Paper guide: Section 2, currents and rigid backgrounds

Section 2 explains why supersymmetry on a curved space needs more sources than a metric. Its route is from conserved operators, through their linear source couplings, to the transformation of a gravitino. Freezing those sources then turns the gravitino transformation into an equation for a surviving supersymmetry parameter.

Read Section 2 of Supersymmetric Field Theories on Three-Manifolds, by Cyril Closset, Thomas T. Dumitrescu, Guido Festuccia and Zohar Komargodski, in arXiv:1212.3388v3, dated 22 October 2013. The paper entrance records the bibliography and version history. Printed pages 6–10 are PDF viewer pages 7–11.

For the current/source passages, prepare with superspace, source variations and stress-tensor improvements. The later Rigid construction part of this guide uses the completed R-multiplet lesson, background-source construction, background-field dictionary and Killing spinor equations.

The setting is a three-dimensional theory with a conserved continuous R-symmetry and $\mathcal N=2$ supersymmetry: four real Lorentzian Poincaré supercharges before continuation. Euclidean spinor families are independent and complex. The matter gauge group is unrestricted; its rank is not a supercharge count. A particular source profile may preserve fewer symmetries than the parent theory.

Read from operators to sources

The symbols in (2.1) describe constraints on operator superfields, not equations of motion for gravity. The components in (2.2) tell you which observable each source will probe. The improvement (2.3) changes a complete current multiplet when the R-current mixes with a flavor current. It is not a change of only one coefficient in a covariant derivative.

Read in v3	Question to resolve	Canonical calculation
(2.1)–(2.2), printed 6/PDF 7	How do a linear-multiplet constraint and a conserved vector fit together?	R-multiplet constraints and components; the simpler linear package
(2.3), printed 6/PDF 7	Which operators change when the R-current is improved?	Flavor improvement of the components and the associated curvature coupling
(2.4)–(2.7), printed 7/PDF 8	What does the full superspace integral pair, and what has Wess–Zumino gauge removed?	The two superfield expansions
(2.8)–(2.9), printed 7/PDF 8	Which source transformations are gauge redundancies, and which fields are dual field strengths?	The dual background fields
(2.10), printed 8/PDF 9	Which combination actually sources the R-current?	The linear component coupling

Equation (2.6) restricts the supergravity gauge-parameter superfields so that (2.4) is gauge invariant. It is distinct from the current conservation conditions in (2.1). The paper notes a larger gauge freedom for the conformal supergravity multiplet; do not impose that special choice on an arbitrary nonconformal R-multiplet.

Keep this notation dictionary beside the source:

Paper notation	Meaning in the lessons
$\mathcal R_\mu$, $\mathcal R_{\alpha\beta}$	The current superfield and its symmetric bispinor form; neither is the scalar curvature.
$\mathcal J^{(Z)}$, $J^{(Z)}$, $j_\mu^{(Z)}$	The scalar linear superfield, its bottom scalar component, and its conserved vector component. The lessons call them $\mathcal J_Z,J_Z,j_{Z\mu}$.
$\mathcal H_\mu$ in (2.4)–(2.7)	The metric superfield; it is different from the scalar dual field strength $H$.
$h_{\mu\nu}$	Half the infinitesimal metric change: $g_{\mu\nu}=\delta_{\mu\nu}+2h_{\mu\nu}$. Thus $-\tfrac12T^{\mu\nu}\delta g_{\mu\nu}=-T^{\mu\nu}h_{\mu\nu}$.
$\psi_\mu,\widetilde\psi_\mu$	Background gravitini, capitalized as $\Psi_\mu,\widetilde\Psi_\mu$ in the lessons to distinguish them from matter fermions.
$D_\alpha,\widetilde D_\alpha$	Superspace differential operators, not vector auxiliaries or spacetime covariant derivatives.
$C_\mu,B_{\mu\nu}$	Potentials whose dual field strengths are $V^\mu=-i\epsilon^{\mu\nu\rho}\partial_\nu C_\rho$ and $H=\tfrac i2\epsilon^{\mu\nu\rho}\partial_\mu B_{\nu\rho}$ in the linearized Cartesian formulas.

The metric is real. Other sources may be complex, but the R and central-charge gauge parameters remain real, as specified between (2.9) and (2.10). “Real linear multiplet” names the Lorentzian reality structure and superspace constraint; it does not identify independent Euclidean tilded spinors with complex conjugates.

There are three useful source combinations:

$$\begin{array}{c|c} A_\mu-V_\mu & \text{top component in (2.7)}\\ A_\mu-\frac32V_\mu & \text{coefficient of }j_R^\mu\text{ in (2.10)}\\ A_\mu-\frac12V_\mu & \text{connection in the later matter derivative} \end{array} \tag{PC01.1}$$

The last belongs to Section 6, not to a redefinition of (2.10); see the matter derivative. The reconstruction below accounts for the first two.

Restore the missing half of the current source

Starting from (2.2), (2.4) and (2.7), recover the coefficient $-3/2$ multiplying $j_R\cdot V$ in (2.10). Work on a Cartesian patch with integrated surface terms zero. Keep only terms that can multiply the R-current or its derivative. Define the odd-coordinate bilinear $s=\theta\widetilde\theta$ and the top monomial $Q=\theta^2\widetilde\theta^2$. The inherited conventions give $s^2=-Q/2$ and $\int d^4\theta\,Q=1$.

Hint 1

Besides the bottom component $j_{R\mu}$, the relevant part of (2.2) is $s\,\epsilon_{\mu\nu\rho}\partial^\nu j_R^\rho$. In (2.7), pair it with $-isC_\mu/2$.

Hint 2

Do not stop at the top component of the source superfield. After the Berezin integral, use $\int C\cdot(\nabla\times j_R)=\int j_R\cdot(\nabla\times C)$ with the stated boundary condition, and then $\nabla\times C=iV$.

Solution

The relevant products in $2\int d^4\theta\,\mathcal R^\mu\mathcal H_\mu$ are

$$2\int d^4\theta\left[ \frac12j_R^\mu Q(A_\mu-V_\mu) -\frac i2s^2 C^\mu\epsilon_{\mu\nu\rho}\partial^\nu j_R^\rho \right] =j_R\cdot(A-V)+\frac i2C\cdot(\nabla\times j_R).$$

The sign in the second term used the square of an odd-coordinate bilinear; replacing the coordinates by commuting variables would lose it. Integration by parts gives

$$\frac i2\int C\cdot(\nabla\times j_R) =\frac i2\int j_R\cdot(iV) =-\frac12\int j_R\cdot V.$$

Adding this contribution yields $j_R\cdot(A-3V/2)$. Both the outer factor two in (2.4) and the factor one-half in (2.7) matter. For a physical boundary, the corresponding surface term must be retained rather than silently discarded.

The complete component multiplication, including the stress, central-charge and odd gravitino terms, is in the current/source derivation. This short calculation reconstructs one of its source-sensitive coefficients.

From source transformations to a rigid symmetry

Now use the four rigid-construction lessons listed above. The second half of Section 2 is a controlled completion of the transformation law, not a presentation of a complete nonlinear gravity action.

Read in v3	Question to resolve	Canonical calculation
(2.11), printed 8/PDF 9	Why must a supersymmetry transformation be accompanied by a transformation restoring Wess–Zumino gauge?	The linearized reconstruction
(2.12)–(2.13), printed 9/PDF 10	Why does making the spinor parameter local produce a derivative in the gravitino variation?	Local parameters and the gravitino source
(2.14)–(2.15), printed 9–10/PDF 10–11	Which covariant terms can survive when the gravitini are zero?	The supplied transformation
(2.15), then (1.4)–(1.5)	Why must a zero source also have zero variation?	Preserving the frozen background and both spinor equations

One factor can be checked before using a spin connection. With odd local parameters $\varepsilon,\widetilde\varepsilon$, use an even variation $\Delta$ and retain the displayed current/source order. The local Noether remainder is $\mathcal S^\mu\partial_\mu\varepsilon- \widetilde{\mathcal S}^{\mu}\partial_\mu\widetilde\varepsilon$. The source terms are $-\mathcal S^\mu\Psi_\mu/2+ \widetilde{\mathcal S}^{\mu}\widetilde\Psi_\mu/2$. Consequently, $\Delta\Psi_\mu=2\partial_\mu\varepsilon$ and $\Delta\widetilde\Psi_\mu=2\partial_\mu\widetilde\varepsilon$ cancel the remainder at zero sources. This is the normalization step from (2.12) to (2.13). Factoring out an external odd constant leaves the commuting spinor label used in the Killing equation.

The dimensional argument in (2.14) then uses $[\Psi]=1/2$, $[\zeta]=-1/2$ and $[A]=[V]=[H]=1$. Within the local supergravity construction under discussion, covariance completes the linear derivative to $\nabla_\mu$; further terms proportional to gravitini disappear on the bosonic background. This does not license eliminating prescribed $H$ or $V$ by gravitational field equations. It also does not show that an arbitrary matter action is invariant: that action must be coupled to the same source multiplet.

Exit calculation: hold one source combination fixed

Can a deformation leave $A-3V/2$ zero and nevertheless remove every solution of the rigid spinor equations? Consider a Cartesian patch with length coordinates $(x^1,x^2,x^3)$, orientation $\epsilon_{123}=+1$, and

$$g_{\mu\nu}=\delta_{\mu\nu},\qquad C=i v x^2\,dx^3,\qquad V=v\,dx^1,\qquad A=\frac{3v}{2}\,dx^1,\qquad H=0, \tag{PC01.2}$$

where $v$ is a nonzero real constant of mass dimension one. The components of $C$ have dimension zero. Verify the dual-field relation and $A-3V/2=0$. Then use (2.15) to test for a nonzero label of either R-charge. This is a local source test; no compact global potential or flux condition is assumed.

Hint 1

For charge $q_R=\pm1$, move every term of the Killing equation to the left: $\mathscr D_\mu=\partial_\mu-iq_RA_\mu+iq_RV_\mu+ \tfrac{q_R}{2}\epsilon_{\mu\nu\rho}V^\nu\gamma^\rho$. Use $\gamma_a\gamma_b=\delta_{ab}+i\epsilon_{abc}\gamma_c$.

Hint 2

It is unnecessary to solve three differential equations individually. Commute $\mathscr D_2$ with $\mathscr D_3$ and ask whether the resulting matrix has a kernel.

Solution

The only nonzero derivative of $C$ is $\partial_2C_3=iv$, so $-i(\nabla\times C)_1=v$. Its divergence is zero, and the displayed R-current source vanishes. Nevertheless the complete operators are

$$\mathscr D_1=\partial_1-\frac{iq_Rv}{2},\qquad \mathscr D_2=\partial_2-\frac{q_Rv}{2}\gamma_3,\qquad \mathscr D_3=\partial_3+\frac{q_Rv}{2}\gamma_2. \tag{PC01.3}$$

Their compatibility condition is

$$[\mathscr D_2,\mathscr D_3]\varepsilon_{q_R} =-\frac{v^2}{4}[\gamma_3,\gamma_2]\varepsilon_{q_R} =\frac{iv^2}{2}\gamma_1\varepsilon_{q_R}=0. \tag{PC01.4}$$

Since $\gamma_1^2=1$ and $v\ne0$, this forces $\varepsilon_{q_R}=0$. The result is the same for both charges because $q_R^2=1$. At $v=0$ all sources vanish and each family instead has two constant complex solutions. The source change therefore removes all solutions of these rigid equations, although the metric and the combination $A-3V/2$ are unchanged.

The lesson is about the full source profile: the central-charge coupling $j_Z\cdot C$ in (2.10) still changes, and the gravitino variation depends explicitly on $V$. Vanishing of one current’s coefficient is not a supersymmetry criterion. We have tested the specified new-minimal rigid equations, not classified possible additional conformal-supergravity transformations of a special SCFT.

You can now follow the paper’s passage from currents to background-preserving equations without equating a superfield constraint, a matter equation of motion and a frozen-source condition. Continue with the Section 3 guide to construct geometric bilinears and test what they retain about a spinor; its canonical starting point is geometry from one spinor.