arXiv subjects

Duco van Straten

Publications and source records attributed to Duco van Straten.

At least 37 records · Page 2Linked to original sources

arXiv2018

Rationalizing roots: an algorithmic approach

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots. In this paper, we give an algorithm for rationalizing roots. The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity $(d-1)$, where $d$ is the degree of the algebraic hypersurface. We show that one can use the algorithm iteratively to rationalize multiple roots simultaneously. Several examples from high energy physics are discussed.

hep-th↗

arXiv2018

A special Calabi-Yau degeneration with trivial monodromy

A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi-Yau threefolds.

math.AG↗

arXiv2018

Bhabha Scattering and a special pencil of K3 surfaces

We study a pencil of K3 surfaces that appeared in the $2$-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apéry--Fermi pencil, that was related to Apéry's proof of the irrationality of $ζ(3)$ through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts.

math.AG↗

arXiv2018

KAM THEORY. Part 3. Applications

We apply the general normal form theorems in Kolmogorov spaces to three classical cases: deformations of hypersurface singularities, normal forms of vector fields and invariant tori in Hamiltonian systems.

math.DS↗

arXiv2018

Hilbert modularity of some double octic Calabi--Yau threefolds

We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.

math.AG↗

arXiv2018

KAM Theory. Part II. Kolmogorov spaces

This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define the exponential and therefore relation between Lie algebra and Lie group actions in the infinite dimensional context. Then we introduce a notion of finite dimensional reduction and use it to prove a fixed point theorem for Kolmogorov spaces. We conclude by proving general normal theorems.

math.DS↗

arXiv2018

KAM Theory. Part I. Group actions and the KAM problem

This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II.

math.DS↗

arXiv2017

Periods of double octic Calabi--Yau manifolds

We compute numerical approximations of the period integrals for eleven rigid double octic Calabi--Yau threefolds and compare them with the periods of corresponding weight our cusp forms and find, as to be expected, commensurabilities. These give information on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.

math.AG↗

arXiv2017

Picard-Fuchs operators for octic arrangements I (The case of orphans)

We report on $25$ families of projective Calabi-Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by C. Meyer. There are seven cases where the Picard-Fuchs operator is of order two and $18$ cases where it is of order four. The birational nature of the Picard-Fuchs operator can be used effectively to distinguish between families whose members have the same Hodge numbers.

math.AG↗

arXiv2017

Calabi--Yau Operators

Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $\mathbb{Q}$-VHS with Hodge numbers \[h^{3 0}=h^{2 1}=h^{1 2}=h^{0 3}=1\] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{11}=1$. We review some of the striking properties of this rich class of operators.

math.AG↗

arXiv2016

Logarithmic vector fields and the Severi strata in the discriminant

The discriminant, $D$, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the {\it Severi strata}. The smallest is the $δ$-constant stratum, $D(δ)$, where the genus of the fibre is $0$. It is well known, by work of Givental' and Varchenko, to be Lagrangian with respect to the symplectic form $Ω$ obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to $Ω$, and moreover that the coefficients of the expression of $Ω^{\wedge k}$ with respect to a basis of $Ω^{2k}(\log D)$ are equations for $D(δ-k+1)$, for $k=1,\ldots,δ$. These equations allow us to show that for $E_6$ and $E_8$, $D(δ)$ is Cohen-Macaulay (this was already shown by Givental' for $A_{2k}$), and that, as far as we can calculate, for $A_{2k}$ all of the Severi strata are Cohen-Macaulay. Our construction also produces a canonical rank 2 maximal Cohen Macaulay module on the discriminant.

math.AG↗

arXiv2014

Resurgent Deformation Quantisation

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. In particular, we exhibit an integral formula for the product of resurgent operators with algebraic singularities. This algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.

math-ph↗

arXiv2013

Some monodromy groups of finite index in $Sp_4(\mathbb{Z})$

We determine the index of five of the $7$ hypergeometric Calabi-Yau operators that have finite index in $Sp_4(\mathbb{Z})$ and in two cases give a complete description of the monodromy group. Furthermore we found six non-hypergeometric Calabi-Yau operators with finite index in $Sp_4(\mathbb{Z})$, most notably a case where the index is one.

math.AG↗

arXiv2013

An abelian surface with (1,6)-polarisation

We use the method of Adler-van Moerbeke and Vanhaecke to show that the general fibre of the hamiltonian system of Dorizzi, Grammaticos and Ramani completes to a (1, 6)-polarised abelian surface.

math.AG↗

arXiv2013

Twisted cubics on cubic fourfolds

We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.

math.AG↗

arXiv2012

Calabi-Yau conifold expansions

We describe examples of computations of Picard-Fuchs operators for families of Calabi-Yau manifolds based on the expansion of a period near a conifold point. We find examples of operators without a point of maximal unipotent monodromy, thus answering a question posed by J. Rohde.

math.AG↗

arXiv2012

Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetry of the pair of genus six curves. The subgroup A_5, of even permutations, is an automorphism of each curve, while the odd permutations interchange the two curves. The ten exceptional curves of dP_5 each intersect each of the genus six curves in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the genus six curves are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold the genus six curves develop six nodes and may be resolved to a P^1. The group A_5 acts on this P^1 and we describe this action.

math.AG↗

arXiv2011

An Index Theorem for Modules on a Hypersurface Singularity

A topological interpretation of Hochster's Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was conjectured by H. Dao.

math.AG↗