# Topology, Geometry And Gauge Fields: Interactions

A study of topology and geometry, beginning with a comprehensible account of the extraordinary and rather mysterious impact of mathematical physics, and especially gauge theory, on the study of the geometry and topology of manifolds. The focus of the book is the Yang-Mills-Higgs field and some considerable effort is expended to make clear its origin and significance in physics. Much of the mathematics developed here to study these fields is standard, but the treatment always keeps one eye on the physics and sacrifices generality in favor of clarity. This second edition has replaced a brief appendix in the first on the Seiberg-Witten equations with a much more detailed survey of Donaldson-Witten Theory and the Witten Conjecture regarding the relationship between Donaldson and Seiberg-Witten invariants.

## Topology, Geometry and Gauge fields: Interactions

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We present a minimal non-Hermitian model where a topologically nontrivial complex energy spectrum is induced by interparticle interactions. Our model consists of a one-dimensional chain with a dynamical non-Hermitian gauge field with density dependence. The model is topologically trivial for a single-particle system, but exhibits nontrivial non-Hermitian topology with a point gap when two or more particles are present in the system. We construct an effective doublon model to describe the nontrivial topology in the presence of two particles, which quantitatively agrees with the full interacting model. Our model can be realized by modulating hoppings of the Hatano-Nelson model; we provide a concrete Floquet protocol to realize the model in atomic and optical settings.

This idea later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as quantum chromodynamics, is a gauge theory with the action of the SU(3) group on the color triplet of quarks. The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.

Gauge theories are usually discussed in the language of differential geometry. Mathematically, a gauge is just a choice of a (local) section of some principal bundle. A gauge transformation is just a transformation between two such sections.

The goal of this network is to facilitate, stimulate, and further promote the many interactions of low-dimensional topology and geometry with various fields including (in no particular order) gauge theory, quantum topology, symplectic topology and geometry, Teichmüller theory, hyperbolic geometry, string theory and quantum field theory. The network is intended to be organized on a European wide scale, reflecting the global nature of the ongoing research in these areas. The planned activities of workshops and conferences, schools and programmes of research visits will reach across international and disciplinary lines to stimulate current and future progress.At the same time, this network will bring together leading experts in the above-mentioned areas with a new generation of researchers, providing them with the interdisciplinary perspective and training which will plant seeds for the breakthroughs of the future.Programme proposal More information can be found on the programme dedicated site.

Condensed matter physics, quantum optics, strong light-matter interactions, physics of atomically thin 2D materials, role of geometry and topology in low-dimensions, Berry phase and artificial gauge fields, open systems.

A closely related theme is to understand the role of geometry and topology in solid- state using light-matter interactions. A non-trivial geometry of electronic bands (wave-functions) in a solid can result in effective electromagnetic fields in the reciprocal space. Can we use light to tune such fields? Is it possible to induce non- trivial geometry and topological states of light/matter using light-matter interactions? Our research hopes to address these questions.

We present a theory of electronic properties of a quadruple quantum dot molecule (QQD) which focuses on geometry, chirality, and electron-electron interactions. The QQD is described by the extended Hubbard model solved using exact diagonalization method in real and Fourier space. The energy spectrum of a QQD is analysed as a function of the number of electrons Ne, for ring, linear, or star geometry. We discuss the interplay of chirality, topology, and Fermi statistics for a half-filled ring QQD charged with either additional electron or hole. We show that the chirality leads to the appearance of a topological phase and an effective gauge field stabilizing the spin polarised state. The spin polarised state with extra electron (hole) and spin unpolarised state at half-filling lead to spin blockade in transport through the ring-like QQD but not through a linear nor star QQD molecule. We demonstrate that the ground state can be tuned between a total spin S=1/2 and S=3/2 by changing the strength of on-site interactions or tuning the tunnelling matrix element.

The topics listed above give a sense of the intellectual terrain in which the Center will operate but there is another crucial aspect to the intellectual focus of the Center. The Center believes that much progress can be made in each domain, geometry and theoretical physics, by serious, substantive interchanges between mathematicians and physicists around fundamental questions of common interest. Enhancing these interactions will be always at the forefront as the Center plans and carries out its activities. Thus, the Center focuses on activities of common interest to mathematicians and physicists and invites members of both groups to each of its programs and workshops. The Center also searches out mathematicians and physicists who have exhibiting an willingness and an ability to work across the subject divide that separates their fields to invite to join the Center as permanent members, as members of its oversight bodies, as organizers of its activities, and as participants in the activities. Our program organizing committees, in almost every case, have both mathematicians and physicists as members and one of their charges will be to construct programs and workshops that are accessible to and appeal to both groups. In the case of those topics that are clearly primarily in one of the two areas, Center will look for ways to broaden the topic in such a way that it has appeal to some members of the other group and then actively recruit the members of the minority group to participate in the activity. 041b061a72