N=4 mechanics, WDVV equations and roots

Abstract

N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.

1+25 pages; v2: major revision (more general analysis, new solutions, additional references); v3: improvements in sects.5,8,9, refs. added

References in corpus (13)

• Black Holes and Calogero Models
• Supersymmetric Calogero models by gauging
• (Super)conformal many-body quantum mechanics with extended supersymmetry
• N=4 superconformal Calogero models
• Deformations of the root systems and new solutions to generalised WDVV equations
• Calogero models and nonlocal conformal transformations
• 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
• On the geometry of V-systems

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