School of Mathematics

The generalized WDVV-system

Jitse Niesen. The generalized WDVV-system.
Master's thesis, Department of Applied Mathematics, University of Twente, the Nederlands, 1998.
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Abstract

In papers on topological field theory in the beginning of the nineties a particular system of partial differential equations appeared, the so-called (classical) WDVV-system. A few years later a generalization of the system made its appearance:

 

If (1) holds for one fixed k, it holds for all k. Furthermore, the matrix  can even be replaced by any linear combination  . There are some equivalent formulations: with or ``the system is consistent''. The WDVV-system is invariant under linear coordinate transformations.

Now consider the function , where  is an element of a Cartan subalgebra of some Lie algebra  and R is a representation of  . If R is the adjoint representation, we have , where  is the root system of  . In this case the (generalized) WDVV-system is equivalent with  if one takes an orthonormal basis. It is proven that for all simple  except  the map F solves the WDVV-system. If this is also true for  (as we conjecture), then the theorem can be extended to semi-simple algebras. The function , where  is a signature, is also a solution if , for other root systems this is not true. The function F is not a solution for other representations R in general.

The infinitesimal symmetries of the WDVV-system in three and four dimensions are  , , , , , and  ; we conjecture that this is also true in higher dimensions. We have found solutions which are invariant under some symmetries, the most interesting is with a and v (almost) arbitrary, which is invariant under .