Supersymmetry and Duality

Use this volume by first identifying the dimension and supersymmetry algebra, then the object—state, field multiplet, vacuum, defect, or protected observable—and finally the strength of claim the evidence can support. Algebraic consequences and cohomological identities are different from controlled semiclassical approximations; matching protected data is different from proving a full duality. The seventeen chapters route those questions from graded symmetry through strong dynamics, lower-dimensional duality webs, localization, and exact correspondences.

Helpful background. The shortest common preparation is spinors and bilinears, ordinary symmetry multiplets, UV and IR fixed points, and gauge redundancy and observables. None is a gate to browsing the volume; the diagnostic below points to the smallest repair.

Choose a route by object and claim

Supersymmetry organizes the subject through a small set of durable relations. Positivity of a supercharge anticommutator produces energy and central-charge bounds; Q-cohomology isolates protected states and operators; holomorphy constrains quantum dynamics; duality compares complete theories only after local and global data are matched; localization computes a specified background observable when its contour and boundary terms are controlled.

Four landmark structures orient the volume:

\{Q,Q^\dagger\}\ge0, \qquad H_Q=\frac{\ker Q}{\operatorname{im}Q}, \qquad M\ge|Z_\gamma|, \qquad \frac{d}{dt}Z[S+tQV]=0.

Each formula has hypotheses. The algebra depends on dimension, signature, and reality; cohomology depends on the chosen Q and boundary behavior; a BPS bound does not guarantee a state exists or is stable; the localization identity requires an invariant measure, a valid deformation, a complete locus, and no omitted boundary contribution.

The volume’s domain map has five connected spines:

Spine	Foundational object	Main operation	Exit capability
Algebra and realization	Supercharges and unitary representations	Shortening, superspace constraints, closure	Construct a multiplet and invariant action
Protected dynamics	Vacua, moduli, central charges, holomorphic data	Quotients, anomalies, decoupling, controlled breaking	Derive a bounded exact result or diagnose its failure
Duality and extended supersymmetry	Complete theory and charge data	Dictionary, deformation, monodromy, S-duality	Test an equivalence claim without dropping global data
Dimension-sensitive theories	2d and 3d multiplets, anomalies, defects	GLSM phases, monopoles, mirror and Chern–Simons dualities	Build and compare lower-dimensional theories
Exact methods and protected interfaces	Rigid supercharge and Q-cohomology	Localization, indices, instanton sums, reduced algebras	Compute and export a protected observable with its limits

The arrows between these spines mean “supplies input to,” not “proves.” In particular, an exact protected observable supplies evidence for a duality but does not turn the equivalence into an algebraic consequence.

Scope and stopping boundaries

This volume develops supersymmetry algebras and representations; supersymmetric quantum mechanics; multiplets, superspace, actions, currents, vacua, BPS sectors, exact constraints, breaking, and strong gauge dynamics; four-dimensional $N=1$ , $N=2$ , and $N=4$ dualities; two- and three-dimensional supersymmetric QFT; rigid backgrounds, localization, indices, instanton counting, protected defects, and exact correspondences.

It uses but does not replace several neighboring domains:

Mathematical Methods supplies spin, Clifford, cohomological, symplectic, determinant, contour, and index foundations.
Symmetry and Gauge Structure supplies general Ward identities, anomalies, generalized symmetries, global forms, gauging, and generic defects.
Renormalization and Effective Field Theory owns general schemes, operator mixing, RG, effective actions, and fixed-point criteria.
Gauge Theories and the Standard Model and Nonperturbative Dynamics own generic phases, confinement, solitons, instantons, and nonsupersymmetric evidence.
Conformal Field Theory and Bootstrap owns conformal blocks, crossing, and bootstrap inference; this volume supplies protected input.
Thermal and Nonequilibrium QFT owns thermal ensembles, and a graded supersymmetric index must not be interpreted as one.
Many-Body QFT and Quantum Matter owns material and platform realizations of descendant duality webs.
QFT in Curved Spacetime owns dynamical and semiclassical gravitational questions; here the supergravity multiplet is a nondynamical source.
Holography and Quantum Gravity owns brane, string, and bulk constructions; they enter here only as explicitly labeled external evidence.
Mathematical QFT owns theorem-first existence and equivalence. A physical construction or protected correspondence is not silently upgraded to a theorem.

Check your preparation

Each item is independent. If one fails, repair that capability and return to the chapter you need.

Spinor translation. Given $\epsilon^{12}=1$ , raise and lower a two-component spinor index and verify one bilinear sign after exchanging two Grassmann spinors. If the sign cannot be reproduced, review spinors, conjugations, and Fierz identities. This unlocks Chapters 1, 3, and every component calculation.

States versus fields. Explain why a covariant gauge field has redundant components while a unitary one-particle representation counts only physical polarizations, and say what an auxiliary field changes. Repair with gauge redundancy and observable content and multiplets and selection rules. This unlocks Chapters 1–4.

Localized symmetry variation. Derive a Ward identity using a spacetime-dependent transformation parameter and identify contact terms at operator insertions. Review localized transformations and Ward–Takahashi identities and contact terms and Schwinger terms. This unlocks supercurrents, anomalies, and localization.

Wilsonian versus 1PI. State which functional integrates out momentum shells and which Legendre-transforms connected correlators; explain why massless infrared effects can make their local terms differ. Repair with the 1PI effective action and renormalization conditions, schemes, and finite parts. This unlocks Chapters 4, 6, and 8.

Complete gauge theory. For one example, give the gauge algebra, global form, matter representations, discrete theta data, and genuine line lattice. If only the algebra is known, review global form and the faithful gauge group. This unlocks strong dynamics, duality, and line defects.

Evidence calibration. Explain why matching ‘t Hooft anomalies is necessary for an infrared duality but cannot determine every spectrum and correlator. Review ‘t Hooft anomaly matching, then enter the volume’s duality evidence framework.

Chapter map

The chapters below appear in their sidebar order, but the order is not a compulsory curriculum.

SUSY Algebras and Unitary Representations. Classifies allowed graded extensions, reality conditions, central charges, unitary multiplets, CPT completion, and shortening. Enter with spinors and Poincaré representations; leave able to distinguish an algebraic bound from the existence of a BPS state. The main blocker is mixing dimensions or signatures without translating reality conditions.
Supersymmetric Quantum Mechanics, Cohomology, and the Witten Index. Turns $H=\{Q,Q^\dagger\}/2$ into a calculable Hilbert complex, with spectral pairing, semiclassics, and continuum caveats. It is the shortest route to understanding Q-cohomology; the common mistake is treating an unregulated index as a count of vacua in a continuous spectrum.
Supermultiplets, Superspace, and Off-Shell Closure. Builds constrained superfields and component multiplets, tracking closure off shell, modulo gauge transformations, or on shell. Leave able to expand and test a multiplet; do not confuse a physical-state multiplet with a covariant field presentation.
Supersymmetric Actions, Supercurrents, and Quantum Effective Theory. Constructs superspace measures and canonical actions, reduces them to components, and connects current multiplets to Wilsonian and 1PI descriptions. Its exit is a sign-checked action and current; its blocker is hiding integration, improvement, or infrared conventions.
Supersymmetric Vacua, Moduli Geometry, and BPS Sectors. Solves F- and D-flatness, quotient geometry, branch singularities, central-charge bounds, defects, and wall crossing. It distinguishes geometric points, low-energy fields, and protected data; the main misconception is that “BPS” means a stable state exists in every chamber.
Holomorphy, Anomalies, and Exact Quantum Constraints. Develops Wilsonian nonrenormalization, spurions, scale matching, Konishi-type relations, and residual ambiguities. Leave able to run an exactness argument with its assumptions; do not transfer a Wilsonian statement unqualified to a massless 1PI functional.
Supersymmetry Breaking and Controlled Deformations. Analyzes F/D order parameters, Goldstini, model laboratories, pseudomoduli, metastability, constrained EFT, and soft limits. It teaches where calculability stops; a small soft parameter alone does not guarantee vacuum continuity.
Four-Dimensional N=1 Gauge Dynamics. Integrates holomorphy, anomalies, instantons, quantum moduli, confinement examples, dynamical breaking, quivers, and compactification in canonical gauge theories. Begin with the SQCD theory card and leave able to classify regimes; never export these controlled mechanisms as an explanation of nonsupersymmetric QCD.
Field-Theory Duality: Dictionaries, Operations, and Global Data. Defines exact versus infrared equivalence and requires operators, parameters, generalized symmetries, lines, global forms, contact terms, flows, and falsifiers. It is the mandatory entry for any named duality; anomaly matching alone is not a complete dictionary.
N=1 Duality, RG Fixed Points, and Protected SCFT Data. Develops Seiberg duality, operator maps, deformations, the conformal window, a-maximization, conformal manifolds, and protected exports. Leave able to test electric and magnetic descriptions through flows; accidental symmetries must be removed before extremization data are trusted.
N=2 Gauge Dynamics and Seiberg–Witten Geometry. Builds the Abelian effective theory, special Kähler geometry, charge local systems, periods, monodromies, BPS chambers, Argyres–Douglas points, and controlled $N=1$ deformations. The pure $SU(2)$ reconstruction is the benchmark; charge-vector and monodromy conventions must remain fixed.
N=4 SYM, S-Duality, and Higher-Dimensional Interfaces. Specifies the complete $N=4$ theory, its protected sectors, global lines, S-duality groupoid, and bounded 5d/6d interfaces. It separates perturbative finiteness and protected evidence from a complete nonperturbative equivalence; the Lie algebra alone is not the theory acted on by duality.
Two-Dimensional Supersymmetric QFT, GLSMs, and Mirror Symmetry. Develops $(p,q)$ algebras, Landau–Ginzburg and sigma models, GLSM phases, twists, mirror dictionaries, elliptic genera, c-extremization, and tt* geometry. Enter through dimension-specific chirality; do not import four-dimensional R-symmetry or anomaly formulas unchanged.
Three-Dimensional Supersymmetric Gauge Theory and Duality Webs. Builds Yang–Mills and Chern–Simons–matter theories, parity/contact terms, monopole operators, mirror and Seiberg-like dualities, and deformation webs. Leave with a contact-term-complete dictionary; ignoring spin structure or half-integer level shifts invalidates it.
Rigid Backgrounds, Topological Twists, and Localization. Establishes background compatibility, generalized Killing spinors, twists, Q-exact deformation, gauge complexes, loci, determinants, contours, boundaries, and residues. It owns the validity chain, not the final matrix-model interpretation; a formal $QV$ term is not enough.
Exact Partition Functions, Indices, and Instanton Counting. Computes counterterm-aware sphere integrals, F-maximization, indices, elliptic genera, instanton sums, factorized blocks, and exact duality tests. Leave able to reproduce one exact observable and state what it forgets; an index is neither a thermal partition function nor a full spectrum.
Protected Operators, BPS Defects, and Exact Correspondences. Resolves Q-cohomology and mixing, quantum rings, protected algebras, BPS defects, geometric Langlands, AGT, and versioned exports. It closes the volume by making information loss explicit; a correspondence between protected objects is not equality of their ambient theories.

Recurring examples across the volume

Seven examples expose how the layers fit without forcing every chapter into one sequence.

Graded algebra to an off-shell action. Start with the 4d $N=1$ algebra, pass through chiral, vector, and linear superfields and D/F-term measures, and finish at supercurrents. Closure class and auxiliary counting remain the invariant checks.
Vacua, exact constraints, and breaking. Follow F/D-flatness through nonperturbative superpotentials to Goldstino identification. The thread ends when strong-coupling vacuum existence is not controlled.
SQCD from ultraviolet fields to an infrared dual. Begin with the SQCD theory card, add quantum moduli, then use the Seiberg dual pair and deformation flows. Anomalies, scales, and operator maps must agree at every step.
Coulomb geometry to BPS spectra. Build the Abelian low-energy theory, special geometry, periods, monodromies, and BPS chambers. The charge lattice and base-point convention are held fixed.
Dimension-sensitive duality webs. Choose the 2d algebra and GLSM/mirror route, or the 3d algebra and monopole route. Anomalies and contact terms determine which apparent maps survive.
Rigid supercharge to exact observable. Move from background compatibility through Q-deformation, loci and determinants, and contours to a sphere matrix model.
Protected data with preserved scope. Define the cohomology and mixing problem, compute a protected algebra, and finish with a versioned export. The consumer receives the exact sector, normalization, uncertainty, and exclusions—not a stronger parent-theory claim.

Convention and evidence card

The volume inherits the site’s $(+---)$ Lorentzian metric, natural units, Hermitian gauge generators, and Fourier convention. Euclidean calculations state their continuation, integration cycle, and reality condition locally.

Domain	Stable baseline	Local declaration and invariant check
Supersymmetry count	Number of real supercharges is primary	State dimension, signature, and reality; check real-component count
4d spinors	Two-component or four-component notation must be translated	Check the supercharge anticommutator and a real kinetic term
Multiplets	Separate physical states, covariant fields, gauge redundancy, and auxiliaries	Check degrees of freedom and closure class
Superspace	Q/D signs and Grassmann measures are page-local	Reproduce one component transformation and action term
Effective actions	Wilsonian and 1PI objects are distinct	Match the declared infrared and scheme limit
Gauge and duality data	Global form and genuine lines accompany the algebra	Check Dirac pairing, anomalies, and screening
BPS and Seiberg–Witten data	Charge vector, symplectic basis, branch cuts, and monodromy order stay fixed	Check $Z_\gamma$ , pairing invariance, and monodromy product
2d and 3d theories	R-symmetry, chirality, spin structure, and contact terms are dimension specific	Check anomaly polynomial or large-gauge invariance
Localization and indices	Background, Q², contour, regulator, counterterms, and spin structure are part of the observable	Check a free determinant and deformation independence
Omega background and AGT	$\epsilon_{1,2}$ and physical/equivariant masses are distinguished	Match the one-instanton or first conformal-block coefficient

Claim language follows the same discipline. “Algebraic” requires explicit algebra and positivity; “protected” names the supercharge, mixing, and recombination; “BPS” separates bound, existence, chamber, and stability; “localization” includes the contour and determinant; “duality evidence” states independence and scope; “theorem” names its hypotheses and proof source.

Interfaces and exits

Use Start Here for formal sequences, diagnostics, exercises, and capstones; the reading paths here are only subject routes. Executable index, localization, duality, and wall-crossing calculations should use frozen environments and publish their validation checks. Use Research for dated assessments of non-Lagrangian theories, mutable BPS spectra, contour questions, and correspondence status. Use Reference for formula and symbol lookup that returns to the relevant explanatory page.

After a chosen route, you should be able to:

specify a supersymmetry algebra and construct its unitary or field multiplets;
build an invariant action and diagnose its closure and current multiplet;
solve controlled vacuum, BPS, holomorphic, or breaking problems;
state and test a duality with local, global, defect, deformation, and evidence data intact;
reconstruct a Seiberg–Witten solution or navigate $N=4$ , 2d, and 3d duality structures;
decide whether a rigid background and localization calculation is valid;
compute an index, partition function, or instanton sum and state what it cannot determine;
export a protected result without strengthening its claim.

There is no universal next volume. Exit to Conformal Field Theory and Bootstrap for crossing, Nonperturbative Dynamics for generic strong-coupling mechanisms, Many-Body QFT and Quantum Matter for physical duality-web applications, Holography and Quantum Gravity for string or bulk constructions, or Mathematical QFT for theorem-first formulations.