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Register a new userElectric-Magnetic Duality and WDVV Equations
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We consider the associativity (or WDVV) equations in the form they appear in Seiberg-Witten theory and prove that they are covariant under generic electric-magnetic duality transformations. We discuss the consequences of this covariance from various perspectives.
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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.
In this paper we present a construction of a new class of explicit solutions to the WDVV (or associativity) equations. Our construction is based on a relationship between the WDVV equations and Whitham (or modulation) equations. Whitham equations appear in the perturbation theory of exact algebro-geometric solutions of soliton equations and are defined on the moduli space of algebraic curves with some extra algebro-geometric data. It was first observed by Krichever that for curves of genus zero the tau-function of a ``universal Whitham hierarchy gives a solution to the WDVV equations. This construction was later extended by Dubrovin and Krichever to algebraic curves of higher genus. Such extension depends on the choice of a normalization for the corresponding Whitham differentials. Traditionally only complex normalization (or the normalization w.r.t. a-cycles) was considered. In this paper we generalize the above construction to the real-normalized case.
We show that reductions of KP hierarchies related to the loop algebra of $SL_n$ with homogeneous gradation give solutions of the Darboux-Egoroff system of PDEs. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten--Dijkgraaf--E. Verlinde--H. Verlinde equations.
We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form begin{equation*} int f(u_{xx},u_{xy},u_{yy}) dxdy. end{equation*} By deriving integrability conditions for the Lagrangian density $f$, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.
We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).
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