Frobenius Structure Research Articles

Comparison of nonarchimedean and logarithmic mirror constructions via the Frobenius structure theorem

AbstractFor a log Calabi Yau pair () with smooth affine, satisfying either a maximal degeneracy assumption or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor that the structure constants defining a trace form on the mirror algebra constructed by Gross–Siebert are given by the naive curve counts defined by Keel–Yu. As a corollary, we deduce that the equality of the mirror algebras constructed by Gross–Siebert and Keel–Yu in the case contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel for Fano pairs satisfying the maximal degeneracy assumption.

Monodromie unipotente maximale, congruences “à la Lucas” et indépendance algébrique

Let f ( z ) f(z) be in 1 + z Q [ [ z ] ] 1+z\mathbb {Q}[[z]] and S \mathcal {S} be an infinite set of prime numbers such that, for all p ∈ S p\in \mathcal {S} , we can reduce f ( z ) f(z) modulo p p . We let f ( z ) ∣ p f(z)_{\mid p} denote the reduction of f ( z ) f(z) modulo p p . Generally, when f ( z ) f(z) is D-finite, f ∣ p ( z ) f_{\mid p}(z) is algebraic over F p ( z ) \mathbb {F}_p(z) . It turns out that if f ∣ p ( z ) f_{\mid p}(z) is a solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , we can use this type of equations to obtain results of transcendence and algebraic independence over Q ( z ) \mathbb {Q}(z) . In the present paper, we look for conditions on the differential operators annihilating f ( z ) f(z) to guarantee the existence of these particular equations. Suppose that f ( z ) f(z) is solution of a differential operator H ∈ Q ( z ) [ d / d z ] \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p ∈ S p\in \mathcal {S} and we also suppose that f ( z ) f(z) annihilates a Fuchsian differential operator D ∈ Q ( z ) [ d / d z ] \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of D \mathcal {D} and the exponents of D \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p ∈ S p\in \mathcal {S} , f ∣ p ( z ) f_{\mid p}(z) is solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , where A p ( z ) A_p(z) is a rational function with coefficients in F p \mathbb {F}_p of height less than or equal to C p 2 l Cp^{2l} with C C a positive constant that does not depend on p p . We also study the algebraic independence of these power series over Q ( z ) \mathbb {Q}(z) .

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M is an endofunctor S on M together with a natural isomorphism Hom_(M,N)⁎≅Hom_(N,S(M)) for M,N∈M, where Hom_ is the internal Hom functor of M. In this paper, we discuss the case where C and M are the category of finite-dimensional left modules over a finite-dimensional Hopf algebra H and a finite-dimensional left H-comodule algebra L, respectively. We give an explicit description of the relative Serre functor of M and its twisted module structure in terms of the Frobenius structure of L. We also study pivotal structures on M and give some concrete examples.

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Abstract Cluster varieties come in pairs: for any 𝒳 {\mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 {\mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.

Cluster structures for the A∞$A_\infty$ singularity

AbstractWe study a category of ‐graded maximal Cohen‐Macaulay (MCM) modules over the curve singularity and demonstrate that it has infinite type cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, has cluster tilting subcategories modelled by certain triangulations of the (completed) ‐gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed ‐gon.

A Gromov–Witten Theory for Simple Normal-Crossing Pairs Without Log Geometry

We define a new Gromov–Witten theory relative to simple normal crossing divisors as a limit of Gromov–Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism, Virasoro constraints (genus zero) and a partial cohomological field theory. Furthermore, we use the degree zero part of the relative quantum cohomology to provide an alternative mirror construction of Gross and Siebert (Intrinsic mirror symmetry, arXiv:1909.07649) and to prove the Frobenius structure conjecture of Gross et al. (Publ Math Inst Hautes Études Sci 122:65–168, 2015) in our setting.

Unitless Frobenius Quantales

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices \(M_n\) and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.

String diagram rewrite theory III: Confluence with and without Frobenius

AbstractIn this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

Unlikely intersections for curves in products of Carlitz modules

The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are almost exclusively restricted to zero characteristic. Not so long ago the second author made a start on removing this restriction by studying multiplicative groups over positive characteristic, and recently both authors went further for additive groups with extra Frobenius structure. Here we study additive groups with extra structure coming instead from the Carlitz module. We state a conjecture for curves in general dimension and we prove it in three dimensions. The main tool is a new relative version (for cyclotomic fields) of Denis’s analogue of Dobrowolski’s classical lower bound for heights, as well as a suitable upper bound. We also work out a couple of special cases in two dimensions: for example with respect to prime fields there are exactly 23 Carlitz roots of unity whose reciprocals are also roots of unity.

String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

A Frobenius manifold for $$\ell $$-Kronecker quiver

We construct a Frobenius structure whose intersection form coincides with the generalized Cartan matrix of the \(\ell \)-Kronecker quiver \(K_{\ell }\), and underlying complex manifold is isomorphic to the space of stability conditions for the bounded derived category of finitely generated modules over the path algebra \({{\mathbb {C}}}K_{\ell }\).

Bessel F-isocrystals for reductive groups

We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal $\mathrm{Be}_{\check{G}}^{\dagger}$ on $\mathbb{G}_{m,\mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $\check{G}$-local system $\mathrm{Kl}_{\check{G}}$ constructed by Heinloth-Ng\^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $\mathrm{Be}_{\check{G}}^{\dagger}$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of $\mathrm{Kl}_{\check{G}}$ due to Katz and Heinloth-Ng\^o-Yun), and establish functoriality between different Kloosterman $\check{G}$-local systems as conjectured by Heinloth-Ng\^o-Yun. We show that the Frobenius Newton polygons of $\mathrm{Kl}_{\check{G}}$ are generically ordinary for every $\check{G}$ and are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ when $\check{G}$ is classical or $G_2$.

Open Frobenius Cluster-Tilted Algebras

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

The differential geometry of the orbit space of extended affine Jacobi group An

We define certain extensions of Jacobi groups of An, prove an analogue of Chevalley Theorem for their invariants and construct a Dubrovin Frobenius structure on its orbit space.

Frobenius algebras and root systems: the trigonometric case

We construct Frobenius structures on the \({\mathbb {C}}^{\times }\)-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a class of Frobenius manifolds parameterizing Frobenius algebras in terms of root systems in a trigonometric setting. We then determine their potential functions, which amounts to giving the trigonometric solutions of WDVV equations.

Scattering diagrams, theta functions, and refined tropical curve counts

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of Fock and Goncharov. We also prove a refined version of the Carl-Pumperla-Siebert result on consistency of theta functions, and we prove the non-degeneracy of the trace-pairing for the Gross-Hacking-Keel Frobenius structure conjecture.

Theta bases and log Gromov-Witten invariants of cluster varieties

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse.

The dually flat structure for singular models

The dually flat structure introduced by Amari-Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, and such an excellent geometric structure is rarely defined on the entire space. To fix this trouble, in the present paper, we propose a novel generalization of the dually flat structure for a certain class of singular models from the viewpoint of Lagrange and Legendre singularity theory - we introduce a quasi-Hessian manifold endowed with a possibly degenerate metric and a particular symmetric cubic tensor, which exceeds the concept of statistical manifolds and is adapted to the theory of (weak) contrast functions. In particular, we establish Amari-Nagaoka's extended Pythagorean theorem and projection theorem in this general setup, and consequently, most of applications of these theorems are suitably justified even for such singular cases. This work is motivated by various interests with different backgrounds from Frobenius structure in mathematical physics to Deep Learning in data science.

Semi-invariants of low-dimensional Lie algebras

The aim of this work is to explicitly compute the semi-invariants of low-dimensional Lie algebras by reducing the amount of work, i.e., we can prove that almost every irreducible Lie algebra of dimension less than or equal to 5, satisfies the following: It is either a contact Lie algebra or there exists a torus such that is a contact Lie algebra. Therefore, the semi-invariants found by using the contact structure are the same found by using the Frobenius structure.