Paper guide: Appendices A and B, conventions and geometry

The appendices answer two different questions. Appendix A fixes how an index, spinor derivative or curvature sign is to be read. Appendix B explains which additional geometric structure a chosen spinor direction determines. A convention error in the first can look like a geometric obstruction in the second.

Keep P v3, Appendices A and B beside this guide. The paper entrance supplies the bibliographic record. The preparation for Appendix A is three-dimensional spinors, frames, Euclidean continuation, spinor derivatives and round-sphere geometry. The separate geometric entrance below specifies the preparation for Appendix B.

All calculations here are three-dimensional and Riemannian. They are kinematic tests; no dynamical gauge group, matter action or preserved charge is assumed. When these formulas are used in the paper’s physical construction, the parent is three-dimensional $\mathcal N=2$ with four real Lorentzian Poincaré supercharges. A geometric identity alone does not count its background-preserved charges.

Read the conventions by operation

Source passage	Operation to check	Canonical explanation
(A.1)–(A.2), printed 38/PDF 39	Raise an index without conjugating; reverse odd products under adjunction.	Index contraction and independent Euclidean variables
(A.3)–(A.4), printed 38/PDF 39	Keep the oriented Clifford product and the lowered gamma indices distinct.	Gamma multiplication and ordered Fierz identity
(A.5)–(A.6), printed 38/PDF 39	Identify which Grassmann coordinates are differentiated and which top coefficient is integrated.	The superspace expansion; this is a later application, not extra preparation for the curvature calculation
(A.7)–(A.9), printed 38–39/PDF 39–40	Translate the input-first connection and the Ricci contraction.	Frame covariance and the explicit curvature trace
(A.10)–(A.12), printed 39/PDF 40	Reduce spin curvature to the Ricci tensor in three dimensions.	Sphere spin connection, followed by the reconstruction below
(A.13)–(A.14), printed 39/PDF 40	Distinguish transport of a spinor from transport of a mixed tensor.	The spin rotation in the curved algebra and the tensor Lie derivative

For lower spinor columns, use

$$\gamma=(\sigma_3,-\sigma_1,-\sigma_2),\qquad C_{\rm spin}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad uv=u^T C_{\rm spin}v. \tag{PC08.1}$$

The transpose does not conjugate. The matrices $C_{\rm spin}\gamma^a$ are symmetric, while $C_{\rm spin}$ is antisymmetric. Consequently commuting spinors have antisymmetric scalar contractions and symmetric vector contractions; for two odd spinors those exchange signs reverse. The all-lower gamma array in the second line of (A.4) is instead $(\gamma^a)_{\alpha\beta}=\epsilon_{\beta\lambda}(\gamma^a)_\alpha{}^\lambda$. It cannot be replaced by $C_{\rm spin}\gamma^a$ without doing the index contractions.

The coordinates in (A.5) are written with lower indices, but differentiation there is with respect to the raised coordinates. For the component labels, $\theta^{\alpha=1}=\theta_2$ and $\theta^{\alpha=2}=-\theta_1$, hence $\partial/\partial\theta^{\alpha=1}=\partial/\partial\theta_2$ and $\partial/\partial\theta^{\alpha=2}=-\partial/\partial\theta_1$, using left derivatives. In the measure formula (A.6), however, $\theta^2$ means the contracted square $\theta^\alpha\theta_\alpha=-2\theta_1\theta_2$. Thus the normalization $\int d^2\theta\,\theta^2=1$ corresponds to $d^2\theta=-\frac12d\theta_2d\theta_1$ for the ordered coefficient-extraction measure. The superspace lesson carries these choices through an actual action.

Restore the spin-curvature contraction

Use an oriented orthonormal frame at a point. Write $R_{ab cd}$ with $c$ the input vector index and $d$ the output index, as in (A.9). The paper’s Ricci tensor is $R_{ac}=\sum_b R_{ab c}{}^b$. If $\Omega^a{}_b$ is the Cartan connection defined by $\nabla E_b=\Omega^a{}_bE_a$, the source uses $\omega_a{}^b=\Omega^b{}_a$. This is an index-order translation, not a different metric.

Let $\mathscr R_{ab}$ denote curvature acting on a spinor. In a coordinate frame it is $[\nabla_a,\nabla_b]$; in a general frame one must subtract $\nabla_{[E_a,E_b]}$. Equation (A.11) gives

$$\mathscr R_{ab} =-\frac i4 R_{ab cd}\epsilon_{cdj}\gamma_j. \tag{PC08.2}$$

To obtain (A.12), contract another epsilon tensor. Define $E_{ij}=\epsilon_{iab}R_{ab cd}\epsilon_{cdj}$. The three-dimensional decomposition (A.10) is

$$\begin{aligned} R_{ab cd}={}&R_{ac}\delta_{bd}-R_{ad}\delta_{bc} -R_{bc}\delta_{ad}+R_{bd}\delta_{ac}\\ &-\frac R2(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}). \end{aligned}$$

Each of its four Ricci terms contributes $R\delta_{ij}-R_{ij}$ to $E_{ij}$; for example $\epsilon_{iab}\epsilon_{cbj}R_{ac}=R\delta_{ij}-R_{ij}$. The scalar-curvature term contributes $-2R\delta_{ij}$, since the contraction of the two antisymmetric delta products is $4\delta_{ij}$. Therefore

$$E_{ij}=2R\delta_{ij}-4R_{ij},\qquad \boxed{\epsilon_{iab}\mathscr R_{ab}\psi =\frac i2(2R_{ij}-R\delta_{ij})\gamma_j\psi.} \tag{PC08.3}$$

This reconstructs the spinor identity (A.12), including its sign and factor of two. It uses the Levi-Civita connection and the special three-dimensional curvature decomposition. It is not a dimension-independent gamma identity, and it is not automatically the curvature identity for a connection with torsion.

Recover the sphere sign from a spinor

The global round-sphere frame already constructed in the geometry lesson has spin connection $\Gamma_a=i\gamma_a/(2\ell)$. A nonzero constant column in that frame thus obeys $\nabla_a\psi=\lambda\gamma_a\psi$ with $\lambda=i/(2\ell)$. This is a differential ansatz, not yet a statement about a supergravity background.

Because the gamma matrices are covariantly constant and $\lambda$ is constant,

$$\mathscr R_{ab}\psi =\lambda^2(\gamma_b\gamma_a-\gamma_a\gamma_b)\psi =-2i\lambda^2\epsilon_{abj}\gamma_j\psi.$$

Contracting gives $-4i\lambda^2\gamma_i\psi$. A round sphere is isotropic, so $R_{ij}=R\delta_{ij}/3$, and (PC08.3) gives $-iR\gamma_i\psi/6$. Hence

$$R=24\lambda^2=-\frac6{\ell^2}. \tag{PC08.4}$$

The imaginary coefficient was squared, not replaced by its absolute square. This recovers the source’s sign using the spinor connection rather than repeating the metric trace. Replacing $R$ by a positive-sphere convention while keeping (PC08.2) unchanged would fail this check.

Read the geometric definitions with their hypotheses

For this entrance, prepare with Fierz identities, almost contact metric structures, integrability and adapted coordinates. The earlier spinor-curvature calculation does not require this section.

Source passage	What is, and is not, asserted	Canonical calculation
(B.1)–(B.2), printed 40/PDF 41	A one-form $\eta$, vector $\xi$ and endomorphism $\Phi$ obey pointwise almost contact identities; compatibility adds a metric.	The algebraic definitions, M02.1–M02.3
(B.3)–(B.6), printed 40–41/PDF 41–42	A metric-compatible connection preserving $\eta$ generally has torsion and a free one-form $W$.	The connection dictionary immediately below; spinor connection construction
(B.7), printed 41/PDF 42	The paper selects one representative of that freedom for its later background construction.	Adapted-frame derivatives, M05.4–M05.6, and the subsequent source solution
(B.8)–(B.11), printed 41–42/PDF 42–43	A projector condition preserves the transverse complex line and supplies local holomorphic coordinates.	The projector and the local coordinate argument, M03.9 and M03.12–M03.13
(B.12)–(B.14), printed 42–43/PDF 43–44, footnote 14	Compatible adapted metrics have a conformal transverse plane, with a fixed symmetric-product and volume convention.	The metric components, M04.2–M04.5, and overlaps, M04.6–M04.7
(B.15) and the three closing comments, printed 43/PDF 44	A local choice of a new distinguished direction is different from preserving a given direction; global extension is additional. Normality is stronger than the projected condition.	Scope of integrability and exit 3 below

Here $\eta(\xi)=1$ and $\Phi^2=-1+\xi\otimes\eta$; after metric compatibility, $\xi=\eta^\sharp$ is unit and $\Phi^\mu{}_\nu=\epsilon^\mu{}_{\nu\rho}\xi^\rho$ in the chosen orientation. A holomorphic vector obeys $\Phi X=iX$, whereas a holomorphic covector obeys $\Omega\Phi=i\Omega$. Raising a covector with the metric reverses this eigenvalue because $\Phi$ is antisymmetric. Do not use the two definitions interchangeably.

“Almost contact” does not require $\eta\wedge d\eta\ne0$. The latter is the separate contact condition. The existence of a compatible almost contact structure on an oriented Riemannian three-manifold, stated in B.1, does not imply the differential condition $\Phi\mathcal L_\xi\Phi=0$. The projector permits a nonzero longitudinal part of $\mathcal L_\xi\Phi$, while the normality condition quoted at the end of B.3 sets the full Lie derivative to zero.

In (B.12), $dz\,d\bar z$ denotes the symmetric tensor product divided by two. For $z=x-iy$ and transverse scale $c$, the real-coordinate determinant is $c^4$, while the absolute complex-coordinate determinant is $c^4/4$. Thus $\epsilon^{\tau z\bar z}=2i/c^2$ agrees with footnote 14’s $i/\sqrt{|g_{\rm complex}|}$. The factor of two belongs to the coordinate transformation, not to a change of orientation.

The compatible connection is additional data

The source writes $\widehat\Gamma^\mu{}_{\nu\rho}=\Gamma^\mu{}_{\nu\rho}+K^\mu{}_{\nu\rho}$, with $\nu$ the derivative index. Its family is

$$K_{\mu\nu\rho} =\eta_\mu\nabla_\nu\eta_\rho-\eta_\rho\nabla_\nu\eta_\mu -2W_\nu\Phi_{\mu\rho}. \tag{PC08.5}$$

Antisymmetry in the first and third indices gives metric compatibility, and contraction with $\eta^\rho$ gives $K^\mu{}_{\nu\rho}\eta^\rho=-\nabla_\nu\eta^\mu$. Thus $\widehat\nabla\eta=0$ for every $W$. This is the reason that metric and almost contact structure do not determine a unique compatible connection.

For a commuting spinor whose Hermitian direction is $\eta$, (B.6) reads

$$\widehat\nabla_\mu\zeta =\nabla_\mu\zeta-iW_\mu\zeta -\frac12(\nabla_\mu\eta_\nu)\gamma^\nu\zeta. \tag{PC08.6}$$

No Killing equation was assumed. The paper’s choice $W=-\eta\,*(\eta\wedge d\eta)/4$ in (B.7) is one representative, not a definition of all compatible connections. This $W$ is a geometric one-form, neither a superpotential nor an independently specified R gauge field. The physical sources still have to be reconstructed and patched. In particular, a curvature computed from $\widehat\nabla$ cannot be substituted for the Levi-Civita curvature in (PC08.3) without accounting for its torsion.

Exit calculations

1. Change the curvature without changing the Clifford algebra

Consider a local orthonormal frame on $S^2_\ell\times\mathbb R$, with $E_1,E_2$ tangent to the sphere and $E_3$ tangent to the line. Put $k=\ell^{-2}>0$. In the source convention the nonzero independent curvature component is $R_{12\,12}=-k$, and $R_{ij}=\operatorname{diag}(-k,-k,0)$. Compute the three matrices $\epsilon_{iab}\mathscr R_{ab}$ in two ways. Can this Levi-Civita connection admit a nonzero parallel spinor?

Hint 1

Include both $(a,b)=(1,2)$ and $(2,1)$ in the epsilon contraction. In (PC08.2), both antisymmetric values of $(c,d)$ contribute too.

Hint 2

The scalar curvature is $-2k$. A parallel spinor is annihilated by every curvature matrix, not just their scalar trace.

Solution

Equation (PC08.2) gives $\mathscr R_{12}=ik\gamma_3/2$, while $\mathscr R_{23}=\mathscr R_{31}=0$. Therefore the three contracted matrices are

$$(0,\ 0,\ ik\gamma_3).$$

On the Ricci side, $2R_{ij}-R\delta_{ij}=\operatorname{diag}(0,0,2k)$, giving exactly the same result. The flat line does not make the full connection flat.

A parallel spinor would satisfy $\gamma_3\psi=0$. Since $\gamma_3^2=1$, its only solution is $\psi=0$. This excludes a nonzero Levi-Civita parallel spinor. It does not exclude a spinor parallel under a suitably twisted connection or a solution to a sourced Killing equation.

2. Preserve the geometry while changing its compatible holonomy

Use a flat three-torus with coordinates $(\tau,x,y)$, $\tau$ of length $L$, $\eta=d\tau$, $\xi=\partial_\tau$ and $\Phi^\mu{}_\nu=\epsilon^\mu{}_{\nu\tau}$. Compare $W=0$ with $W=w\,d\tau$, where $w$ is real and constant. Determine the compatible vector and spinor connections and their parallel-transport matrices around the $\tau$ circle. What happens at $wL=\pi$ for a periodic spin structure? Does (B.7) choose this nonzero $W$?

Hint 1

Here $\nabla\eta=0$. Equation (B.4) translates the contorsion to the spin connection; on the two spinor components it is $-iW\gamma_1$. Parallel transport solves $\partial_\tau\psi=-\widehat\Gamma_{\tau,\rm spin}\psi$.

Hint 2

The vector connection is $-2W\Phi$. Its exponential rotates the transverse plane by twice the spinor phase. A periodic parallel spinor requires a fixed vector of the spin holonomy.

Solution

The two connection one-forms are

$$\widehat\Gamma_{\rm vector}=-2W\Phi,\qquad \widehat\Gamma_{\rm spin}=-iW\gamma_1,$$

so the respective holonomies are $\exp(2wL\Phi)$ and $\exp(iwL\gamma_1)$. Both connections are flat for constant $w$, but their global transport need not be trivial. At $wL=\pi$, the vector holonomy is the identity and the spin holonomy is $-I$. No nonzero spinor parallel for this connection is periodic; choosing the antiperiodic spin structure around this circle changes that conclusion.

The metric, unit direction and transverse complex structure did not change. The connection generally has torsion, despite its zero curvature in this example. Since $d\eta=0$, the paper’s selected representative (B.7) is $W=0$, not $w\,d\tau$. The example concerns the larger family (B.5); it does not silently replace the representative used in the physical reconstruction.

3. Rescale the unit direction and keep the projector

Start from any compatible triple $(g,\eta,\xi,\Phi)$ satisfying $\Phi\mathcal L_\xi\Phi=0$. For a real smooth function $f$, set $g'=e^{2f}g$, $\eta'=e^f\eta$, $\xi'=e^{-f}\xi$, and $\Phi'=\Phi$. Does the projected integrability condition survive? Is normality necessarily preserved? Work out a local example starting from flat $\mathbb R^3$, $\eta=d\tau$, with $f=\alpha(x^2+y^2)$ and nonzero real $\alpha$ of dimension inverse length squared. Find a coordinate $t$ with $\xi'=\partial_t$ and write the new adapted metric.

Hint 1

For a scalar function $a$, compute $\mathcal L_{a\xi}\Phi$ using the tensor formula (A.14), retaining the derivatives of $a$. Use $\Phi\xi=0$.

Hint 2

Here $f$ is independent of $\tau$. Try $t=e^f\tau$, but differentiate its full product when transforming $d\tau$. Set $z=x-iy$.

Solution

The compatibility identities survive, and

$$\mathcal L_{a\xi}\Phi =a\,\mathcal L_\xi\Phi-\xi\otimes(da\circ\Phi), \qquad \Phi\mathcal L_{\xi'}\Phi=e^{-f}\Phi\mathcal L_\xi\Phi=0.$$

The extra term is longitudinal, so the projector removes it. It generally prevents the unprojected derivative from vanishing: normality is not preserved by every such rescaling.

In the flat example, the only nonzero row of $\mathcal L_{\xi'}\Phi$ is the $\tau$ row, $(0,-e^{-f}f_y,e^{-f}f_x)$. It is nonzero away from the axis when $\alpha\ne0$, but its product with $\Phi$ vanishes. Taking $t=e^f\tau$ gives

$$\eta'=dt-t\,df,\qquad g'=(dt-t\,df)^2+e^{2\alpha|z|^2}dz\,d\bar z.$$

Thus the adapted coefficients are $h=-\alpha t\bar z$ and $c=e^{\alpha|z|^2}$. Dropping the transverse term $-t\,df$ would change the metric. This is a local coordinate construction on $\mathbb R^3$; no fixed-period global circle coordinate has been asserted.

The calculation tests the geometry, not a transformation rule for prescribed physical sources $A,V,H$. Those sources cannot simply be held fixed and assumed to preserve a supercharge after changing the metric.

Return to the main paper with three distinct objects identified: the spinor convention, the Levi-Civita curvature, and the chosen almost contact structure with its compatible connection. The appendices make their relations precise; they do not remove the separate source and global-patching questions.