Paper guide: Section 6, curved multiplets and actions

Section 6 turns a supplied supersymmetric background into transformations and action densities. Its long component lists become easier to use when you keep three questions separate: what algebra must close, which components form a multiplet, and which neutral combination can be integrated?

Begin with the curved chiral multiplet and the curved algebra. Open P v3, Section 6, printed pages 20–29, PDF pages 21–30. Read (6.1)–(6.2), then the chiral example (6.7)–(6.8), before returning to the general list (6.3)–(6.6). This gives the list a purpose before you track all its components.

The parent is three-dimensional $\mathcal N=2$, with four real Lorentzian Poincaré supercharges. The paper uses its Euclidean complexification; the supplied Killing spinors determine the preserved transformations. Its explicit gauge formulas in this section are Abelian. The worked calculation below uses a background flavor $U(1)$ and no dynamical gauge group.

Read the component names literally

Paper notation	What to keep distinct
$C$ in (6.3)	The bottom scalar of a general multiplet, not the graviphoton one-form $C_\mu$
$a_\mu,\sigma,D$ in a general multiplet	Components, not automatically a gauge connection, a real mass and an independent auxiliary
$a_\mu,\sigma,D$ in (6.18)	Entries of a vector multiplet in Wess–Zumino gauge
$A_\mu,V_\mu,H$	Fixed rigid sources; $V_\mu$ is not the vector superfield $\mathcal V$
$r,z,q$	Scalar R-charge, central-charge eigenvalue and Abelian gauge or flavor charge; $z$ has mass dimension one
$\widetilde\Phi,\widetilde\psi,\widetilde F$	Independent Euclidean partners; the tilde alone imposes no integration contour
$D_\mu$ and the modified derivative in (6.31)	A component derivative and a useful rearrangement of the action, not interchangeable prescriptions

For a component $X$ with charges $(r_X,z_X,q_X)$, (6.24) means

$$D_\mu X=\left[\nabla_\mu-ir_X\mathcal A_\mu-iz_XC_\mu-iq_Xa_\mu\right]X, \qquad \mathcal A_\mu=A_\mu-\frac12V_\mu . \tag{PC05.1}$$

The current-source combination is instead $A-3V/2$. Equation (6.31) does not replace $\mathcal A$ everywhere: it defines $\mathscr D_\mu X=D_\mu X+i r_{0X}V_\mu X$, with $r_{0\phi}=1/2$, $r_{0\psi}=-1/2$ and opposite partner values. These fixed numbers organize the canonical kinetic action. They are not a determination of an interacting R-charge.

Keep the charge of the actual component or product in every derivative. In (6.8), $D_\mu(\widetilde\zeta\gamma^\mu\psi)$ differentiates the label as well as the matter field and carries R-charge $r-2$. Also retain $z-rH$: it is a source-dependent coefficient, not permission to make the central-charge eigenvalue $z$ a function of position.

The commuting labels generate odd operations. Their bracket in (6.1) is an anticommutator. The paper motivates and checks that algebra, and gives a dimensional-reduction construction in Appendix D; it does not claim to start from a fully available gauge-fixed three-dimensional supergravity component action.

Systematic actions

For this part prepare with F- and D-terms, the vector multiplet, gauge and matter actions, flavor backgrounds, CS and FI terms, and the complete sphere action. The following map covers the whole section; its targets distinguish a complete derivation from a selected component check.

P v3 landmark	What to establish	Canonical teaching target
(6.1)–(6.2), printed 21/PDF 22	Isometry plus R and central transformations	The algebra and its scope, R07.5–R07.9
(6.3)–(6.6), printed 22/PDF 23	The general package and its charge-dependent rules	The component package; the following scalar calculation checks the needed part of (6.5)
(6.7)–(6.10), printed 23/PDF 24	Constrain the general package to a chiral or antichiral multiplet	The chiral rules and their subsequent closure calculation
(6.11)–(6.12), printed 23–24/PDF 24–25	A neutral linear multiplet has a conserved vector component	Its embedded components, then the field-strength example below
(6.13)–(6.14), printed 24/PDF 25	Neutrality and the background completion make an invariant integral	D-term prescription and F-term charge conditions
(6.15), printed 24/PDF 25	Complete an R-current improvement nonlinearly	Mixed coupling to the R source
(6.16), printed 25/PDF 26	Multiply components with graded signs and summed charges	Chiral product and charged product expansion
(6.17)–(6.22), printed 25–26/PDF 26–27	Gauge freedom, its compensation and the field-strength multiplet	Vector rules, gauge closure, and the linear field strength
(6.23)–(6.25), footnote 10, printed 27/PDF 28	Charged matter requires the gauge completion	Expand the gauge-invariant product, including $\widetilde\Phi e^{-2q\mathcal V}\Phi$
(6.26), printed 28/PDF 29	A frozen source must itself preserve the retained labels	Freeze the whole flavor multiplet, A05.1
(6.27)–(6.29), printed 28/PDF 29	FI, gauge CS and gauge–R CS have different completions	Curved CS/FI construction, A06.6–A06.7
(6.30), printed 29/PDF 30	The field-strength square supplies Yang–Mills terms	Complete gauge action
(6.31)–(6.32), printed 29/PDF 30	Assemble the matter action and check its limits	Complete matter action and the direct sphere variation

The general multiplet’s $16+16$ component count is an off-shell package, not a particle count. Similarly, the scalar and highest entries in the embeddings (6.7), (6.9) are composites of the chiral fields and sources, not additional dynamical fields.

Three checks should accompany any use of the action list. The F-term in (6.14) requires bottom charges $(r,z)=(2,0)$ and the opposite antichiral assignment. The D-term in (6.13) requires a neutral general multiplet and includes both $a\cdot V$ and $\sigma H$. Finally, “variation is a divergence” requires global fields and a vanishing boundary flux to imply invariance of the action. None of these local prescriptions chooses a quantum regulator, compact CS level lattice or Euclidean integration contour.

Restore an auxiliary coefficient from the matter action

Here is a useful calculation across (6.15), (6.24) and (6.32). Suppose a theory has an unbroken Abelian flavor symmetry. A local change of generators can be written $\mathsf R'=\mathsf R+t\mathsf F_{\rm flavor}$, where $t$ is a real dimensionless constant; the plain $R$ below continues to mean Ricci scalar. A chiral scalar of flavor charge $q$ then has $r'=r+tq$. Changing its label alone changes its background couplings. Can we compensate by changing the flavor sources?

Use a frozen vector multiplet $(a_F,\sigma_F,0,0,D_F)$ and hold the rigid sources fixed. Put $M=z-q\sigma_F$. Preserving both the connection (PC05.1) and the chiral coefficient $M-rH$ forces

$$r'=r+tq,\qquad a'_F=a_F-t\mathcal A,\qquad \sigma'_F=\sigma_F-tH,\qquad M'=M+tqH. \tag{PC05.2}$$

The same cancellation works for $r-1$ and $r-2$, since every component has the same flavor charge. The labels have flavor charge zero and keep their R-charges $\pm1$.

The remaining unknown is the auxiliary shift. Extract the scalar coefficient from (6.32):

$$\begin{aligned} U(r,M,D_F,\sigma_F)={}&q(D_F+\sigma_FH)+M^2-2(r-1)HM-\frac r4R\\ &+\frac12\left(r-\frac12\right)V^2 +r\left(r-\frac12\right)H^2 . \end{aligned} \tag{PC05.3}$$

Here $V^2=V_\mu V^\mu$, without complex conjugation. Write $\delta=tq$ and $\Delta D_F=D'_F-D_F$. In the three terms quadratic in $M,r$, the $\delta M$ terms cancel and the $\delta^2$ terms cancel, leaving $3\delta H^2/2$. The change of $q\sigma_FH$ contributes $-\delta H^2$. Therefore

$$\begin{aligned} U'-U &=q\Delta D_F+\delta\left[-H^2+\frac32H^2-\frac R4+\frac{V^2}{2}\right]\\ &=q\left[\Delta D_F-\frac t4(R-2V^2-2H^2)\right]. \end{aligned} \tag{PC05.4}$$

For a nonzero flavor charge, equality fixes

$$\boxed{D'_F=D_F+\frac t4(R-2V^2-2H^2).} \tag{PC05.5}$$

The fermion coefficient $i[M-(r-\tfrac12)H]$ is unchanged already by (PC05.2); so are $\mathscr D$ and the matter auxiliary term. With source gauginos zero, the whole local canonical matter density agrees, not just its scalar mass-squared.

This is our reconstruction from the source formulas, rather than an additional quoted equation of P. As a check, the geometric entries $a_R=\mathcal A$, $\sigma_R=H$ and $D_R=-(R-2V^2-2H^2)/4$ reproduce the bosonic mixed density $i\epsilon a\,\partial a_R-D\sigma_R-\sigma D_R$ in P (6.29). Its $-2H^2$ must not be copied as the $+2H^2$ in the linear-multiplet prescription (6.15); inserting the field-strength component $K=D+\sigma H$ explains the difference.

Equations (PC05.2)–(PC05.5) establish local component identities. To use the transformed values as fixed backgrounds, their gaugino variations and global transitions must also be valid. We now check those requirements on a supplied background. This calculation alone is not a theorem equating partition functions in independently chosen contact-term schemes.

Worked changed input: an R shift on the sphere

Take the established round sphere with $A=V=C=0$, $H=-i/\ell$, $R=-6/\ell^2$ and $\ell>0$. Start with $a_F=0$, $\sigma_F=\mu$, $D_F=i\mu/\ell$, where $\mu$ is real. This is (6.26): $D_F+H\sigma_F=0$. It preserves the sphere’s two labels in each family, the continuation of the four-real-charge parent.

The compensated change becomes

$$r'=r+tq,\qquad \sigma'_F=\mu+\frac{it}{\ell},\qquad D'_F=\frac{i\mu}{\ell}-\frac{t}{\ell^2},\qquad a'_F=0. \tag{PC05.6}$$

Indeed $D'_F+H\sigma'_F=0$. With constant $\sigma'_F$ and zero field strength, both gaugino matrices in (6.20) vanish. This is a complex Euclidean flavor background, which the paragraph after (6.26) explicitly permits; its imaginary shift is not a new real Lorentzian mass.

Set $z=0$ and define the dimensionless combination

$$\varrho=r+iq\ell\mu,\qquad \varrho'=r'+iq\ell\sigma'_F=\varrho . \tag{PC05.7}$$

The scalar operator and the fermion operator can be written

$$\mathcal O_B=-\nabla^2+\frac{\varrho(2-\varrho)}{\ell^2}, \qquad \mathcal O_F=\gamma^\mu\nabla_\mu-\frac{i}{\ell} \left(\varrho-\frac12\right). \tag{PC05.8}$$

For the second formula the independent fermion row is $\chi=-i\widetilde\psi^{\mathsf T}C_{\rm spin}$, with $C_{\rm spin}=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$, so the density is $\chi\mathcal O_F\psi$. At fixed $r$ the flat signed Dirac mass is $q\mu=-M$. This translation prevents a sign in the paper’s scalar spinor contraction from being mistaken for a physical mass sign.

For example, $q=1$, $r=1/2$ and $t=1/3$ give $r'=5/6$, $\sigma'_F=\mu+i/(3\ell)$, yet both descriptions have

$$\mathcal O_B=-\nabla^2+\mu^2+\frac{i\mu}{\ell} +\frac{3}{4\ell^2}, \qquad \mathcal O_F=\gamma^\mu\nabla_\mu+\mu . \tag{PC05.9}$$

On the scalar contour $\widetilde\phi=\phi^*$ the real part of this quadratic scalar form is positive. The independent matter auxiliary can use $\widetilde F=-F^*$, making $-\widetilde F F=|F|^2$. The fixed $D_F$ is not integrated over. These statements identify a convergent bosonic Gaussian direction; they do not establish all regulated quantum Ward identities.

For a mode check, the constant scalar has eigenvalue $\varrho(2-\varrho)/\ell^2$. A dimensionless ambient coordinate $Y=y^0$ on the unit sphere satisfies $-\nabla^2Y=3Y/\ell^2$, so its eigenvalue is $[3+\varrho(2-\varrho)]/\ell^2$ in either description. This tests a nonconstant field as well as the scalar coefficient. Sending $\ell\to\infty$ at fixed $r,t,\mu$ gives the common flat operators $-\partial^2+q^2\mu^2$ and $\gamma\cdot\partial+q\mu$.

The agreement concerns a change of source description together with an R assignment. Varying a real $\mu$ at fixed $r$, or changing $r$ while holding every source fixed, is a different operation. The sphere has no two-cycle obstruction to the local comparison; arbitrary mixing on a manifold with compact flux requires the further test below.

Exit calculations

1. Keep a superpotential neutral through the change

Take two chirals with flavor charges $q_1=2$, $q_2=-1$, R-charges $r_1=1$, $r_2=1/2$, and zero central charge. Consider $W=g\Phi_1\Phi_2^2$ with a fixed neutral coefficient. On the sphere family, apply $t=1/4$. Find both new R-charges and both invariant combinations $\varrho_i$. Check the F-term charge conditions before and after the change. Would assigning both flavor charges $+1$ allow the same fixed coupling and source construction?

Hint 1

Add charges with the powers appearing in the monomial. The source scalar changes once for the common flavor symmetry, not separately for each field.

Hint 2

Use $r_i'=r_i+tq_i$ and $\sigma'_F=\mu+it/\ell$ in each $\varrho_i=r_i+iq_i\ell\mu$.

Solution

The new charges are $r'_1=3/2$, $r'_2=1/4$. The unchanged combinations are $\varrho_1=1+2i\ell\mu$ and $\varrho_2=1/2-i\ell\mu$. Their sum $\varrho_1+2\varrho_2=2$ also exhibits cancellation of the flavor mass in this neutral product.

Before the change, $q_W=2-2=0$ and $r_W=1+2(1/2)=2$. Afterward $r'_W=3/2+2(1/4)=2$, while $q_W=z_W=0$ remain unchanged. Thus the product auxiliary is a neutral F-term integrand under (6.14). Its antichiral partner has the opposite bottom R-charge. The canonical product rule supplies the fermion terms; no new uncharged constant is being assigned an artificial transformation.

With both flavor charges $+1$, $q_W=3$. The fixed nonzero neutral coefficient would break that proposed flavor symmetry. A formal spurion charge for $g$ would describe a transforming source, not an unbroken symmetry of this fixed theory. One could not apply the same frozen flavor background without supplying and checking a different coupling multiplet.

2. Change the scalar source but forget its auxiliary

On the sphere start from $a_F=0$, $\sigma_F=\mu$, $D_F=i\mu/\ell$. Make $r'=r+tq$ and $\sigma'_F=\mu+it/\ell$, but incorrectly keep $D'_F=D_F$. Calculate both gaugino variations and the difference in the scalar coefficient from the compensated answer. Can the discrepancy be a boundary term for a constant nonzero scalar on the closed sphere?

Hint 1

For constant sources with $a_F=0$, (6.20) reduces to $\delta\lambda_F=i(D_F+H\sigma_F)\zeta$ and $\delta\widetilde\lambda_F=-i(D_F+H\sigma_F)\widetilde\zeta$.

Hint 2

Compare the retained auxiliary value with (PC05.6). The matter density depends on it through $qD_F\widetilde\phi\phi$.

Solution

The incorrect shift gives $D'_F+H\sigma'_F=t/\ell^2$. Hence

$$\delta\lambda_F=\frac{it}{\ell^2}\zeta,\qquad \delta\widetilde\lambda_F=-\frac{it}{\ell^2}\widetilde\zeta,\qquad U_{\rm wrong}-U=\frac{qt}{\ell^2}.$$

For $t\ne0$ neither nonzero label leaves this frozen source unchanged. Its local scalar discrepancy is not a change of the fermion pole-mass convention; the fermion coefficient was unchanged. For $\widetilde\phi=\phi^*=v^*$ constant, integration gives $\Delta S=2\pi^2qt\ell|v|^2$. For nonzero $qtv$ this cannot be the integral of a globally defined divergence on the closed sphere. The wrong result diagnoses the omitted source auxiliary, not a freedom to eliminate a fixed background field by a matter equation.

3. A local redefinition that fails the compact transition

Use the supplied product background $S^1\times S^2_\ell$ with periodic circle spin structure, $H=V=0$, and $R=-2/\ell^2$. On the sphere take $A_N=(1-\cos\theta)d\varphi/2$ and $A_S=-(1+\cos\theta)d\varphi/2$. Their flux is $2\pi$, and $A_N-A_S=d\varphi$. In regular spin frames $\Gamma=iA\gamma_1$; the selected charge-$+1$ and charge-$-1$ labels are $u_+$ and $u_-$ with $\gamma_1u_\pm=\pm u_\pm$.

Start with the trivial compact flavor line, zero flavor sources and a unit-charge probe. Apply the local redefinition. Verify the flavor gaugino equations on these labels, then determine which real $t$ give a valid compact flavor transition. Explain the failure at $t=1/2$.

Hint 1

The new sources are $a'_F=-tA$, $\sigma'_F=0$ and $D'_F=-t/(2\ell^2)$. In the oriented frame with direction 1 along the circle, $b'_F=*da'_F=-t e^1/(2\ell^2)$.

Hint 2

The two matrices are $i(D'_F-\gamma\cdot b'_F)$ and $-i(D'_F+\gamma\cdot b'_F)$. A compact unit-charge transition must be single-valued as $\varphi$ increases by $2\pi$.

Solution

The first matrix annihilates $u_+$ because $D'_F=b'_{F1}$; the second annihilates $u_-$ because $\gamma_1u_-=-u_-$. Thus the local source check succeeds for every real $t$ on the specified labels. It does not require four independent sphere-product labels.

Across the overlap $a'_{FN}-a'_{FS}=-t\,d\varphi$, so the flavor transition is $e^{-it\varphi}$ and its first Chern number is $-t$. With the specified compact group and unit-charge probe, it defines a line bundle exactly when $t\in\mathbb Z$. At $t=1/2$ the transition changes sign after one turn. A local gaugino cancellation has not made that source global.

This restriction is about the chosen compact charge lattice and flux. Changing the global group or additional bundle data would change the problem. It is absent from the contractible local calculation and is not detected by the algebraic equality of the matter densities.

You can now use the section as a construction sequence: identify the component charges, use the correct product or gauge compensation, form a neutral invariant, and test the resulting source dependence without losing its global assumptions. The original-paper reading guide places this action construction between background existence and the paper’s later current-response checks.

Continue with the Section 7 guide to compare the completed curved couplings with their first-order current description.