Generalized Killing Spinors and Global Spin-R Bundles

A generalized Killing-spinor equation is local, whereas a supercharge is global. Its parameter must be a section of the correct combined spin–R bundle, local representatives must satisfy a cocycle condition, and the R flux must lie in the charge lattice of the actual global symmetry group. Bilinears in $Q^2$ must also lift consistently to every gauge and background bundle. Differential integrability does not perform any of 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 live in combined bundles

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 transition function of the unit-charge R line. Solving the differential equation separately on every $U_i$ is insufficient unless these relations hold on double overlaps and satisfy the cocycle condition on triple overlaps.

To see what can change on a non-spin manifold, choose local lifts $S_{ij}$ of the oriented frame transitions. On a triple overlap their product may be

$$S_{ij}S_{jk}S_{ki}=(-1)^{w_{ijk}},$$

where the cocycle $w_{ijk}$ represents $w_2(TM)$. The spinor still patches if an order-two central element of the R symmetry supplies the same sign, so that it cancels in the combined representation. The relevant structure group is then

$$\operatorname{Spin}^{G_R}(d)= \frac{\operatorname{Spin}(d)\times G_R}{\mathbb Z_2},$$

provided the identified diagonal pair acts trivially on every field. This is theory data: if the global form of $G_R$ has no suitable central element, R flux cannot repair the spin obstruction.

For $G_R=U(1)$ this is the spin-c construction. Its honest determinant line $L_{\det}$ must satisfy

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

The local “square root” seen by a spin-c spinor need not be an honest line bundle: on a non-spin four-manifold the separate bundles $S_\pm$ and $L_R$ may not exist even though $S_+\otimes L_R$ and $S_-\otimes L_R^{-1}$ do. In the present convention, where $\zeta$ has R charge $+1$, the honest square is

$$L_{\det}=L_R^2, \qquad \left[\frac{2\,\operatorname{Re}F_R}{2\pi}\right] =c_1(L_{\det}).$$

Thus $\operatorname{Re}F_R/(2\pi)$ can have half-integral periods on a non-spin manifold, with their parity fixed by $w_2(TM)$ through the determinant-line condition above. This distinction and the new-minimal bundle assignments are stated explicitly in Dumitrescu, Festuccia, and Seiberg 2012, §2. A different minimal R charge or global form of $G_R$ changes which power is honest; “turn on half the spin connection” is not complete global data.

Flux quantization uses the actual charge lattice

For an honest unit-charge $U(1)$ line bundle, use the physics convention $D=\nabla-iqA_R$ and let $F_R=dA_R$. On every closed two-cycle $\Sigma$,

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

If all fields have integral R charge in this normalization, their transition phases are single valued. Fractional-looking fluxes can occur when the actual symmetry is a quotient by fermion parity or when the R connection participates in a spin-c structure, but then the quotient and allowed charge lattice must be stated explicitly.

There is one Euclidean subtlety. In the four-dimensional new-minimal equations, $A^{(R)}$ may be complex. Its real part is the connection on the unitary R bundle and carries the quantized Chern class; its imaginary part is a globally defined one-form and carries no independent bundle topology Dumitrescu, Festuccia, and Seiberg 2012, §2. Applying the flux condition indiscriminately to a complex local potential is therefore incorrect.

Background flavor bundles enter in the same way. A flavor-neutral supersymmetry parameter may patch even when a charged matter multiplet does not, so every matter representation must be checked separately. Dynamical gauge bundles add topological sectors that later contribute distinct localization loci; their fluxes are not fixed by the R bundle.

Bilinears inherit global information

Spinor bilinears are useful 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. In the smooth new-minimal system on a connected manifold, any nontrivial solution is nowhere zero because the first-order equation determines it by its value at one point. Moreover,

$$K^\mu K_\mu=0, \qquad K^\mu\overline K_\mu =2\lvert\zeta\rvert^2\lvert\widetilde\zeta\rvert^2.$$

Thus $K$ is complex and null but is nowhere zero when both spinors are nontrivial. These facts follow from Dumitrescu, Festuccia, and Seiberg 2012, §2, eqs. (2.9)–(2.11). A claimed isolated zero of $\zeta$, $\widetilde\zeta$, or this particular $K$ therefore signals singular coefficients, a failed patching ansatz, or departure from the stated smooth new-minimal system.

The local equations also imply $\mathcal L_Kg=0$, but a global algebra still requires:

1. a real flow, when one is required, whose vector field is complete and preserves the background;
2. a lift of that flow to every gauge, R, and flavor bundle matching the transformations in $Q^2$;
3. preservation of defects, boundaries, field-space contours, and operator insertions.

On a compact manifold without boundary, every smooth real vector field is complete. With a boundary, completeness instead requires a boundary-preserving tangent flow (or another explicitly specified global evolution). The vector $K$ above is generally complex, so exponentiating it to a compact bosonic symmetry may in any case require choosing an appropriate real combination or reality structure. In other dimensions or other rigid-supergravity systems, the vector in $Q^2$ can have fixed points; their smooth zero loci must then be analyzed rather than imported into this example.

One chiral solution produces Hermitian geometry

For a single nontrivial solution $\zeta$, define

$$J^\mu{}_{\nu} =\frac{2i}{\lvert\zeta\rvert^2} \zeta^\dagger\sigma^\mu{}_{\nu}\zeta.$$

Fierz identities give $J^2=-1$, so $J$ is an almost-complex structure. Substituting the new-minimal equation into $\nabla J$ makes its Nijenhuis tensor vanish; $J$ is therefore integrable and the Riemannian metric is Hermitian. Conversely, on any smooth oriented Hermitian four-manifold, compatible complex new-minimal fields $A^{(R)}$ and $V$ can be constructed so that one solution exists. The background fields are not unique. This local-to-global theorem does not require compactness, but it does require the global spin-c/R data used in its construction Dumitrescu, Festuccia, and Seiberg 2012, §3, especially eqs. (3.1)–(3.3) and §3.2.

The theorem is specific to the displayed four-dimensional new-minimal system. Old-minimal, conformal, and higher-dimensional generalized Killing-spinor equations have different geometric classifications.

Global construction test

For each proposed supercharge, record the following scientific data:

• the principal spin–R bundle, its quotient, and the embedding of fermion parity;
• the representation and charge of every supersymmetry parameter and field;
• transition functions, triple-overlap cocycles, and quantized characteristic classes;
• the complete generalized Killing-spinor equation and its integrability conditions;
• the allowed zero set implied by that particular first-order system;
• the lift of the even symmetry to gauge and background bundles;
• Euclidean reality or contour data;
• allowed singularities at defects and boundary restrictions.

At a codimension-two defect, for example, the R and spin connections may each have monodromy; only their combined holonomy decides whether the supersymmetry parameter is single valued. At a fixed boundary $Y$, the even transformation must preserve $Y$, the surviving spinor components must obey the boundary projector, and the boundary conditions must form a closed supersymmetry orbit. Explicit rigid bulk-plus-boundary constructions illustrating the latter requirement are given in Belyaev and van Nieuwenhuizen 2008, §§2–4.

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.

Fixed points are system dependent. The vector built from two nontrivial opposite-charge solutions of the smooth four-dimensional new-minimal equations is nowhere zero. Fixed-point arguments from another localization background cannot be transferred without recomputing the relevant bilinear.

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.

3. Can the mixed bilinear vanish? Assume $\zeta$ and $\widetilde\zeta$ are nontrivial smooth solutions of the displayed new-minimal equations on a connected manifold. Can $K^\mu=\zeta\sigma^\mu\widetilde\zeta$ vanish at an isolated point?

Solution

No. Each nontrivial solution is nowhere zero, and

$$K^\mu\overline K_\mu =2\lvert\zeta\rvert^2\lvert\widetilde\zeta\rvert^2>0.$$

Therefore $K$ is nowhere zero, even though the complex contraction $K^\mu K_\mu$ vanishes. An apparent isolated zero indicates that at least one assumption—smooth coefficients, nontrivial global solutions, correct patching, or the new-minimal equation—has failed.

References

• Belyaev, Dmitry V., and Peter van Nieuwenhuizen. “Rigid Supersymmetry with Boundaries.” Journal of High Energy Physics 2008, no. 4 (2008): 008. doi:10.1088/1126-6708/2008/04/008. Open preprint.
• 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.