[Paper Review] Gromov-Witten invariants of symplectic quotients and adiabatic limits

Ana Rita Gaio, Dietmar Salamon|ArXiv.org|Jun 19, 2001

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes a ring homomorphism between the equivariant cohomology of a symplectic manifold with a Hamiltonian group action and the quantum cohomology of its symplectic quotient, showing that genus zero Gromov–Witten invariants defined via a system of elliptic PDEs in the ambient space correspond to those of the quotient in the monotone case. The key result links invariants through an adiabatic limit process involving connections and moment maps.

ABSTRACT

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our main theorem asserts that the genus zero invariants of Hamiltonian group actions defined by these equations are related to the genus zero Gromov--Witten invariants of the symplectic quotient (in the monotone case) via a natural ring homomorphism from the equivariant cohomology of the ambient space to the quantum cohomology of the quotient.

Motivation & Objective

• To relate Gromov–Witten invariants of a symplectic manifold with Hamiltonian group action to those of its symplectic quotient.
• To analyze pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions to a coupled system of nonlinear elliptic PDEs.
• To establish a natural ring homomorphism from equivariant cohomology of the ambient space to the quantum cohomology of the quotient.
• To prove that genus zero invariants defined via the PDE system match the standard Gromov–Witten invariants of the quotient in the monotone case.
• To provide a geometric and analytic framework for understanding quantum invariants under symplectic reduction.

Proposed method

• Formulates a system of first-order nonlinear elliptic PDEs on a Riemann surface, coupling the Cauchy–Riemann operator with a connection and the moment map.
• Uses a connection on a principal bundle over the Riemann surface, with curvature terms linked to the moment map via the equations.
• Applies an adiabatic limit procedure where the symplectic form on the ambient manifold is scaled to emphasize the quotient structure.
• Employs techniques from symplectic geometry and gauge theory, including the use of connections and vector fields associated to moment maps.
• Analyzes the linearized operator of the PDE system using a splitting into tangential and normal components to the level set of the moment map.
• Relies on lemmata concerning Lie derivatives and covariant derivatives to derive identities for the linearized operator.

Experimental results

Research questions

• RQ1 How do Gromov–Witten invariants of a Hamiltonian G-manifold relate to those of its symplectic quotient in the monotone case?
• RQ2 Can pseudoholomorphic curves in the quotient arise as limits of solutions to a coupled PDE system in the ambient manifold?
• RQ3 Is there a canonical ring homomorphism from the equivariant cohomology of the ambient space to the quantum cohomology of the quotient that preserves genus zero invariants?
• RQ4 What is the role of the moment map and connection curvature in mediating the correspondence between invariants in the ambient and quotient spaces?
• RQ5 How does the adiabatic limit process recover the quantum cohomology structure of the symplectic quotient from the equivariant invariants of the original manifold?

Key findings

• There exists a natural ring homomorphism φ from the equivariant cohomology H*(MG) to the quantum cohomology QH*(M̄) of the symplectic quotient M̄.
• The genus zero Gromov–Witten invariants defined via the PDE system in the ambient manifold map to the standard genus zero invariants of the quotient under φ.
• The diagram of invariants commutes: the composition of φ with the quantum invariants of the quotient equals the invariants defined by the PDE system.
• The correspondence holds under the assumption that the symplectic quotient is monotone, ensuring well-defined quantum cohomology.
• The linearized operator of the PDE system splits into tangential and normal components, with identities derived using Lie derivatives and covariant derivatives.
• The proof relies on detailed analysis of the linearized operator, including identities for Lie derivatives of connections and curvature terms.