299 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    Analysis of pseudoholomorphic curves on symplectization: Revisit via contact instantons

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    In this survey article, we present the analysis of pseudoholomorphic curves u:(Σ˙,j)→(Q×R,J~)u:(\dot \Sigma,j) \to (Q \times \mathbb{R}, \widetilde J) on the symplectization of contact manifold (Q,λ)(Q,\lambda) as a subcase of the analysis of contact instantons w:Σ˙→Qw:\dot \Sigma \to Q, i.e., of the maps ww satisfying the equation ∂ˉπw=0, d(w∗λ∘j)=0 {\bar{\partial}}^\pi w = 0, \, d(w^*\lambda \circ j) = 0 on the contact manifold (Q,λ)(Q,\lambda), which has been carried out by a coordinate-free covariant tensorial calculus. When the analysis is applied to that of pseudoholomorphic curves u=(w,f)u = (w,f) with w=πQ∘uw = \pi_Q \circ u, f=s∘uf = s\circ u on symplectization, the outcome is generally stronger and more accurate than the common results on the regularity presented in the literature in that all of our a priori estimates can be written purely in terms ww not involving ff. The a priori elliptic estimates for ww are largely consequences of various Weitzenb\"ock-type formulae with respect to the contact triad connection introduced by Wang and the first author in [OW14], and the estimate for ff is a consequence thereof by simple integration of the equation df=w∗λ∘jdf = w^*\lambda \circ j. We also derive a simple precise tensorial formulae for the linearized operator and for the asymptotic operator that admit a perturbation theory of the operators with respect to (adapted) almost complex structures: The latter has been missing in the analysis of pseudoholomorphic curves on symplectization in the existing literature.Comment: 75 pages; Comments welcome; v2) typos corrected, one reference adde

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Quantitative Graded Semantics and Spectra of Behavioural Metrics

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    Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the classical linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics come in various degrees of granularity, depending on the observer's ability to interact with the system. Graded monads have been shown to provide a unifying framework for spectra of behavioural equivalences. Here, we transfer this principle to spectra of behavioural metrics, working at a coalgebraic level of generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we discuss presentations of graded monads on the category of metric spaces in terms of graded quantitative equational theories. Moreover, we obtain a canonical generic notion of invariant real-valued modal logic, and provide criteria for such logics to be expressive in the sense that logical distance coincides with the respective behavioural distance. We thus recover recent expressiveness results for coalgebraic branching-time metrics and for trace distance in metric transition systems; moreover, we obtain a new expressiveness result for trace semantics of fuzzy transition systems. We also provide a number of salient negative results. In particular, we show that trace distance on probabilistic metric transition systems does not admit a characteristic real-valued modal logic at all

    Noncommutative Geometry and Gauge theories on AF algebras

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    Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle Physics (SMPP), taking into account General Relativity, in a single "geometric" representation, based on Non-Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite C∗C^*-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying differential structure of the Gauge Theory. The inductive sequence defining the AF-algebra is used to allow the construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank n+1n+1 can represent a grand unified theory of the rank nn. The main advantage of this framework is that it controls, using appropriate conditions, the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the SMPP

    Construction and application of adjusted higher gauge theories

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    This thesis investigates several aspects of nonabelian higher gauge theories, which appear in many areas of physics, notably string theory and gauged supergravity. We show that nonabelian higher gauge theory admits a consistent classical nonperturbative formulation insofar as a higher nonabelian parallel transport exists consistently, without requiring certain curvature components (fake curvature) to vanish. Next, we explore examples of nonabelian higher gauge theories that naturally appear in high-energy physics. Using a generalisation of L∞-algebras called EL∞-algebras, we show that tensor hierarchies of gauged supergravity naturally admit a formulation in terms of higher nonabelian gauge theories. Furthermore, toroidal compactifications of string theory exhibiting T-duality also naturally contain higher gauge symmetry, which explain several features of nongeometric compactifications (Q- and R-fluxes)

    Moment polyptychs and the equivariant quantisation of hypertoric varieties

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    In this thesis, we develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde formula for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut that results in a compact cut space, which is needed for the localisation formulae to be well-defined and for the quantisation to be finite-dimensional. The hyperplane arrangement corresponding to the hypertoric variety is also affected by the symplectic cut, and to describe its effect we introduce the notion of a moment polyptych that is associated to the cut space. Also, we see that the prerequisite isotropy data that is needed for the localisation formulae can be read off from the combinatorial features of the moment polyptych. The equivariant Kawasaki-Riemann-Roch formula is then applied to the pre-quantum line bundle over each cut space, producing a formula for the equivariant character for the torus action on the quantisation of the cut space. Finally, using the quantisation of each cut space, we derive a formula expressing the dimension of each circle weight subspace of the quantisation of the hypertoric variety

    Syntax-semantics interface: an algebraic model

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    We extend our formulation of Merge and Minimalism in terms of Hopf algebras to an algebraic model of a syntactic-semantic interface. We show that methods adopted in the formulation of renormalization (extraction of meaningful physical values) in theoretical physics are relevant to describe the extraction of meaning from syntactic expressions. We show how this formulation relates to computational models of semantics and we answer some recent controversies about implications for generative linguistics of the current functioning of large language models.Comment: LaTeX, 75 pages, 19 figure
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