Frobenius Structure Research Articles

Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.

On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental–Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit.

Subring Depth, Frobenius Extensions, and Towers

The minimum depthd(B,A)of a subringB⊆Aintroduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show thatd(B,A)< ∞ ifAis a finite-dimensional algebra andBehas finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. IfA⊇Bis a QF extension, minimum left and right even subring depths are shown to coincide. IfA⊇Bis a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depthnextensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.

Stabilité de l'holonomie sans structure de Frobenius: cas des courbes

By using Christol and Mebkhout's algebrization and finiteness theorem, we prove that in the case of smooth curves, Berthelot’s strongest conjecture on the stability of holonomicity is still valid without Frobenius structure but under some non-Liouville type hypotheses.

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

In this paper we study the Fuchsian RiemannHilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this RiemannHilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.

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.

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer’s calculus of ‘conditional density operators’. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.

Isomonodromic tau function on the space of admissible covers

The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor. This yields an explicit formula for the pullback of the Hodge class to the space of admissible covers in terms of the classes of compactification divisors.

Degenerations and limit Frobenius structures in rigid cohomology

AbstractWe introduce a ‘limiting Frobenius structure’ attached to any degeneration of projective varieties over a finite field of characteristicpwhich satisfies ap-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens–Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.

Images directes I : Espaces rigides analytiques et images directes

This article is the first one of a series of three articles devoted to direct images of isocrystals: here we consider isocrystals without Frobenius structure; in the second one [Et 6] (resp. the third one [Et 7]), we will introduce a Frobenius structure in the convergent (resp. overconvergent) context.

Riemann–Hilbert Problems for Hurwitz Frobenius Manifolds

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.

Coherence of certain overconvergent isocrystals without Frobenius structures on curves

In this paper, we will prove the coherence as \({\fancyscript{D}^\dagger}\)-modules of overconvergent isocrystals on curves which do not contain tame parts at each singular points of the given isocrystal, even without Frobenius structures. In the proof, we will construct finite presentations concretely using special differential operators defined by Garnier, and the decomposition theorem of Christol–Mebkhout.

Soliton equations, vertex operators, and simple singularities

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto (Proc. Symp. Pure Math. 48:138–177, 1989) using the principal realization of the basic representations of the corresponding affine Kac–Moody algebra. The second was defined in Givental and Milanov (The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, vol. 232, pp. 173–201, Birkhäuser, Basel, 2005) using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard–Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.

Arithmetic, mutually unbiased bases and complementary observables

Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.

Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of \({\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}\), i.e., the complex projective line with a orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy (Carlet, The extended bigraded toda hierarchy. arXiv preprint arXiv:math-ph/0604024). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P 1(x) + P 2(y) + P 3(z) which contains as special cases the ones constructed on the space of Laurent polynomials (Dubrovin, Geometry of 2D topologica field theories. Integrable systems and quantum groups, Springer Lecture Notes in Mathematics 1620:120–348, 1996; Milanov and Tseng, The space of Laurent polynomials, \({\mathbb{P}^1}\)-orbifolds, and integrable hierarchies. preprint arXiv:math/0607012v3 [math.AG]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial \({\mathbb{P}^1}\)-orbifolds. Finally we link rational SFT of Seifert fibrations over \({\mathbb{P}^1_{a,b,c}}\) with orbifold Gromov-Witten invariants of the base, extending a known result (Bourgeois, A Morse-Bott approach to contact homology. Ph.D. dissertation, Stanford University, 2002) valid in the smooth case.

Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal (axiomatic) Gromov–Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction (due to van de Leur) describes the action of the same group on the space of Frobenius structures via the multi-component KP hierarchies. Our main theorem states that the genus zero restriction of the Y.-P. Lee formulas coincides with the tangent van de Leur action.

Formula omitted] formal genus zero Gromov–Witten theories and Givental’s formalism

In [A. Givental, Symplectic geometry of Frobenius structures. arxiv: math.AG/0305409] Givental introduced and studied a space of formal genus zero Gromov–Witten theories G W 0 , i.e. functions satisfying string and dilaton equations and topological recursion relations. A central role in the theory plays the geometry of certain Lagrangian cones and a twisted symplectic group of hidden symmetries. In this note we show that the Lagrangian cones description of the action of this group coincides with the genus zero part of Givental’s quantum Hamiltonian formalism. As an application we identify explicitly the space of N = 1 formal genus zero GW theories with lower-triangular twisted symplectic group modulo the string flow.

Logarithmic growth and Frobenius filtrations for solutions ofp-adic differential equations

AbstractFor a ∇-moduleMover the ringK[[x]]0of bounded functions over ap-adic local fieldKwe define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, ifMalso admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case ofMa ϕ–∇-module of rank 2, the Frobenius polygon forMand the log-growth polygon for its dual,Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Frobenius modules and Galois representations

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for p-adic differential equations with (strong) Frobenius structure over p-adic differential fields with algebraically closed residue field.

Semistable reduction for overconvergent F-isocrystals, II: A valuation-theoretic approach

AbstractWe introduce a valuation-theoretic approach to the problem of semistable reduction (i.e. existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann–Zariski space associated to the function field of a variety. We also make some initial reductions, which allow attention to be focused on valuations of height 1 and transcendence degree 0.