On the L^2-metric of vortex moduli spaces

Abstract

We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. These include linear non-abelian vortices and vortices in gauged toric sigma-models.

34 pages; v3: typos fixed and reference added

References in corpus (24)

• Phases of $N=2$ Theories In Two Dimensions
• Vortices, Instantons and Branes
• Nonabelian Superconductors: Vortices and Confinement in $\mathcal{N}=2$ SQCD
• Supersymmetric vacua and Bethe ansatz
• Solitons in the Higgs phase -- the moduli matrix approach --
• Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Moduli Space of Non-Abelian Vortices
• Asymptotic Interactions of Critically Coupled Vortices
• Manifestly Supersymmetric Effective Lagrangians on BPS Solitons
• Volume of Vortex Moduli Spaces
• Statistical Mechanics of Vortices from D-branes and T-duality
• Vortex equations in abelian gauged sigma-models
• J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions
• One-vortex moduli space and Ricci flow
• Non-abelian vortices on compact Riemann surfaces
• The Moduli Space Metric for Well-Separated Non-Abelian Vortices
• The Kähler Potential of Abelian Higgs Vortices
• Exact moduli space metrics for hyperbolic vortices
• Vortex invariants and toric manifolds
• Twisting gauged non-linear sigma-models
• Gauged vortices in a background
• The volume of the space of holomorphic maps from S^2 to CP^k
• Some special Kahler metrics on SL(2,C) and their holomorphic quantization

Cited by in corpus (26)

• Decomposing Instantons in Two Dimensions
• Fractional and semi-local non-Abelian Chern-Simons vortices
• Vortices and magnetic impurities
• Group Theory of Non-Abelian Vortices
• Dynamics of Non-Abelian Vortices
• On the curvature of vortex moduli spaces
• Moduli of vortices and Grassmann manifolds
• Vortex counting from field theory
• Vortices as degenerate metrics
• Vortices and Jacobian varieties
• Vortices on Orbifolds
• The volume of the space of holomorphic maps from S^2 to CP^k
• Ginzburg-Landau equations on Riemann surfaces of higher genus
• The Volume of the Quiver Vortex Moduli Space
• Ricci magnetic geodesic motion of vortices and lumps
• Vortex counting and the quantum Hall effect
• On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$
• Moduli Spaces of Abelian Vortices on Kahler Manifolds
• Dynamics of Abelian Vortices Without Common Zeros in the Adiabatic Limit
• Kähler quantization of vortex moduli
• Quantum Statistical Mechanics of Vortices
• The Einstein-Hilbert action of the space of holomorphic maps from S^2 to CP^k
• Abelian Vortices with Singularities
• The Berry connection of the Ginzburg-Landau vortices
• The Gromov Limit for Vortex Moduli Spaces
• Quantum Statistical Mechanics of Dissolving Vortices