Celebratio Mathematica — Schwarz

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Y. Chen, M. Kontsevich, and A. Schwarz: “Symmetries of WDVV equations,” Nuclear Phys. B 730 : 3 (2005), pp. 352–363. MR 2180009 Zbl 1276.81096 ArXiv hep-th/0508221 article

We say that a function \( F(\tau) \) obeys WDVV equations, if for a given invertible symmetric matrix \( \eta^{\alpha \beta} \) and all \( \tau \in \mathcal{T} \subset \mathbb{R}^n \), the expressions \[ c_{\beta\gamma}^{\alpha}(\tau) = \eta^{\alpha\lambda}\partial_{\lambda}\partial_{\beta}\partial_{\gamma}F \] can be considered as structure constants of commutative associative algebra; the matrix \( \eta_{\alpha \beta} \) inverse to \( \eta^{\alpha \beta} \) determines an invariant scalar product on this algebra. A function \( x^{\alpha}(z,\tau) \) obeying \[ \partial_{\alpha}\partial_{\beta} \,x^{\gamma}(z,\tau) = z^{-1}c_{\alpha\beta}^{\epsilon}\partial_{\epsilon}\,x^{\gamma}(z,\tau) \] is called a calibration of a solution of WDVV equations. We show that there exists an infinite-dimensional group acting on the space of calibrated solutions of WDVV equations (in different form such a group was constructed in [Givental 2004]. We describe the action of Lie algebra of this group.

Yujun Chen	Related
Maksim Lvovich Kontsevich	Related