arXiv subjects

Jordan Kostiuk

Publications and source records attributed to Jordan Kostiuk.

8 recordsLinked to original sources

arXiv2023

The Doran-Harder-Thompson conjecture for toric complete intersections

Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and $I$-functions of the mirror Calabi-Yau family and those of the two mirror Landau-Ginzburg models. Our proof makes explicit the "gluing/splitting" of fibrations in the Doran-Harder-Thompson mirror conjecture. Our gluing formula implies an identity, obtained by composition with their respective mirror maps, that relates the absolute Gromov-Witten invariants for the Calabi-Yaus and relative Gromov-Witten invariants for the quasi-Fanos.

math.AG↗

arXiv2022

Eigenvalues and Critical Groups of Adinkras

Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers critical groups over polynomial rings.

math.CO↗

arXiv2021

Degenerations, fibrations and higher rank Landau-Ginzburg models

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations.

math.AG↗

arXiv2019

Geometric Variations of Local Systems and Elliptic Surfaces

Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of Kähler manifolds for which each fibre itself is fibred by codimension one Kähler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livné in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the universal families of $M_N$-polarized K3 surfaces. Finally, we explain the connection between geometric variations of local systems and geometric isomonodromic deformations, which were originally considered by the first author in 1999.

math.AG↗

arXiv2019

Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras

The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincaré algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as {\em Adinkras}. In this paper we {\em canonically} realize these graphs as Grothendieck ``dessins d'enfants,'' or Belyi curves uniformized by certain normal torsion-free subgroups of the $(N,N,2)$-triangle group. We exhibit an explicit algebraic model over $\mathbb{Q}(ζ_{2N})$, as a complete intersection of quadrics in projective space, and use Galois descent to prove that the curves are, in fact, definable over $\mathbb{Q}$ itself. The stage is thereby set for the geometric interpretation of the remaining Adinkra decorations in Part II.

hep-th↗

arXiv2019

The Signed Monodromy Group of an Adinkra

An ordering of colours in an Adinkra leads to an embedding of this Adinkra into a Riemann surface $X$, and a branched covering map $\beta_X:X\to\mathbb{CP}^1$. This paper shows how the dashing of edges in an Adinkra determines a signed permutation version of the monodromy group, and shows that it is isomorphic to a Salingaros Vee group.

math.CO↗

arXiv2018

Enumerative Gadget Phenomena for $(4,1)$-Adinkras

Adinkras are combinatorial objects developed to study supersymmetry representations. Gates et al. introduced the "gadget" as a function of pairs of adinkras, obtaining some mysterious results for $(n=4, k=1)$ adinkras with computer-aided computation. Specifically, very few values of the gadget actually appear, suggesting a great deal of symmetry in these objects. In this paper, we compute gadgets symbolically and explain some of these observed phenomena with group theory and combinatorics. Guided by this work, we give some suggestions for generalizations of the gadget to other values of the $n$ and $k$ parameters.

math.CO↗

arXiv2017

Geometrization of $N$-Extended $1$-Dimensional Supersymmetry Algebras II

The problem of classifying off-shell representations of the $N$ -extended one-dimensional super Poincaré algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as Adinkras. In previous work we canonically embedded these graphs into explicitly uniformized Riemann surfaces via the "dessins d'enfant" construction of Grothendieck. The Adinkra graphs carry two additional structures: a selection of dashed edges and an assignment of integral helghts to the vertices. In this paper, we complete the passage from algebra, through discrete structures, to geometry. We show that the dashings correspond to special spin structures on the Riemann surface, defining thereby super Riemann surfaces. Height assignments determine discrete Morse functions, from which we produce a set of Morse divisors which capture the topological properties of the height assignments.

hep-th↗