Cyclic Stratum of Frobenius Manifolds, Borelā€“Laplace (Ī±,Ī²)(\boldsymbol{\alpha}, \boldsymbol{\beta})-Multitransforms, and Integral Representations of Solutions of Quantum Differential Equations

  • Giordano Cotti

    Universidade de Lisboa, Portugal
Cyclic Stratum of Frobenius Manifolds, Borelā€“Laplace (šœ¶, šœ·)-Multitransforms, and Integral Representations of Solutions of Quantum Differential Equations cover
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In the first part of this paper, we introduce the notion of cyclic stratum of a Frobenius manifold M\smash{M}. This is the set of points of the extended manifold Cāˆ—Ć—M\smash{\mathbb C^*\times M} at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. We show that at points of the cyclic stratum, the isomonodromic system attached to M\smash{M} can be reduced to a scalar differential equation, called the master differential equation of M\smash{M}. In the case of Frobenius manifolds coming from Gromovā€“Witten theory, namely quantum cohomologies of smooth projective varieties, such a construction reproduces the notion of quantum differential equation.

In the second part of the paper, we introduce two multilinear transforms, called Borelā€“Laplace (Ī±,Ī²)(\boldsymbol{\alpha}, \boldsymbol{\beta})-multitransforms, on spaces of Ribenboim formal power series with exponents and coefficients in an arbitrary finite-dimensional C\smash{\mathbb C}-algebra A\smash{A}. When A\smash{A} is specialized to the cohomology of smooth projective varieties, the integral forms of the Borelā€“Laplace (Ī±,Ī²)(\boldsymbol{\alpha}, \boldsymbol{\beta})-multitransforms are used in order to rephrase the Quantum Lefschetz theorem. This leads to explicit Mellinā€“Barnes integral representations of solutions of the quantum differential equations for a wide class of smooth projective varieties, including Fano complete intersections in projective spaces.

In the third and final part of the paper, as an application, we show how to use the new analytic tools, introduced in the previous parts, in order to study the quantum differential equations of Hirzebruch surfaces. For Hirzebruch surfaces diffeomorphic to P1ƗP1\smash{\mathbb P^1\times \mathbb P^1}, this analysis reduces to the simpler quantum differential equation of P1\smash{\mathbb P^1}. For Hirzebruch surfaces diffeomorphic to the blow-up of P2\smash{\mathbb P^2} in one point, the quantum differential equation is integrated via Laplace (1,2;12,13)\smash{(1,2;\frac{1}{2},\frac{1}{3})}-multitransforms of solutions of the quantum differential equations of P1\smash{\mathbb P^1} and P2\smash{\mathbb P^2}, respectively. This leads to explicit integral representations for the Stokes bases of solutions of the quantum differential equations, and finally to the proof of the Dubrovin conjecture for all Hirzebruch surfaces.