---
title: '3D N=2 Gauged Supergravity: Off-Shell Branches'
url: https://www.emergentmind.com/topics/d-3-n-2-gauged-supergravity
type: topic
---

# 3D N=2 Gauged Supergravity: Off-Shell Branches

Searching arXiv for recent and foundational papers on 3D \( \mathcal N=2 \) gauged supergravity and related AdS / Chern–Simons formulations.
In the literature represented here, \(D=3,\; \mathcal N=2\) gauged supergravity is not a single theory but a family of inequivalent constructions whose distinction is already visible off shell, in superspace, in higher-derivative completions, in matter-coupled \(U(1)\)-gauged models, and in Chern–Simons formulations. A central organizing fact is that three-dimensional \(\mathcal N=2\) supergravity admits two inequivalent AdS branches, \(\mathcal N=(1,1)\) and \(\mathcal N=(2,0)\), associated with different compensators and different realizations of \(R\)-symmetry. In addition, there are matter-coupled \(U(1)\)-gauged models with Fayet–Iliopoulos extensions, string-derived gauged truncations in which \(c\)-extremization is realized as extremization of a \(T\)-tensor, and asymptotically flat \(\mathcal N=(2,0)\) Chern–Simons supergravities whose “gauged” description refers instead to a boundary chiral Wess–Zumino–Witten reduction [1109.0496].

## 1. Off-shell branches and the meaning of the \((p,q)\) labels

Three-dimensional \(\mathcal N=2\) supergravity comes in two distinct off-shell versions, \(\mathcal N=(1,1)\) and \(\mathcal N=(2,0)\). The \(\mathcal N=(p,q)\) notation refers to the underlying AdS superalgebras \(OSp(2,p)\oplus OSp(2,q)\), with bosonic \(AdS_3\) isometry \(O(2,2)\) and \(R\)-symmetry \(SO(p)\times SO(q)\). This means that the total supersymmetry count does not determine the theory off shell; the left/right split does [1412.3118].

Both theories are obtained from the same \(3D\ \mathcal N=2\) Weyl multiplet by coupling a compensating multiplet and then fixing conformal symmetries. The \(\mathcal N=(1,1)\) theory uses a compensating scalar multiplet and is also referred to as Type I minimal or 3D old minimal. The \(\mathcal N=(2,0)\) theory uses a compensating vector multiplet and is also referred to as Type II minimal or 3D new minimal. In superspace there is also an improved complex linear compensator, but it furnishes a dual realization of \((1,1)\) AdS supergravity rather than a third inequivalent AdS branch [1109.0496].

| Formulation | Compensator | AdS realization |
|---|---|---|
| Type I minimal | chiral scalar multiplet | \((1,1)\) |
| Type II minimal | vector multiplet | \((2,0)\) |
| Improved non-minimal | improved complex linear multiplet | dual to \((1,1)\) |

The distinction has direct implications for gauging. In the \((1,1)\) construction, the compensating scalar is complex, so gauge fixing removes both dilatations and \(U(1)_R\). In the \((2,0)\) construction, the compensating scalar is real, so \(U(1)_R\) is not gauge-fixed away. This is the branch in which the \(R\)-symmetry gauge field remains explicitly in the Poincaré multiplet and can be viewed dynamically [1412.3118].

## 2. Superspace geometry, cosmological terms, and \(U(1)_R\) gauging

The superspace structure group is
\[
{\rm SL}(2,\mathbb R)\times U(1)_R,
\]
with covariant derivatives
\[
{\cal D}_A = E_A{}^M \partial_M + \frac12 \Omega_A{}^{bc} M_{bc} + i \Phi_A J .
\]
The geometry is encoded in the dimension-1 torsion superfields \({\cal S}\), \(R\), \(\bar R\), and \({\cal C}_a\), together with their Bianchi identities and super-Weyl transformations. The local symmetries are local superspace diffeomorphisms, local Lorentz \(SL(2,\mathbb R)\), local \(U(1)_R\), and super-Weyl symmetry before gauge fixing [1109.0496].

In the Type I formulation the compensator is a covariantly chiral scalar \(\Phi\),
\[
\bar{\cal D}_\alpha \Phi = 0,
\]
and with cosmological term the action is
\[
S = -4 \int d^3x\, d^4\theta\, E\, \bar\Phi \Phi + \mu \int d^3x\, d^2\theta\, {\cal E}\, \Phi^4 + \bar\mu \int d^3x\, d^2\bar\theta\, \bar{\cal E}\, \bar\Phi^4 .
\]
In the gauge \(\Phi=1\), the on-shell AdS conditions are
\[
R=\mu=\text{const},\qquad {\cal S}=0,\qquad {\cal C}_{\alpha\beta}=0,
\]
which is precisely \((1,1)\) AdS superspace.

In the Type II formulation the compensator is a real prepotential \(G\) with gauge-invariant field strength
\[
\mathbb G = i {\cal D}^\alpha \bar{\cal D}_\alpha G = \bar{\mathbb G},
\]
and the AdS deformation is a supersymmetric Chern–Simons term,
\[
S_{\rm AdS} = \int d^3x\, d^4\theta\, E \left( 4\mathbb G \ln \mathbb G - \mathbb G\, {\cal S} + \frac12 \rho\, G \mathbb G \right).
\]
In the gauge \(\mathbb G=1\), the background is characterized by
\[
R=\bar R=0,\qquad {\cal S}=\rho=\text{const},\qquad {\cal C}_{\alpha\beta}=0,
\]
which is \((2,0)\) AdS superspace. Here the \(U(1)_R\) generator appears directly in the supersymmetry anticommutator, making the gauged \(U(1)_R\) structure explicit [1109.0496].

This split controls matter couplings and supercurrents. In \((1,1)\) AdS, every theory admits a Ferrara–Zumino multiplet by improvement, while in \((2,0)\) AdS the natural supercurrent is \(R\)-multiplet-like. The rigid matter sectors also differ sharply. In \((1,1)\) AdS, the sigma-model target space must have exact Kähler form and therefore be noncompact. In \((2,0)\) AdS, only \(R\)-invariant models are allowed in the generic case, Fayet–Iliopoulos terms are allowed, compact Kähler targets are allowed in the \(r_I=0,\ W=0\) class, and a frozen vector multiplet exists, enabling central-charge real masses [1109.0496].

## 3. Component multiplets and higher-derivative massive deformations

The \(3D\ \mathcal N=2\) Weyl multiplet contains
\[
(e_\mu{}^a,\ \psi_\mu,\ V_\mu,\ b_\mu,\ \omega_\mu{}^{ab},\ f_\mu{}^a,\ \phi_\mu),
\]
where \(V_\mu\) is the \(U(1)\) \(R\)-symmetry gauge field. The off-shell scalar compensator has components
\[
(A,\ \chi,\ F),
\]
while the off-shell vector compensator has
\[
(\rho,\ C_\mu,\ \lambda,\ D).
\]
After gauge fixing, the \((1,1)\) Poincaré multiplet is
\[
(e_\mu{}^a,\ \psi_\mu,\ V_\mu,\ S),
\]
with \(S=A+iB\), whereas the \((2,0)\) Poincaré multiplet is
\[
(e_\mu{}^a,\ \psi_\mu,\ V_\mu,\ C_\mu,\ D).
\]
The corresponding bosonic Einstein–Hilbert Lagrangians are
\[
e^{-1}\mathcal L_{\rm EH} = R + 2V^2 - 2|S|^2 - \left(\bar\psi_\mu\gamma^{\mu\nu\rho}D_\nu(\omega)\psi_\rho + {\rm h.c.}\right)
\]
for \((1,1)\), and
\[
e^{-1}\mathcal L_{\rm EH} = R - 2G^2 - 8D^2 - 8\epsilon^{\mu\nu\rho}C_\mu\partial_\nu V_\rho
\]
for \((2,0)\) [1412.3118].

All off-shell invariants up to four derivatives were constructed for both branches. A common ingredient is the superconformal Lorentz Chern–Simons invariant
\[
L_{\rm CS} = \frac14 \epsilon^{\mu\nu\rho} \left( R_{\mu\nu}{}^{ab}(\omega)\,\omega_{\rho ab} +\frac23 \omega_\mu{}^{ab}\omega_{\nu b}{}^{\ \ c}\omega_{\rho ca} \right) +\epsilon^{\mu\nu\rho}F_{\mu\nu}V_\rho -\bar R^\mu \gamma_\nu\gamma_\mu R^\nu.
\]
In the \((1,1)\) branch, the invariant structure includes an off-diagonal \(RS^2\)-type term, and this is the crucial ingredient permitting a special higher-derivative combination with a supersymmetric AdS vacuum and a ghost-free massive spin-2 multiplet. For the parameter choice
\[
\nu=0,\qquad c_1=-\frac{3}{8m^2},\qquad c_2=\frac{1}{8m^2},
\]
the theory becomes the supersymmetric extension of generalized massive gravity, and the paper identifies a maximally supersymmetric AdS background together with a ghost-free and unitary massive spectrum for suitable noncritical parameters [1412.3118].

The \((2,0)\) branch behaves differently. Its four-derivative invariants include \(RD\), \(R^2\), and Ricci-squared structures, but the analysis does not find an independent \(RD^2\)-type invariant without curvature-squared contamination. This prevents the cancellation mechanism that is available in \((1,1)\), and the conclusion is explicit: the \((2,0)\) model does not seem to have a supersymmetric AdS vacuum with ghost-free spectrum. The same branch does, however, admit a ghost-free massive theory around Minkowski. For the parameter choice
\[
M=0,\qquad \nu=0,\qquad c=\frac1{8m^2},
\]
the maximally supersymmetric Minkowski vacuum has
\[
D=0,\qquad V_\mu=0,\qquad C_\mu=0,
\]
and the bosonic spectrum organizes into two massive spin-2 multiplets of \(\mathcal N=(2,0)\) [1412.3118].

## 4. Matter-coupled \(U(1)\)-gauged models and Fayet–Iliopoulos deformations

A distinct and more conventional gauged-supergravity sector is provided by the \(D=3,\;N=2\) \(U(1)\)-gauged matter-coupled supergravity of Abou-Zeid–Nash–Roček with Fayet–Iliopoulos extension. In the model studied in [2510.25652], the bosonic field content is the dreibein \(e_\mu{}^a\), one \(U(1)\) gauge field \(A_\mu\), and \(n\) complex scalars; the analysis specializes to \(n=1\) with
\[
\phi = R e^{i\theta}.
\]
The scalar target space must be Kähler, and the maximally symmetric two-dimensional choices considered are
\[
\mathbb{S}^2,\qquad \mathbb{H}^2,\qquad \mathbb{R}^2.
\]

The bosonic Lagrangian is
\[
\mathcal{L}= \sqrt{-g} \left( \frac{1}{4}\mathcal{R} -G\,(\partial_\mu R \partial^\mu R + R^2 D_\mu\theta D^\mu\theta) - \frac{g_0}{8} \epsilon^{\mu\nu\rho} A_\mu F_{\nu\rho} + g_0^2\, V \right),
\]
with
\[
D_\mu\theta=\partial_\mu\theta+g_0A_\mu.
\]
The scalar potential is generated by
\[
V= 2W^2- \frac{G^{-1}}{4} (\partial_R W)^2,
\qquad
W = \mathcal{C}^2 + b,
\qquad
\mathcal{C} = -\frac{1}{2}R\partial_R K + c.
\]
The FI extension introduces the additional constant \(c\), while \(b\) parameterizes freedom in defining the superpotential. For \(\mathbb S^2\) and \(\mathbb H^2\), the parameters may be taken as \(\{a^2,b,c\}\) or equivalently \(\{a^2,b,x_0\}\), with
\[
x_0 = c - \frac{\varepsilon}{2a^2}, \qquad \varepsilon=\pm1.
\]
Supersymmetric extrema satisfy
\[
\partial_\rho W = -\frac{2}{a}\mathcal{C}\mathcal{S}=0.
\]
There is always an extremum at \(\rho=0\), and for \(c>0\) there is an additional pair at \(\mathcal C=0\). The FI term therefore creates extra supersymmetric extrema and is crucial for interpolating string solutions between two vacua [2510.25652].

The supersymmetric analysis uses Killing-spinor bilinears and splits all solutions into null and timelike classes. In the null class, the general supersymmetric metric is
\[
ds^2=dr^2+2e^{2U(r)}dudv+e^{2\beta(u,r)}du^2,
\qquad
\partial_r U = 2g_0 W.
\]
The solution space includes pp-waves in Minkowski or AdS, electromagnetic waves, null \(z\)-warped \(AdS_3\), and string solutions with waves. For constant nonzero \(\rho\) at \(\mathcal C=0\), the null warped geometry takes the form
\[
ds^2= \frac{1}{4g_0^2b^2} \left( \frac{dy^2}{4y^2} + 2y\, dtd\phi +\left[\beta y + \alpha^2 + \frac{Q^2}{4(b-\eta)}y^{\eta/b}\right] d\phi^2\right),
\qquad
A= \frac{Q}{2g_0b} y^{\eta/(2b)} d\phi,
\]
with
\[
z=\frac{\eta}{b}.
\]
For suitable parameter ranges, periodic identification \(\phi\sim \phi+2\pi\) yields a physically sensible supersymmetric null \(z\)-warped AdS black hole [2510.25652].

In the timelike class, the metric is a timelike fibration over a two-dimensional base,
\[
ds^2=-4f^2\bigl[dt+M(x,y)dx+N(x,y)dy\bigr]^2+e^{2\sigma(x,y)}(dx^2+dy^2).
\]
The solution space includes global \(AdS_3\), Ricci-flat or Euclidean Rindler-type geometries, timelike warped flat, timelike stretched warped \(AdS_3\), and string-like solutions. The timelike stretched warped \(AdS_3\) solution has
\[
ds^2= \frac{1}{16g_0^2bW} \left[ - \nu^2 \left(dt + \frac{dx}{y}\right)^2+\frac{dx^2+dy^2}{y^2}\right],
\qquad
\nu^2= 1+ \frac{\mathcal{C}^2}{b} >1.
\]
Charged string and string-like solutions interpolate between supersymmetric AdS or Minkowski extrema of the scalar potential; one endpoint controls the asymptotic geometry, while the other is a horizon, and the FI term is essential for the existence of the second vacuum and therefore for the horizon [2510.25652].

## 5. String-derived \(D=3,\mathcal N=2\) gauged supergravities and \(T\)-extremization

Wrapped D3-brane \(AdS_3\) solutions in type IIB provide a string-theoretic origin for a broad class of three-dimensional gauged supergravities. For D3-branes wrapping Kähler two-cycles in a \(CY_4\), the dual two-dimensional SCFT has \(\mathcal N=(0,2)\), and the reduced three-dimensional theory preserves four real supercharges, exactly those of \(D=3,\mathcal N=2\) supergravity [1307.2086].

In the cleanest example, one starts from \(5D\ U(1)^3\) gauged supergravity and reduces on a Riemann surface \(\Sigma_{\mathfrak g}\) with
\[
ds_5^2=e^{-4C}ds_3^2+e^{2C}ds^2(\Sigma_{\mathfrak g}),
\qquad
F^i = G^i - a_i\,\mathrm{vol}(\Sigma_{\mathfrak g}).
\]
After dualizing the three-dimensional vectors, one defines
\[
e^{W_i}=e^{2C}X_i^{-1},
\qquad
z_i = e^{W_i}+ iY_i,
\]
and the scalar target manifold becomes
\[
\big[SU(1,1)/U(1)\big]^3
\]
with Kähler potential
\[
\mathcal K = -\sum_{i=1}^3 \log (\Re z_i).
\]
This realizes the standard \(3D\ \mathcal N=2\) requirement that the scalar manifold be Kähler [1307.2086].

The scalar potential is expressed entirely in terms of a \(T\)-tensor,
\[
T = -\frac14\left[ a_1 e^{-W_2-W_3} +a_2 e^{-W_1-W_3} +a_3 e^{-W_1-W_2} \right] +\frac12\left[e^{-W_1}+e^{-W_2}+e^{-W_3}\right],
\]
because when the \(R\)-symmetry is gauged, consistency requires the holomorphic superpotential to vanish,
\[
W=0.
\]
Supersymmetry imposes the twist condition
\[
a_1+a_2+a_3 = -\kappa,
\]
and supersymmetric \(AdS_3\) vacua are critical points of \(T\), located at
\[
W_1 = \ln\frac{a_2+a_3-a_1}{a_2 a_3}, \qquad
W_2 = \ln\frac{a_1+a_3-a_2}{a_1 a_3}, \qquad
W_3 = \ln\frac{a_1+a_2-a_3}{a_1 a_2}.
\]
The same framework yields a direct supergravity realization of \(c\)-extremization:
\[
T=2\mathcal V^i \Theta_{ij}\mathcal V^j,\qquad
c_R=3\eta_\Sigma d_G T^{-1}, \qquad
R=2\mathcal V^i T^{-1}Q_i.
\]
In this sense, the exact central charge and the exact \(R\)-symmetry of the dual \(\mathcal N=(0,2)\) SCFT are determined by extremizing the \(T\)-tensor [1307.2086].

The same paper also studies less generic but explicit reductions. For an \(H^2\times KE_4\) geometry descending from a universal \(SE_5\) truncation, a consistent \(\mathcal N=2\) truncation yields scalar manifold \([SU(1,1)/U(1)]^4\), while for intersecting D3-branes or \(S^2\times T^4\) one can extract \(\mathcal N=2\) subsectors with scalar manifolds \([SU(1,1)/U(1)]^2\) and \([SU(1,1)/U(1)]^3\). The same three-dimensional framework also accommodates null-warped \(AdS_3\) sectors in suitable truncations [1307.2086].

## 6. Asymptotically flat \((2,0)\) Chern–Simons supergravity and boundary gauging

A separate line of development concerns asymptotically flat \(\mathcal N=(2,0)\) Chern–Simons supergravity with internal \(R\)-symmetry. Here the bulk gauge superalgebra is the \(\mathcal N=(2,0)\) super-Poincaré algebra with bosonic generators
\[
J_a,\quad P_a,\quad T,\quad Z,
\]
fermionic generators \(Q^i_\alpha\), and defining brackets
\[
[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,
\]
\[
[J_a,Q^i_\alpha]=\frac12(\Gamma_a)^\beta{}_\alpha Q^i_\beta,\qquad [Q^i_\alpha,T]=\epsilon^{ij}Q^j_\alpha,
\]
\[
\{Q^i_\alpha,Q^j_\beta\} =-\frac12 \delta^{ij}(C\Gamma^a)_{\alpha\beta}P_a + C_{\alpha\beta}\epsilon^{ij} Z.
\]
The \(R\)-symmetry generator \(T\) acts on the two supercharges, and the central element \(Z\) is required for the existence of a nondegenerate invariant bilinear form [1907.05866].

The Chern–Simons connection is
\[
A= e^a P_a + \hat\omega^a J_a + \psi_i^\alpha Q^i_\alpha + B\,T + C\,Z,
\qquad
\hat\omega^a=\omega^a+\gamma e^a,
\]
and the action is
\[
I[A]=\frac{k}{4\pi}\int_M \left\langle A,dA+\frac23 A^2\right\rangle, \qquad k=\frac1{4G}.
\]
In component form,
\[
I_{\mu,\bar\mu,\gamma}^{(2,0)} =\frac{k}{4\pi}\int \Big[ 2 e^a \hat R_a +\mu\,L(\hat\omega_a) -\bar\Psi^i_\beta \nabla \Psi_i^\beta -2 B\,dC +\bar\mu\,B\,dB \Big].
\]
The field equation is the flatness condition
\[
F=dA+A\wedge A=0.
\]

The paper’s main result is not the construction of a conventional matter-coupled gauged supergravity with scalar potentials and AdS vacua. Rather, it studies asymptotically flat \(\mathcal N=(2,0)\) supergravity at null infinity and derives equivalent reduced descriptions: a chiral Wess–Zumino–Witten model, a gauged chiral Wess–Zumino–Witten model, and a reduced phase-space theory that is the flat limit of a generalized Liouville theory, up to zero modes. The word “gauged” therefore refers to boundary gauging of a subalgebra of chiral WZW symmetries associated with first-class constraints, not to gauged supergravity in the standard higher-dimensional sense [1907.05866].

The asymptotic symmetry algebra is quantum \(\mathcal N=2\) super-BMS\(_3\). The gauged boundary action introduces Lagrange multipliers that enforce current constraints, while the reduced phase-space action becomes
\[
I=\frac{k}{4\pi}\int du\,d\phi \Big[ \mu \varphi'\dot\varphi +\xi'\dot\varphi -(\varphi')^2 -2i\dot a\, D' +\bar\mu\,a'\dot a -4(a')^2 +2(\dot\chi_1\chi_2+\dot\chi_2\chi_1) \Big].
\]
The gauged chiral WZW model and the reduced Liouville-like theory possess identical global and gauge symmetries. This sharpens a common terminological distinction: the presence of an internal \(R\)-symmetry gauge field in the bulk Chern–Simons superalgebra is a genuine bulk gauge feature, but the “gauged” model constructed in the dual description is a gauged boundary WZW theory rather than a standard bulk gauged supergravity [1907.05866].

Source: https://www.emergentmind.com/topics/d-3-n-2-gauged-supergravity