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Pure Mathematics

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9932

This is the collection for the University of Waterloo's Department of Pure Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Browsing Pure Mathematics by Author "Charbonneau, Benoit"

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    Deformation theory of nearly G₂-structures and nearly G₂ instantons
    (University of Waterloo, 2021-08-30) Singhal, Ragini; Charbonneau, Benoit; Karigiannis, Spiro
    We study two different deformation theory problems on manifolds with a nearly G₂-structure. The first involves studying the deformation theory of nearly G₂ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G₂-structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G₂-structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the de Rham cohomology of nearly G₂ manifolds. In the second problem we study the deformation theory of G₂ instantons on nearly G₂ manifolds. We make use of the one-to-one correspondence between nearly parallel G₂-structures and real Killing spinors to formulate the deformation theory in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to explicitly describe the deformation space of the canonical connection on the four normal homogeneous nearly G₂ manifolds. We also describe the infinitesimal deformation space of the SU(3) instantons on Sasaki–Einstein 7-folds which are nearly G₂ manifolds with two Killing spinors. A Sasaki–Einstein structure on a 7-dimensional manifold is equivalent to a 1-parameter family of nearly G₂-structures. We show that the deformation space can be described as an eigenspace of a twisted Dirac operator.
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    Integrality theorems for symmetric instantons
    (University of Waterloo, 2022-08-05) Whitehead, Spencer; Charbonneau, Benoit
    Anti-self-dual (ASD) instantons on R4 are connections A on SU(N)-vector bundles with finite L2-norm and curvature satisfying the ASD equation. Solutions to this non-linear partial differential equation correspond to certain algebraic data via the celebrated ADHM correspondence. While much is known about the space of instantons, it is still difficult to give explicit examples of them, aside from classes of solutions provided by certain ansatze. The perspective in this thesis is that of symmetry: by introducing a suitable notion of a nice group action on an instanton, one expects that the condition of 'equivariance with respect to the symmetry group' to reduce the number of parameters present in the ADHM equations, thus allowing for the creation of solutions not visible to existing ansatze. Through this method of symmetry, a theory of symmetric instantons is developed and applied it in particular to the case of finite-energy ASD solutions on R4 with symmetry a compact subgroup of Spin(4). This theory acts as a framework in which previous work on symmetric instantons may be realized, and in particular allows for a number of '(algebraic) integrality' results for solutions to the symmetric instanton equations. Using the equivariant index theorem the 'SU(2) restriction' ansatz used in previous work is proved to give the only non-trivial class of solutions to the symmetric instanton equations for certain symmetry subgroups of SU(2). Additionally, a question of Allen and Sutcliffe on the existence of a non-trivial instanton with symmetries of the 600-cell occurring at a charge lower than that of the JNR bound of 119 is resolved in the negative. Finally, ADHM data for two new instantons symmetric under the binary icosahedral group occurring at charges 13 and 23 are presented, as well as the software package used to generate them.
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    Solitons with continuous symmetries
    (University of Waterloo, 2024-08-29) Lang, Christopher James; Charbonneau, Benoit
    In this thesis, we develop a framework for classifying symmetric points on moduli spaces using representation theory. We utilize this framework in a few case studies, but it has applications well beyond these cases. As a demonstration of the power of this framework, we use it to study various symmetric solitons: instantons as well as hyperbolic, singular, and Euclidean monopoles. Examples of these objects are hard to come by due to non-linear constraints. However, by applying this framework, we introduce a linear constraint, whose solution greatly simplifies the non-linear constraints. This simplification not only allows us to easily find a plethora of novel examples of these solitons, it also provides a framework for classifying such symmetric objects. As an example, by applying this method, we prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we study the symmetry breaking of monopoles, a part of their topological classification. We prove a straightforward method for determining the symmetry breaking of a monopole and explicitly identify the symmetry breaking for all Lie groups with classical, simply Lie algebras. We also identify methods for doing the same for the exceptional simple Lie groups.