Frobenius Manifolds Research Articles

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.

Grothendieck–Verdier duality patterns in quantum algebra

After a brief survey of the basic definitions of Grothendieck– Verdier categories and dualities, I consider in this context dualities introduced earlier in the categories of quadratic algebras and operads, largely motivated by the theory of quantum groups. Finally, I argue that Dubrovin's `almost duality' in the theory of Frobenius manifolds and quantum cohomology must also fit a (possibly extended) version of Grothendieck–Verdier duality.

Complex reflection groups, logarithmic connections and bi-flat F-manifolds

We show that bi-flat $F$-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's $\vee$-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank $2$ and $3$. On the Veselov's $\vee$-systems side, we provide a generalization of the notion of $\vee$-systems that gives rise to a dual connection which coincides with a Dunkl-Kohno-type connection associated with such groups. In particular, this allows us to treat on the same ground several different examples including Coxeter and Shephard groups. Remarkably, as a byproduct of our results, we prove that in some examples basic flat invariants are not uniquely defined. As far as we know, such a phenomenon has never been pointed out before.

DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

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.

G-Frobenius manifolds

The goal of this paper is to introduce the notion of G-Frobenius manifolds for any finite group G. This work is motivated by the fact that any G-Frobenius algebra yields an ordinary Frobenius algebra by taking its G-invariants. We generalize this on the level of Frobenius manifolds.To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative “rings of functions”, generalizing the commutative rings of power series on ordinary vector spaces.As the genus zero part of any ordinary cohomological field theory of Kontsevich–Manin contains a Frobenius manifold, we show that any G-cohomological field theory defined by Jarvis–Kaufmann–Kimura contains a G-Frobenius manifold up to a rescaling of its metric.Finally, we specialize to the case of G=Z/2Z and prove the structure theorem for (pre-)Z/2Z-Frobenius manifolds. We also construct an example of a Z/2Z-Frobenius manifold using this theorem, that arises in singularity theory in the hypothetical context of orbifolding.

Generalized Legendre transformations and symmetries of the WDVV equations

The Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class—the so-called Legendre transformations—were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.

Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebra {mathcal {A}}, one can arrive at such a theory. More generally, one can apply the same idea to construct an {mathcal {A}}-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an {mathcal {A}}-valued modified Camassa–Holm equation is constructed.

Mixed trTLEP-structures and mixed Frobenius structures

We introduce the notion of mixed trTLEP-structures and prove that a mixed trTLEP-structure with some conditions naturally induces a mixed Frobenius manifold. This is a generalization of the reconstruction theorem of Hertling and Manin. As a special case, we also show that a graded polarizable variation of mixed Hodge structure with $H^2$-generation condition gives rise to a family of mixed Frobenius manifolds. It implies that there exist mixed Frobenius manifolds associated to local B-models.

Flat coordinates for Saito Frobenius manifolds and string theory

It was shown in \cite{DVV} for $2d$ topological Conformal field theory (TCFT) \cite{EY,W} and more recently in \cite{BSZ}-\cite{BB2} for the non-critical String theory \cite{P}-\cite{BAlZ} that a number of models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin\cite{Dub}. More precisely these models are connected with a special case of FMs, so called Saito Frobenius manifolds (SFM)\cite{Saito} (originally called Flat structure together with the Flat coordinate system), which arise on the space of the versal deformations of the isolated Singularities after choosing of a suitabe so-called Primitive form, and which also arises on the quotient spaces by reflection groups. In this paper we explore the connection of the models of TCFT and non-critical String theory with SFM. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of SFMs as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of Gauss-Manin system connected with the corresponding SFM for a simple singularity. Also, we address the possible generalization of our approach for the models investigated in \cite{Gep} which are ${SU(N)_k}/({SU(N-1)_{k+1} \times U(1)})$ Kazama-Suzuki theories \cite{KS}.

Flat structures on Frobenius manifolds in the case of irrelevant deformations

In this paper we use a recently suggested conjecture about the integral representation for flat coordinates on Frobenius manifolds, connected with isolated singularities, to compute the flat coordinates and Saito primitive form on the space of deformations of Gepner chiral ring . We verify this conjecture comparing the expressions for flat coordinates obtained from the conjecture with the one found by direct computation. The considered case is of a particular interest since apart from the relevant and marginal deformations, it also has an irrelevant one.

Minimal string theory and the Douglas equation

We use the connection between the Frobenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.

Flat structures on the deformations of Gepner chiral rings

We propose a simple method for the computation of the flat coordinates and Saito primitive forms on Frobenius manifolds of the deformations of Jacobi rings associated with isolated singularities. The method is based on using a conjecture about integral representations for the flat coordinates and on the Saito cohomology theory. This reduces the computation to a simple linear problem. We consider the case of the deformed Gepner chiral rings. The knowledge of the flat structures of Frobenius manifolds can be used for exact solution of the models of the topological conformal field theories corresponding to these chiral rings.

On exact solution of topological CFT models based on Kazama–Suzuki cosets

We compute the flat coordinates on the Frobenius manifolds arising on the deformation space of Gepner chiral rings. The explicit form of the flat coordinates is important for exact solutions of models of topological CFT and 2D Liouville gravity. We describe the case k = 3, which is of particular interest because apart from the relevant chiral fields it contains a marginal one. Whereas marginal perturbations are relevant in different contexts, their analysis requires additional care compared to the relevant perturbations.

Hermitian metrics on F-manifolds

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.

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.

Flat coordinates of topological conformal field theory and solutions of the Gauss–Manin system

It was shown many years ago by Dijkgraaf, Velinde, and Verlinde for two-dimensional topological conformal field theory and more recently for the non-critical String theory that some models of these two types can be solved using their connection to the special case of Frobenius manifolds—the so-called Saito Frobenius manifolds connected to a deformed singularity. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of Saito Frobenius manifolds as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of the Gauss–Manin system connected to the corresponding Saito Frobenius manifold for the singularity.

Mixed Frobenius Structure and Local Quantum Cohomology

In a previous paper, the authors introduced the notion of mixed Frobenius structure (MFS) as a generalization of the structure of a Frobenius manifold. Roughly speaking, the MFS is defined by replacing a metric of the Frobenius manifold with a filtration on the tangent bundle equipped with metrics on its graded quotients. The purpose of the current paper is to construct a MFS on the cohomology of a smooth projective variety whose multiplication is the nonequivariant limit of the quantum product twisted by a concave vector bundle. We show that such a MFS is naturally obtained as the nonequivariant limit of the Frobenius structure in the equivariant setting.

Enhanced homotopy theory for period integrals of smooth projective hypersurfaces

The goal of this paper is to reveal hidden structures on the singular cohomology and the Griffiths period integral of a smooth projective hypersurface in terms of BV(Batalin-Vilkovisky) algebras and homotopy Lie theory (so called, L∞-homotopy theory). Let XG be a smooth projective hypersurface in the complex projective space P n defined by a homogeneous polynomial G(x) of degree d ≥ 1. Let H = H n−1 prim(XG ,C) be the middle dimensional primitive cohomology of XG . We explicitly construct a BV algebra BVX = (AX , ·,KX ,` KX 2 ) such that its 0-th cohomology H 0 KX (AX ) is canonically isomorphic to H. We also equip BVX with a decreasing filtration and a bilinear pairing which realize the Hodge filtration and the cup product polarization on H under the canonical isomorphism. Moreover, we lift C[γ ] : H → C to a cochain map Cγ : (AX ,KX )→ (C, 0), where C[γ ] is the Griffiths period integral given byω 7→ ∫ γ ω for [γ ] ∈Hn−1(XG ,Z). We use this enhanced homotopy structure on H to study a new type of extended formal deformation of XG and the correlation of its period integrals. As an application, if XG is in a formal family of hypersurfaces XGT , we provide an explicit formula and algorithm (based on a Grobner basis) to compute the period matrix of XGT in terms of the period matrix of XG and an L∞-morphism κ which enhances C[γ ]. Furthermore, if XG is Calabi-Yau, we explicitly construct a formal Frobenius manifold structure on ? e-mail: jaesuk@ibs.re.kr. The work of Jae-Suk Park was supported by the IBS (CA1305-01). ?? e-mail: jeehoonpark@postech.ac.kr. The work of Jeehoon Park was partially supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2013023108) and was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2013053914). 2 Jae-Suk Park, Jeehoon Park H induced from the special (quantum) solution to the Maurer-Cartan equation of the differential graded Lie algebra (AX ,KX ,` KX 2 ).

Explicit isomorphisms between pairings in quantum singularity theory

In this paper, we discuss the relations between various pairings appearing in FJRW theory and in the Frobenius manifold theory of singularities. These pairings include the intersection pairing, the Witten index, the Poincare dual and the residue pairing.