Topics in Enumerative Algebraic Geometry Lecture 14
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چکیده
Definition 2. AFrobenius structure on a manifold H is a Frobenius structure on each tangent space TtH such that (1) The metric <,> is flat (∇ = 0) (2) The vector field 1 is covariantly constant (∇1 = 0) (3) The system of PDE’s ~∇ws = w ◦ s is integrable ∀~ 6= 0, where w and s are vector fields and ◦ denotes the Frobenius multiplication. In ∇-flat coordinates {t}, this means that the family of connections ∇~ := ~d − Aα(t)dt ∧ is flat for all ~ 6= 0, where Aα = ∂α◦. A Frobenius structure is called conformal of dimension D if H is equipped with a vector field E(uler) such that 1, ◦ and <,> are eigenvectors of the Lie derivative operator LE with eigenvalues -1, 1 and 2−D, respectively. Example. For X compact Kähler, let H = H(X)/2πiH(X,Z). Then H is a Frobenius manifold of conformal dimension D = dimCX. Here ◦ is the quantum cup product, <,> is the Poincaré pairing, α integration over the fundamental cycle, 1 the fundamental class, and
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