Four-Dimensional N=4 Super Yang-Mills

Adding supersymmetry to a gauge theory can constrain its interactions and make selected observables accessible beyond ordinary weak-coupling perturbation theory. Four-dimensional N=4 Yang–Mills is a central example: sixteen real Poincaré supercharges relate a gauge field, six real scalars and four Weyl fermions, all in the adjoint representation.

The central question of this volume is how a specified interacting theory leads to calculable physical quantities. The route begins with the action, then develops local operators and an exact circular Wilson loop. “Exact” always refers to a particular quantity with stated conventions; it does not mean that every correlation function is fixed by symmetry.

Begin with the actual theory

Read What is N=4 super Yang–Mills? after N=2 vector multiplets and hypermultiplets. The four-dimensional foundations supply the needed spinors, superspace, gauge representations and multiplet counting. Neither the three-dimensional lessons nor a curved-space construction is an entrance requirement.

The first calculation gives eight bosonic and eight fermionic physical polarizations per gauge generator. Then organize the same fields into one N=1 vector and three adjoint chirals. Only four supercharges are manifest in that notation, while the specified action has sixteen.

The default interacting local theory here has gauge group $SU(N)$. A $U(N)$ matrix integral contains an additional central sector. Keep that distinction explicit when comparing formulas, even when a normalized trace hides part of the difference.

From fields to local observables

First, build the action from three adjoint chiral multiplets. Differentiate its ordered cubic interaction, eliminate the auxiliaries, and recover the potential for six real scalars with matching scalar and fermion masses in a simple background. Knowing those interactions then raises questions about their symmetries, the choice of vacuum, and the difference between classical engineering dimensions and quantum scaling dimensions.

Symmetries and different multiplet descriptions packages the fields into SU(4) representations and checks the fermion interactions. Follow a rotation that mixes two descriptions, then compare that change of notation with an actual hypermultiplet mass.

Couplings and conformal invariance then checks the one-loop gauge cancellation. Vary the cubic coupling independently to see why counting fields alone does not establish a conformal theory, and identify the extra supersymmetry argument.

Commuting matrices and the Coulomb branch then solves the vacuum conditions, identifies joint eigenvalues and the residual gauge quotient, and calculates the masses of fields that become heavy away from a collision.

Operators, color and mixing

Protected operators and unprotected operators compares a traceless scalar bilinear with the Konishi singlet. Derive a positive descendant norm, identify the additional representation-theory input that protects a dimension, and see why a generic sum is not one scaling eigenoperator.

Next, compute a first gauge-invariant correlator with both Wick pairings and every finite-color subtraction visible. Normalize the operator, then remove the Abelian center from a U(N) calculation to recover the SU(N) answer. Operators with the same quantum numbers can mix; a matrix of their anomalous dimensions contains more information than a single correction. Large N and what a ‘t Hooft limit holds fixed prepares another useful organization: specify the coupling, observable normalization and neglected corrections before taking a color limit.

These questions lead to the spectral branch: translating a specified operator-mixing calculation into a spin-chain problem. Its spin labels are not the gauge-group rank.

Double lines and color counting turns each matrix propagator into a ribbon and derives the color power of a connected graph. Compare an exact cubic trace in the U(N) bookkeeping model with the actual SU(N) result, then test when a planar approximation is accurate.

Why composite operators mix adds the interaction logarithms. Renormalize a two-operator basis, include its two-point inner product, and find the combinations with definite scaling dimensions.

A first cyclic spin chain then builds the nearest-neighbor operator on two holomorphic scalars. Project onto trace-cyclic states, normalize words with different stabilizers, and distinguish a vanishing one-loop eigenvalue from the additional argument for protection.

From a loop to a finite integral

Supersymmetric Wilson lines and circles defines the scalar coupling, derives the preserved line and circle spinors, and computes the first correction with finite-color normalization. The exact-observable branch asks which supersymmetry the loop preserves, how the theory is placed on a four-sphere, and how localization reduces that observable to a finite matrix integral.

Put N=4 SYM on the four-sphere constructs the actual spinor and its bosonic square, then derives the curvature coupling and tests it with a scalar propagator. Then find the localization locus and its measure: keep the auxiliary shift, use the Bianchi identity, and distinguish the surviving Cartan coordinate from normal fluctuations and polar sectors. Why the matrix model becomes Gaussian evaluates those factors and fixes the shifted mass in the instanton formula. Its finite-color evaluation and its planar limit then answer different questions.

Evaluate the circular Wilson loop at finite N carries the Gaussian sphere integral to a finite polynomial answer. Derive the eigenvalue marginal, retain its normalization, and remove the central Gaussian to obtain the physical SU(N) result at the same coupling.

The planar limit and strong coupling then derives the normalized semicircle and its Bessel-function loop. Resolve the strong-coupling endpoint, calculate the first finite-rank correction, and test which approximation controls a chosen observable.

From a circular loop to a defect observable turns the normalized coupling response into a displacement correlation. Keep the Euclidean cusp sign, the finite-color central subtraction and the slow-probe radiation limit distinct.

Use the Pestun paper companion to read the four-sphere construction in its original conventions. Translate the physical mass, restore a potentially singular local limit, and change the loop representation without changing its measure.

Duality, global data and gravity

Electric–magnetic duality from familiar gauge theory offers another entrance from the complex coupling and compact connections. Quantize Maxwell fluxes in a box, exchange their integer labels with the coupling, and test the energy. This exact free-field calculation explains which charge units and global data a non-Abelian duality statement must retain; it does not prove that statement.

Global gauge groups and line operators follows that duality question beyond the local gauge algebra. Derive the three SU(2) and SO(3) line spectra, then use an explicit gauge bundle to see why a theta shift can change the global theory.

A first AdS5 times S5 dictionary derives the gauge/string coupling, common radius and Newton coefficient. Distinguish the limits controlling curvature and loops, then test a vacuum split whose smaller throat loses the original approximation.

Check the theory behind the formula

Your first checkpoint: for gauge group SU(2), recover the count $24/24$, explain why the gauge rank is one rather than three, and identify the cubic interaction that distinguishes N=4 Yang–Mills from arbitrary adjoint matter. Explain also why an auxiliary field does not add a particle polarization.

Use the learning paths for the guided N=4 entrance. Return to the first checkpoint whenever a formula leaves unclear which fields, gauge group, vacuum or observable it describes.