Laboratories

A useful computation answers a question you can state before running it. These laboratories connect executable calculations to their analytic lessons: identify the inputs, reproduce a benchmark, then change one assumption and explain the result. Each page includes the defining equations, interpretable output, complete solutions, and editable downloads with their package requirements.

You can read the calculations without running a notebook service or browser simulation. Choose a lab that follows the theory you already know; the preparation links below lead to the necessary derivations.

Algebra and transformation rules

• Gamma matrices and bilinears. Begin with three-dimensional spinors and Fierz identities. Test the Clifford algebra, then rearrange products of commuting and anticommuting spinors. Changing a gamma-matrix sign tests orientation; changing the entry statistics tests a different assumption.
• Chiral multiplet closure. Follow off-shell closure through ordered Grassmann products. Verify the transformation algebra on every component, then remove the auxiliary field and identify the precise equation-of-motion residue. More successful matrix identities cannot compensate for a wrong parity convention.

Curved geometry and sphere sources

• Geometry of the round sphere. After the sphere-frame calculation, reconstruct the connection and curvature in a noncoordinate frame. The target convention gives scalar curvature $-6/\ell^2$. Check the connection sign and the frame bracket: constant connection coefficients do not imply flatness.
• Verify a Killing spinor. Use the round-sphere solution to evaluate every component of the spinor equation. Pair a small local residual with the declared global frame or overlap data; a local solution alone does not establish a global supercharge.
• Hopf fibers and transverse geometry. Prepare with circle bundles and the Hopf fibration and almost contact metric structures. Generate linked fibers, check the connection and integer bundle number, and distinguish numerical transport error from a change of global geometry.
• Free-chiral squashing response. Follow the free-chiral response calculation. Recover the round-point squashing Hessian $\pi^2/8$ and the stated current coefficient $\tau_{RR}=1/4$, while keeping the source family and finite regularization term fixed. Agreement at the round point does not make distinct finite-squashing families identical.
• Responses on deformed spheres. Prepare with ABJM masses and squashing and superconformal current correlators. Reproduce the fourth mass derivative in rank-one, level-one ABJM, then change the sphere’s squashing at a fixed source normalization. A mass derivative at fixed geometry is different from differentiating the geometry itself.

Gauge-invariant observables

• N=4 Gaussian Wilson loop. Begin with the finite-rank circular loop. Sample the Hermitian Gaussian measure, separate the central mode, and compare with the exact finite-rank result. Statistical uncertainty and finite-rank corrections answer different questions. Its planar comparison additionally uses the planar and strong-coupling limit.
• A short N=4 spin chain. Start from the cyclic one-loop chain. Build cyclic scalar words with their correct trace norms and distinguish protected from unprotected one-loop combinations. A wrong inner product can leave the eigenvalues unchanged. The computation holds length fixed in the planar approximation; it is not a finite-gauge-rank Hamiltonian.
• ABJM small-N partition functions. Follow the regulated one-brane integral to recover its absolute normalization, separating regulator bias, domain cutoffs and numerical error. The optional spectral checks require the Fermi-gas reformulation and add a rank-two calculation that tests the exchange contribution.
• ABJM large N and the Airy approximation. Begin with the Airy function and its corrections. Compare independently calculated finite-rank data with the leading growth and the normalized perturbative Airy expression at fixed level. Distinguish numerical errors from omitted exponential contributions; a short sequence does not prove an asymptotic law.

Vacuum identifications and effective couplings

• ABJM one-brane quotient. Follow the one-brane moduli-space derivation. Track compact fluxes and the periodic dual photon into the residual discrete identification. A local gauge choice does not determine the global quotient without the original charge lattice.
• Seiberg–Witten periods and couplings. Prepare with periods and the effective coupling. Compare oriented contour quadrature with transport by the period equation, retaining branch, scale and charge conventions. Its first-instanton extension additionally needs the instanton calculation.
• Seiberg–Witten monodromy and BPS charges. Start with BPS masses and stability. Continue periods around oriented loops with a common base point, then compensate the period-basis change in the charge row. Preserving the central charge checks the convention; it does not establish which charges occur as stable particles.

Read the output as a physical argument

For each experiment, first predict what the changed input should do. Then compare with a check that could reveal a different error: an analytic limit, a separate representation, or a calculation with different numerical failure modes. Several wrappers around one formula do not provide independent confirmation.

Exact arithmetic checks the algebra that was represented. It does not prove that those equations describe the intended theory. Numerical precision, integration resolution, domain cutoffs and physical approximations must likewise be tested separately. A small rounding error cannot repair an incorrect contour, missing exchange term or wrong normalization.

You have reached the lab’s exit calculation when you can reproduce its benchmark, account for the changed-input result and state the limits of the conclusion. Use the learning paths to continue the physical argument.