Preprints

My preprints can also be found on Google Scholar and on arXiv. They have all been submitted for publication.

1. Infinite-dimensional Dubrovin-Frobenius manifolds and the Stokes phenomenon, with F. Hernández Iglesias. (2022)
   • Abstract: We study the Dubrovin equation of the infinite-dimensional 2D Toda Dubrovin-Frobenius manifold at its irregular singularity. We first revisit the definition of the canonical coordinates, proving that they emerge naturally as generalized eigenvalues of the operator of multiplication by the Euler vector field. We then show that the formal solutions to the Dubrovin equation with exponential type behaviour at the irregular singular point are not uniquely determined by their leading order, but instead depend on an infinite number of parameters, contrary to what happens in the finite-dimensional case. Next, we obtain a large family of solutions to the Dubrovin equation given by integrals along the unit circle of certain combinations of the superpotentials. Observing that such a family is not complete and has trivial monodromy, we study a larger family of weak solutions obtained via Borel resummation of some distinguished formal solutions. These resummed solutions naturally appear in monodromy-related pairs, finally allowing us to compute the infinite-dimensional analogue of the Stokes matrices.

Papers

The titles link to the published version of the pdf file.

1. Enumeration of hypermaps and Hirota equations for extended rationally constrained KP, with J. van de Leur, H. Posthuma and S. Shadrin. Comm. Number Th. Phys. 17, 3 (2023) 643-708.
   • Abstract: We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin--Frobenius manifold is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.
2. Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy, with J. van de Leur, H. Posthuma and S. Shadrin. Lett. Math. Phys. 111, 63 (2021).
   • Abstract: We consider the Dubrovin--Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin-Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev-Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental--Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.
3. Normal forms of dispersive scalar Poisson brackets with two independent variables, with M. Casati and S. Shadrin. Lett. Math. Phys. 108 (2018) 2229-2253.
   • Abstract: We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.
4. Central invariants revisited, with R. Kramer and S. Shadrin. J. École polytechnique - Mathématiques 5 (2018) 149-175.
   • Abstract: We give a new proof of the statement of Dubrovin-Liu-Zhang that the Miura-equivalence classes of the deformations of semi-simple bi-Hamiltonian structures of hydrodynamic type are parametrized by the so-called central invariants.
5. Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed, with H. Posthuma and S. Shadrin. J. Differential Geom. 108 (2018) 63-89.
   • Abstract: We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.
6. Rational reductions of the 2D-Toda hierarchy and mirror symmetry, with A. Brini, S. Romano and P. Rossi. J. Eur. Math. Soc. 19 (2017) 835-880.
   • Abstract: We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed Gelfand-Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki-Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz-Ladik hierarchy to Gromov-Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi-Yau threefolds obtained from minimal resolutions of the local orbifold line.
7. Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets, with M. Casati and S. Shadrin. J. Geom. Phys. 114 (2017) 404-419.
   • Abstract: We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D>1. Hence, in contrast with the D=1 case, the deformation theory in the multivariable case is non-trivial.
   • Remark: A Mathematica package to explicitly compute the dimension of Poisson cohomology groups is available here.
8. The bi-Hamiltonian cohomology of a scalar Poisson pencil, with H. Posthuma and S. Shadrin. Bull. London Math. Soc. 48 (2016) 617-627.
   • Abstract: We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p\sub{d}(\hat{F}, d\sub{1},d\sub{2})$ is isomorphic to $\R$ for (p,d)=(0,0), to $C^\infty (\R)$ for (p,d)=(1,1), (2,1), (2,3), (3,3), and vanishes otherwise.
9. Bihamiltonian cohomology of KdV brackets, with H. Posthuma and S. Shadrin. Comm. Math. Phys. 341 (2016) 805-819.
   • Abstract: Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.
10. Principal hierarchies of infinite-dimensional Frobenius manifolds: The extended 2D Toda lattice, with L. Ph. Mertens. Adv. Math. 278 (2015) 137-181.
    • Abstract: We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\infty$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinite-dimensional Frobenius manifold $M\sub{0}$ introduced by Carlet, Dubrovin and Mertens in Math. Ann. 349 (2011) 75-115 and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of $M\sub{0}$.
11. Towards Lax formulation of integrable hierarchies of topological type, with J. van de Leur, H. Posthuma and S. Shadrin. Comm. Math. Phys. 326 (2014) 815-849.
    • Abstract: To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [BurPosSha1, BurPosSha2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.
12. Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of $\mathbb{C}P^1$ orbifolds, with J. van de Leur. J. Phys. A: Math. Theor. 46 (2013) 405205-405220.
    • Abstract: We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this confirms a conjecture that relates the total descendent potential of the orbifold $C\sub{k,m}$ with a tau function of the extended bigraded Toda hierarchy.
13. Integrable hierarchies and the mirror model of local $\mathbb{C}P^1$, with A. Brini and P. Rossi. Physica D 241 (2012) 2156-2167.
    • Abstract: We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of local $\mathbb{C}P^1$. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local bi-Hamiltonian structure of the Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make precise the relation between this canonical Frobenius structure and the one that underlies the Gromov-Witten theory of the resolved conifold in the equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of "almost duality" of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local $\mathbb{C}P^1$ in terms of a dual logarithmic Landau-Ginzburg model.
14. Infinite-dimensional Frobenius manifolds for 2+1 integrable systems, with B. Dubrovin and L. Ph. Mertens. Math. Ann. 349 (2011) 75 -115.
    • Abstract: We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0/\infty$ respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.
15. On the Lax representation of the 2-component KP and 2D Toda hierarchies, with M. Mañas. J. Phys. A: Math. Theor. 43 (2010) 434011.
    • Abstract: We identify a set of free dependent variables of the 2-component KP and 2D Toda hierarchies by explicitly solving the constraints appearing in their Lax formulation.
16. Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures, with P. Lorenzoni and A. Raimondo. J. Phys. A: Math. Theor. 43 (2010) 045201.
    • Abstract: We study the finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.
17. The Extended Bigraded Toda Hierarchy. J. Phys. A: Math. Gen. 39 (2006) 9411-9435.
    • Abstract: We generalize the Toda lattice hierarchy by considering $N+M$ dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [CDZ04]. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups $\tilde{W}^{(N)}(A\sub{N+M-1})$ of the $A$ series, defined in [DZ98].
18. The Hamiltonian structures of the two-dimensional Toda lattice and R-matrices. Lett. Math. Phys. 71 (2005) 209-226.
    • Abstract: We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.
19. The Extended Toda Hierarchy, with B. Dubrovin and Y. Zhang. Moscow Math. J. 4 (2004) 313-332.
    • Abstract: Using construction of logarithm of a difference operator, we present the Lax pair formalism for certain extension of the continuous version of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the $\mathbb{C}P^1$ topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy.
20. Extended Toda Lattice. Theor. Math. Phys. 137 (2003) 1390-1395.
    • Abstract: We introduce nonlocal flows that commute with those of the classical Toda hierarchy. We define a logarithm of the difference Lax operator and use it to obtain a Lax representation of the new flows.

PhD and Master thesis

20. Extended Toda hierarchy and its Hamiltonian structure, PhD thesis, supervisor B. Dubrovin, International School for Advanced Studies (2003).
21. M5-brana: confronto tra l'approccio PST e quello doppiamente supersimmetrico, Master thesis, supervisor M. Tonin, University of Padova (1998).