Intersecting Solitons, Amoeba and Tropical Geometry

Abstract

We study generic intersection (or web) of vortices with instantons inside, which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1 supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1} \times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (Nf=Nc=1), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Ampere measure with respect to a plurisubharmonic function on (C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin function. The general form of the Kahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.

39 pages, 11 figures

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• Non-Abelian String Junctions as Confined Monopoles
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• Quantum Foam and Topological Strings
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• Moduli Space of Non-Abelian Vortices
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• Brane Tilings
• Monopoles in the Higgs Phase
• Instantons in the Higgs Phase
• TASI Lectures on Solitons
• Non-Abelian Meissner Effect in Yang--Mills Theories at Weak Coupling
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• Non-Abelian Vortices of Higher Winding Numbers
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• Nonabelian Monopoles and the Vortices that Confine Them
• The Moduli Space of BPS Domain Walls
• D-brane Solitons in Supersymmetric Sigma-Models
• Non-Abelian Semilocal Strings in N=2 Supersymmetric QCD
• Localization of Non-Abelian Gauge Fields on Domain Walls at Weak Coupling (D-brane Prototypes)
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• Multi-Domain Walls in Massive Supersymmetric Sigma-Models
• Manifest Supersymmetry for BPS Walls in N=2 Nonlinear Sigma Models
• Non-Abelian Vortices on Cylinder -- Duality between vortices and walls
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• Constructing Non-Abelian Vortices with Arbitrary Gauge Groups
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• Non-Abelian Vortices in N=1* Gauge Theory
• Integrability of Vortex Equations on Riemann Surfaces
• Non-Abelian Webs of Walls
• Global Structure of Moduli Space for BPS Walls
• Composite non-Abelian Flux Tubes in N=2 SQCD
• Domain Walls with Non-Abelian Clouds
• Quiver Gauge Theory of Nonabelian Vortices and Noncommutative Instantons in Higher Dimensions
• Statistical Mechanics of Vortices from D-branes and T-duality
• Exact Wall Solutions in 5-Dimensional SUSY QED at Finite Coupling
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• BPS Wall in N=2 SUSY Nonlinear Sigma Model with Eguchi-Hanson Manifold
• Non-Abelian Vortices on Riemann Surfaces: an Integrable Case
• Effective Theory on Non-Abelian Vortices in Six Dimensions
• Dynamics of Domain Wall Networks
• Massive Nonlinear Sigma Models and BPS Domain Walls in Harmonic Superspace
• Vortex equations in abelian gauged sigma-models
• Kinky D-Strings
• An Index Theorem for Domain Walls in Supersymmetric Gauge Theories
• Massless Localized Vector Field on a Wall in D=5 SQED with Tensor Multiplets
• Effective Action of Domain Wall Networks
• Moduli Space of BPS Walls in Supersymmetric Gauge Theories
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• Exactly marginal deformations of quiver gauge theories as seen from brane tilings
• Sigma-model soliton intersections from exceptional calibrations
• On Monopoles and Domain Walls
• Anomalies and O-plane charges in orientifolded brane tilings
• New BPS Objects in N=2 Supersymmetric Gauge Theories
• Estimates on Monge-Ampère operators derived from a local algebra inequality
• Global symmetries and 't Hooft anomalies in brane tilings
• Amoebas and Instantons
• Twisting gauged non-linear sigma-models
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Cited by in corpus (35)

• Vortices and Other Topological Solitons in Dense Quark Matter
• Color Magnetic Flux Tubes in Dense QCD. II: Effective World-Sheet Theory
• Neutral bions in the $\mathbb{C}{P}^{N-1}$ model
• Josephson vortices and the Atiyah-Manton construction
• Correspondence between Skyrmions in 2+1 and 3+1 Dimensions
• Jewels on a wall ring
• Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes
• Incarnations of Skyrmions
• BPS States in Supersymmetric Chiral Models with Higher Derivative Terms
• Emergent Calabi-Yau Geometry
• Fractional instantons and bions in the O(N) model with twisted boundary conditions
• Confined Monopoles Induced by Quantum Effects in Dense QCD
• Non-Abelian Sine-Gordon Solitons: Correspondence between $SU(N)$ Skyrmions and $\mathbb{C}{P}^{N-1}$ Lumps
• Classifying BPS States in Supersymmetric Gauge Theories Coupled to Higher Derivative Chiral Models
• Relations among topological solitons
• Non-Abelian Monopoles in the Higgs Phase
• Higher Derivative Corrections to Manifestly Supersymmetric Nonlinear Realizations
• SO and USp Kahler and Hyper-Kahler Quotients and Lumps
• Crystal Melting and Wall Crossing Phenomena
• Dynamics of Strings between Walls
• Integrable vortex-type equations on the two-sphere
• Josephson instantons and Josephson monopoles in a non-Abelian Josephson junction
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Dynamics of Non-Abelian Vortices
• 2d Partition Function in Omega-background and Vortex/Instanton Correspondence
• Instantons confined by monopole strings
• Incarnations of Instantons
• Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions
• Exact solutions of domain wall junctions in arbitrary dimensions
• Knotted instantons from annihilations of monopole-instanton complex
• BPS pion domain walls in the supersymmetric chiral Lagrangian
• Higher Derivative Supersymmetric Nonlinear Sigma Models on Hermitian Symmetric Spaces, and BPS States Therein
• Gauge symmetry breaking in ten-dimensional Yang-Mills theory dynamically compactified on S^6
• Asymptotic Dynamics of Monopole Walls
• Holographic QCD Matter: Chiral Soliton Lattices in Strong Magnetic Field