A uniqueness theorem for Frobenius manifolds and Gromov–Witten theory for orbifold projective lines

Abstract

We prove that the Frobenius structure constructed from the Gromov-Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten-Dijkgraaf-Verlinde-Verlinde equations with certain natural initial conditions.

On the Frobenius manifolds for cusp singularities

“…For the cases that χ A > 0 by [9,10,7] and the cases that χ A = 0 by [17], same statements as Theorem 4.1 are shown. Therefore, combining them with Theorem 4.1, it is shown that, for arbitrary triplet of positive integers A, there exists the classical mirror symmetry between the orbifold projective line with at most three orbifold points P 1 A and the pair of the cusp polynomial f A and the primitive form ζ A associated to it.…”

“…In our previous paper [6], it is shown that a Frobenius structure with certain conditions can be reconstructed uniquely and the one constructed from the Gromov-Witten theory of the orbifold projective line P 1 A with arbitrary triplet of positive integers A satisfies the conditions. We shall see the Frobenius structure constructed from the pair (f A , ζ A ) also satisfies the conditions, i.e., the following Theorem 4.2 holds:…”

“…For simplicity, we shall denote by M the Frobenius manifold M (f A ,ζ A ) constructed from the pair of the cusp polynomial f A and the primitive form ζ A , where ζ A is the one given in Theorem 2.38 for χ A < 0 and the one in Theorem 3.1 in [7] for χ A > 0. We shall systematically calculate the intersection form of M (f A ,ζ A ) by the period mappings of the primitive form.…”

“…Next, we shall show that the Frobenius structure associated to the pair (f A , ζ A ) given above satisfies the conditions in Theorem 3.1 of [6]:…”

“…Here ζ A is the one given in Theorem 2.38 for χ A < 0 and the one in Theorem 3.1 in [7] for χ A > 0. Let T A be the following Coxeter-Dynkin diagram:…”