Supersymmetry from First Principles

A symmetry becomes useful when you can apply it to an explicit object. This volume starts with supercharges acting on fields and states, then develops the algebra and component fields needed to construct a three-dimensional supersymmetric theory.

Start with two shared ideas

Read What a supercharge does to distinguish an odd charge operator from an infinitesimal transformation. Its simple state pair lets you check the algebra by matrix multiplication.

Then read Grassmann variables and their calculus. Anticommuting parameters allow a fermionic generator to produce a field variation with the correct parity. The lesson fixes the order of differentiation and integration before using either.

These two lessons are also the opening for four-dimensional readers. The subsequent three-dimensional calculations are a separate branch.

For a quantum-mechanical example directly after the first lesson, solve a supersymmetric oscillator. Construct its two partner Hamiltonians, pair the positive-energy states, and test which formal zero-energy wavefunction actually belongs to the Hilbert space.

Then calculate what the Witten index can tell us. Compare a signed count with the total number of ground states, and follow a normalized state as it escapes to infinity. The examples show why a zero index and a nonexistent thermal trace require different conclusions.

For the three-dimensional field-theory branch, continue with Spinors in three dimensions and Euclidean continuation and independent spinors. You will verify the explicit gamma matrices, distinguish index raising from conjugation, and carry a fermion kinetic term between signatures with every factor of $i$ visible.

From algebra to a model

The next questions are concrete. Which spinor components describe a charge in three dimensions? Which fields must transform together? Why does closing the algebra before imposing equations of motion require an auxiliary field? How do the kinetic terms cancel under a supersymmetry variation?

The chiral multiplet supplies a running example for these questions. Interactions and a gauge multiplet then lead to a simple Abelian gauge theory, whose classical vacua and quantum operators support the later SQED–XYZ comparison. Optional quantum-mechanics lessons offer another way to study paired states and zero-energy states.

Begin this calculation with The three-dimensional N=2 algebra, which composes two transformations on a scalar and identifies the translation and central term. Then use The chiral multiplet to transform a scalar, fermion, and auxiliary field with their dimensions and charges checked.

Why auxiliary fields make the algebra close compares the two transformation orders on the fermion and isolates the term that would otherwise require its equation of motion. Build the free chiral theory then determines the relative kinetic coefficients and checks that the complete action varies only by boundary terms. These are separate tests: composing transformations and varying an action answer different questions.

R-symmetry and flavor symmetry separates charge assignments from the conditions on an interaction. Superpotentials and a first interacting model then derives the Yukawa terms and scalar potential of a quadratic-plus-cubic model. Find its two classical supersymmetric vacua, compare their boson and fermion masses, and distinguish them from a stationary saddle.

The vector multiplet adds the gauge sector. Count its modes, recover the gauge transformation in its algebra, and compare Yang–Mills and Chern–Simons dynamics. Its charged-matter check shows why introducing a covariant derivative also requires revisiting the superpotential.

Superspace as a way to organize known fields recovers the kinetic and superpotential terms by multiplying finite Grassmann polynomials. Fix the integration measure, derive the component signs, and distinguish a chiral package from a linear conserved-current package before using them to construct a gauge theory.

Build a simple supersymmetric Abelian gauge theory then fixes one compact U(1) model with a pair of oppositely charged chirals. Expand every component interaction, eliminate the auxiliary fields, and test its classical vacuum equations. A Higgs mass calculation and the gauge parity-anomaly condition distinguish checks of the local action from quantum consistency.

Higgs and Coulomb branches identifies the gauge-equivalent vacuum configurations and derives their low-energy coordinates. Compute the Higgs quotient metric, replace the massless photon by a periodic scalar, and test its normalization by matching magnetic flux energy to the energy of a quantum rotor. The calculation also explains why these classical descriptions cannot decide the quantum branch intersection.

Monopole operators turn magnetic flux into a local operator boundary condition. Count the charged angular fermion modes and derive the axial and trial R-charges of the two elementary flux insertions. The calculation separates those charges from an exact interacting scaling dimension.

Build the SQED–XYZ duality dictionary matches the meson and two monopoles to a cubic chiral theory. Derive its three branches and chiral-ring relations, determine the symmetric infrared R-charge, and distinguish a quantum equivalence from an equality of ultraviolet fields or metrics.

Follow a deformation through the duality adds a complex mass and an FI parameter. Solve the resulting vacuum equations, normalize the topological source, and calculate how the FI parameter smooths the classical Higgs-branch metric.

Your first checkpoint

After the opening pair of lessons, explain why an odd symmetry generator does not turn an ordinary commuting scalar variation into an anticommuting scalar. Show the role of its odd parameter, and verify one sign by writing the product in both orders. Then perform a two-variable Grassmann integral with its measure order stated. These are the basic operations on which the component calculations will rely.