Theories and running models

A theory becomes easier to recognize when the same fields return in several calculations. A chiral multiplet can first illustrate a supersymmetry transformation, then an interacting potential, then a sphere observable. A gauge theory can lead from vacuum equations to monopoles and a comparison with a different description of the same infrared physics.

Choose a model below and follow its question through the linked lessons. Each entrance states its own preparation. The four-dimensional routes begin with four-dimensional supersymmetry; they do not require completing the three-manifold route first. If you want a complete ordered course instead, use the learning paths.

A supersymmetric oscillator: paired states and a ground state

Solve the supersymmetric oscillator after the elementary supercharge lesson. This one-dimensional system has two real supercharges and no gauge group. Factor its Hamiltonian, normalize the paired excited states, and find the single square-integrable zero mode. Reversing the oscillator superpotential exchanges its grading; removing confinement changes whether a ground state exists. These are explicit Hilbert-space questions that require no field-theory spinor preparation.

A chiral multiplet: from a symmetry to an observable

Begin with the three-dimensional chiral multiplet and its free action. The theory has four real Poincaré supercharges, conventionally called 3d N=2. The auxiliary field makes the transformations close without using the equations of motion; it does not add a propagating particle.

Then add a Wess–Zumino interaction. The same holomorphic function determines both a Yukawa coupling and the scalar potential after eliminating the auxiliary field. Your checkpoint is to locate the classical vacua of the stated quadratic-plus-cubic example and compare the scalar and fermion masses about one supersymmetric vacuum. Keep that interacting model distinct from the free chiral used in the next calculation.

For an observable, calculate the free chiral determinant on its specified sphere background. Pair the modes, retain the unpaired factors and state how their infinite product is regulated. The background and the regulator are part of the question. This sequence connects a field transformation, an action and a quantum quantity without changing models silently.

One-flavor SQED and XYZ: follow the vacua through a duality

Build three-dimensional Abelian gauge theory with gauge group U(1), one charge-positive and one charge-negative chiral, zero gauge Chern–Simons level and initially vanishing masses, FI parameter and superpotential. Its four real supercharges are all visible in the N=2 description. “One flavor” here means that pair of charged chirals.

The Higgs and Coulomb branches give different local coordinates on the vacuum space. Monopole operators supply the flux-defined operators needed alongside the ordinary meson. The SQED–XYZ dictionary compares them with three neutral chiral fields and a cubic superpotential.

A concrete checkpoint is to add the complex meson mass, translate it to the XYZ superpotential, and derive the changed vacuum equations in the deformation lesson. A different calculation constructs the one-flavor sphere matrix integral, retaining its real-mass and FI sources. Defining that integral alone is not a duality test. The vacuum dictionary compares the specified infrared descriptions; it does not identify their elementary ultraviolet fields one by one.

Four-dimensional SQCD: keep color and flavor counts separate

The N=1 quantum-dynamics entrance uses four-dimensional theories with four real Poincaré supercharges. It first separates a holomorphic superpotential calculation from the information that requires kinetic terms. That distinction remains useful in the interacting gauge examples.

For a small model, study SU(2) with one flavor and its dynamically generated superpotential. Adding a flavor mass changes the vacuum question in a way you can calculate. To see a different regime, use the SU(3), five-flavor duality dictionary: the magnetic gauge group is SU(2), and its gauge-singlet mesons are part of its field content.

The anomaly and mass-deformation checks ask whether those two descriptions can have the same infrared physics. Match a flavor anomaly with its multiplicities visible, then follow what an electric flavor mass does to the magnetic vacuum. The conformal-window lesson supplies the setting for interpreting the superconformal meson dimension. The color rank, number of flavors and supersymmetry label answer three different questions.

Pure N=2 SU(2): a vacuum-dependent effective coupling

Start with what the Seiberg–Witten solution solves. The theory is four dimensional, has eight real Poincaré supercharges and no matter hypermultiplet. Its generic Coulomb vacuum leaves an Abelian low-energy gauge field. A chosen N=1 notation makes four of those supercharges manifest.

The prepotential relates the scalar and gauge kinetic couplings. The curve and its normalized differential then encode periods that can be calculated in the period lesson. Your checkpoint is to obtain the effective coupling as the ratio of the two period derivatives, with the period ordering and charge basis fixed.

BPS masses and stability explains why a central-charge formula is not a list of all stable particles. The further N=1 deformation changes the theory and produces a calculable monopole-condensation problem in its stated small-deformation regime. Keep that deformation distinct from simply changing the Coulomb vacuum of the original N=2 theory.

N=4 Yang–Mills: local operators and line probes

Define N=4 super Yang–Mills in four dimensions with the specified SU(N) gauge group and adjoint fields. Its sixteen real Poincaré supercharges can be organized in a notation displaying four or eight. The defining interactions matter as much as the count of fields.

After studying the commuting-scalar vacua, return to the conformal origin for a first free-field gauge-invariant correlator. An appropriate checkpoint is to retain the finite-color subtraction that distinguishes SU(N) from the U(N) bookkeeping calculation. The short cyclic spin chain develops a different observable, with its planar and one-loop assumptions explicit.

For an extended probe, construct the supersymmetric Wilson loop and then place the theory on the four-sphere. Its curved background and selected charge are additional data. Global gauge groups and line operators asks which lines exist before choosing their scalar couplings or preserved supercharges. A local action and a numerical coupling alone do not specify that global information.

ABJM: matter interactions with compact Chern–Simons gauge fields

Define ABJM in three dimensions with equal-rank U(N) gauge factors at levels k and minus k. The generic theory has twelve real Poincaré supercharges, with four manifest in N=2 notation; the standard levels one and two admit sixteen. Each gauge factor has rank N, so the product has rank 2N. Neither rank is a supersymmetry count. Its pure Chern–Simons vector sectors do not supply Yang–Mills photon polarizations.

Build the quartic superpotential and eliminate the auxiliary fields to recover the sextic scalar potential. The extended-symmetry lesson relates the specified interactions to the larger symmetry of the model.

At rank one, derive the residual discrete quotient rather than treating a locally decoupled connection as globally absent. The checkpoint is to explain the residual order-k identification from the compact first-order action. The many-brane and global-data lesson then distinguishes a controlled diagonal vacuum sector from a complete classification and shows how changing the compact group changes allowed fluxes. That global information is needed before constructing a monopole operator or comparing partition functions.

Prepare a field-to-gravity comparison

A first AdS/CFT dictionary from QFT and GR solves a scalar boundary problem before specializing to either supersymmetric model. Distinguish its prescribed source from its interior-selected response, subtract a local divergence and recover a conformal two-point function. The exercise also separates the classical, low-curvature and weak-probe approximations.

Carry the model specification into the next calculation

Before moving to an observable, write down the spacetime dimension, gauge group, matter content, couplings and chosen vacuum or background. State both the full supersymmetry of the theory and the symmetry used by the calculation. Then name the quantity being computed: a vacuum equation, correlation function, index, period or line expectation value.

This short specification lets you change one assumption deliberately. A mass deformation, a new global quotient and a different preserved supercharge are distinct changes, even when part of the notation stays the same. Return to the relevant model entrance whenever a formula leaves one of those inputs unclear.