Mathematical and Field-Theory Tools

Putting a field theory on a curved space raises two distinct questions: how do its fields vary from point to point, and how does its action respond to the background? This volume supplies the geometry and field-theory methods needed to answer those questions. Later branches develop conformal and matrix methods for quantum observables.

Use these lessons when a calculation needs them. The chapter navigation groups related methods; the learning paths select the preparation for a particular physical result. You do not need to complete this entire volume before studying a theory.

Start from a metric

Frames and spin connections starts from familiar general relativity. It constructs an orthonormal frame for flat space in cylindrical coordinates and finds its connection in two ways: from Cartan’s torsion equation and from the coordinate Christoffel symbols. A rotation to a Cartesian frame then explains why nonzero connection coefficients can occur even in flat space.

This entrance requires no supersymmetry or three-dimensional spinor calculus. It also explains which parts of the construction survive in four dimensions and how Lorentzian signature changes the lowering of frame indices.

Your first checkpoint is concrete: reconstruct the metric from the coframe, recover the sign of the cylindrical connection, and show that a constant Cartesian vector is parallel even though its cylindrical frame components change. That distinction between a changing component and a changing geometric object is the basis for covariant differentiation.

Then derive the spinor covariant derivative and apply it to the round three-sphere. The first calculation checks a spinor in rotating and Cartesian frames; the second constructs a frame smooth over the whole sphere and recovers its curvature in the target paper’s convention.

Spinors as bundle sections explains how those local formulas join across patches. Compare two spin structures on punctured flat space and determine which one admits a global parallel spinor. The loop’s return map includes the transition between endpoint frames.

Connections, flux, and holonomy provides an independent gauge-theory entrance to global geometry. Construct a monopole from two regular potentials, derive its flux quantization, and determine how a flat circle connection changes a charged scalar’s spectrum.

Circle bundles and the Hopf fibration makes the phase direction into a geometric circle. Join its two coordinate descriptions, compute the connection and curvature, and recover the three-sphere metric with the correct fiber and base radii.

Fierz identities from explicit matrices supplies the algebra behind spinor bilinears. Rearrange four spinors with their actual statistics, test a quartic interaction, and show how a commuting spinor determines a unit direction.

Continue a fermion to Euclidean space

Euclidean continuation and independent spinors answers a different question. Starting from an explicit Lorentzian kinetic term, it follows the time coordinate, derivative, integration measure, and spinor bilinear into Euclidean signature. It then distinguishes the algebra of independent fields from a chosen integration contour.

For this three-dimensional example, first read Spinors in three dimensions. That lesson fixes the matrices and index contractions used in the continuation. It is preparation for this example, rather than a requirement for every geometric method in the volume.

At the end, reproduce the continued kinetic term with its factors of $i$, and explain why a tilded Euclidean spinor cannot be replaced by a Hermitian conjugate without specifying additional structure. The finite Gaussian example gives a small calculation in which an integration contour can be checked directly.

Probe a theory with background fields

Sources, currents, and generating functionals has an independent entrance from ordinary QFT. Vary a complex scalar action with respect to a flavor connection and a metric to obtain the current and stress tensor, then differentiate its partition function. The second derivative contains both a connected correlation and an explicit contact term; a finite Gaussian lets you test the signs.

For the round-sphere route, these source variations explain how the geometry meets the field theory. They will identify the backgrounds needed when a supersymmetric flat-space action is placed on the sphere. The twenty-lesson route keeps that sequence separate from optional geometry.

Stress tensors and improvement terms then shows why two actions that agree in flat space can select different local stress tensors. Derive the scalar curvature coupling by varying the metric, find the conformal coefficient in three and four dimensions, and check when an improvement leaves the total energy unchanged.

Read a scaling law

Fixed points, scaling, and conformal symmetry starts from the ordinary free scalar. Derive its two-point function from a unit point source, then see how a scaling dimension determines the decay with distance. The lesson explains anomalous dimensions, relevant deformations, and the extra condition needed for conformal symmetry without assuming prior CFT knowledge.

Its first checkpoint is to distinguish a changed normalization from a changed exponent. A massive propagator and a logarithmic approximation then show how a theory can depart from a fixed-point power law.

Local operators, states, and the OPE explains why the same scaling dimension is an energy gap on a cylinder. Construct a primary state, count its derivatives, and calculate the first two terms in a free-field operator product. A descendant norm then gives the scalar unitarity bound and identifies the free equation of motion as a null state.

What BPS and protected mean adds supercharge norms to the radial picture. Derive a scalar shortening condition, compare it with a particle mass bound, and see why a supersymmetric observable can still depend on a coupling. Its three-dimensional and four-dimensional examples have separate entry requirements.

Follow the color indices

Matrix-valued matter, representations, and traces starts directly from ordinary gauge theory. Transform an adjoint matrix, follow a bifundamental between two color spaces, and close an invariant product in the correct order. The examples distinguish an ordinary trace, a normalized trace, and a choice of generator normalization.

This method serves both the four-dimensional gauge-theory routes and three-dimensional ABJM. Its first checkpoint is to explain why an adjoint trace is invariant while the trace of a single bifundamental generally is not. A rectangular example makes the two color spaces impossible to confuse.

Matrix integrals, roots, weights, and eigenvalues turns that matrix notation into an integration measure. Derive the squared eigenvalue difference and permutation factor, check their absolute normalization, and calculate how removing the central direction changes a trace observable.

Large N and what a ‘t Hooft limit holds fixed asks which parameters and observable normalization stay fixed as the color space grows. Compare Yang–Mills and ABJM sequences, then use an exactly defined Gaussian to calculate finite-rank fluctuations and bound a truncated expansion’s error.

Saddle points and eigenvalue densities begins with a Gaussian correction whose error can be bounded. It then derives the semicircle density from the matrix measure, checks the force outside its support, and compares the result with an exact finite-rank density.

Decide what can be compared

Universal observables and scheme dependence compares a normalized loop with a sphere free energy. Calculate an allowed curvature counterterm, follow its effect on source responses, and identify the constants, logarithmic coefficients or mixed derivatives that survive the stated changes of prescription.

When comparing two conventions, write their definitions beside each other before comparing a sign. When testing a result, choose a check that could fail for a different reason: a coordinate calculation alongside a frame calculation, or a convergent integral alongside its formal algebra. Each lesson gives a worked example and complete exercise solutions so that you can perform that comparison yourself.