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assets/basis-plus-lattice/basis-plus-lattice.tex

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\documentclass[tikz,border=5pt]{standalone}
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.18}
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\usetikzlibrary{calc}
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\begin{document}

assets/closed-string-topologies/closed-string-topologies.yml

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> Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with $g$ handles ($g$ for genus) and $b$ boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic $\chi = 2 - 2g - b$.
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What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles $g$ in case of the closed string and the number of boundaries $b$ for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the open sector, see [open string topologies](https://janosh.github.io/tikz/open-string-topologies).
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What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles $g$ in case of the closed string and the number of boundaries $b$ for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the open sector, see [open string topologies](https://janosh.github.io/diagrams/open-string-topologies).

assets/higgs-potential/higgs-potential.yml

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- quantum field theory
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- symmetry
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description: |
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The Higgs mechanism plays a vital role in the Standard Model for explaining how gauge bosons obtain mass. The Standard Model would otherwise predict these particles to be massless. Through interactions with the Higgs field that permeates all space and whose elevated potential at zero field leads to spontaneous symmetry breaking (SSB), gauge bosons also experience symmetry breaking, causing them to acquire mass. See <https://janosh.github.io/tikz/maxican-hat> for a very similar image.
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The Higgs mechanism plays a vital role in the Standard Model for explaining how gauge bosons obtain mass. The Standard Model would otherwise predict these particles to be massless. Through interactions with the Higgs field that permeates all space and whose elevated potential at zero field leads to spontaneous symmetry breaking (SSB), gauge bosons also experience symmetry breaking, causing them to acquire mass. See <https://janosh.github.io/diagrams/maxican-hat> for a very similar image.

assets/matsubara-contour-2/matsubara-contour-2.yml

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- physics
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- quantum field theory
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- Matsubara
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description: Deformation of contour $C$ in [Matsubara contour 1](https://janosh.github.io/tikz/matsubara-contour-1) into a circle followed by taking the radius to infinity. This will enclose the poles of $(-p_0^2 + x^2)^{-1}$ scattered throughout the complex plane. Their contribution is removed again by enclosing them in clockwise contours.
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description: Deformation of contour $C$ in [Matsubara contour 1](https://janosh.github.io/diagrams/matsubara-contour-1) into a circle followed by taking the radius to infinity. This will enclose the poles of $(-p_0^2 + x^2)^{-1}$ scattered throughout the complex plane. Their contribution is removed again by enclosing them in clockwise contours.

assets/mexican-hat/mexican-hat.yml

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- quantum field theory
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- symmetry
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description: |
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The [Mexican hat potential](https://wikipedia.org/wiki/Spontaneous_symmetry_breaking) exhibits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry is broken even though the Lagrangian as a whole retains it. See <https://janosh.github.io/tikz/higgs-potential> for a very similar image.
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The [Mexican hat potential](https://wikipedia.org/wiki/Spontaneous_symmetry_breaking) exhibits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry is broken even though the Lagrangian as a whole retains it. See <https://janosh.github.io/diagrams/higgs-potential> for a very similar image.
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In this image, the system starts out in the naive $O(N)$-invariant vacuum (blue dot) but quantum fluctuations quickly push it into the real vacuum (red dot) where $O(N)$ is broken down to $O(N-1)$.

assets/open-string-topologies/open-string-topologies.yml

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> Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with $g$ handles ($g$ for genus) and $b$ boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic $\chi = 2 - 2g - b$.
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What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles $g$ in case of the closed string and the number of boundaries $b$ for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the closed sector, see [closed string topologies](https://janosh.github.io/tikz/open-string-topologies).
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What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles $g$ in case of the closed string and the number of boundaries $b$ for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the closed sector, see [closed string topologies](https://janosh.github.io/diagrams/open-string-topologies).

citation.cff

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affiliation: University of Arizona
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license: MIT
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license-url: https://github.com/janosh/tikz/blob/main/license"
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repository-code: https://github.com/janosh/tikz
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license-url: https://github.com/janosh/diagrams/blob/main/license"
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repository-code: https://github.com/janosh/diagrams
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type: software
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url: https://github.com/janosh/tikz
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url: https://github.com/janosh/diagrams
2020
doi: 10.5281/zenodo.7486911
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version: 0.1.0
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date-released: 2022-12-27

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