arXiv subjects
Publications and source records attributed to John Alexander Cruz Morales.
We give a reformulation of the Dubrovin conjecture about the semisimplicity of quantum cohomology in terms of the so-called second structure connection of quantum cohomology. The key ingredient in our work is the notion of a twisted reflection vector which allows us to give an elegant description of the monodromy data of the quantum connection in terms of the monodromy data of its Laplace transform.
Motivated by Aganagic's equivariant mirror symmetry for certain Coulomb branches of a $3d$ $\mathcal{N}= 4$ gauge quiver theory, we would like to propose a set of ideas towards an extension of Aganagic's proposal to Hitchin systems. At the end, there are two main points in our proposal; namely, that the equivariant mirror of the Hitchin systems should be a Landau-Ginzburg model (with twisted masses) and that the dichotomy between additive and multiplicative varieties in the context of mirror symmetry for Nakajima quiver varieties should be considered in the case of Hitchin systems.
Starting from Greg Moore's description about Physical Mathematics, a framework is proposed in order to understand it, based on Gilles Châtelet's philosophy. It will be argued that Châtelet's ideas of inverting, splitting, augmenting and virtuality are crucial in the discussion about the nature of Physical Mathematics. Along this line, it will be proposed that mirror symmetry is a natural study case to test Châtelet's ideas in this context. This should be considered as a first step in a long term project aiming to study the relations among mathematics, physics and philosophy in the construction of a global understanding of the structure of the universe, as it was envisioned by Grothendieck in the late 80's of the last century and it was started to be developed independently by Châtelet in the beginning of the 90's. The main suggestion of the essay is that it is in the relations between mathematics, physics and philosophy that new knowledge arises.
In this paper, first we introduce a new approach to the notion of $F$-algebroids, which is a generalization of $F$-manifold algebras and $F$-manifolds, and show that $F$-algebroids are the corresponding semi-classical limits of pre-Lie formal deformations of commutative associative algebroids. Then we use the deformation cohomology of pre-Lie algebroids to study pre-Lie infinitesimal deformations and extension of pre-Lie $n$-deformations to pre-Lie $(n+1)$-deformations of a commutative associative algebroid. Next we develop the theory of Dubrovin's dualities of $F$-algebroids with eventual identities and use Nijenhuis operators on $F$-algebroids to construct new $F$-algebroids. Finally we introduce the notion of pre-$F$-algebroids, which is a generalization of $F$-manifolds with compatible flat connections. Dubrovin's dualities of pre-$F$-algebroids with eventual identities, Nijenhuis operators on pre-$F$-algebroids and their applications to integral systems are discussed.
In this essay we give a general picture about the evolution of Grohendieck's ideas regarding the notion of space. Starting with his fundamental work in algebraic geometry, where he introduces schemes and toposes as generalizations of classical notions of spaces, passing through tame topology and ending with the formulation of a geometry of forms, we show how the ideas of Grothendieck evolved from pure mathematical considerations to physical and philosophical questions about the nature and structure of space and its mathematical models.
In this article we present and describe a notion of "logical perfection". We extract the notion of "perfection" from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a "mathematically perfect" structure is.
We define F-algebra--Rinehart pairs and super F-algebroids and study the connection between them.
In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.
Many distinct problems give birth to Darboux-Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in $\mathbb{R}^3$. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss-Manin connection in disguise developed by one of the authors and finally in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.
The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\text{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov.
The classical Stokes matrices for the quantum differential equation of projective n-space are computed, using multisummation and the so-called monodromy identity. Thus, we recover the results of D. Guzzetti that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in projective n-space and for weighted projective spaces.
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].