⚛️ high-energy theory

3d Lovelock gravity and the holographic c-theorem: Proof at all orders and resummation

This paper proves that 3d Lovelock gravity (Horndeski theory) admits a holographic c-theorem at all orders for arbitrary couplings, demonstrating that the dual theory is scale-invariant but not conformal, and provides closed-form resummed expressions for the Lagrangians, c-functions, and central charges that match the UV c-function value.

Original authors: Gokhan Alkac, Luis Guajardo, Hikmet Ozsahin

Published 2026-09-29

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CC BY 4.0

Original authors: Gokhan Alkac, Luis Guajardo, Hikmet Ozsahin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity, as we experience it daily, is the force that keeps our feet on the ground and the planets in their orbits. For over a century, our best description of this force has been Einstein's theory of general relativity, which portrays gravity not as a pull, but as the bending of space and time itself. However, physicists suspect that this elegant theory is only a partial picture, valid at the low energies of our everyday world. When we look at the universe at its most extreme scales—inside black holes or at the very moment of the Big Bang—Einstein's equations likely need to be corrected by adding new, more complex terms that account for the curvature of space itself. The challenge is that adding these corrections often breaks the theory, introducing mathematical ghosts or impossible behaviors. A specific family of theories, known as Lovelock gravity, was designed to avoid these pitfalls in higher dimensions, but a direct translation to our three-dimensional space seemed impossible because the mathematical ingredients simply vanished.

Recent breakthroughs have shown that by carefully redefining these higher-dimensional ingredients, a new, consistent theory of gravity can emerge in three dimensions. This theory, which includes a mysterious scalar field alongside the usual geometry of space, offers a unique laboratory for testing deep ideas about how gravity and quantum mechanics might fit together. One of the most profound questions in this field is whether gravity can explain the flow of information in the universe, specifically through a principle called the holographic c-theorem. This principle suggests that as the universe evolves, a specific measure of its complexity, known as the central charge, must always decrease or stay the same, never increase, as it moves from a high-energy state to a low-energy one. Proving that this rule holds true for these new, complex theories of gravity is essential to understanding if they are viable descriptions of reality.

In a recent study, a team of researchers has provided a complete proof that this holographic rule holds for the newly defined three-dimensional Lovelock gravity, no matter how many complex corrections are added to the theory. They demonstrated that for any possible strength of the forces involved, the theory respects this fundamental law of evolution. The researchers focused on a specific type of solution called a domain wall, which acts like a bridge connecting different regions of space. By analyzing the equations that govern how space stretches and how the scalar field changes along this bridge, they discovered a remarkable simplification: the behavior of the scalar field is locked directly to the stretching of space. This lock allows the complex equations to be solved exactly, revealing that the measure of complexity always behaves correctly, flowing in the right direction as the universe evolves.

The team also calculated the exact value of this complexity measure at the very beginning of the flow, known as the ultraviolet limit, where the theory is most active. They found that this value is determined by a simple, linear relationship with the strength of the various forces in the theory. To ensure their result was not just a mathematical trick, they performed an independent calculation using the energy of the system on a spherical surface. The two methods matched perfectly, confirming that the theory is consistent and that the complexity measure is well-defined. Interestingly, they found that while the universe described by this theory looks the same at all scales, it does not possess the full symmetry of a perfect mirror image, meaning the dual description of this gravity is scale-invariant but not fully conformal. This subtle difference does not break the rules of the theory but adds a new layer of understanding to how these gravitational models relate to the quantum world.

The researchers then took their findings a step further by applying them to specific versions of the theory that have been proposed to solve the problem of singularities, the infinitely dense points found inside black holes. They showed that for these specific versions, the theory remains healthy and stable, provided the strength of the forces stays within certain bounds. If the forces become too strong, the theory breaks down, and the universe described by it becomes unstable. By mapping out these boundaries, the team identified exactly which versions of the theory are physically possible and which are not. They found that in the most promising versions, the complexity measure is always positive, a necessary condition for the theory to make physical sense, and that the universe described by these theories has a unique, stable starting point.

This work fills a critical gap in our understanding of gravity in three dimensions. It confirms that the new, regularized Lovelock theories are not just mathematical curiosities but robust frameworks that obey the fundamental laws of information flow. The proof holds for every possible order of correction, meaning the result is not an approximation but a complete and exact description of how these theories behave. The researchers have also provided a way to sum up the infinite series of corrections into a single, manageable form for the most interesting cases, allowing for a clearer view of the theory's structure. By establishing that these theories respect the holographic c-theorem and possess a well-defined central charge, the study strengthens the case that these models are the true three-dimensional analogues of the higher-dimensional Lovelock gravity, offering a promising path toward resolving the deepest mysteries of black holes and the early universe.

Technical Summary: 3d Lovelock Gravity and the Holographic c-Theorem

Problem Statement
Three-dimensional (3d) gravity serves as a simplified laboratory for studying gravitational phenomena and the AdS/CFT correspondence. While New Massive Gravity (NMG) has been a prominent toy model, it suffers from fourth-order field equations, which are generally undesirable due to potential ghost-like degrees of freedom. True analogues of Lovelock gravity in 3d were recently identified as Horndeski theories obtained by regularizing Lovelock invariants in lower dimensions. These theories possess second-order field equations and inherit properties from higher-dimensional parents, such as the resolution of singularities in resummed versions. However, a critical gap remained: it was unknown whether these 3d Lovelock theories admit a holographic c-theorem at all orders and for arbitrary coupling constants. Furthermore, the nature of the dual field theory and the precise definition of its central charge in the presence of a non-trivial scalar profile required clarification.

Methodology
The authors employ a minisuperspace approach, restricting the analysis to domain wall solutions with a symmetric ansatz. This reduction allows for the computation of field equations from a reduced action, which is tractable for the infinite tower of regularized Lovelock invariants.

Field Equations: The study utilizes the domain wall metric $ds^2 = e^{2A(r)}(-dt^2 + dx^2) + dr^2$ and a radial scalar profile $\phi(r)$ . By varying the reduced action, the authors derive the equations of motion for the warp factor $A(r)$ and the scalar $\phi(r)$ .
Null Energy Condition (NEC): The NEC is imposed on the matter sector, leading to an inequality involving the derivatives of the warp factor and the scalar.
c-Function Construction: The authors construct a candidate c-function $c(r)$ and demonstrate that its derivative is proportional to the NEC quantity, ensuring monotonicity along the renormalization group (RG) flow.
Central Charge Verification: To independently verify the UV value of the c-function, the authors compute the logarithmic divergence of the Euclidean on-shell action on a sphere ( $S^2$ ). This involves solving the equations of motion for the scalar and metric on the hyperbolic space $H_3$ and extracting the coefficient of the logarithmic divergence.
Resummation: For specific choices of coupling constants that allow the infinite tower of corrections to be summed in closed form, the authors utilize generating functions to express the Lagrangian, c-function, and central charge analytically.

Key Contributions and Results

Proof of the Holographic c-Theorem at All Orders: The paper proves that 3d Lovelock gravity admits a holographic c-theorem for arbitrary coupling constants. A crucial finding is that the equation of motion for the Horndeski scalar is a total derivative at every order. This forces a "locking" condition where the derivative of the scalar is equal to the derivative of the warp factor ( $\phi' = A'$ ) along the flow (assuming a vanishing conserved charge). Consequently, the NEC reduces to a form that can be integrated to yield a monotonic c-function.
Nature of the Dual Theory: The vacuum solution features a linear scalar profile ( $\phi \propto r$ ). This profile breaks the special conformal transformations of $AdS_3$ , implying that the dual field theory is scale-invariant but not conformally invariant. The c-theorem and the computation of the central charge remain valid, but the central charge is defined strictly as the coefficient of the logarithmic divergence of the on-shell action, rather than as a Weyl anomaly coefficient of a conformal field theory.
Central Charge Calculation: The authors compute the central charge $c_\infty$ exactly for the Gauss-Bonnet case and asymptotically for the full tower. They find that $c_\infty$ is linear in the coupling constants and matches the UV limit of the constructed c-function.
Resummed Theories and Generating Functions: Since the results depend on the couplings only through two generating functions, $h(x)$ and $g(x)$ , the authors provide closed-form expressions for the resummed Lagrangians, c-functions, and central charges for five specific choices of couplings (including those leading to regular black holes).
Vacuum Structure and Positivity: The analysis classifies the resummed theories based on the positivity of the effective Newton constant (related to $1+g$ $1 + g$ ) and the uniqueness of the vacuum (related to the function $P = 1-h'$ $P = 1 - h^{'}$ ).
- For couplings where all orders contribute with positive signs (e.g., $c_n=1$ ), the central charge is positive, the vacuum is unique, and higher-order terms raise the central charge.
- For couplings involving only odd orders (where alternating signs are lost), the theory exhibits singularities at finite coupling values, and the central charge can become negative beyond a critical value of the cosmological constant, requiring the discarding of such vacua.
- A deformation parameter $q$ in the couplings $c_n = q^{-n}$ allows for a transition between unique and multiple vacuum branches.

Significance and Claims
The paper claims to fill a significant gap in the study of 3d Lovelock gravity by establishing the existence of a holographic c-theorem at all orders, thereby supporting the view that these Horndeski theories are the true 3d analogues of Lovelock gravity, superior to NMG in this regard. The work demonstrates that the "locking" of the scalar to the warp factor is a robust feature of the regularized invariants, enabling the integration of the NEC into a monotonic function.

The authors modestly note that while the central charge is well-defined, the dual theory's lack of conformal invariance raises questions about unitarity and the applicability of standard Cardy formulas for black hole entropy. They suggest that the unitarity of the bulk theory and the spectrum of the boundary theory require further investigation, particularly given that a 2d scale-invariant theory that is not conformal must either be non-unitary or possess a continuous spectrum. Additionally, the extension of entropy formulas to the regular black holes of these resummed theories is identified as a natural next step, distinct from standard conformal expectations.

Technical Summary: 3d Lovelock Gravity and the Holographic c-Theorem

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