Learn supersymmetric field theory

Choose a destination by the calculation you want to understand. The volumes organize subjects; a learning path chooses the preparation needed for one result. You do not have to read every earlier volume before entering a later theory.

Open the learning-path browser to see each route’s first lessons, core sequence, and optional extensions. Why study these theories? compares the scientific questions behind the destinations.

The common beginning

The opening sequence is What a supercharge does followed by Grassmann variables and their calculus. It introduces the meaning of a fermionic symmetry generator and the calculus needed to manipulate its parameters and fields. The foundations overview explains the questions that follow.

If you need a refresher in ordinary field theory, start with four diagnostic calculations. The course teaches additional geometry, complex curves, conformal methods, and matrix methods when a calculation first needs them.

Curved spaces and three-dimensional theories

The first curved-space route constructs a free chiral theory on a round three-sphere. Its twenty lessons connect the supercharge algebra, the chiral multiplet, covariant derivatives, Killing spinors, and the complete curved action. Quantum mechanics and the full geometry of general three-manifolds are optional branches for that route.

After Grassmann calculus, continue with Spinors in three dimensions and Euclidean continuation and independent spinors. They establish the matrix and conjugation conventions used in the three-dimensional supersymmetry algebra and the chiral multiplet. You can then compose transformations on a scalar and apply the complete component rules to an explicit field configuration.

Continue through off-shell closure and the free chiral action to check the fermion algebra and all kinetic-term cancellations. For the geometric entrance, the tools overview starts from a metric and a rotating frame without requiring supersymmetry.

The rigid-background overview brings those ingredients together. It starts with the explicit failure of a naive covariantization, identifies the sources that repair the construction, and derives the equations for preserved supersymmetry parameters.

The curved-action overview connects the solved sphere background to its matter theory. At the route’s endpoint, you can verify both independent supersymmetry variations off shell and explain why the curvature terms disappear in the fixed-theory flat limit. The background-field reference provides a compact source and connection dictionary.

Later three-dimensional routes study SQED and its infrared dual description, protected traces, and ABJM theory. The ABJM entrance specifies the two compact gauge groups, opposite Chern–Simons levels and four matrix-valued chiral multiplets. These routes answer different questions: defining a theory, comparing its operators and vacua, or calculating a particular observable. Knowing a partition function does not by itself specify all of a theory’s correlation functions.

Four-dimensional entrances

After the shared opening, the four-dimensional entrance starts with its own Weyl spinors and supersymmetry algebra. The multiplet comparison shows how the fields organize under one or more copies of that algebra. From this foundation, choose quantum dynamics with one supersymmetry, Seiberg–Witten theory with extended supersymmetry, or N=4 super Yang–Mills. None of these entrances requires the three-dimensional spinor or curved-space course.

Their first substantial checkpoints differ. Four-dimensional N=1 quantum dynamics begins by removing a heavy field and deriving its effective interactions, then develops the constraints and anomalies used to compare two descriptions. Seiberg–Witten theory begins by calculating a charged-vector mass and identifying the low-energy question; normalized periods will encode its effective couplings. N=4 super Yang–Mills begins with its actual fields and interactions, then asks how an ordinary operator calculation differs from a protected circular Wilson loop. The mathematics needed for each checkpoint belongs to its route. The spinor and curvature reference keeps the dimension-specific convention packages together when you move between these subjects.

From 4d N=2 to 3d N=4 follows a periodic spatial compactification. Derive both angular coordinates from the global gauge group and test the range in which the leading metric is useful.

Exact observables through localization

Localization and protected observables begins with a finite integral that needs only Grassmann calculus. Derive why its value stays fixed under a supersymmetric deformation, check the boundary term, and compute the answer from a Gaussian limit. This common method has its own entrance before any three- or four-dimensional field-theory specialization.

Localization in a field theory transfers the argument to a regulated path integral. Its general entrance keeps the measure, contour and boundary assumptions explicit; a separate sphere example constructs the deformation in components.

After completing both sphere calculations, What localization shares across three and four dimensions compares their selected charges, backgrounds, loci and measures. A coupling-response calculation distinguishes the common symmetry identity from the physical information that remains.

How to use a lesson

For reading a primary source, use the paper companions. Each guide fixes the source version, connects its notation to the teaching calculations, and asks you to reconstruct a step before changing one of its assumptions.

Read the visible derivation before opening the solutions. State its assumptions aloud: dimension, signature, fields, and the quantity being calculated. Then change one assumption and predict which step would fail. For the opening sequence, a useful exit task is to calculate a two-state supercharge action and a two-variable Grassmann integral without confusing a state vector, a field, and a formal parameter.

The laboratories pair analytic lessons with editable calculations and expected output. Begin with gamma matrices and bilinears: predict the effect of changing orientation or entry statistics, then compare the executable result with a hand calculation.

From a checked result to a new question

From reproduction to a research question begins after any one of six completed capstones. Work through a known period integral, certify a truncation error and distinguish a controlled training extension from a novelty claim.

Compare exact, planar, and gravitational predictions explains what agreement can establish when two calculations use different approximations. Its rank diagnostics separate numerical uncertainty from omitted physical terms.