gromov-Witten thought of the symplectic quotient. The morphism might be a deformation of the Kirwan map. the belief, as a result of D. A. Salamon, is to outline this sort of deformation through counting gauge equivalence periods of symplectic vortices over the advanced aircraft C. the current memoir is a part of a undertaking whose aim is to make this definition rigorous. Its major effects take care of the symplectically aspherical case

**Read or Download A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane PDF**

**Similar differential geometry books**

This advent to the most rules of algebraic and geometric invariant thought assumes just a minimum historical past in algebraic geometry, algebra and illustration concept. themes lined comprise the symbolic strategy for computation of invariants at the area of homogeneous varieties, the matter of finite-generatedness of the algebra of invariants, and the speculation of covariants and buildings of express and geometric quotients.

**Differential Geometry: Bundles, Connections, Metrics and Curvature **

Bundles, connections, metrics and curvature are the 'lingua franca' of contemporary differential geometry and theoretical physics. This booklet will offer a graduate pupil in arithmetic or theoretical physics with the basics of those gadgets. a few of the instruments utilized in differential topology are brought and the fundamental effects approximately differentiable manifolds, gentle maps, differential kinds, vector fields, Lie teams, and Grassmanians are all awarded right here.

**Surveys in Differential Geometry, Vol. 13: Geometry, Analysis, and Algebraic Geometry**

Contents exact Lagrangian fibrations, wall-crossing, and reflect symmetry (Denis Auroux) Sphere theorems in geometry (Simon Brendle and Richard Schoen) Geometric Langlands and non-Abelian Hodge conception (Ron Donagi and Tony Pantev) advancements round confident sectional curvature (Karsten Grove) Einstein metrics, four-manifolds, and conformally KÃ¤hler geometry (Claude LeBrun) lifestyles of Faddeev knots (Fengbo cling, Fanghua Lin, and Yisong Yang) Milnor K2 and box homomorphisms (Fedor Bogomolov and Yuri Tschinkel) Arakelov inequalities (Eckart Viehweg) A survey of Calabi-Yau manifolds (Shing-Tung Yau)

- Differential Geometry of Submanifolds
- Differential manifolds and mathematical physics
- Elliptic operators, topology and asymptotic methods
- Supersymmetry and Equivariant de Rham Theory

**Extra info for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane**

**Example text**

Let z ∈ A(4r, rR). 73) ew (z) ≤ 32 E w, B |z| (z) . 2 π|z|2 We deﬁne a := |z|/(2r). Then a ≥ 2 and B|z|/2 (z) is contained in A(ar, a−1 R). 61) we have E w, B |z| (z) ≤ 16r 2−ε |z|−2+ε E(w). 73) and the fact |dA u|(z) ≤ 1− ε2 −2+ ε2 |dA u(z)v| ≤ Cr |z| E(w)|v|, √ where C := 32/ π. 75) |dA u(z)v| ≤ CR−1+ 2 |z|− 2 ε ε E(w)|v|, 2ew (z), it follows that √ ∀z ∈ A(4r, rR), v ∈ C. √ ∀z ∈ A( rR, R/4). Let now a ≥ 4 and z, z ∈ A(ar, a−1 R). Assume that ε ≤ 1. ) We deﬁne γ : [0, 1] → C to be the radial path of constant speed, such that γ(0) = z and |γ(1)| = |z |.

Furthermore, if α, β ∈ T1 α ∈ T∞ the J¯-holomorphic map u are such that α = β and E(Wα ) = 0, and ϕ ∈ TC , then ϕ∗ Wα = Wβ . Moreover, if ¯α is nonconstant, and if ϕ ∈ PSL(2, C), then α, β ∈ T∞ are such that α = β and u ¯β . We denote by u ¯α ◦ ϕ = u M∗ (T ) := M∗ T0 , T1 , T∞ , E ⊆ M(T ) the subset of all simple stable maps. The action of GT on M(T ) leaves M∗ (T ) invariant. 31. Proposition. The action of GT on M∗ (T ) is free. The proof of this result uses the following lemma. 32. Lemma. The action of TC on M<∞ is free.

Such a map is needed for the deﬁnition of the quantum Kirwan map. Note here that we cannot deﬁne evaluation of a vortex class W at some point z ∈ Σ by choosing a representative (P, A, u) of W and evaluating u at some point in the ﬁber over z, thus obtaining a point in M , since this point depends on the choices. 12 Another reason for allowing only translations as reparametrizations (for α ∈ T1 ) is that the action of Isom+ (Σ) on the set of vortex classes with positive energy is not always free.