Paper guide: Section 4, transverse holomorphic geometry

Section 3 turned a spinor into geometric tensors. Section 4 asks the reverse question: when can that geometry support a Killing spinor, and which background fields make it possible? Read its existence statement with the quantifiers intact. The construction chooses compatible sources and bundle data. It does not say that every source profile on a permitted metric preserves supersymmetry.

Use P v3, Section 4, printed pages 12–16, PDF pages 13–17. Begin with integrability and transverse holomorphic geometry. The Section 3 companion helps separate the preceding pointwise algebra from the differential conditions introduced here.

The setting remains a real positive three-dimensional metric and the complex background fields of an $\mathcal N=2$ theory. The Lorentzian parent has four real Poincaré supercharges. This section constructs one positive-charge Euclidean spinor direction; it does not assume an independent opposite-charge solution. No dynamical gauge group or gauge rank is chosen.

Read the necessity argument first

Read (4.1), then (4.4)–(4.7). Differentiating the normalized spinor direction removes its norm and the scalar R-connection term. The transverse symmetric traceless derivative must vanish. The result is

$$\Phi^\mu{}_\nu(\mathcal L_\xi\Phi)^\nu{}_\rho=0, \qquad \xi=\eta^\sharp. \tag{PC03.1}$$

The projection is essential: the unprojected Lie derivative can have a component along $\xi$. The canonical calculation displays that remainder. Neither a Killing vector nor a contact form is required.

Next read (4.8)–(4.12), following the reference to Appendix B. The additional local input is the two-dimensional complex-coordinate existence theorem. It converts a preserved transverse complex structure into charts, rather than following merely from the algebra $\Phi^2=-1+\xi\otimes\eta$.

P v3 landmark	Reading question	Canonical calculation
(4.1), printed 12/PDF 13	Which sources are fixed in the differential equation?	The Killing-spinor equations
(4.2)–(4.3), printed 13/PDF 14	Which shifts preserve this spinor, and which constraint remains?	Source freedom and its divergence, M05.14–M05.16
(4.4)–(4.7), printed 13/PDF 14	Why is there no transverse shear?	The bilinear derivative and projector, M03.3–M03.9
(4.8)–(4.12), printed 14/PDF 15	What does an adapted overlap preserve?	Coordinate transitions, M04.6–M04.7
(4.13), printed 14/PDF 15, footnote 7	Why use a metric connection with torsion?	Compare its preservation of $\eta$ with the Levi-Civita connection, M05.6; they are different connections.
(4.14), printed 15/PDF 16	Which combination becomes the reduced R connection?	Component reconstruction, M05.7–M05.9; keep the source's $\widehat A$ distinct from $A$.
Bundle paragraph and (4.15)–(4.16), printed 15/PDF 16	What cancels the adapted frame phase?	The bilinear bundle constraint, M06.8–M06.9
(4.17)–(4.19), printed 15–16/PDF 16–17	How does the charge-two equation determine a connection locally?	The reconstruction below, followed by the full overlap equation, M06.6–M06.7
(4.20)–(4.21), printed 16/PDF 17, footnotes 8–9	In which frame is the solution a simple column, and what flat data remain?	The explicit adapted spinor and global transport

For (4.2), “holomorphic vector” means the $+i$ eigenspace of $\Phi$, not that every component is a holomorphic function. The separate equation $\nabla_\mu(U^\mu+\kappa\xi^\mu)=0$ cannot be dropped. An allowed shift generally changes physical sources; it is not an ordinary R-gauge transformation.

Construction and patching

Before this part, work through adapted coordinates, background reconstruction and global patching. We use their derivations to read the compact source formulas, then test a new overlap.

The paper’s $\widehat\nabla$ preserves both the metric and $\eta$. In general it has torsion; it is not a replacement for the Levi-Civita derivative in the original Killing equation without the compensating source change. Equation (4.14) defines

$$\widehat A_\mu=A_\mu-\frac12 (2\delta_\mu{}^\nu-i\Phi_\mu{}^\nu)V_\nu +\frac i2\eta_\mu H-W_\mu. \tag{PC03.2}$$

Here $W_\mu$ is the geometric one-form in (4.13), not a superpotential or a Wilson loop. The source shifts (4.2) leave $\widehat A$ unchanged. On the selected spinor line the equation becomes $(\widehat\nabla-i\widehat A)\zeta=0$. The resulting reduced holonomy explains the R-symmetry compensation.

Keep three quantities in (4.15)–(4.21) separate: the covector component $p=P_{\bar z}$, the metric scale $c$, and the local spinor multiplier $\chi$. In the oriented frame $e^2-ie^3=c\,dz$, with $z=x-iy$, the canonical contraction gives

$$\zeta=\chi\binom10,\qquad P=c\chi^2d\bar z,\qquad s=\frac pc=\chi^2. \tag{PC03.3}$$

This fixes the component normalization directly. To compare literally with the paper’s prefactor $s=pg^{-1/4}/\sqrt2$, use the positive magnitude of the complex-coordinate metric determinant:

$$g_{\rm C}=\left|\det g_{(\tau,z,\bar z)}\right|=\frac{c^4}{4}, \qquad g_{\rm R}=\det g_{(\tau,x,y)}=c^4. \tag{PC03.4}$$

Then $pg_{\rm C}^{-1/4}/\sqrt2=p/c$. This is an explicit density conversion for reading (4.15); the two determinants are not equal. Their logarithmic derivatives agree, so using $g_{\rm R}$ inside $d\log g$ does not alter (4.19). A constant normalization of a homogeneous spinor solution is arbitrary, but it must also be applied to its bilinear.

Under $z'=f(z)$, write $f'=|f'|e^{i\alpha}$. Before an R transformation, $p'=p/\overline{f'}$ and $c'=c/|f'|$, hence $s'=e^{i\alpha}s$. The square root in (4.21) still needs a spin lift. Compensating with an R phase makes the selected spinor a scalar under the combined transformation. It does not trivialize each bundle separately.

Worked overlap: a cylinder written as an annulus

Take $M=\mathbb R_\tau\times\mathbb R_u\times S^1_v$, with $v\sim v+2\pi L$, $L>0$, and

$$g=d\tau^2+du^2+dv^2,\qquad A=V=H=0.$$

Choose the periodic spin structure in the global coframe $(d\tau,du,dv)$ and the trivial R line with zero connection. The constant column $u_+=(1,0)^{\mathsf T}$ is a global solution. Put $w=u-iv$ and use the adapted transverse coordinate

$$z=L e^{w/L}=r e^{-i\varphi},\qquad r=L e^{u/L},\quad \varphi=v/L. \tag{PC03.5}$$

The cylinder is the punctured plane as a complex manifold. Its metric becomes $g=d\tau^2+(L/r)^2(dx^2+dy^2)$, where $z=x-iy$. Thus $h=0$, $c=L/r$ and $\eta=d\tau$. This is still a flat cylindrical metric; it is not the Euclidean plane metric and the omitted origin is not a finite-distance point.

The adapted coframe $(c\,dx,c\,dy)$ rotates by $\varphi$ relative to $(du,dv)$. Its Cartan connection and spin connection are

$$\Omega_{23}=d\varphi,\qquad \Gamma=\frac i2\gamma_1d\varphi,\qquad \zeta_z=e^{-i\varphi/2}u_+. \tag{PC03.6}$$

The derivative of the phase cancels $\Gamma\zeta_z$, so the same source $A=0$ still works. Equivalently the reconstruction formula gives $A=\Omega_{23}/2-i\,d\log\chi=d\varphi/2-d\varphi/2=0$. The phase also follows from $f'=z/L$: $s_z=e^{-i\varphi}$.

Now retain the actual overlap. Two branches of the angle obey $\varphi_j=\varphi_i+2\pi n$. Their spinor components satisfy $\zeta_j=(-1)^n\zeta_i$ although the orthonormal coframe is single-valued. This is the lift of the rotating frame for the original periodic cylinder spin structure. Requiring periodic components in this new frame would select different spin data.

One may instead use patchwise R changes $e^{i\varphi_i/2}$ to make the spinor components constant. They produce

$$\zeta'_i=u_+,\qquad A'_i=\frac12d\varphi_i, \qquad h'_{ji}=(-1)^n. \tag{PC03.7}$$

The R transition cancels the spin sign. This is a valid change of local description, not a single-valued R-gauge function on the whole annulus. Its R holonomy is still trivial: at the seam, the return map is $h'^{-1}\exp(i\int_0^{2\pi}A')=(-1)^{-1}e^{i\pi}=1$. Using the line integral alone would wrongly assign holonomy $-1$ to the original zero connection.

The coordinate formula, local KSE and global seam now agree. This is a concrete way to read “scalar supercharge” after (4.21). Footnote 8 retains flat connection data; it does not permit discarding them because the curvature vanishes.

Exit calculations

1. Hold the frame fixed and change the spin data

In the annulus frame (PC03.6), impose a spin seam $\varepsilon=\pm1$ and an R seam equal to one. Set $A=\nu\,d\varphi$ with real constant $\nu$, keeping $V=H=0$. For which $\nu$ does the upper-component solution return with the prescribed seam? Which choice reproduces the original cylinder with $A=0$?

Hint 1

The upper equation is $[\partial_\varphi+i/2-i\nu]\chi=0$.

Hint 2

Compare $\chi(2\pi)/\chi(0)$ with $\varepsilon$, and quotient only by single-valued gauge functions $e^{im\varphi}$.

Solution

The solution is $\chi=e^{i(\nu-1/2)\varphi}\chi_0$. Its return factor is $-e^{2\pi i\nu}$, so

$$\nu\in\mathbb Z\quad(\varepsilon=-1),\qquad \nu\in\mathbb Z+\tfrac12\quad(\varepsilon=+1).$$

Integer gauge shifts of $\nu$ preserve these classes. The original cylinder uses $\varepsilon=-1$ in this rotating frame and $\nu=0$. Choosing periodic components while retaining $\nu=0$ loses the solution. Choosing $\varepsilon=+1$, $\nu=1/2$ restores a solution on different separately specified spin/R data. This is distinct from (PC03.7), which changed the R seam along with the local connection.

2. A flat metric need not have trivial connection holonomy

On the punctured plane choose $\eta=d\tau$, $c=(r/L)^\beta$ and $\chi=1$, with real $\beta$. Reconstruct $A$ for $V=H=0$, check its curvature away from the origin and compute its holonomy for trivial R seam. For $\beta\ne-1$, identify the transverse cone metric. What changes at $\beta=-1$?

Hint 1

Use $A=(\partial_y\log c\,dx-\partial_x\log c\,dy)/2$ and $d\varphi=(x\,dy-y\,dx)/r^2$.

Hint 2

Integrate $dR=c\,dr$ locally. Curvature zero on the annulus does not fill in the omitted origin or make every closed one-form exact.

Solution

One finds $A=-\beta\,d\varphi/2$ and $dA=0$ for $r>0$. The R holonomy is $e^{-i\pi\beta}$. It can be removed by a single-valued unit-charge gauge function precisely when $\beta\in2\mathbb Z$; local exactness alone is insufficient.

For $\beta\ne-1$, let $R=L(r/L)^{\beta+1}/(\beta+1)$. Then

$$c^2(dr^2+r^2d\varphi^2)=dR^2+(\beta+1)^2R^2d\varphi^2.$$

The circumference-to-radial-distance ratio is $2\pi|\beta+1|$. These local flat metrics may have a conical completion; the lesson has not included its apex. At $\beta=-1$, use $u=L\log(r/L)$ instead, obtaining $du^2+L^2d\varphi^2$: a cylinder, with the origin at infinite distance. The constant-spinor description then has $A=d\varphi/2$. To represent the original zero-connection cylinder rather than new R holonomy, retain the nontrivial R seam in (PC03.7).

3. A local source repair that cannot be periodic

On a flat three-torus with lengths $L_\tau,L_x,L_y$, let $\eta=d\tau$ and use the reference spinor $u_+$. In the source shifts (4.2), prescribe $\kappa=\kappa_0\cos(2\pi\tau/L_\tau)$ with nonzero $\kappa_0$ of inverse-length dimension. Can a smooth periodic holomorphic vector $U=w(\tau,x,y)\partial_z$ repair the divergence condition? Compare with a contractible patch.

Hint 1

The condition is $\partial_\tau\kappa+\partial_z w=0$, where $\partial_z=(\partial_x+i\partial_y)/2$.

Hint 2

Integrate this equation over the transverse two-torus at fixed $\tau$. Periodic derivatives have zero integral.

Solution

Integration gives $L_xL_y\,\partial_\tau\kappa=0$, which the prescribed nonconstant cosine violates. No smooth periodic $w$ can repair it. Equivalently its Fourier mode has nonzero longitudinal momentum and zero transverse momentum; the transverse derivative cannot cancel that mode.

On a contractible patch, $w=-z\,\partial_\tau\kappa$ works because $\partial_z z=1$. It is not periodic on the torus. Both attempted shifts cancel algebraically in the selected spinor equation, but the globally periodic proposal fails the independent field-strength constraint. This obstruction appears before questions about graviphoton flux quantization or globally patched potentials.

The section’s conclusion now has concrete content: transverse integrability permits a construction with compatible sources, and the local scalar description includes compensating bundle transitions. To ask for an additional solution on those same sources, continue with the Section 5 guide.