What each background field does

Before this lesson: Supergravity as a system of background sources.

A metric tells a field theory how to measure distances and differentiate spinors. It does not specify all the sources needed by the curved supersymmetric theory. We must also say which connection acts on R-charged fields and which additional prescribed fields accompany it. There is a particularly instructive trap: the connection in the paper’s matter transformations differs from the source multiplying the R-current.

The preparation is Supergravity as a system of background sources. Here we give those sources names and calculate with them. The theory is three-dimensional $\mathcal N=2$ , with four real Poincaré supercharges in flat Lorentzian signature before continuation. We work with a real, positive-definite Euclidean metric. The background fields are fixed functions, and an ungauged chiral field will serve as a probe. We do not yet assume that a proposed background preserves any supercharge; that requires a separate spinor equation.

Identify the fields and their units

Use local coordinates with dimensions of length and set $\hbar=c=1$ . Brackets denote engineering mass dimension, so $[\partial_\mu]=1$ and $[g_{\mu\nu}]=0$ . These are units of the source components in this coordinate convention, not quantum scaling dimensions at an interacting fixed point.

The bosonic fields used in the rigid equations are:

Field	Dimension	Euclidean reality assumption	Role
$g_{\mu\nu}$	$0$	Real, positive definite	Determines lengths, the volume element, and the Levi-Civita and spin connections
$A_\mu$	$1$	May be complex	Connection for local R-symmetry transformations
$V_\mu$	$1$	May be complex	Gauge-invariant dual field strength, constrained by $\nabla_\mu V^\mu=0$ ; also enters matter and spinor couplings
$H$	$1$	May be complex	Scalar dual field strength supplying algebraic terms in the spinor and matter transformations

The dimensions are those of equation (2.14). The source’s reality choice appears after (1.3) and before (2.10). A complex source does not mean that the metric is complex, or that its complex conjugate is an additional independently named background field.

There are potentials behind $V$ and $H$ . The one-form $C=C_\mu dx^\mu$ is called the graviphoton: it couples to the current for the central charge $Z$ . The two-form $B=\frac12B_{\mu\nu}dx^\mu\wedge dx^\nu$ has a three-form field strength. Their component dimensions are

[C_\mu]=[B_{\mu\nu}]=0.

These follow from the derivative definitions below. The bosonic potentials have zero R-charge, although the connection $A_\mu$ itself transforms inhomogeneously under an R-gauge transformation.

The central charge is an operator appearing beside momentum in the supersymmetry algebra. If $Z$ acts on a probe field $X$ as $ZX=zX$ , its eigenvalue has dimension $[z]=1$ . This differs from the dimensionless R-charge $r$ . Thus $zC_\mu$ and $rA_\mu$ both have the dimension required in a derivative.

Distinguish a connection from a dual field strength

Fix the orientation and let $\epsilon^{\mu\nu\rho}$ be the antisymmetric tensor with $\epsilon^{123}=1/\sqrt g$ in an oriented coordinate chart, where $g=\det g_{\mu\nu}$ . The definitions in equation (1.1) are

V^\mu=-i\epsilon^{\mu\nu\rho}\partial_\nu C_\rho, \qquad H=\frac{i}{2}\epsilon^{\mu\nu\rho}\partial_\mu B_{\nu\rho}.

Only the antisymmetric derivatives contribute. For example, $V^\mu=-\frac{i}{2}\epsilon^{\mu\nu\rho} (\partial_\nu C_\rho-\partial_\rho C_\nu)$ . In form notation these statements are $V^\flat=-i\,*dC$ and $H=i\,*dB$ . Here $V^\flat=V_\mu dx^\mu$ , and the Hodge star $*$ uses the metric and orientation to turn a two-form into a one-form, or a three-form into a scalar.

The potential transformations are

C\longmapsto C+d\lambda_C,\qquad B\longmapsto B+d\Lambda_B,

where $\lambda_C$ is a scalar and $\Lambda_B$ a one-form. Since $d^2=0$ , $V$ and $H$ are invariant. They are not further gauge connections with transformations of the form $V\mapsto V+d\lambda$ .

There is also a local constraint on $V$ . Write $\varepsilon^{\mu\nu\rho}$ for the constant alternating symbol, so $\epsilon^{\mu\nu\rho}=\varepsilon^{\mu\nu\rho}/\sqrt g$ . Then

\begin{aligned} \nabla_\mu V^\mu &=\frac1{\sqrt g}\partial_\mu(\sqrt g V^\mu)\\ &=-\frac{i}{\sqrt g}\varepsilon^{\mu\nu\rho} \partial_\mu\partial_\nu C_\rho=0. \end{aligned}

The last equality follows because the derivatives are symmetric in $\mu,\nu$ and the symbol is antisymmetric. This is a field-strength identity, not an equation of motion. No analogous differential constraint on $H$ follows from $d(dB)=0$ : a four-form vanishes identically in three dimensions.

These are local potential formulas. On a manifold with nontrivial topology, potentials can require patches. For example, if $B$ were a single smooth two-form on a closed three-manifold, Stokes’ theorem would give

\int H\,\mathrm{vol}_g=i\int dB=0.

A nonzero constant $H$ on a compact sphere therefore cannot come from such a single global $B$ . One must address potential patching when constructing that global background. The local field dictionary alone does not settle its global data.

Read the two connection combinations in their proper places

For a matter component of R-charge $r$ and central charge $z$ , the derivative defined in equation (6.6) is

D_\mu X= \left[\nabla_\mu-ir\left(A_\mu-\frac12V_\mu\right) -izC_\mu\right]X.

The Levi-Civita derivative acts on whatever tensor or spinor indices $X$ has; for a scalar it is simply $\partial_\mu$ . The charge in this formula belongs to that particular component. For a chiral multiplet with scalar charge $r$ , its fermion instead has charge $r-1$ .

Let us name the two combinations without identifying them:

a_\mu^{\rm mat}=A_\mu-\frac12V_\mu, \qquad a_\mu^{R}=A_\mu-\frac32V_\mu.

The first occurs in the stated matter derivative. The second is the R-current source in the linearized bosonic coupling. In the paper’s normalization $g_{\mu\nu}=\delta_{\mu\nu}+2h_{\mu\nu}$ , that coupling is

\delta\mathcal L =-T^{\mu\nu}h_{\mu\nu} +j_R^\mu a_\mu^R +j_Z^\mu C_\mu +J_Z H.

This is equation (7.1), equivalently the bosonic part of (2.10). Here $T^{\mu\nu}$ is the stress tensor, $j_R^\mu$ the R-current, $j_Z^\mu$ the central-charge current, and $J_Z$ its scalar source partner. This equation identifies which operators the sources probe; reconstructing their complete supersymmetry multiplet is a separate calculation.

Subtracting the two definitions gives a useful translation:

a_\mu^{\rm mat}=a_\mu^R+V_\mu, \qquad D_\mu=\nabla_\mu-ir a_\mu^R-irV_\mu-izC_\mu.

Thus replacing $a^{\rm mat}$ by $a^R$ in the first derivative would omit the term $-irV_\mu$ . Conversely, using $a^{\rm mat}$ as the only R-current source would omit $-j_R^\mu V_\mu$ from the displayed linearized coupling. The coefficients describe different appearances of the same background fields.

Why does the connection in a derivative not determine the whole source coupling? A supersymmetric matter action contains further couplings to $V$ and $H$ , beyond the terms obtained by inserting $D_\mu$ into a flat-space kinetic term. The current is read from the variation of the complete action. Also, authors may regroup those additional terms into a different derivative; the formula above specifically means the derivative defined in (6.6).

Both combinations transform as R-connections. Under $A_\mu\mapsto A_\mu+\partial_\mu\lambda_R$ , the invariant $V$ does not change, so both $a^{\rm mat}$ and $a^R$ acquire $d\lambda_R$ . Indeed, any constant shift $A-cV$ has that property. Gauge covariance therefore cannot decide whether the coefficient is $1/2$ or $3/2$ ; its role in the transformation or action must decide it.

Calculate an explicit local background

Work in flat $\mathbb R^3$ with oriented Cartesian coordinates and take constants $v,h$ of dimension one. On the patch, prescribe

A_\mu=0,\qquad C_3=i v x^2,\qquad B_{23}=-i h x^1,

with all other independent components of $C$ and $B$ zero and $B_{32}=-B_{23}$ . Direct differentiation gives

V^1=-i\,\partial_2 C_3=v,\qquad V^2=V^3=0.

For $H$ , both antisymmetric entries contribute:

H=\frac{i}{2} \left(\partial_1 B_{23}-\partial_1 B_{32}\right) =i(-ih)=h.

This checks the factor $1/2$ in the definition. The divergence of the constant vector vanishes. Even if $v,h$ are real, the potentials chosen here are imaginary, consistent with the allowed complex Euclidean fields.

Now probe the background with a scalar of R-charge $r$ , central charge zero, and profile $\phi=e^{ipx^1}$ . Since $a_1^{\rm mat}=-v/2$ ,

D_1\phi=i\left(p+\frac{rv}{2}\right)\phi.

Meanwhile $a_1^R=-3v/2$ , so the R-current part of the linearized deformation is $-\frac32v j_R^1$ . Using the source connection alone in the matter derivative would incorrectly give $i(p+3rv/2)\phi$ , differing by $irv\phi$ .

This is a check of source definitions and the probe derivative. The linearized action formula retains only first order in the source amplitudes; it is not a full finite-background action. No claim that this choice of $v,h$ preserves supersymmetry follows from these calculations.

The scalar $H$ also has a role beyond the vector connections. In the chiral transformation (6.8), an algebraic term contains $z-rH$ . It has dimension one, just as $z$ does. Calling this combination masslike identifies its place in the transformation; it does not identify a real, positive physical mass for arbitrary complex Euclidean sources.

Keep the backgrounds fixed

The fields $V$ and $H$ descend from auxiliary fields of the gravity multiplet. In the rigid construction their values remain prescribed, subject to their definitions and, when supersymmetry is requested, the background spinor equations. They are not set to zero by varying a dynamical supergravity action.

They are also different objects from the chiral matter auxiliary field $F_\Phi$ . That field belongs to the matter path integral. Whether one retains or eliminates $F_\Phi$ is a choice in describing the matter theory; it supplies no equation for the fixed background scalar $H$ .

There is one more Euclidean convention to retain. Although $A$ and $C$ can be complex, the paper restricts their gauge parameters $\lambda_R$ and $\lambda_C$ to be real, as stated immediately before (2.10). One cannot erase an imaginary source component by silently allowing a complex gauge transformation. In particular, nonzero $V$ is invariant even under a change of the potential $C$ .

Exercises

1. Check the charges in a local transformation

Take real, constant charges $r,z$ and a scalar $X$ . Let

X'=e^{i(r\lambda_R+z\lambda_C)}X,\quad A'=A+d\lambda_R,\quad C'=C+d\lambda_C.

Keep $V$ fixed. Prove $D'_\mu X'=e^{i(r\lambda_R+z\lambda_C)}D_\mu X$ . What dimensions must the gauge parameters have? Would the same covariance check distinguish $A-V/2$ from $A-3V/2$ ?

First hint

Differentiate the phase before applying the shifted connections.

Second hint

The terms proportional to $\partial_\mu\lambda_R$ and $\partial_\mu\lambda_C$ must cancel separately. The extra $V$ term has no inhomogeneous gauge variation.

Complete solution

Writing $\Theta=r\lambda_R+z\lambda_C$ , we have

\partial_\mu(e^{i\Theta}X) =e^{i\Theta} \left(\partial_\mu X+ir\,\partial_\mu\lambda_R X +iz\,\partial_\mu\lambda_C X\right).

The new connection terms subtract exactly the last two terms. Consequently $D'_\mu X'=e^{i\Theta}D_\mu X$ . This is a local check; it makes no assumption about large transformations or charge quantization on a compact manifold.

Because the phase is dimensionless, $[\lambda_R]=0$ and $[\lambda_C]=-1$ . This agrees with $[d\lambda_R]_{\rm component}=1$ and $[d\lambda_C]_{\rm component}=0$ . Replacing $1/2$ by $3/2$ , or by any constant coefficient of $V$ , leaves the cancellation intact. Covariance is necessary but does not identify which derivative a particular transformation law uses.

2. Reject an impossible dual vector

On flat $\mathbb R^3$ , let $C_3=i\kappa x^1x^2$ , with other components zero and $[\kappa]=2$ . Find $V$ . Compare it with the proposed vector $W=(\kappa x^1,0,0)$ for $\kappa\ne0$ . Can $W$ have the local form $-i*dC$ for a smooth potential? Explain without trying to guess a potential.

First hint

The first and second vector components involve $\partial_2C_3$ and $-\partial_1C_3$ , respectively.

Second hint

Check the divergence before solving for any potential. A necessary identity cannot be repaired by a gauge transformation.

Complete solution

The two nonzero components are

V^1=-i(i\kappa x^1)=\kappa x^1,\qquad V^2=+i(i\kappa x^2)=-\kappa x^2.

Thus $\partial_\mu V^\mu=\kappa-\kappa=0$ . Both components have dimension one, as required.

For $W$ , instead $\partial_\mu W^\mu=\kappa$ . Since the divergence of a smooth curl vanishes identically on the patch, there is no such local potential for $W$ . Dropping the second component changes a permitted field strength into an impossible one; neither a matter equation of motion nor a choice of gauge alters this conclusion.

3. Say what is held fixed

Consider a small variation of $C$ that induces $\delta V=-i*d(\delta C)$ at fixed metric. In the R-current term $j_R^\mu a_\mu^R$ of the linearized source coupling, compare:

Keeping $A$ fixed.
Keeping $a^{\rm mat}$ fixed by varying $A$ as well.

Treat the flat-space current operator as fixed at this order. Why do the two answers differ? What additional term from the displayed bosonic source coupling must be retained if one asks for its complete variation under this change of $C$ ?

First hint

Use $\delta a^R=\delta A-\frac32\delta V$ .

Second hint

In the second case, $0=\delta a^{\rm mat}=\delta A-\frac12\delta V$ . Also remember that $C$ itself is a source, not only a way to specify $V$ .

Complete solution

At fixed $A$ ,

\delta\mathcal L_R=-\frac32j_R^\mu\delta V_\mu.

At fixed $a^{\rm mat}$ , the compensating variation is $\delta A=\frac12\delta V$ , hence

\delta\mathcal L_R=-j_R^\mu\delta V_\mu.

They are different physical variations of the sources: the second includes an R-connection change that the first excludes. There is no contradiction in the response coefficients. In both cases the complete variation also contains $j_Z^\mu\delta C_\mu$ , with $g$ and $H$ held fixed. One cannot vary $V$ as an unconstrained independent vector while forgetting its relation to $C$ in this description.

A related check is to set $A=V/2$ . Then $a^{\rm mat}=0$ , while $a^R=-V$ . Vanishing of the connection in the stated matter derivative does not mean that all source couplings have disappeared.

Specify a background before testing supersymmetry

You can now give a candidate background by stating a metric, the R-connection, the constrained dual vector, and the scalar, together with potential and patch data when needed. Check its units and reality assumptions, and distinguish the connections $a^{\rm mat}$ and $a^R$ before using any matter or current formula.

The next question is more restrictive: which of these choices leave a nonzero supersymmetry parameter invariant? Derive and interpret the Killing spinor equations answers it by setting the appropriate gravitino variation to zero and testing the resulting equations.

The paper introduction guide collects these conventions beside the original equations and tests the distinction between a complex source and a real gauge transformation.