There is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

PB - SISSA UR - http://preprints.sissa.it/handle/1963/35304 U1 - 35610 U2 - Mathematics U4 - 1 U5 - MAT/03 ER - TY - JOUR T1 - Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations JF - Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 Y1 - 2011 A1 - Boris Dubrovin A1 - M.V. Pavlov A1 - Sergei A. Zykov KW - Frobenius manifold AB - We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions. PB - Springer UR - http://hdl.handle.net/1963/6430 U1 - 6367 U2 - Mathematics U4 - -1 ER -