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Each route starts with its full preparation. Open a calculation checkpoint to see the reading steps that lead to it. If you already know a step, use its exercises to check your understanding before continuing.

First calculation on a curved spaceBuild and verify a free chiral theory on the round three-sphere.20 articles · 4 checkpoints

Begin: What a supercharge does

1. Make the algebra act on a fieldReading steps 1–5

Use Grassmann variables and three-dimensional spinors to identify the translation in two successive supersymmetry transformations.

2. Build your first supersymmetric theoryReading steps 6–9

Check off-shell closure and the free chiral action, then assign its R-charges.

3. Find the condition for a preserved superchargeReading steps 10–17

Compute the sphere geometry and derive the background Killing spinor equation.

4. Complete the round-sphere actionReading steps 18–20

Solve the Killing spinor equation and check the free chiral action and its flat limit.

Reconstruct the three-manifold paperReconstruct the three-manifold paper’s backgrounds and actions, then reproduce its free-chiral response.62 articles · 6 checkpoints

Begin: What a supercharge does

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Complete the round-sphere actionReading steps 10–20

Solve the Killing spinor equation and check the free chiral action and its flat limit.

3. Test a proposed curved backgroundReading steps 21–38

Use current sources, curved closure, and spinor integrability to check a background locally.

4. Build and patch three-manifold backgroundsReading steps 39–48

Construct geometry from spinors, check global patching, and recover the round limit of a squashed sphere.

5. Turn a field theory into a matrix integralReading steps 49–57

Construct curved interactions and combine localized determinants with the gauge measure.

Build invariant F-terms and D-terms
The curved vector multiplet
Gauge and matter actions from multiplets
Real masses and flavor backgrounds
Chern-Simons and Fayet-Iliopoulos terms · Flat space entry; Curved completion
A finite-dimensional localization example
Localization in a field theory · General localization; Three dimensional field example
The free chiral determinant
A first gauge-theory matrix integral

6. Recover the free-chiral current responseReading steps 58–62

Differentiate the sphere free energy in the stated background and reproduce the current coefficient with a numerical check.

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Four-dimensional appendices23 articles

Follow the four-dimensional background construction and its twisted reduction to three dimensions.

1. Construct a four-dimensional interacting modelReading steps 1–6

Expand superspace interactions and eliminate auxiliary fields to obtain the scalar potential.

2. Find preserved four-dimensional spinorsReading steps 7–11

Use the fixed new minimal background and its source conventions to derive the spinor condition.

3. Identify the three-dimensional source fieldsReading steps 12–21

Check the flat chiral theory and distinguish the background fields entering matter and current couplings.

4. Track the twisted reductionReading steps 22–23

Follow a background-field combination through the paper’s appendices and compare the resulting three-dimensional sources.

Learn 4d N=4 SYM and its exact circular loopConstruct N=4 Yang–Mills, calculate a scalar correlator, and evaluate its exact finite-rank circular Wilson loop.31 articles · 5 checkpoints

Begin: What a supercharge does

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct N=4 Yang–MillsReading steps 9–13

Derive its scalar potential, organize the symmetries, and check the one-loop beta cancellation.

3. Calculate a gauge-invariant correlatorReading steps 14–18

Distinguish protected from unprotected operators and perform explicit Wick and color contractions.

Commuting matrices and the Coulomb branch
Local operators, radial quantization, and the OPE
What BPS and protected mean · Four dimensional particle bounds
Protected operators and unprotected operators
Compute a first gauge-invariant correlator

4. Prepare the loop for localizationReading steps 19–26

Define the supersymmetric loop and check the four-sphere background and curvature couplings.

Supersymmetric Wilson lines and circles
Frames and spin connections
Covariant derivatives of spinors · Generic spinor derivative
Sources, currents, and generating functionals
Stress tensors and improvement terms
A finite-dimensional localization example
Localization in a field theory · General localization
Put N=4 SYM on the four-sphere

5. Evaluate the finite-rank circular loopReading steps 27–31

Derive the Gaussian matrix result and check the distinct SU(N) and U(N) normalizations.

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Planar and strong-coupling loop35 articles

Take the planar limit of the circular loop and interpret its strong-coupling asymptotics.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct N=4 Yang–MillsReading steps 9–13

Derive its scalar potential, organize the symmetries, and check the one-loop beta cancellation.

3. Prepare the four-sphere loopReading steps 14–23

Fix the loop’s trace and preserved symmetry, then construct its curved-space setting.

4. Evaluate the finite-rank circular loopReading steps 24–28

Derive the Gaussian matrix result and check the distinct SU(N) and U(N) normalizations.

5. Take controlled large-rank limitsReading steps 29–35

Use color counting and the eigenvalue density to derive the planar loop and its weak- and strong-coupling limits.

Operator mixing and spin chain22 articles

Build a small scalar-operator mixing problem and solve its first cyclic spin chain.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct N=4 Yang–MillsReading steps 9–13

Derive its scalar potential, organize the symmetries, and check the one-loop beta cancellation.

3. Normalize a scalar correlatorReading steps 14–18

Identify protected operators and compute the Wick and color factors of a short trace.

4. Diagonalize a cyclic mixing problemReading steps 19–22

Use planar color counting to build a short cyclic basis and find protected and nonzero one-loop eigenvalues.

Global dualities and holography27 articles

Track global line-charge data under duality and identify the controlled limits of the AdS5 × S5 dictionary.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct N=4 Yang–MillsReading steps 9–13

Derive its scalar potential, organize the symmetries, and check the one-loop beta cancellation.

3. Follow electric and magnetic line chargesReading steps 14–16

Transform the coupling and charge lattice while keeping the global gauge group explicit.

4. Count powers of the gauge rankReading steps 17–23

Compute a scalar correlator and compare planar with nonplanar color contractions.

5. Translate the holographic parametersReading steps 24–27

Match rank and coupling to the bulk scales and state which limits suppress string and quantum corrections.

Learn ABJM and its first sphere partition functionConstruct ABJM’s quiver theory and derive its first exact sphere partition function.44 articles · 1 section return · 6 checkpoints

Begin: What a supercharge does

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Build a three-dimensional gauge theoryReading steps 10–13

Expand superspace interactions and derive the one-flavor Abelian theory’s classical vacuum equations.

3. Construct the ABJM theoryReading steps 14–22

Write the quiver interactions and sextic potential, and distinguish manifest N=2 from full N=6 supersymmetry.

4. Complete the round-sphere actionReading steps 23–33

Solve the Killing spinor equation and check the free chiral action and its flat limit.

5. Assemble a localized gauge integralReading steps 34–42

Complete the curved gauge and Chern–Simons terms, then combine determinants with the gauge measure.

Build invariant F-terms and D-terms
The curved vector multiplet
Gauge and matter actions from multiplets
Return to Chern-Simons and Fayet-Iliopoulos terms · Curved completion
A finite-dimensional localization example
Localization in a field theory · General localization; Three dimensional field example
The free chiral determinant
Matrix integrals, roots, weights, and eigenvalues
A first gauge-theory matrix integral

6. Evaluate the rank-one partition functionReading steps 43–45

Assemble the ABJM matrix integral and derive Z(1,k)=1/(4k) for positive level k, with the regulator specified.

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Vacua and enhanced supersymmetry31 articles

Derive ABJM’s vacuum identifications and explain the role of monopoles in enhanced supersymmetry.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Construct the Abelian gauge exampleReading steps 10–13

Eliminate auxiliary fields and write the zero-mass, zero-FI vacuum equations.

3. Construct the ABJM theoryReading steps 14–22

Write the quiver interactions and sextic potential, and distinguish manifest N=2 from full N=6 supersymmetry.

4. Identify the vacuum quotientReading steps 23–25

Derive the rank-one discrete identification and the permutation quotient in a controlled many-brane sector.

5. Dress monopoles and examine enhancementReading steps 26–31

Determine a flux insertion’s gauge charge and follow the additional symmetry multiplets at levels one and two.

Fermi gas and fixed-k large N50 articles · 1 section return

Recast ABJM as a Fermi gas, derive the fixed-level large-rank free energy, and identify what Airy resummation omits.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Construct the ABJM theoryReading steps 10–22

Write the quiver interactions and sextic potential, and distinguish manifest N=2 from full N=6 supersymmetry.

3. Complete the round-sphere actionReading steps 23–33

Solve the Killing spinor equation and check the free chiral action and its flat limit.

4. Evaluate the rank-one partition functionReading steps 34–45

Assemble the ABJM matrix integral and derive Z(1,k)=1/(4k) for positive level k, with the regulator specified.

5. Recover partition functions from spectral tracesReading steps 46–49

Distinguish the large-rank limits, derive the density operator, and check its first two canonical partition functions.

6. Derive the growth and its perturbative resummationReading steps 50–51

Calculate the leading free-energy coefficient at fixed level, invert the cubic grand potential, and identify the omitted exponential corrections.

Enter research after one completed capstoneReproduce the free-chiral response, then turn one controlled variation into a literature-grounded investigation.54 articles · 6 checkpoints

Begin: What a supercharge does

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Complete the round-sphere actionReading steps 10–20

Solve the Killing spinor equation and check the free chiral action and its flat limit.

3. Control the geometric deformationReading steps 21–40

Build and patch spinor backgrounds and specify a squashed-sphere family with a checked round limit.

4. Construct the localized observableReading steps 41–48

Combine curved gauge interactions, localization, and determinants into a sphere matrix integral.

5. Reproduce the free-chiral responseReading steps 49–53

Separate contact terms from current data and check the response analytically and numerically.

6. Pose one controlled research questionReading step 54

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Other capstone entrances

Choose the entrance that matches your completed calculation. Each sequence includes its own preparation.

Enter research after the N=4 circular loop29 articles

Use the finite-rank circular Wilson loop as the starting point for a controlled research exercise.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct N=4 Yang–MillsReading steps 9–13

Derive its scalar potential, organize the symmetries, and check the one-loop beta cancellation.

3. Set up the protected loopReading steps 14–23

Specify the loop and its four-sphere background before applying localization.

Local operators, radial quantization, and the OPE
What BPS and protected mean · Four dimensional particle bounds
Supersymmetric Wilson lines and circles
Frames and spin connections
Covariant derivatives of spinors · Generic spinor derivative
Sources, currents, and generating functionals
Stress tensors and improvement terms
A finite-dimensional localization example
Localization in a field theory · General localization
Put N=4 SYM on the four-sphere

4. Evaluate the finite-rank circular loopReading steps 24–28

Derive the Gaussian matrix result and check the distinct SU(N) and U(N) normalizations.

5. Pose one controlled research questionReading step 29

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Enter research after the ABJM rank-one integral45 articles

Use the rank-one ABJM sphere integral as a reproducible starting point for a controlled investigation.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Complete the round-sphere actionReading steps 10–26

Solve the Killing spinor equation and check the free chiral action and its flat limit.

3. Construct the ABJM theoryReading steps 27–36

Write the quiver interactions and sextic potential, and distinguish manifest N=2 from full N=6 supersymmetry.

4. Learn the localized gauge measureReading steps 37–41

Derive the chiral determinant and assemble a simple gauge-theory sphere integral.

A finite-dimensional localization example
Localization in a field theory · General localization; Three dimensional field example
The free chiral determinant
Matrix integrals, roots, weights, and eigenvalues
A first gauge-theory matrix integral

5. Evaluate the rank-one partition functionReading steps 42–44

Assemble the ABJM matrix integral and derive Z(1,k)=1/(4k) for positive level k, with the regulator specified.

6. Pose one controlled research questionReading step 45

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Enter research after Seiberg-Witten periods22 articles

Use normalized Seiberg–Witten periods and their physical interpretation to formulate a bounded investigation.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Constrain the strong-coupling singularitiesReading steps 9–17

Relate the low-energy prepotential to duality and derive a monodromy with its charge basis stated.

3. Calculate normalized periodsReading steps 18–20

Integrate the Seiberg–Witten differential and recover the effective coupling from period derivatives.

4. Interpret a BPS central chargeReading step 21

Calculate charge-dependent central charges and explain why a mass bound alone does not establish a stable particle.

BPS masses and the stability of particles

5. Pose one controlled research questionReading step 22

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Enter research after the SQCD duality checks18 articles

Use the SQCD duality checks to design a controlled and independently checked research exercise.

1. Check the vacuum before studying dynamicsReading steps 1–7

Construct four-dimensional interactions and test the vacuum equations and transverse stability.

2. Constrain quantum dynamicsReading steps 8–13

Use chiral operators and anomalies, construct SQCD vacua, and match scales across a flavor threshold.

3. Check the SQCD duality pairReading steps 14–17

Match anomalies for SU(3) with five flavors and its magnetic dual, then follow a mass deformation.

4. Pose one controlled research questionReading step 18

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Enter research after SQED-XYZ deformations23 articles

Use an SQED–XYZ deformation to formulate a bounded investigation with explicit sources and checks.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Solve the SQED vacuum equationsReading steps 10–15

Construct the gauge theory, take its Higgs quotient, and identify the semiclassical Coulomb coordinates.

3. Match the infrared operatorsReading steps 16–20

Compute monopole charges and compare the SQED operators with the XYZ chiral-ring relations.

4. Follow a deformation through the dualityReading steps 21–22

Map a complex mass between SQED and XYZ and compare the resulting vacuum equations.

5. Pose one controlled research questionReading step 23

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

Learn 4d N=2 and the Seiberg–Witten solutionCompute Seiberg–Witten periods, the effective coupling, and central charges with their stability qualifications.21 articles · 5 checkpoints

Begin: What a supercharge does

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Find the Coulomb branch and its chargesReading steps 9–12

Solve the classical SU(2) vacuum conditions and derive the monopole energy bound in the stated normalization.

3. Follow the low-energy fields around a singularityReading steps 13–17

Use the prepotential and electric–magnetic duality to derive a local monodromy.

4. Calculate normalized periodsReading steps 18–20

Integrate the Seiberg–Witten differential and recover the effective coupling from period derivatives.

5. Interpret a BPS central chargeReading step 21

Calculate charge-dependent central charges and explain why a mass bound alone does not establish a stable particle.

BPS masses and the stability of particles

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Confinement deformation22 articles

Use a small N=1 deformation of the Seiberg–Witten solution to derive monopole condensation.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct the low-energy dual descriptionReading steps 9–13

Derive couplings from a prepotential and transform periods and charges together.

3. Normalize the Seiberg–Witten curveReading steps 14–19

Use singularities, monodromies, and oriented cycles to fix the differential and its weak-coupling limit.

4. Interpret a BPS central chargeReading steps 20–21

Calculate charge-dependent central charges and explain why a mass bound alone does not establish a stable particle.

5. Solve for the monopole condensateReading step 22

Use the local monopole description to solve the deformed F- and D-term equations in the small-mass regime.

How an N=1 deformation produces confinement

Matter and N=2-star24 articles

Add matter to the Seiberg–Witten setting and distinguish fundamental-mass and adjoint-mass deformations.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Construct the low-energy dual descriptionReading steps 9–13

Derive the prepotential couplings and transform electric and magnetic variables consistently.

3. Build the normalized pure-theory curveReading steps 14–19

Identify its singularities and cycles and check the differential’s dimensions and weak-coupling normalization.

4. Interpret a BPS central chargeReading steps 20–21

Calculate charge-dependent central charges and explain why a mass bound alone does not establish a stable particle.

5. Compare matter and adjoint-mass deformationsReading steps 22–24

Check a heavy-flavor decoupling limit and identify the eight-supercharge dynamics retained by N=2-star.

First instanton coefficient23 articles

Calculate a first instanton coefficient and match it to the weak-coupling period expansion.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Define the prepotential and dual variablesReading steps 9–13

Derive the Abelian couplings and the compatible period and charge transformations.

3. Calculate normalized periodsReading steps 14–20

Integrate the Seiberg–Witten differential and recover the effective coupling from period derivatives.

4. Match two calculations of one coefficientReading steps 21–23

Use localization to evaluate the first instanton contribution and compare it with the period-derived prepotential in matched conventions.

A finite-dimensional localization example
Localization in a field theory · General localization
Recover a prepotential coefficient by instanton counting

Eight-supercharge circle compactification22 articles

Compactify the Abelian Coulomb description on a circle and derive its leading scalar metric and angular periods.

1. Assemble four-dimensional multipletsReading steps 1–8

Construct N=1 interactions, derive a BPS bound, and organize N=2 fields.

2. Identify the low-energy electric and magnetic fieldsReading steps 9–13

Construct the prepotential couplings and their dual description.

3. Calculate normalized periodsReading steps 14–20

Integrate the Seiberg–Witten differential and recover the effective coupling from period derivatives.

4. Interpret a BPS central chargeReading step 21

Calculate charge-dependent central charges and explain why a mass bound alone does not establish a stable particle.

BPS masses and the stability of particles

5. Reduce and dualize on a circleReading step 22

Count four real scalars per rank, derive the Wilson-line and dual-photon periods, and state the leading metric’s approximation.

From 4d N=2 to 3d N=4: Wilson lines and dual photons

4d N=1 quantum dynamics and vacuum physicsAnalyze supersymmetry breaking, test a concrete SQCD duality, and determine a superconformal R-symmetry.24 articles · 5 checkpoints

Begin: What a supercharge does

1. Construct a four-dimensional interacting modelReading steps 1–6

Expand the chiral and gauge interactions and eliminate auxiliary fields to obtain its scalar potential.

2. Diagnose a supersymmetry-breaking vacuumReading steps 7–10

Check the O’Raifeartaigh F-term obstruction, transverse stability, Goldstino, and local quantum lifting.

3. Constrain the quantum theoryReading steps 11–15

Compute chiral-ring and anomaly data, construct SQCD vacua, and match scales across a flavor threshold.

4. Check the SQCD duality pairReading steps 16–19

Match anomalies for SU(3) with five flavors and its magnetic dual, then follow a mass deformation.

5. Determine the superconformal R-symmetryReading steps 20–24

Check the conformal window and operator dimensions, then extremize the trial anomaly with its assumptions stated.

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Continue with a duality benchmark project18 articles

Turn the SQCD duality benchmark into a bounded research exercise with independent checks.

1. Construct the four-dimensional modelReading steps 1–6

Use superspace and explicit trace conventions to derive the F- and D-term scalar potential.

2. Prepare the quantum constraintsReading steps 7–12

Compute chiral-ring and anomaly data and derive the heavy-flavor scale-matching relation.

3. Check the SQCD duality pairReading steps 13–17

Match anomalies for SU(3) with five flavors and its magnetic dual, then follow a mass deformation.

4. Pose one controlled research questionReading step 18

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

3d SQED and infrared dualityCompare the branches and operators of SQED and XYZ, then follow a deformation through their infrared duality.22 articles · 4 checkpoints

Begin: What a supercharge does

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Solve the SQED vacuum equationsReading steps 10–15

Construct the gauge action, take the Higgs quotient, and identify semiclassical Coulomb coordinates.

3. Build the infrared operator dictionaryReading steps 16–20

Compute monopole charges and compare the meson and monopoles with the XYZ chiral-ring relations.

4. Follow a deformation through the dualityReading steps 21–22

Map a complex mass between SQED and XYZ and compare the resulting vacuum equations.

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Sphere duality benchmark45 articles

Compare the sourced SQED and XYZ sphere partition functions with their contact phases and approximation errors explicit.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Follow a deformation through the dualityReading steps 10–22

Map a complex mass between SQED and XYZ and compare the resulting vacuum equations.

3. Pose one controlled research questionReading step 23

Specify one variation of your reproduced calculation, an independent check, and the literature needed to assess it.

From reproduction to a research question

4. Complete the round-sphere actionReading steps 24–35

Solve the Killing spinor equation and check the free chiral action and its flat limit.

5. Keep the source phases in the observableReading steps 36–44

Construct the localized gauge integral and separate current responses from local contact contributions.

6. Reproduce the sphere duality benchmarkReading step 45

Match both source parameters and the full complex partition functions, then test a controlled large-mass limit.

Reproduce a duality with its contact terms

Protected-index comparison23 articles

Check SQED–XYZ duality by comparing protected indices through the first nonzero flux sectors.

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Count the free chiral indexReading steps 10–14

Identify the paired states and allowed fugacities, then derive low-order terms from the product formula.

3. Identify the operators that the index will countReading steps 15–22

Construct SQED, assign monopole charges, and map its chiral operators to XYZ.

4. Compare SQED and XYZ indicesReading step 23

Apply gauge projection and include the first nonzero magnetic-flux sectors in a low-order duality check.

Gauge projection and an index check of SQED-XYZ

A first field-theory indexCount the free chiral index and carry out a low-order gauge-theory duality check with magnetic flux included.23 articles · 4 checkpoints

Begin: What a supercharge does

1. Build your first supersymmetric theoryReading steps 1–9

Check off-shell closure and the free chiral action, then assign its R-charges.

2. Count a free chiral by handReading steps 10–14

Use radial quantization and shortening to identify cancellations and derive low-order index coefficients.

3. Identify the gauge-invariant operatorsReading steps 15–22

Construct SQED, compute monopole charges, and match its infrared operators to XYZ.

4. Compare SQED and XYZ indicesReading step 23

Apply gauge projection and include the first nonzero magnetic-flux sectors in a low-order duality check.

Gauge projection and an index check of SQED-XYZ

Optional extensions

These sequences include their own preparation, including articles shared with the core route. They are not extra requirements for the result above.

Optional quantum-mechanics preparation3 articles

Practice state pairing and the Witten index in a solvable quantum-mechanical model.

1. Solve the supersymmetric oscillatorReading steps 1–2

Construct the partner Hamiltonians and identify the unpaired normalizable ground state.

2. Test what an index tells youReading step 3

Compute the signed trace and explain how normalizability limits conclusions about supersymmetric vacua.

What the Witten index can tell us

An optional quantum-mechanics entranceSolve a supersymmetric oscillator and learn what its Witten index can and cannot establish.3 articles · 2 checkpoints

Begin: What a supercharge does

1. Solve the supersymmetric oscillatorReading steps 1–2

Construct the partner Hamiltonians and identify the unpaired normalizable ground state.

2. Test what an index tells youReading step 3

Compute the signed trace and explain how normalizability limits conclusions about supersymmetric vacua.

What the Witten index can tell us

How to choose and begin

Choose a calculation you would like to reproduce. Each route above supplies its preparation in reading order and divides the work into calculation checkpoints: results you can explain and check before continuing. You can begin with basic QFT, gauge theory, and GR; the routes introduce supersymmetry and the additional tools as needed. Use the starting-knowledge questions to identify a useful refresher.

Find your entrance

For a first curved-space calculation, choose the twenty-lesson round-three-sphere route. You will build a free chiral theory, solve for its preserved spinors, and check its sphere action. The longer three-manifold route begins with this example and continues to general backgrounds and a measurable response to squashing.

For four-dimensional quantum dynamics, Seiberg–Witten theory, or N=4 Yang–Mills, take the direct four-dimensional entrance. These routes share the first two conceptual lessons, then teach their own spinor conventions. Their preparation does not include the three-dimensional spinor course or the full curved-space course.

For three-dimensional gauge theories, SQED–XYZ offers a small model of infrared duality. ABJM first develops the quiver, interactions, and supersymmetry, then prepares its sphere partition function. The index route instead computes a signed protected state count. The optional quantum-mechanics entrance lets you practice state pairing in a solvable oscillator.

Begin with a small calculation

For the first sphere route, start with:

What a supercharge does: distinguish the symmetry generator, its parameter, and its action on a field or state.
Grassmann variables and their calculus: differentiate and integrate a short polynomial with the ordering convention stated.
Spinors in three dimensions: multiply the explicit gamma matrices and check a spinor contraction.

Your first check is already in the first lesson: act with the two-state supercharge and verify its anticommutator. For a four-dimensional destination, keep the first two lessons and take Weyl spinors and the N=1 algebra as the third.

Work through the checkpoints

Open a destination above, then follow its lessons in order. At each checkpoint, reproduce the stated calculation and explain its assumptions. If a step is unfamiliar, use the linked prerequisite lesson before advancing. Milestones and self-assessment shows how to test your understanding.

Optional extensions answer a further question. Each lists its complete preparation, including lessons shared with the core route; only study the shared material again when you need it. Keep any explicit return to a later section in its assigned place: a flat-space construction may need new tools before its curved completion.

The research route starts from one completed capstone. Choose the entrance matching a calculation you can reproduce, then formulate one controlled variation and check what the literature already establishes.