Born-Infeld New Massive Gravity (BINMG)

• The paper introduces BINMG as a three-dimensional higher-curvature gravity theory that nonlinear completes NMG using a determinantal (Born-Infeld) action.
• It employs a low-curvature expansion that recovers Einstein gravity at leading order and the specific NMG invariant at quadratic order, linking to holographic c-theorems.
• The framework enables auxiliary metric reformulations and exact black-hole solutions, offering insights into unique vacuum structure and supersymmetric extensions.

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1. Determinantal construction and three-dimensional status

The original BINMG proposal is a determinantal completion of cosmological NMG. In the conventions of the 2010 construction, the action is written as

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$

with the cosmological Einstein tensor

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$

The same theory is also written in later conventions as

$$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$

where $m$ is the mass scale, $\sigma=\pm1$, and $\lambda_0$ is the dimensionless cosmological parameter (Gullu et al., 2010, Tekin, 18 Jun 2026).

The determinant structure is the essential Born-Infeld feature. Rather than specifying a finite polynomial in curvature, BINMG packages an infinite tower of higher-curvature terms into a single square root of a determinant. This is why the theory is repeatedly described as an all-orders extension of NMG (Gullu et al., 2010, Gullu et al., 2010).

A broader Born-Infeld gravity analysis later embedded the three-dimensional theory into a $D\ge 3$ family. In that framework, the $D=3$ specialization takes the form

$c$0

with the Schouten tensor

$c$1

Because the Weyl tensor vanishes identically in three dimensions, the would-be free parameters of the higher-dimensional family disappear, and the final $c$2 theory has no remaining free couplings beyond the overall parameters already shown. In that sense, BINMG appears as the unique three-dimensional member of a wider Born-Infeld gravity class (Alkac et al., 2018).

2. Curvature expansion and relation to NMG

The low-curvature expansion is the central mechanism by which BINMG reproduces known three-dimensional massive gravity. The determinant identity used in the original construction generates the series

$c$3

The quadratic term is exactly the NMG invariant, while the cubic term matches the specific deformation previously obtained from AdS/CFT and a holographic $c$4-theorem (Gullu et al., 2010).

In later notation, the same point is summarized by the small-curvature determinant expansion

$c$5

which again isolates the special NMG combination

$c$6

This is the curvature-squared structure that yields a pure massive spin-2 theory without an extra scalar at the linearized level (Tekin, 18 Jun 2026).

The expansion is not merely a perturbative convenience. It is the reason BINMG is interpreted as a nonlinear completion of NMG rather than as an unrelated higher-curvature model. The order-by-order structure also underlies later claims that Born-Infeld gravity generates an infinite number of higher-derivative models admitting a holographic $c$7-function when suitably truncated (Alkac et al., 2018).

3. Vacuum structure, auxiliary-metric reformulation, and linearized degrees of freedom

The vacuum structure of BINMG differs sharply from ordinary NMG. For maximally symmetric spaces, the exact field equations reduce to an algebraic vacuum condition. In the conventions of the 2010 $c$8-function analysis, one finds

$c$9

and the physical interpretation given there is that, unlike ordinary NMG, BINMG has a unique vacuum for viable $$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$0 (Gullu et al., 2010).

A later auxiliary-metric reformulation makes this uniqueness especially transparent. Introducing an independent auxiliary metric $$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$1, the action becomes

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$2

with

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$3

Variation with respect to $$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$4 yields the algebraic equation

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$5

so the auxiliary formulation is an exact rewriting of the determinant theory on the regular branch. In the densitized variable $$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$6, the action becomes polynomial, and all derivative dependence is isolated in the single coupling $$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$7 (Tekin, 18 Jun 2026).

On a maximally symmetric background,

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$8

the auxiliary equation gives

$$I_{CBI}=4m^2\int d^3x\left[\sqrt{-\det\left(g+\frac{1}{m^2}G\right)}-(2\lambda+1)\sqrt{-\det g}\right],$$9

and the metric equation fixes

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$0

with $$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$1 required for a nondegenerate massive-gravity branch. This is the unique maximally symmetric vacuum in those conventions (Tekin, 18 Jun 2026).

The same reformulation also clarifies the linearized spectrum. Expanding around the unique vacuum, the quadratic action splits into a pure Einstein piece for the metric perturbation and a Pauli-Fierz action for the auxiliary fluctuation. The massive mode has

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$2

and the mass term takes the exact Fierz-Pauli form

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$3

The reported spectrum therefore contains no propagating massless graviton in three-dimensional Einstein gravity, one healthy massive spin-2 field, and two local helicities. A notable point is that the Fierz-Pauli tuning is generated by the Born-Infeld structure rather than imposed by hand (Tekin, 18 Jun 2026).

4. Holographic $$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$4-theorem and AdS/CFT interpretation

BINMG acquired much of its original motivation from holography. In a domain-wall geometry

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$5

the null-energy condition translates into a monotonicity statement for the warp factor and hence into a candidate holographic $$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$6-function (Gullu et al., 2010).

The first detailed BINMG analysis derived the exact equations of motion for globally and asymptotically (anti-)de Sitter spaces and showed that the null-energy condition leads to two simple $$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$7-functions. One is Einstein-like,

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$8

while the second is specific to the Born-Infeld structure,

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 g_{\mu\nu}R-m^2 g_{\mu\nu}.$$9

At an AdS fixed point, the fixed-point value reproduces the central charge of the asymptotic Virasoro algebra and the Weyl-anomaly coefficient,

$$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$0

using the paper’s relation between $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$1, $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$2, and $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$3 (Gullu et al., 2010).

A later treatment of the holographic $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$4-theorem emphasized that Born-Infeld gravity theories admit such a monotonic flow only for a highly restricted parameter choice. In the three-dimensional case, the tuned theory yields the second-order inequality

$$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$5

and hence the monotonic holographic $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$6-function

$$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$7

equivalently written there as

$$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$8

At an AdS fixed point with $$I_{\text{BINMG}[g]}=-\frac{4m^2}{\kappa^2}\int d^3x\Big(\sqrt{-\det\!\big(g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)\big)}-(1-\lambda_0)\sqrt{-g}\Big),$$9, the result becomes

$m$0

up to normalization conventions, and matches the central charge extracted independently from the Weyl anomaly calculation (Alkac et al., 2018).

The significance assigned to these results is explicit: generic higher-curvature theories usually produce higher-than-second-order flow equations, so no universal monotonic quantity exists. BINMG is special because its determinant structure preserves a second-order inequality in the domain-wall sector. This is the main holographic evidence that the Born-Infeld completion is not arbitrary (Gullu et al., 2010, Alkac et al., 2018).

5. Exact solutions, special spacetimes, and black-hole thermodynamics

The nonlinear solution space of BINMG is substantially richer than the perturbative expansion alone suggests. A detailed black-hole analysis constructed warped $m$1 black holes in three versions of the theory: the four-derivative expansion, the six-derivative expansion, and the full unexpanded Born-Infeld action. In each case the entropy, angular momentum, mass, and central charges of the putative dual conformal field theory were computed (Ghodsi et al., 2010).

A central conclusion of that work is that the four-derivative, six-derivative, and full Born-Infeld theories behave differently, and the properties of the black holes cannot be inferred simply by reading the lower-order truncations as if they were equivalent to the full action. In the expanded theories, uncharged, Maxwell-charged, and Maxwell-Chern-Simons-charged warped black holes all appear. In the full BI theory, by contrast, the Maxwell-charged branch collapses to a solution with

$m$2

so there is no genuinely charged Maxwell black hole of that type (Ghodsi et al., 2010).

On locally AdS backgrounds, the auxiliary-metric formulation yields a much simpler pattern. Since the auxiliary metric becomes proportional to the physical metric,

$m$3

the full higher-curvature structure reduces to an overall factor multiplying the Einstein results. For any asymptotic Killing vector $m$4,

$m$5

and in particular the BTZ quantities obey

$m$6

with Brown-Henneaux central charge

$m$7

and entropy

$m$8

The same paper states that this agrees with the direct Wald-entropy computation in the determinant formulation (Tekin, 18 Jun 2026).

BINMG also admits a particularly simple reduction on algebraic type N spacetimes. There the traceless Ricci tensor has the canonical form

$m$9

and the full Born-Infeld field equations reduce to the massive tensorial Klein-Gordon-type equation

$\sigma=\pm1$0

supplemented by the scalar-curvature relation

$\sigma=\pm1$1

This allows the general type N solution of ordinary NMG to be imported into BI-NMG with the appropriate parameter relation. At the same time, BI-NMG does not admit the critical-point solutions that correspond to logarithmic AdS pp-wave-type modes in finite-order curvature-corrected NMG: the would-be critical point $\sigma=\pm1$2 makes $\sigma=\pm1$3, while the other critical branch requires $\sigma=\pm1$4, which is also excluded (Ahmedov et al., 2012).

6. Supersymmetric, higher-dimensional, and related extensions

The supersymmetric status of BINMG is constrained and should be stated carefully. A detailed $\sigma=\pm1$5 off-shell supergravity construction in three dimensions built the most general parity-even higher-derivative action with terms up to six derivatives, starting from the Poincaré multiplet

$\sigma=\pm1$6

At four derivatives, supersymmetry plus ghost freedom around an $\sigma=\pm1$7 vacuum restrict the coefficients in

$\sigma=\pm1$8

to the unique ghost-free supersymmetric NMG combination, with

$\sigma=\pm1$9

At six derivatives, after excluding explicit-derivative curvature-squared terms such as $\lambda_0$0 and $\lambda_0$1, the same ghost-free AdS analysis leaves a unique invariant whose purely gravitational sector is

$\lambda_0$2

in that paper’s normalization. This matches the six-derivative truncation of Born-Infeld gravity and earlier holographic $\lambda_0$3-theorem results (Bergshoeff et al., 2014).

The same work then proposed a bosonic Born-Infeld supergravity structure,

$\lambda_0$4

whose perturbative truncations reproduce the bosonic sectors of Einstein supergravity, supersymmetric NMG, and supersymmetric cubic extended NMG. That paper explicitly does not present a fully completed all-orders supersymmetric BINMG action in closed form; instead, it proposes a bosonic Born-Infeld supergravity structure motivated and verified through the six-derivative level. It also states that the term BINMG is used there primarily for the known bosonic Born-Infeld extension of NMG, while the supersymmetric construction is a supergravity model whose bosonic sector contains BINMG as its purely gravitational truncation (Bergshoeff et al., 2014).

Beyond three dimensions, a Born-Infeld type extension of (non-)critical gravity was formulated as a natural extension of BI-NMG to arbitrary dimensions. Its action is built from the Einstein tensor inside a determinant, and for the special choice

$\lambda_0$5

the theory satisfies a holographic $\lambda_0$6-theorem, has no scalar graviton mode, and reduces to BINMG when $\lambda_0$7. After consistent truncation of ghost modes by AdS boundary conditions, the resulting non-critical Born-Infeld theory is argued to be classically equivalent to Einstein gravity at the nonlinear level (Yi, 2012).

A further related development is the 2025 Chern-Simons Inspired Massive Gravity model. There, after eliminating an auxiliary field, the action becomes a Born-Infeld-type determinant expression whose second-order expansion is equivalent to NMG. The construction is described as a Chern-Simons origin of a BINMG-like action rather than as BINMG itself, but it shows that the determinant logic associated with BI-NMG continues to organize later $\lambda_0$8-dimensional massive gravity model building (Chagoya et al., 7 Mar 2025).