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M. J. D. Hamilton

Publications and source records attributed to M. J. D. Hamilton.

18 recordsLinked to original sources
arXiv2024

Born geometry via Künneth structures and recursion operators

We propose a simple definition of a Born geometry in the framework of Künneth geometry. While superficially different, this new definition is equivalent to the known definitions in terms of para-quaternionic or generalized geometries. We discuss integrability of Born structures and their associated connections. In particular we find that for integrable Born geometries the Born connection is obtained by a simple averaging under a conjugation from the Künneth connection. We also give examples of integrable Born geometries on nilmanifolds.

math.DG↗
arXiv2021

A lift of the Seiberg-Witten equations to Kaluza-Klein 5-manifolds

We consider Riemannian 4-manifolds $(X,g_X)$ with a Spin^c-structure and a suitable circle bundle $Y$ over $X$ such that the Spin^c-structure on $X$ lifts to a spin structure on $Y$. With respect to these structures a spinor $ϕ$ on $X$ lifts to an untwisted spinor $ψ$ on $Y$ and a U(1)-gauge field $A$ for the Spin^c-structure can be absorbed into a Kaluza-Klein metric $g_Y^A$ on $Y$. We show that irreducible solutions $(A,ϕ)$ to the Seiberg-Witten equations on $(X,g_X)$ for the given Spin^c-structure are equivalent to irreducible solutions $ψ$ of a Dirac equation with cubic non-linearity on the Kaluza-Klein circle bundle $(Y,g_Y^A)$. As an application we consider solutions to the equations in the case of Sasaki 5-manifolds which are circle bundles over Kaehler-Einstein surfaces.

math.DG↗
arXiv2020

The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking

These notes form part of a lecture course on gauge theory. The material covered is standard in the physics literature, but perhaps less well-known to mathematicians. The purpose of these notes is to make spontaneous symmetry breaking and the Higgs mechanism of mass generation for elementary particles more easily accessible to mathematicians interested in theoretical physics. We treat the general case with an arbitrary compact gauge group G and an arbitrary number of Higgs bosons and explain the situation in the classic case of the electroweak interaction where G=SU(2)xU(1). Prerequisites are only a basic knowledge of Lie groups and manifolds. No prior knowledge of gauge theory or bundle theory is assumed.

math.DG↗
arXiv2019

J-holomorphic curves and Dirac-harmonic maps

Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by the complex structure and the target space is a Kaehler manifold. If the underlying map f is a J-holomorphic curve, we determine a space of spinors on the Riemann surface which form Dirac-harmonic maps together with f. For suitable complex structures on the target manifold the tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps. We also discuss the relation to the A-model of topological string theory.

math.DG↗
arXiv2019

Bi-Lagrangian structures on nilmanifolds

We study bi-Lagrangian structures (a symplectic form with a pair of complementary Lagrangian foliations, also known as para-Kähler or Künneth structures) on nilmanifolds of dimension less than or equal to 6. In particular, building on previous work of several authors, we determine which 6-dimensional nilpotent Lie algebras admit a bi-Lagrangian structure. In dimension 6, there are (up to isomorphism) 26 nilpotent Lie algebras which admit a symplectic form, 16 of which admit a bi-Lagrangian structure and 10 of which do not. We also calculate the curvature of the canonical connection of these bi-Lagrangian structures.

math.SG↗
arXiv2016

The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation

We review the actions of the supergravity theory in eleven dimensions as well as the type IIA and IIB supergravities in ten dimensions and derive the bosonic equations of motion in a coordinate-free notation. We also consider the existence of supersymmetries and the associated generalized Killing spinor equations. The aim of this note is to serve as a formulary and make the equations of supergravity more easily accessible to mathematicians.

math.DG↗
arXiv2015

Cyclic group actions and embedded spheres in 4-manifolds

In this note we derive an upper bound on the number of 2-spheres in the fixed point set of a smooth and homologically trivial cyclic group action of prime order on a simply-connected 4-manifold. This improves the a priori bound which is given by one half of the Euler characteristic of the 4-manifold. The result also shows that in some cases the 4-manifold does not admit such actions of a certain order at all or that any such action has to be pseudofree.

math.GT↗
arXiv2014

The closure of the symplectic cone of elliptic surfaces

The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplectic cone. We prove this conjecture in the case of the spin surfaces E(2m) using inflation and the action of self-diffeomorphisms of the elliptic surface. An additional obstruction appears in the non-spin case.

math.GT↗
arXiv2013

Iterated fibre sums of algebraic Lefschetz fibrations

Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculate the Seiberg-Witten invariants of the fibre sum M(n) using a formula of Morgan-Szabo-Taubes. As an application we derive an obstruction for self-diffeomorphisms of the boundary of the tubular neighbourhood of a general fibre in M(n) to extend over the complement of the neighbourhood. Similar obstructions are known in the case of elliptic surfaces.

math.GT↗
arXiv2013

The minimal genus problem for elliptic surfaces

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.

math.GT↗
arXiv2013

Homology classes of negative square and embedded surfaces in 4-manifolds

Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or characteristic. In particular, for genus zero, there is a lower bound on the self-intersection of embedded spheres in these kinds of homology classes. This question is related to a problem from the Kirby list.

math.GT↗
arXiv2013

Generalized fibre sums of 4-manifolds and the canonical class

In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the homologies of the 4-manifolds are torsion free and the surfaces represent indivisible homology classes, we derive a formula for the intersection form of X. If in addition the 4-manifolds M and N are symplectic and the surfaces symplectically embedded we derive a formula for the canonical class of the symplectic fibre sum.

math.SG↗
arXiv2012

Inequivalent contact structures on Boothby-Wang 5-manifolds

We consider contact structures on simply-connected 5-manifolds which arise as circle bundles over simply-connected symplectic 4-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequivalent contact structures in the same equivalence class of almost contact structures with non-zero first Chern class.

math.SG↗
arXiv2011

On certain exotic 4-manifolds of Akhmedov and Park

In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homology of these manifolds adapted to their construction as fibre sums. We also calculate the canonical classes of the symplectic structures. This gives a new proof for a formula derived by A. Akhmedov, R. I. Baykur and B. D. Park.

math.GT↗
arXiv2009

On the geography of symplectic 4-manifolds with divisible canonical class

In this article we consider a version of the geography question for simply-connected symplectic 4-manifolds that takes into account the divisibility of the canonical class as an additional parameter. We also find new examples of 4-manifolds admitting several symplectic structures, inequivalent under deformation and self-diffeomorphisms of the manifold.

math.SG↗
arXiv2006

On the conformal systoles of four-manifolds

We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy the so-called 5/4-conjecture.

math.DG↗
arXiv2006

Minimality and irreducibility of symplectic four-manifolds

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.

math.SG↗