Semisimple Frobenius Manifolds Research Articles

On real forms of Dubrovin-Ugaglia Poisson brackets

Dubrovin and Ugaglia introduced a holomorphic Poisson structure on the space of semisimple Frobenius manifolds, which is identified with the entries of an upper triangular matrix with ones on the diagonal. In this paper, we compute explicitly the real forms of the Dubrovin-Ugaglia Poisson structures, arising from the real forms of the Lie algebra son(C).

Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

This series of papers is devoted to the study of Virasoro symmetries of deformations of the Principal Hierarchy associated with a semisimple Frobenius manifold. The main tool we use is a generalization of the bihamiltonian cohomology called the variational bihamiltonian cohomology. In the present paper, we give its definition and compute the associated cohomology groups that will be used in our study of Virasoro symmetries. As an application of the variational bihamiltonian cohomology theory, we classify conformal bihamiltonian structures with semisimple hydrodynamic limits.

Gamma conjecture II for quadrics

The Gamma conjecture II for the quantum cohomology of a Fano manifold F, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of \({\mathcal {D}}^b(F)\) and the \({\widehat{\Gamma }}\)-integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and Gamma II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin’s theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov–Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov’s exceptional collections and an application of our criterion.

Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro Symmetries

We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite set of Virasoro symmetries.

A Fock sheaf for Givental quantization

We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov-Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic-Bouchard-Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the (3g-2)-jet condition of Eguchi-Xiong; they also satisfy a certain anomaly equation, which generalizes the Holomorphic Anomaly Equation of Bershadsky-Cecotti-Ooguri-Vafa. We interpret Givental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When X is a variety with semisimple quantum cohomology, a theorem of Teleman implies that the canonical section coincides with the geometric descendant potential defined by Gromov-Witten invariants of X. We use our formalism to prove a higher-genus version of Ruan's Crepant Transformation Conjecture for compact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that the total descendant potential for compact toric orbifold X is a modular function for a certain group of autoequivalences of the derived category of X.

A construction of Frobenius manifolds from stability conditions

A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to each mutation of $Q$ and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple Frobenius manifold. In particular we check that this holds in the case of $A_n$, $n \leq 5$.

Frobenius manifolds near the discriminant and relations in the tautological ring

We give a criterion for extending a generically semisimple (not necessarily conformal) Frobenius manifold locally near a smooth point of the discriminant to a cohomological field theory. As an application, we show that a large set of tautological relations related to the Givental--Teleman classification for any generically semisimple cohomological field theories follow from Pixton's generalized Faber--Zagier relations.

Analytic geometry of semisimple coalescent Frobenius structures

We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov–Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is an integrable deformation of the Korteweg–de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

The Darboux–Egoroff system of PDEs with any number n ≥ 3 of independent variables plays an essential role in the problems of describing n -dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an ( n − 1 ) -component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.

4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension $$n=4$$ and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painleve $$\hbox {VI}\mu $$ equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension $$n=3$$ , this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces.

Analyticity of the total ancestor potential in singularity theory

K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard–Orantin recursion.

Proof of a conjecture on the genus two free energy associated to the [formula omitted] singularity

In a recent paper Dubrovin et al. (1998), it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two G-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two G-functions vanish. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.

On adding a variable to a Frobenius manifold and generalizations

Let \(\pi :V\rightarrow M\) be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure \((\circ _{M},e_{M},g_{M})\) and typical fiber has the structure of a Frobenius algebra \((\circ _{V},e_{V},g_{V})\). Using a connection \(D\) on the bundle \(\pi : V{\,\rightarrow \,}M\) and a morphism \(\alpha :V\rightarrow TM\), we construct an almost Frobenius structure \((\circ , e_{V},g)\) on the manifold \(V\) and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on \(V\) obtained in this way, when \(M\) is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure \(k_{M}\) on \(M\) and a real structure \(k_{V}\) on the bundle \(\pi : V \rightarrow M\). Using \(k_{M}\), \(k_{V}\) and \(D\) we define a real structure \(k\) on the manifold \(V\). We study when \(k\), together with an almost Frobenius structure \((\circ , e_{V}, g) \), satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and \(tt^{*}\)-geometry.

On the genus two free energies for semisimple Frobenius manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called "genus two G-function". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for ${\bf P}^1$-orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in certain particular cases.

Pole distribution of PVI transcendents close to a critical point

The distribution of the poles of Painlevé VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. As an example, the Frobenius manifold given by the quantum cohomology of CP2 is considered. The general PVI is also considered.

Flat 3-webs via semi-simple Frobenius 3-manifolds

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are hexagonal and admit at least one infinitesimal symmetry at each singular point. Singularities of the web are also discussed.

Integrable hierarchies and the mirror model of local [formula omitted

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 CP1. 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. Second, 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 CP1 in terms of a dual logarithmic Landau–Ginzburg model.

A Remark on Deformations of Hurwitz Frobenius Manifolds

In this note we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a Kähler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.