On geometry of a special class of solutions to generalised WDVV equations
نویسنده
چکیده
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions (∨conditions) on such a set which are necessary and sufficient for the corresponding function to satisfy the generalised WDVV equations. These conditions are satisfied for all Coxeter systems but there are also other examples discovered in the theory of the generalised Calogero-Moser systems. As a result some new solutions for the generalized WDVV equations are found. Introduction. TheWDVV (Witten-Dijgraaf-Verlinde-Verlinde) equations have been introduced first in topological field theory as some associativity conditions [1, 2]. Dubrovin has found an elegant geometric axiomatisation of the these equations introducing a notion of Frobenius manifold (see [3]). What we will discuss in this paper are their generalised versions appeared in the Seiberg-Witten theory [4, 5] The generalised WDVV equations are the following overdetermined system of nonlinear partial differential equations: FiF −1 k Fj = FjF −1 k Fi, i, j, k = 1, . . . , n, (1)
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A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (∨-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by ...
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