Generalized Killing Spinors and Global Spin-R Bundles

A generalized Killing-spinor equation is local, whereas a supercharge is global. The parameter must be a section of the spin bundle tensored with the appropriate R-symmetry and flavor bundles; its local representatives must patch, its fluxes must be quantized, and every bilinear appearing in $Q^2$ must extend across zeros, fixed points, defects, and boundaries. Integrability of the differential equation is necessary but does not perform these checks.

Required background. Rigid supersymmetry from nondynamical supergravity supplies the local Killing-spinor equation and algebra. Spin structures and Dirac operators supplies spin, spin-c, and chiral spinor bundles.

Helpful background. Vector, principal, and associated bundles supplies transition functions, connections, and characteristic classes.

Supersymmetry parameters are bundle sections

On a spin four-manifold in the new-minimal example, the two parameters have bundle assignments

$$\zeta\in\Gamma(S_+\otimes L_R), \qquad \widetilde\zeta\in\Gamma(S_-\otimes L_R^{-1}),$$

where $L_R$ is the line bundle of unit R charge. On an overlap $U_i\cap U_j$, local representatives must obey

$$\zeta_i=S_{ij}\,r_{ij}\zeta_j, \qquad \widetilde\zeta_i=\widetilde S_{ij}\,r_{ij}^{-1}\widetilde\zeta_j.$$

Here $S_{ij},\widetilde S_{ij}$ are the two chiral spin lifts and $r_{ij}$ is the R-bundle transition function. Solving the differential equation separately on every $U_i$ is insufficient unless these relations hold on all overlaps and satisfy the cocycle condition on triple overlaps.

The most invariant structure group is often not a direct product but

$$G_{\text{spin-R}}= \frac{\operatorname{Spin}(d)\times G_R}{\mathbb Z_2},$$

with the quotient chosen so that the common central element acts trivially on every field. This can permit a theory on a manifold that is not spin. In the familiar spin-c case, a determinant line bundle $L_{\det}$ must satisfy

$$c_1(L_{\det})\equiv w_2(TM)\pmod 2.$$

Which power of the physical R bundle plays the role of $L_{\det}$ depends on the normalization and global form of $G_R$. One must therefore state the charge lattice, not merely write “turn on half the spin connection.”

Flux and charge quantization

For an honest $U(1)$ line bundle and a closed two-cycle $\Sigma$,

$$\frac{1}{2\pi}\int_\Sigma F_R\in\mathbb Z.$$

If all fields have integral R charge in the chosen normalization, this is enough for their phases around overlaps to be single valued. Fractional-looking fluxes can occur when the actual symmetry is a quotient by fermion parity or when an R connection participates in a spin-c structure, but then the quotient and the allowed charge lattice must be given explicitly. A local gauge potential $A^{(R)}$ cannot decide this global question.

Background flavor bundles enter in the same way. If a supersymmetry parameter is flavor neutral but matter has flavor charge, the supercharge may be global while the matter multiplet fails to patch unless the background flavor flux is allowed. Dynamical gauge bundles add topological sectors that later contribute distinct localization loci.

From a spinor to the square of Q

Spinor bilinears are useful precisely because their bundle charges can cancel. For parameters of opposite R charge,

$$K^\mu=\zeta\sigma^\mu\widetilde\zeta$$

is R neutral and patches as a complex vector field. The local Killing-spinor equations may imply $\mathcal L_Kg=0$, but three global questions remain:

1. Does $K$ generate a complete action on the manifold?
2. Where does $K$ vanish, and are those zeros smooth fixed components?
3. Does the lift of the action to every gauge, R, and flavor bundle agree with the transformation in $Q^2$?

A zero of $K$ is not a singularity of the algebra. It is commonly a fixed point of the equivariant action, and it can support extra local saddles or unpaired modes. A zero of $\zeta$ itself is more delicate: a field redefinition that divides by $\zeta$ is then only local and may hide additional degrees of freedom.

With a single nowhere-zero chiral solution in four-dimensional new-minimal supergravity, normalized bilinears define an almost-complex structure; the Killing-spinor equation forces its integrability. On compact backgrounds this yields the Hermitian-manifold characterization proved in Dumitrescu, Festuccia, and Seiberg 2012, §§2–3. The theorem assumes the relevant spinor data globally; it is not a license to infer a supercharge from an arbitrary local complex chart.

A global construction test

For each proposed supercharge, record the following scientific data:

• the principal spin-R bundle and its global form;
• the representation and charge of every supersymmetry parameter and field;
• transition functions and quantized characteristic classes;
• the complete generalized Killing-spinor equation and its integrability conditions;
• zeros of the spinors and of the bilinear generating $Q^2$;
• the lift of the even symmetry to gauge and background bundles;
• allowed singularities at defects and boundary restrictions.

At a codimension-two defect, for example, the R connection or spin connection may have prescribed monodromy. A spinor can be single valued only after combining those holonomies. At a boundary $Y$, the preserved even vector must be tangent to $Y$ unless the boundary condition is itself moved by the symmetry, and the surviving spinor components must be compatible with the boundary projector.

Common pitfalls

Local cancellation is not a global twist. Setting $A^{(R)}$ equal to a component of the spin connection in one frame does not establish that the corresponding bundles are isomorphic. Compare transition functions and characteristic classes.

A bilinear can lose information. A global vector $K$ does not prove that its constituent spinors are global; opposite patching failures can cancel in the product.

Flux normalization is theory dependent. A statement such as “one unit of R flux” has no meaning until the smallest allowed R charge and the global symmetry group are fixed.

Exercises

1. Neutrality of the bilinear. Use the overlap relations above to show that $K^\mu$ is R neutral.

Solution

On an overlap the factors $r_{ij}$ and $r_{ij}^{-1}$ multiply and cancel. The spin transition functions combine with $\sigma^\mu$ to give the vector transition function. Thus $K$ patches as a vector rather than as a section carrying R charge.

2. Spin-c obstruction. Why can a complex line bundle with $c_1(L_{\det})\not\equiv w_2(TM)\pmod2$ not repair the absence of a spin structure?

Solution

The obstruction to lifting oriented frame transitions to spin transitions is $w_2(TM)$. In a spin-c lift, the failure of the spin lifts on triple overlaps must be cancelled by the parity of the determinant-line transitions. If their mod-two classes differ, the combined cocycle still fails.

References

• Dumitrescu, Thomas T., Guido Festuccia, and Nathan Seiberg. “Exploring Curved Superspace.” Journal of High Energy Physics 2012, no. 8 (2012): 141. doi:10.1007/JHEP08(2012)141. Open preprint.

Further reading

• Lawson, H. Blaine, Jr., and Marie-Louise Michelsohn. Spin Geometry. Princeton Mathematical Series 38. Princeton, NJ: Princeton University Press, 1989. Publisher.

Next step

A spin-R bundle can sometimes be built directly into the field representations. Continue to topological and holomorphic twists.