Use symmetry in a theory.
Connect group-theoretic statements to matter, interactions, fields, topology, and geometry. Each guide leads to the relevant records and supported workbenches.
Choose the group, fields, and vacuum.
Physical setup libraryGroup theory for gauge theories
An entrance to gauge groups, matter, charges, and invariant interactions.
Read the guide Gauge and matterMatter representations and charge conventions
Keep conjugation, chirality, normalization, and global-group data in field multiplets.
Read the guide Gauge and matterSymmetry breaking, stabilizers, and vacuum orbits
Find the unbroken subgroup from an order parameter and its actual action.
Read the guideSpacetime and state representations.
Follow the learning pathSpinors, covers, and reality conditions
Read spin representations with dimension and signature visible.
Read the guide Fields and statesConformal symmetry and representation categories
A bounded guide to conformal groups, local operators, and state representations.
Read the guide Fields and statesAdS isometries and holographic symmetry
Identify real forms, covers, and representation labels in AdS/CFT.
Read the guideFinite internal symmetries.
Follow the learning pathFrom an A₄ triplet to an invariant potential
Construct the quadratic, cubic, and quartic interactions, impose identical-field statistics, and distinguish a candidate vacuum stabilizer from a dynamical prediction.
Read the guideFrom fields to interactions.
Enumerate allowed interactionsSpecify the complete field representation, including Lorentz spin and flavor. Then impose field statistics and project onto gauge and Lorentz singlets at a fixed monomial.
Standard Model operator sectors
Read the Higgs, Yukawa, Weinberg, and four-fermion sectors with explicit conjugates and baryon/lepton selection rules.
Read the guide Neutrino massesThe Weinberg operator
Derive the unique dimension-five field structure and its symmetric flavor coefficients.
Read the guide Model buildingYukawa and four-fermion interactions
Track chirality, gauge charges, flavor, and index identities in concrete Standard Model sectors.
Read the guideFrom unified multiplets to a global gauge group.
Open a workbenchThe Lie algebra determines infinitesimal transformations. The global group also constrains which representations and bundles are allowed. Follow the SU(5) block map, then test the embedded central quotient against the matter.
From SU(5) to Standard Model multiplets
Follow an explicit homomorphism, keep hypercharge, and identify the quotient subgroup.
Read the guide Global structureWhich Standard Model gauge group?
Check the diagonal center against matter and distinguish the four compatible global forms.
Read the guideMore ways to use symmetry.
Learning pathsWhich global forms admit the matter?
Derive the common-kernel criterion and use it in worked SU(4) and SU(6) examples.
Read the guide Gauge theoryThe global group matters
From local Lie algebra data to allowed matter and line operators.
Read the guide Model buildingA singlet is only the beginning
How field species and statistics change the number of interactions.
Read the guide Symmetry breakingChoose the map before branching
Two su(2) embeddings in su(3) give different physical multiplets.
Read the guide Geometry & topologyFrom symmetry breaking to defects
Keep the vacuum orbit, global group, and homotopy maps together.
Read the guide Fields & particlesWhich representation describes the object?
Lorentz fields, Poincaré particles, and Euclidean spinors.
Read the guide Finite groups & geometryFrom a finite group to an ADE diagram
The quaternion group connects SU(2) tensor products to affine D₄.
Read the guide