Givental Graphs and Inversion Symmetry

Abstract

Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarchy.2010 Mathematics Subject Classification. 53D45, 37K10.

Ramifications of Hurwitz theory, KP integrability and quantum curves

“…Step 3: From topological recursion to intersection numbers. Once we have spectral curve topological recursion, we can immediately conclude that the corresponding combinatorial problem can be solved in terms of some intersection numbers on the moduli space of curves that represent the correlators of a semi-simple cohomological field theory with a possibly non-flat unit [19,20,16], and therefore, have expressions in terms of the Givental graphs [17,16].…”

4D limit of melting crystal model and its integrable structure

“…as → 0 (with Λ normalized to 1) is the spectral curve of topological recursion in this case [7,8,22]. We have thus re-derived the quantum spectral curve of CP 1 from the 4D limit of the melting crystal model.…”

“…This is inspired by the work of Dunin-Barkowski et al [6] on Gromov-Witten theory of CP 1 . They derived a quantum spectral curve of CP 1 in the perspective of topological recursion [7,8,22]. Since the deformed 4D Nekrasov function Z 4D (T , s) of U(1) gauge theory coincides with a generating function of all genus Gromov-Witten invariants of CP 1 [12,24], it will be natural to reconsider this issue from the point of view of the melting crystal model.…”

Dubrovin’s Superpotential as a Global Spectral curve

“…A n -singularities, 1 ≤ n ≤ 3 Consider the vanishing cycles β i ∈ H 0 (f −1 (λ), C) given by β i := p 0 − p i , where λ = f (p 0 , τ ) = f (p i , τ ), and p 0 ∈ D 0 , p i ∈ D i . Then the system of solutions of Equation (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) given by φ (i) (λ, τ ) := φ βi (λ, τ ) satisfies the properties given by Equations (2-18)- (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21). In particular, G ij = 1/2 for i = j and G ii = 1.…”

“…Choose β to be the cycle given by p 0 − p 1 , for p 0 and p 1 the pre-images of λ in each of the two disks. In normalized canonical coordinates I β (λ; u) is represented by a solution φ β (λ; u) = i φ β i (λ; u)∂ vi of the Gauss-Manin system (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) which has components given by…”