Background fields

A background field is a prescribed source held fixed while the matter fields fluctuate. “Auxiliary” instead means that a field has an algebraic equation of motion in a particular action. A background field can descend from an auxiliary field of supergravity without being integrated out in the rigid matter theory.

For the derivations, use the source construction and the background-field lesson. This page provides the compact definitions and the translation between the original three-manifold paper and its sequel.

Which variables are held fixed?

Variable	In the rigid matter theory	What equation is relevant?
Metric and bosonic gravity sources	Prescribed	Background-preservation conditions if supersymmetry is wanted
Background gravitini	Set to zero on a bosonic background	Their supersymmetry variations must also vanish
Matter scalar and fermion	Dynamical	Matter field equations; these need not hold for an off-shell symmetry check
Matter auxiliary field, such as $F_\Phi$	Part of the off-shell matter variables	It may be eliminated with its own algebraic equation, changing how closure is expressed
Flavor connection	Prescribed when used as an external probe	Its full source multiplet must preserve the desired supercharge
Dynamical gauge connection	Integrated over	Gauge-field equations and gauge redundancy

The source is not allowed to follow each fluctuating matter configuration. Fixing a source $u=u_0$ and imposing its would-be equation $u=-\lambda x$ are different operations. The Gaussian comparison calculates their different effects.

This distinction applies in one-dimensional quantum mechanics as well as in three- and four-dimensional field theory. It does not identify their supergravity multiplets. In four dimensions, for example, the old-minimal formulation uses a scalar pair and a vector, while the new-minimal formulation with R-symmetry uses an R-connection and a conserved dual vector. These are different source systems with different gravitino equations. Festuccia–Seiberg, sections 2 and 6, explains the two four-dimensional constructions; do not transfer the three-dimensional coefficients below by changing the index range.

Three-dimensional bosonic dictionary

This table uses three-dimensional $\mathcal N=2$ supergravity sources for a theory with a continuous R-symmetry. The parent theory has four real Lorentzian Poincaré supercharges. In Euclidean signature the two spinor families are independent. The table specifies the sources, not the number of supercharges preserved by a particular background.

Use length coordinates, $[\partial_\mu]=1$, and a real positive-definite metric. Dimensions are engineering mass dimensions of components. Fix $\epsilon^{123}=1/\sqrt g$ in an oriented chart.

Field	Dimension	Definition or role
$g_{\mu\nu}$	$0$	Metric and Levi-Civita spin connection
$A_\mu$	$1$	Original paper’s R-gauge potential
$C_\mu$	$0$	Potential coupled to the central-charge current
$V^\mu$	$1$	$-i\epsilon^{\mu\nu\rho}\partial_\nu C_\rho$
$B_{\mu\nu}$	$0$	Two-form potential, $B=\frac12B_{\mu\nu}dx^\mu\wedge dx^\nu$
$H$	$1$	$\frac i2\epsilon^{\mu\nu\rho}\partial_\mu B_{\nu\rho}$

The invariant identities are

$$V^\flat=-i*dC,\qquad H=i*dB,\qquad \nabla_\mu V^\mu=0.$$

Here $*$ is the metric Hodge dual and $V^\flat=V_\mu dx^\mu$. The divergence follows from the definition; it is not a gravitational equation of motion. $A,C,B,V,H$ can be complex in the Euclidean construction, while the original paper keeps the gauge parameters for $A$ and $C$ real. Potentials can require patches. In particular, nonzero $\int H\,\mathrm{vol}_g$ on a closed three-manifold rules out a single smooth global $B$.

Three connections with different jobs

For a matter component $X$ of R-charge $r_X$ and central-charge eigenvalue $z_X$, define

$$a_\mu^{\mathrm{mat}}=A_\mu-\frac12V_\mu,\qquad A_\mu^{(R)}=A_\mu-\frac32V_\mu.$$

Expression	Where it appears
$\nabla_\mu-isA_\mu$	Original paper’s spinor equation; $s=+1$ for $\zeta$, $s=-1$ for $\widetilde\zeta$
$\nabla_\mu-ir_Xa_\mu^{\mathrm{mat}}-iz_XC_\mu$	The matter derivative defined in the original paper’s (6.6)
$j_R^\mu A_\mu^{(R)}$	The linear R-current source coupling, also the sequel’s choice of connection symbol

The component charge matters: a chiral scalar of charge $r$ has a fermion of charge $r-1$ and an auxiliary of charge $r-2$. The central charge $z_X$ has dimension one; $r_X$ is dimensionless. All terms in the matter derivative therefore have dimension one.

At first order around flat space, with $g_{\mu\nu}=\delta_{\mu\nu}+2h_{\mu\nu}$, the bosonic source coupling is

$$\Delta\mathcal L=-T^{\mu\nu}h_{\mu\nu} +j_R^\mu A_\mu^{(R)}+j_Z^\mu C_\mu+J_ZH.$$

$j_Z^\mu$ is the central-charge current and $J_Z$ its scalar source partner. This is the original paper’s (7.1). The sequel’s (1.18) instead writes the metric term as $-\frac12\Delta g_{\mu\nu}T^{\mu\nu}$, the same term because $\Delta g_{\mu\nu}=2h_{\mu\nu}$.

All three connection combinations acquire $d\lambda_R$ under $A\mapsto A+d\lambda_R$, since $V$ is invariant. Gauge covariance alone cannot tell you which one belongs in a given formula.

Translate the sequel without losing a vector term

The sequel, The Geometry of Supersymmetric Partition Functions, explicitly defines $A^{(R)}=A-\frac32V$ in Appendix A, equation (A.1). Therefore

$$A=A^{(R)}+\frac32V,\qquad a^{\mathrm{mat}}=A^{(R)}+V,$$

and the original matter derivative becomes

$$D_\mu X= \left[\nabla_\mu-ir_XA_\mu^{(R)}-ir_XV_\mu-iz_XC_\mu\right]X.$$

The extra $-ir_XV_\mu$ remains. For a quick numerical coefficient check, take $r_X=2$, $z_X=0$, $A_1=0$, $V_1=v$ and $X=e^{ipx^1}$ in a flat local frame. Then $A_1^{(R)}=-3v/2$ and $a_1^{\mathrm{mat}}=-v/2$. The original expression gives $D_1X=i(p+v)X$. The translated expression gives $[ip+3iv-2iv]X$, the same result. Omitting the extra vector term would give $i(p+3v)X$.

The spinor equations also change appearance. Starting from the original equations and substituting for $A$, one obtains

$$(\nabla_\mu-isA_\mu^{(R)})\varepsilon_s =-\frac H2\gamma_\mu\varepsilon_s +\frac{is}{2}V_\mu\varepsilon_s -\frac s2\epsilon_{\mu\nu\rho}V^\nu\gamma^\rho\varepsilon_s.$$

Here $\varepsilon_+=\zeta$ and $\varepsilon_-=\widetilde\zeta$. The change from $-isV_\mu$ in the original right-hand side to $+isV_\mu/2$ is exactly the connection shift. This agrees with the sequel’s equations (1.19)–(1.20); it is the same background equation in different variables.

Before using a background formula

Identify its dimension, signature, source conventions, and chosen R-symmetry. State which variables are prescribed and which are integrated over. Then specify local fields and global information such as the spin structure, gauge period, holonomies, and potential patches. Vanishing source fermions are not enough: their variations select the preserved spinors. The Killing spinor lesson provides both the local test and a global circle example.