Spinor and curvature conventions

Use this page to identify the convention package before moving an index, continuing an action, or importing a curvature coupling. The three-dimensional formulas follow Closset–Dumitrescu–Festuccia–Komargodski, arXiv:1212.3388v3. The four-dimensional Lorentzian spinor matrices follow Bilal, hep-th/0101055v1, with the volume-orientation translation stated below. The Euclidean continuations are the explicitly derived choices used in the linked lessons.

For an introduction, start with three-dimensional spinors or the independent four-dimensional Weyl-spinor lesson. The curvature calculation belongs to the round-sphere geometry lesson.

Choose the dimension and signature

Setting	Metric and coordinate labels	Spinor representation used here
Three-dimensional Euclidean	$\delta_{ab}$, $a=1,2,3$	Two-component lower columns; both partner spinors transform under $\operatorname{Spin}(3)=SU(2)$
Three-dimensional Lorentzian continuation example	$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1)$, $\mu=0,1,2$	Two-component Dirac column and its Lorentzian adjoint row
Four-dimensional Lorentzian	$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$, $\mu=0,1,2,3$	Undotted Weyl column and its dotted conjugate representation
Four-dimensional Euclidean continuation	$\delta_{ab}$, $a=1,2,3,4$	Independent chiral sectors of $SU(2)_L\times SU(2)_R$

These are kinematic choices. A curved background’s preserved supersymmetries require solving its spinor equations. The parent counts are four real Lorentzian Poincaré supercharges for both $3d\ \mathcal N=2$ and $4d\ \mathcal N=1$. Complexifying their variables does not change that count into eight real Lorentzian generators.

Matrices and contractions

Let the ordinary Pauli matrices be

$$\tau_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \tau_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad \tau_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

Use the following matrices with the indicated index placements:

Object	Three-dimensional Euclidean package	Four-dimensional Lorentzian package
Vector matrices	$(\gamma^a)_\alpha{}^\beta=(\tau_3,-\tau_1,-\tau_2)$	$\sigma^\mu_{\alpha\dot\alpha}=(I,-\tau_i)$
Partner matrix	The same $\gamma^a$ acts on either lower-column spinor	$\bar\sigma^{\mu\dot\alpha\alpha}=(I,+\tau_i)$
Clifford identity	$\{\gamma^a,\gamma^b\}=2\delta^{ab}I$	$\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu=2\eta^{\mu\nu}I$
Orientation-sensitive identity	$\gamma^a\gamma^b=\delta^{ab}I+i\epsilon^{abc}\gamma_c$, $\epsilon^{123}=1$	The ordered products use the displayed sigma pair and the Lorentz metric
Undotted scalar contraction	$\psi\chi=\psi^\alpha\chi_\alpha$	$\psi\chi=\psi^\alpha\chi_\alpha$
Other scalar contraction	$\widetilde\psi\chi=\widetilde\psi^\alpha\chi_\alpha$; no dotted index	$\bar\psi\bar\chi=\bar\psi_{\dot\alpha}\bar\chi^{\dot\alpha}$

The explicit raising and lowering matrices are the same numerical matrices in these two packages:

$$(\epsilon^{\alpha\beta})= \begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad (\epsilon_{\alpha\beta})= \begin{pmatrix}0&-1\\1&0\end{pmatrix}\equiv C, \qquad \psi^\alpha=\epsilon^{\alpha\beta}\psi_\beta .$$

The four-dimensional dotted tensors have the same entries. Thus $\psi^1=\psi_2$, $\psi^2=-\psi_1$, and $\psi\chi=\psi^TC\chi$ for lower undotted columns. Raising an index is a linear contraction, whereas complex conjugation is an antilinear operation.

In four dimensions the barred sigma matrix means

$$\bar\sigma^{\mu\dot\alpha\alpha} =\epsilon^{\dot\alpha\dot\beta}\epsilon^{\alpha\beta} \sigma^\mu_{\beta\dot\beta}.$$

Its bar specifies the other index placement. Entrywise conjugation is a different operation. In particular, the four-dimensional sigma pair connects two chiral index spaces; it cannot be substituted into a three-dimensional identity involving one mixed-index gamma matrix. P, (A.1)–(A.4); Bilal, (2.9)–(2.16).

For spinors of definite Grassmann parity $p_u,p_v$, the three-dimensional contractions obey

$$uv=-(-1)^{p_up_v}vu,\qquad u\gamma^a v=(-1)^{p_up_v}v\gamma^a u.$$

Consequently an odd $\psi$ has $\psi\psi=-2\psi_1\psi_2$, while a commuting $\zeta$ has $\zeta\zeta=0$. The undotted four-dimensional scalar contraction has the same parity rule. Its dotted contraction uses the reversed index placement displayed in the table. Adjoint conjugation reverses the order of odd factors.

For the four-dimensional volume tensor, choose $\epsilon_{0123}=+1$, hence $\epsilon^{0123}=-1$ in Lorentzian signature, and $\epsilon_{1234}=+1$ after Euclidean continuation. Define $\widetilde f^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}/2$. With the displayed Weyl matrices, the bosonic chiral coefficient is $[W^\alpha W_\alpha]_{\theta^2}=D^2-f_{\mu\nu}f^{\mu\nu}/2 +if_{\mu\nu}\widetilde f^{\mu\nu}/2$. This requires a translation of the upper-epsilon sign printed beside Bilal (4.49): for $f_{03}=E,f_{12}=B$, direct matrix multiplication gives $D^2+E^2-B^2-2iEB$. Under $x^0=-ix^4$ and $v_0=iv_4$, the dual contraction continues as $f_L\widetilde f_L=i f_E\widetilde f_E$. A positive Lorentzian theta coefficient consequently becomes the negative imaginary Euclidean coefficient in the action. The prepotential lesson derives the corresponding scalar and photon coefficients.

Two short matrix checks

In three dimensions,

$$\gamma^1\gamma^2 =\begin{pmatrix}0&-1\\1&0\end{pmatrix} =i\gamma^3,\qquad \gamma^2\gamma^1=-i\gamma^3.$$

The anticommutator vanishes, and $\gamma^1\gamma^2\gamma^3=iI$ fixes the orientation sign. In the four-dimensional Lorentzian package,

$$\sigma^1\bar\sigma^1=(-\tau_1)(+\tau_1)=-I, \qquad \sigma^0\bar\sigma^i+\sigma^i\bar\sigma^0=\tau_i-\tau_i=0.$$

The first result is the spatial metric sign. Multiplying $\sigma^1$ by itself instead would give $+I$ and would be the wrong index contraction for this Clifford identity. The complete derivations and rearrangement identities are in the three-dimensional spinor lesson and four-dimensional Weyl-spinor lesson.

Continuation and normalization dictionaries

The following choices include the coordinate order and every spatial factor of $i$.

Item	Three-dimensional continuation	Four-dimensional continuation
Time rotation	$t=-i\tau$	$x^0=-ix^4$
Euclidean coordinate order	$(x_E^1,x_E^2,x_E^3)=(\tau,x,y)$	$(x_E^1,x_E^2,x_E^3,x_E^4)=(x^1,x^2,x^3,x^4)$
Lorentzian matrices	$\Gamma^0=\gamma^1$, $\Gamma^1=i\gamma^2$, $\Gamma^2=i\gamma^3$	$\gamma_L^\mu=\begin{pmatrix}0&\sigma^\mu\\\bar\sigma^\mu&0\end{pmatrix}$
Euclidean matrices	The $\gamma^a$ above	$\gamma_E^4=\gamma_L^0$, $\gamma_E^i=-i\gamma_L^i$
Continued derivative	$\Gamma^\mu\partial_\mu\to i\gamma^a\partial_a$	$\gamma_L^\mu\partial_\mu\to i\gamma_E^a\partial_a$

The four-dimensional Euclidean chiral blocks are therefore

$$\sigma_E^a=(i\tau_1,i\tau_2,i\tau_3,I),\qquad \bar\sigma_E^a=(-i\tau_1,-i\tau_2,-i\tau_3,I).$$

They satisfy the sigma-pair identity with $\delta^{ab}$. The Lorentzian blocks are Bilal’s (2.16); this Euclidean pair is the site’s derived continuation. See the four-dimensional continuation for its independent-chirality test.

In the three-dimensional example, $\bar\psi_L=\psi_L^\dagger\Gamma^0$. Its continued row $\chi_E$ is independent of the column $\psi_E$:

$$\int d^3x_L\,\bar\psi_L(i\Gamma^\mu\partial_\mu-m)\psi_L \ \longrightarrow\ i\int d^3x_E\,\chi_E(\gamma^a\partial_a+m)\psi_E.$$

Thus $e^{iS_L}$ becomes $e^{-S_E}$ with this prescription. The continuation lesson derives the action and vector-bilinear factors.

To match that lesson’s kinetic normalization to the three-manifold paper’s $-i\widetilde\psi_P\gamma^\mu\nabla_\mu\psi_P$, use

$$\psi_E=\psi_P,\qquad \chi_E=-i\,\widetilde\psi_P^{\,T}C.$$

The extra phase is required by the action coefficient. The contraction-only definition $\chi_E=\widetilde\psi_E^{\,T}C$ instead gives $\widetilde\psi_E=-i\widetilde\psi_P$. An action’s interaction coefficients and transformations must be translated along with that field redefinition. Free-action normalization dictionary; P, (6.32).

Distinguish a momentum eigenvalue on a c-number mode from the action of a translation generator on a field. The positive-frequency mode obeys

$$i\partial_\mu e^{-ip\cdot x}=p_\mu e^{-ip\cdot x}, \qquad [P_\mu,\Phi(x)]=-i\partial_\mu\Phi(x).$$

The second equation states the field-adjoint action. Bilal’s superspace differential representation also uses $P_\mu=-i\partial_\mu$, as does the site’s three-dimensional component convention. These contexts must be identified before comparing signs; the mode calculation is not a different dimension-dependent choice of physical translation generator. The Lorentzian relation $\bar Q=Q^\dagger$ remains fixed. Bilal, (4.10).

Euclidean partner variables, a Hermitian conjugate, and an integration contour serve different purposes. For example, the three-dimensional antilinear spin map $\mathcal J\psi=C\psi^*$ satisfies $\mathcal J^2=-1$, so it does not define a nonzero single-spinor Majorana condition. In four Euclidean dimensions the two chiral sectors transform under independently chosen $SU(2)$ factors. Their independence does not supply a bosonic path-integral contour. These statements concern complexified kinematics, before specifying a quantum integration prescription.

Curvature and the round three-sphere

Let $e^a$ be an orthonormal coframe, $E_a$ its dual frame, and $\nabla E_b=\Omega^a{}_b E_a$. The torsion-free and curvature equations are

$$de^a+\Omega^a{}_b\wedge e^b=0,\qquad \mathcal R^a{}_b=d\Omega^a{}_b+\Omega^a{}_c\wedge\Omega^c{}_b.$$

The three-manifold paper puts the input frame index first:

$$\omega_{\mu a}{}^b=\Omega_\mu{}^b{}_a,\qquad \mathcal R^b{}_a=\frac12R_{\mu\nu a}{}^b\,dx^\mu\wedge dx^\nu.$$

With Euclidean frame indices lowered, $\omega_{\mu ab}=-\Omega_{\mu ab}$. Its spinor derivative is

$$\nabla_\mu\psi =\left(\partial_\mu+\frac14\Omega_{\mu ab}\gamma^a\gamma^b\right)\psi =\left(\partial_\mu-\frac i4\omega_{\mu ab}\epsilon^{abc}\gamma_c\right)\psi.$$

These are the same connection in two index arrangements. The source definitions are P, (A.8)–(A.11); their derivation is in covariant spinor derivatives.

For the global oriented sphere frame of radius $\ell$,

$$de^a=\frac1\ell\epsilon_{abc}e^b\wedge e^c,\qquad \Omega_{ab}=\frac1\ell\epsilon_{abc}e^c,\qquad \mathcal R^a{}_b=\frac1{\ell^2}e^a\wedge e^b.$$

The compact structure equation sums both orders of $b,c$; in particular $de^1=2e^2\wedge e^3/\ell$. The source curvature components and trace are

$$R_{cd a}{}^b =\frac{\delta_c^b\delta_{da}-\delta_d^b\delta_{ca}}{\ell^2}, \qquad R_{ab}=\sum_{c=1}^3R_{ac b}{}^c =\frac{1-3}{\ell^2}\delta_{ab}.$$

Consequently,

$$\boxed{R_{ab}=-\frac2{\ell^2}\delta_{ab},\qquad R=-\frac6{\ell^2}.}$$

This verifies the scalar sign using the declared contraction. A convention that contracts this same commutator tensor with the first two indices reversed gives $R_+=+6/\ell^2$. If $R_P=-R_+$, an action coefficient must also change sign when translated:

$$c_P R_P=c_+R_+,\qquad c_+=-c_P.$$

For example, the conformal free-chiral coefficient is $-R_P/8=+R_+/8=3/(4\ell^2)$. The numerical curvature sign alone is insufficient to translate an action.

The same sphere frame gives $\nabla_a=E_a+i\gamma_a/(2\ell)$ on spinors. The rigid background $A=V=C=0$, $H=-i/\ell$ then admits constant-component solutions of $\nabla_a\zeta=i\gamma_a\zeta/(2\ell)$. The sphere solution supplies the complete background and global argument. Curvature by itself does not fix that source choice.

Boundaries of the dictionary

The curvature entries above belong to a three-dimensional sphere and P’s exact trace convention. A four-dimensional source must state its own curvature definition before its couplings are compared. A one-dimensional quantum-mechanics model likewise specifies its time algebra and fermionic variables directly; the spatial Clifford triples here are not a prerequisite for that setting.

In component supersymmetry calculations, keep parameter parity explicit as well. The three-dimensional lessons use commuting labels multiplying odd operations $\delta$; a physical even variation is $\Delta=\rho\delta$ with an independent odd parameter $\rho$. When comparing an odd-parameter component formula, translate the graded and ordinary product rules with the parameters. The matrix and curvature identities in this reference do not change that bookkeeping.