299 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
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 Bringâs 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 Bringâs 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
In this survey article, we present the analysis of pseudoholomorphic curves
on the
symplectization of contact manifold as a subcase of the analysis
of contact instantons , i.e., of the maps satisfying
the equation on
the contact manifold , which has been carried out by a
coordinate-free covariant tensorial calculus. When the analysis is applied to
that of pseudoholomorphic curves with , 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 not
involving . The a priori elliptic estimates for 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 is a consequence thereof by simple integration of the equation .
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
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Quantitative Graded Semantics and Spectra of Behavioural Metrics
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
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 -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 can represent a grand unified
theory of the rank . 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
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 Lâ-algebras called ELâ-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)
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Monopoles and Dehn twists on contact 3-manifolds
In this dissertation, we study the isotopy problem for a certain three-dimensional contactomorphism which is supported in a neighbourhood of an embedded 2-sphere with standard characteristic foliation. The diffeomorphism which underlies it is the Dehn twist on the sphere, and therefore its square becomes smoothly isotopic to the identity. The main result of this dissertation gives conditions under which any iterate of the Dehn twist along a non-trivial sphere is not contact isotopic to the identity.
This provides the first examples of exotic contactomorphisms with infinite order in the contact mapping class group, as well as the first examples of exotic contactomorphisms of 3-manifolds with b_1 = 0. The proof crucially relies on the construction of an invariant for families of contact structures in monopole Floer homology which generalises the Kronheimer--Mrowka--OzsvĂĄth--SzabĂł contact invariant, together with the nice interaction between this families invariant and the U map in Floer homology
Moment polyptychs and the equivariant quantisation of hypertoric varieties
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
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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