N=4 Multi-Particle Mechanics, WDVV Equation and Roots

Abstract

We review the relation of N=4 superconformal multi-particle models on the real line to the WDVV equation and an associated linear equation for two prepotentials, F and U. The superspace treatment gives another variant of the integrability problem, which we also reformulate as a search for closed flat Yang-Mills connections. Three- and four-particle solutions are presented. The covector ansatz turns the WDVV equation into an algebraic condition, for which we give a formulation in terms of partial isometries. Three ideas for classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and matroids. Various examples and counterexamples are displayed.

References in corpus (26)

• WDVV-like equations in N=2 SUSY Yang-Mills Theory
• N=4 Supersymmetric Mechanics in Harmonic Superspace
• Supersymmetric Calogero models by gauging
• (Super)conformal many-body quantum mechanics with extended supersymmetry
• N=4 superconformal Calogero models
• Nonperturbative Renormalization Group Equation and Beta Function in N=2 SUSY Yang-Mills
• Deformations of the root systems and new solutions to generalised WDVV equations
• WDVV Equations from Algebra of Forms
• N=4 Supersymmetric Multidimensional Quantum Mechanics, Partial SUSY Breaking and Superconformal Quantum Mechanics
• Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation
• N=4 mechanics, WDVV equations and roots
• Gauging N=4 supersymmetric mechanics II: (1,4,3) models from the (4,4,0) ones
• New insight into WDVV equation
• Solutions of WDVV equations in Seiberg-Witten theory from root systems
• N=4 supersymmetric 3-particles Calogero model
• Locus configurations and $\vee$-systems
• Many-Body Superconformal Systems from Hamiltonian Reductions
• N=4 superconformal n-particle mechanics via superspace
• Supersymmetric Mechanics in Superspace
• Weyl groups and Elliptic Solutions of the WDVV equations
• On the geometry of V-systems
• WDVV Equations in Seiberg-Witten theory and associative algebras
• A new class of solutions to the WDVV equation
• Intensity of parametric fluorescence pumped by ultrashort pulses
• WDVV solutions from orthocentric polytopes and Veselov systems
• The number of matroids on a finite set

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• Superconformal mechanics
• Many-particle mechanics with D(2,1;alpha) superconformal symmetry
• Quantum SU(2$|$1) supersymmetric Calogero-Moser spinning systems
• Minimal unitary representation of D(2,1;λ) and its SU(2) deformations and d=1, N=4 superconformal models
• Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and $\mathcal{N}=4$ mechanics
• Superconformal Quantum Mechanics from M2-branes
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• On deformation and classification of V-systems
• V-systems, holonomy Lie algebras and logarithmic vector fields