Supersymmetric Three-Manifolds

A curved-space supersymmetry is specified by a spinor that solves an equation involving the metric and background fields. The spinor depends on a local frame, but the geometry it selects must make sense independently of that frame. Which direction does it pick out? What structure remains perpendicular to that direction? When do those local constructions fit together across the manifold?

This volume turns those questions into calculations. The physical setting is three-dimensional $\mathcal N=2$, with four real supercharges in the flat parent theory. The number preserved by a particular background must be established from its spinor solutions. A manifold’s name alone does not determine that number or specify all the background sources.

Begin with one nonzero spinor

Construct geometry from one spinor starts with a smooth commuting spinor and a positive Riemannian metric. Form its Hermitian bilinear, divide by its norm, and obtain a unit direction. Orientation then supplies a rotation in the perpendicular two-plane.

Prepare with the Killing-spinor equation, spinor bundles, and the commuting-spinor projector. These provide the differential equation, the overlap rules, and the short matrix identity used in the first calculation. Complex geometry is developed from the construction; it is not an entrance requirement.

Your first checkpoint: given a two-component commuting spinor, calculate the unit direction and transverse projector. Check the complex bilinear’s norm and charge, then explain why lowering a transverse vector’s index changes the sign in its eigenvalue equation. The lesson includes a spinor that passes every pointwise algebraic test but fails the differential equation.

Keep three kinds of statement separate

Almost contact metric structures turns the spinor’s algebraic data into an explicit metric and frame. Compare contact and noncontact examples, follow a horizontal loop, and calculate how the Reeb vector changes when the one-form is rescaled. These calculations distinguish a transverse rotation from the differential conditions needed for a supersymmetric background.

Integrability and transverse holomorphic geometry derives that differential restriction directly from the spinor equation. Follow its meaning through a projected Lie derivative, solve a transverse complex-coordinate equation, and test a global identification that obstructs the chosen flow despite valid local geometry.

Work in adapted coordinates then derives the metric, horizontal vectors and chart transitions. A nonlinear overlap and a scalar kinetic term show how the same geometry can have different-looking components, including cross terms.

An algebraic identity holds at one point. Squaring a transverse rotation to obtain minus the identity on a plane is such a statement. It follows from the spinor matrices and metric, without differentiating the spinor.

A differential condition compares nearby points. A smooth choice of transverse planes need not be tangent to surfaces. A unit vector need not generate an isometry. To establish either claim, calculate the relevant derivatives rather than infer it from the existence of a normalized bilinear.

A global statement compares overlapping descriptions and closed paths. A spinor carries both spin-frame and R-symmetry transition data. Its neutral bilinear can be an ordinary global tensor while a charged bilinear still carries a line-bundle phase. Connections, flux, and holonomy supplies the preparation for checking those phases and their consistency.

These distinctions guide the reconstruction of supersymmetric backgrounds. Even when an equation is solved in one chart, the sources and spinor must transform consistently into the next. A small numerical residual in one chart establishes neither the overlap law nor the existence of a global solution.

Reconstruct the background fields carries out the local calculation. Derive the spin connection, solve both spinor components for compatible sources, and check their field-strength constraints. Its worked geometry has both transverse expansion and twist; the exercises expose additional constraints that an algebraic solution alone can miss.

Patch the spinor and the R-symmetry bundle constructs a complete two-patch sphere example. The spin rotation and R phase cancel for the preserved component; a separate circle calculation shows why matching curvature does not fix global holonomy.

Add a second preserved supercharge

Two supercharges and their Killing vector derives the neutral bilinear of opposite-charge solutions. State when this vector can be chosen real, then solve both equations in a warped background. Its isometry has varying length: normalizing the vector changes its flow and generally destroys the Killing property.

Seifert fibrations and explicit examples follows that flow globally. Descend a complete round-sphere background to a lens space, then construct a smooth supersymmetric neighborhood of an exceptional fiber and check both spinor return phases.

Four supercharges and product geometries asks when both complete spinor connections are flat. Solve their equations on a sphere times a circle, then use the circle holonomy to decide how many local solutions survive globally.

Return to a concrete background

Use the round three-sphere as a familiar comparison. Its metric, frame, background sources and preserved spinors are specified together. Check a proposed general statement against that complete example, then identify which input would change on another manifold or in a deformed metric.

Squashed spheres and their full background data stretches the Hopf direction while solving both charge equations globally. Follow the radius factors, auxiliary branch and two-form patches, then compare the complete construction with a toric ellipsoid.

The useful outcome is the ability to move in both directions: extract tensors from a spinor, and test whether proposed geometric data can support the required spinor equation. The learning paths connect this geometry route to the background-field construction and to observables that depend on the full supersymmetric background.