The Mathematical Construction of Supersymmetry
Supersymmetry is something that physics got gifted from string theory, but has so far been kind of discounted outside of it, since it's been disproved experimentally in the realm of particle physics. It's probably still good to at least take a look at this theory, since on paper it's a unifying theory of the four fundamental forces of physics. As a short summary of the problem: The standard model of physics features four fundamental forces that compose all forces that one will be observing in physics. These are the weak and strong interaction, the electromagnetic force, and gravity. The problem is that last one, since it's not been integrated into the standard model so far. It's so unimplemented in the standard model that trying to apply it in places will actually break the model. The idea of Majorana fermions for example can be constructed in a way where they don't interact, but are heavy (for particles, at least). Gravity would then dictate interaction, but these Majorana fermions had been constructed to not interact. Supersymmetry aims to fix this problem, and while it's probably not correct without imposing very heavy constraints, it's an interesting mathematical construction requiring quite a bit of background. My hope is to fill out the gaps in my understanding up to the point where I can explain how and why to get to the Type IIA/B supersymmetry construction used in string theory.
Supersymmetry is a pretty large topic in of itself, but I've never quite had its construction shown to me in a way that I found easy to absorb. I'll presuppose knowledge of spinor-representations, Feynman rules, the basics of QFT and standard model gauge theory.
The central idea of supersymmetry is actually quite simple. It's supposed to act somewhat like a parity operator, but for fermionic and bosonic state. That means we expect to find an operator Q for which:
Because of the way fermionic states are constructed, this operator should be some anticommutating spinor. Because all spinors consist of complex elements, the hermitian conjugate should also fulfill the same function. Supersymmetry always works off a description of bosonic and fermionic particles. These days that means a quantum field theory, on the basis of which the properties of possible symmetry operators can be narrowed down using the supercharges that emerge from their gauge theory. Each supersymmetric theory has an irreducible representation of its algebra on which the computations take place. This representation is called a supermultiplet and will pair up fermion and boson states as superpartners.
Seeing as a single left-handed 2-part Weyl fermion is the minimal fermion content in any 4-dimensional theory, and quantum field theory always includes some complex scalar field, they're very convenient to pair up. For this system, one can construct the Lagrangian quite easily, discounting interaction terms
The simplest operation to fulfill the requirement is something akin to δφ = ϵΨ for some infinitesimal, anticommuting, 2-component Weyl fermion. For global supersymmetry (the easier consideration), the derivative of ϵ has to vanish. Of course we require the Lagrangian to be invariant this transformation, but in a way where extra terms from the fermionic component and extra terms from the scalar component cancel exactly. For the algebra to be closed, the commutator of supersymmetry transformations (in spinor parametrization) yields another symmetry. Occasionally an auxiliary field might be necessary to make sure the algebra closes off-shell. These should be familiar as the Einstein strength tensor. As in standard model physics, these get their own gauge transformation. The through Noether's theorem implied current to the symmetry invariance is a anticommuting 4-vector carrying a spinor index. It's called the supercurrent and is usually written as J that lead to conserved charges Q. The approach is standard fare.
As a side note: Supersymmetry is still a form of symmetry, and true to them, is expected to experience spontaneous breaking (meaning the vacuum state is not invariant under supersymmetry). How exactly that is supposed to work is unclear as of time of writing, so the current approach is to dance around this gap of knowledge in a way that doesn't leave too many parameters empty.
The easiest example for the construction of a supersymmetric model might well be the minimal supersymmetric standard model, because a physicist should better be familiar with the standard model, and - well - it's minimal. In true construction fashion, begin with the superpotential.
Refer to standard model naming convention to associate the chiral fields to each of the objects in the term above. As usual in the standard model, the Yukawa coupling parameters are elements of SU(3) with all the suppressed gauge and family indices, that are usually suppressed. With a little parametrization work, and making the matrix product explicit will give an approximation for the superpotential.
From this representation, the construction works as it was previously for the Lagrangians. It does however begin introducing a bunch of somewhat odd particles that aren't part of the standard model. Instead of writing down the Lagrangians here, I'd like to take a look at those - mostly because the word "squarks" looks funny. This follows from the concept of R-parity (or matter parity, depending on the literature). Certain chiral superfields that one might have expected from the standard model are not represented in the MSSM, due to them violating baryon number or total lepton number as a conserved quantity, and where most general (meaning including the most particle superfields) superpotential would solve this problem using gauge terms, the existence of these would just somewhat accept the existence of processes that would violate those symmetries, without there being experimental proof for them. Even worse, if they did, then the maths would dictate the possibility of proton-decay, the process of which would put the lifetime of a proton on an extremely short timescale. Judging by the prolonged existence of the reader and non-instantaneous implosion of hydrogen bottles, this possibility seems highly unlikely. Fixing the conservation of baryon number and total lepton number as an axiom for the MSSM would fix the problem, but since the standard model does not presuppose it, then the MSSM would not posit any improved theory to the existing ones. Besides, non-perturbative electroweak effects violate those symmetries on a negligible, but provable scale. Instead, one could posit the conservation of a new quantum number
This is in essence the same trick that the standard model does to fix its supercharges to a conserved quantity. By design, its discrete symmetry commutes with supersymmetry, since all supermultiplets have the same matter parity. It can then be an exact symmetry, which neither B-symmetry nor L-symmetry aren't due to their aforementioned symmetry breaking in non-perturbative electroweak regimes. The MSSM does not involve interactions violating them, so the matter parity should hold. The term "R-Parity" comes from the occasionally convenient recasting of the quantum number
Beyond supersymmetry, R-parity has its place for phenomenology, since all standard model particles (and the Higgs boson) have even R-parity (+1) and all of the odd extra particles (squarks, sleptons, gauginos, higgsinos) have odd R-parity (-1). The latter ones are considered "supersymmetric particles" or "sparticles" for short. The lightest sparticle (LSP) has to be absolutely stable, for about the same reason that the lightest massive non-sparticle has to be. Assuming it's electrically neutral, then it only interacts weakly with ordinary matter. Cosmology might investigate it for non-baryonic dark matter. Each non LSP sparticle must decay into a state containing an odd number of LSPs (usually one), and in collisions, sparticles can only be produced in even numbers (usually pairs).
Having collected all the symmetries that the model requires, there's of course the question of how the soft supersymmetry breaking is going to look like in the model. Writing the most general set as the "soft Lagrangian"
M3, M2 and M1 are the mass terms of sparticles named "gluino", "wino" and "bino" respectively. a are matrices in one-to-one correspondence of the Yukawa matrices. The mass terms and the a-matrices are proportional to the Lagrangian mass. Many of the terms in the soft Lagrangian for the MSSM are not present in the standard model. It contains 105 masses, phases and mixing angles that can't be constructed or rotated away through quark/lepton supermultiplets without standard model counterparts.
Following the arguments of QFT, the higgsinos and gauginos mix through electroweak symmetry breaking. Of those, the neutral higgsinos (one for up, one for down) and the neutral gauginos (B, W) are classified as neutralinos, while the two charged higgsinos (the positive has up type, and the negative has down type) and the two winos with a charge of ± 1 can mix to form 2 mass eigenstates called charginos. The lightest neutralino is assumed to be the LSP, unless a lighter gravitino exists or R-parity is not conserved. Both options are not commonly part of the MSSM.
By the same logic, any scalar particles with the same electric charge, colour and R-parity should be able to mix. From this, each quark gets a supersymmetric counterpart (left-handed and right-handed), as well as the neutral neutrinos. Mathematically these might as well be represented by 6x6 squark matrices (one for up-type, and one for the down-type, since they don't mix across types), a 6x6 slepton matrix (selectron, smuon, stauon, for both handedness) and a 3x3 sneutrino matrix. There is a hypothesis for flavour-blind soft parameters stating that their mixing angles come out small. The mixing of these particles will have both a Yukawa coupling and soft coupling dependency, which will be large in third-family squarks and sleptons, and negligible in the first and second family squarks and sleptons. Unfortunately the properties of these particles can't be easily derived from their non-supersymmetric counterparts, so they need to be determined using the same procedure applied within the classical standard model.
These musings are technically interesting, but as of time of writing, neither squarks, sleptons, gluinos and their derived states/particles have been discovered, and so the MSSM is really more of a proof of concept.