Several ways to reduce to a first order ODE to nonlinear PDE’s governing the relativistic motion of an axially symmetric membrane in 4 space-time dimensions, as well as examples for a previously found non-trivial transformation generating solutions, are given.

Neden Teorik Fizik: “İnsan karar vererek aşık olmaz, sadece bir bakar olmuş.” Landau-Lifshtiz: “Niçin çıktım, nasıl çıktım bunu izaha gerek yok.” Memlekette Eğitim: “Türkiye’de hiçbir başarı cezasız kalmaz.”

The Gauge/YBE correspondence claims precise relations between super symmetric quiver gauge theories and integrable models, wherein statements on the one side is translated into those on the other. In this talk I will introduce the topic, starting with the basics of quantum field theories and integrable models.

Quantum Chromodynamics (QCD), a fundamental theory in particle physics, describes the strong nuclear interactions of quarks and gluons. A key feature of QCD is asymptotic freedom, where fundamental particles behave as free entities at high energies. Conversely, at low energies, particles are confined in bound states (protons, neutrons, etc.), which is sometimes referred as infrared slavery. Despite extensive study, the theoretical prediction of confinement in QCD remains elusive. However, confinement has been demonstrated in various toy models, including supersymmetric theories and QCD-like theories in non-flat spacetimes. We will focus on the Yang-Mills-Higgs model in three dimensions, where Polyakov first demonstrated confinement in 1977 through monopole condensation. We aim to understand Polyakov’s mechanism, beginning with foundational field theory concepts, path integral quantization, and approximation methods. We will then explore topological characteristics in field theory, introduce gauge theories, and present criteria for topological solutions. Later we will see the 3-dimensional SU(2) Yang-Mills-Higgs model in detail, and examine the impact of instantons on the low-energy theory. The modification of the low-energy theory due to a dilute gas of instantons will be illustrated, demonstrating the theory’s mass gap and effective potential.

Quantum Chromodynamics (QCD), a fundamental theory in particle physics, describes the strong nuclear interactions of quarks and gluons. A key feature of QCD is asymptotic freedom, where fundamental particles behave as free entities at high energies. Conversely, at low energies, particles are confined in bound states (protons, neutrons, etc.), which is sometimes referred as infrared slavery. Despite extensive study, the theoretical prediction of confinement in QCD remains elusive. However, confinement has been demonstrated in various toy models, including supersymmetric theories and QCD-like theories in non-flat spacetimes. We will focus on the Yang-Mills-Higgs model in three dimensions, where Polyakov first demonstrated confinement in 1977 through monopole condensation. We aim to understand Polyakov’s mechanism, beginning with foundational field theory concepts, path integral quantization, and approximation methods. We will then explore topological characteristics in field theory, introduce gauge theories, and present criteria for topological solutions. Later we will see the 3-dimensional SU(2) Yang-Mills-Higgs model in detail, and examine the impact of instantons on the low-energy theory. The modification of the low-energy theory due to a dilute gas of instantons will be illustrated, demonstrating the theory’s mass gap and effective potential.

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum-generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. This algebra generates a stack of Bargmann spaces in one-to-one correspondence with the Landau levels. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3).The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.

The talk is based on a recent work in collaboration with Prof. Tekin Dereli (Koç University) and my PhD student Philippe Nounahon (Institut de Mathématique et de Sciences Physiques, Porto-Novo, Bénin)

The goal of these lectures is to cover the main aspects of quantum effects on the black-hole back-
ground. I also tried to make the entry threshold as low as possible so that even junior students could
understand the material. In general, these lectures notes reproduce the lectures from outstanding
scientists, so please see the originals.