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Ian A. B. Strachan

Publications and source records attributed to Ian A. B. Strachan.

At least 19 recordsLinked to original sources
arXiv2026

Open WDVV equations and $\bigvee$-systems

The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the WDVV equations. Here, this idea is generalized to open WDVV equations, which are an additional set of PDEs originating from open Gromow-Witten Theory. We develop -- for rank-one extensions -- algebraic/geometric conditions on the covectors that supplement the $\bigvee$-system to give rational solutions to the open WDVV equations. Examples, and the relation to superpotentials and to Dubrovin almost-duality, are given.

math-ph↗
arXiv2025

Dubrovin duality for open Hurwitz flat F-manifolds

We prove that the Dubrovin dual of a Hurwitz Frobenius manifold extends naturally to an F-manifold with compatible flat connection on the universal curve, in the sense of the open WDVV equations. A similar result is proven for the Frobenius manifold itself in arXiv:2503.09258 . This equips the universal curve with two F-manifolds with compatible flat structure, and we study their duality. We show that they combine into a bi-flat F-manifold. Conditions on open WDVV solutions imposed in previous work are retrieved in this setting, thus providing them with a geometrical meaning. Finally, explicit examples are computed. For Saito Frobenius manifolds of types $A$ and $D$, the extended prepotentials coincide with open WDVV solutions computed independently, whereas even the existence of the solution in type $E$ had not been previously discussed. On the other hand, new non-homogeneous solutions are constructed by duality.

math-ph↗
arXiv2025

Diagonal invariants and genus-zero Hurwitz Frobenius manifolds

The Frobenius manifold structure on the space of rational functions with multiple simple poles is constructed. In particular, the dependence of the Saito-flat coordinates on the flat coordinates of the intersection form is studied. While some of the individual flat coordinates are complicated rational functions, they appear in the prepotential in certain combinations known as diagonal invariants, which turn out to be polynomial. Two classes are studied in more detail. These are generalisations of the Coxeter and extended-affine-Weyl orbits space for the group $W=W(A_\ell)\,.$ An invariant theory is also developed.

math-ph↗
arXiv2025

Dubrovin duality and mirror symmetry for ADE resolutions

We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-Ginzburg mirrors for the quantum cohomology of $\mathrm{ADE}$ resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the $\widehat{\mathrm{ADE}}$ affine relativistic Toda chain.

math.AG↗
arXiv2024

Legendre transforms for type $A_{n}$ and $B_{n}$ $\vee$-systems

The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to $A_n$- and $B_n$-type multi-parameter rational solutions, relating them to known and new trigonometric solutions.

math-ph↗
arXiv2024

Regular F -manifolds with eventual identities

Given an F-manifold one may construct a dual multiplication (generalizing the idea of an almost-dual Frobenius manifold introduced by Dubrovin) using a so-called eventual identity, the definition of which ensure that the dual object is also an F-manifold. In this paper we solve the equations for an eventual identity for a regular (so non-semi-simple) F-manifold and construct a dual coordinate system in which dual multiplication is preserved. As an application, families of Nijenhuis operators are constructed.

math.DG↗
arXiv2020

Complex hyperkähler structures defined by Donaldson-Thomas invariants

The notion of a Joyce structure was introduced in arXiv:1912.06504 to describe the geometric structure on the space of stability conditions of a CY3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

math.AG↗
arXiv2020

The Saito determinant for Coxeter discriminant strata

Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in $\mathcal{M}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.

math.DG↗
arXiv2019

Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials

For the root systems of type $B_l, C_l$ and $D_l$, we generalize the result of \cite{DZ1998} by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in \cite{DZ1998}. It also depends on certain additional data. We also construct LG superpotentials for these Frobenius manifold structures.

math.DG↗
arXiv2018

A construction of Multidimensional Dubrovin-Novikov Brackets

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from commutative, associative algebras. Given such an algebra, the construction involves only linear algebra.

nlin.SI↗
arXiv2018

Darboux coordinates for Hamiltonian structures defined by Novikov algebras

The Gauss-Manin equations are solved for a class of flat-metrics defined by Novikov algebras, this generalizing a result of Balinskii and Novikov who solved this problem in the case of commutative Novikov algebras (where the algebraic conditions reduce to those of a Frobenius algebra). The problem stems from the theory of first-order Hamiltonian operators and their reduction to a constant, or Darboux, form. The monodromy group associated with the Novikov algebra gives rise to an orbit space, which is, for a wide range of Novikov algebras, a cyclic quotient singularity.

nlin.SI↗
arXiv2016

Extended V-systems and almost-duality for extended affine Weyl orbit spaces

Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as $\bigvee$-conditions. The simplest examples of such configuration are the root systems of finite Coxeter groups. In this paper conditions are derived which ensure that an extended configuration - a configuration in a space one-dimension higher -satisfy these $\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit, as defined by Serganova. Symmetries of such resulting solutions to the WDVV-equations are studied; in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions and, for certain values of the free data, one obtains a transformation from extended $\bigvee$-systems to the trigonometric almost dual solutions corresponding to the classical extended affine Weyl groups.

nlin.SI↗
arXiv2016

Generalized Legendre transformations and symmetries of the WDVV equations

The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.

nlin.SI↗
arXiv2015

Integrability of the Frobenius algebra-valued KP hierarchy

We introduce a Frobenius algebra-valued KP hierarchy and show the existence of Frobenius algebra-valued $τ$-function for this hierarchy. In addition we construct its Hamiltonian structures by using the Adler-Dickey-Gelfand method. As a byproduct of these constructions, we show that the coupled KP hierarchy defined by P.Casati and G.Ortenzi in \cite{CO2006} has at least $n$-``basic" different local bi-Hamiltonian structures. Finally, via the construction of the second Hamiltonian structures, we obtain some local matrix, or Frobenius algebra-valued, generalizations of classical $W$-algebras.

math-ph↗
arXiv2015

Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \cite{Kaufmann,KMK}. By specializing this construction, using a fixed Frobenius algebra $\mathcal{A},$ one can arrive at such a theory. More generally one can apply the same idea to construct an $\mathcal{A}$-valued Topological Quantum Field Theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed and, as an example, an $\mathcal{A}$-valued modified Camassa-Holm equation is constructed.

math-ph↗
arXiv2014

Novikov algebras and a classification of multicomponent Camassa-Holm equations

A class of multi-component integrable systems associated to Novikov algebras, which interpolate between KdV and Camassa-Holm type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multi-component bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated to the Novikov algebras. The related bilinear forms generating cocycles of first, second and third order are classified. Several examples, including known integrable equations, are presented.

nlin.SI↗
arXiv2011

Polynomial Modular Frobenius Manifolds

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class - so-called modular Frobenius manifolds - lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.

nlin.SI↗