Motivated by the recently obtained Hadrons to Atomic nuclei from Lattice QCD potentials for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mtext>−</mtext><mi>ϕ</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mtext>−</mtext><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mtext>−</mtext><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> interactions, we investigate the structure of the exotic nuclei <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><mrow><mi>ϕ</mi></mrow><mrow><mn>8</mn></mrow></mmultiscripts></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow><mrow><mn>8</mn></mrow></mmultiscripts></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><msub><mi>η</mi><mi>c</mi></msub><mrow><mn>8</mn></mrow></mmultiscripts></mrow></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi><mo>+</mo><mi>α</mi><mo>+</mo><mi>M</mi></math> three-body systems (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math> denotes the meson). The bound and resonant states are calculated consistently using the Gaussian expansion method, with resonances identified via the complex scaling method. For the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi><mi>ϕ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math>-charmonium interactions, a folding potential is constructed based on the Hadrons to Atomic nuclei from Lattice (HAL) QCD potentials and fitted to a Woods-Saxon form. We find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math> meson exhibits a strong “gluelike” effect, binding the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>4</mn><mn>1</mn><mo>+</mo></msubsup></math> resonant states of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>8</mn></mrow></mmultiscripts></mrow></math> into stable states and significantly reducing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>α</mi><mtext>−</mtext><mi>α</mi></mrow></math> distance. In contrast, the interactions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> with the nucleus are weaker, forming only shallow bound states with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></math> state of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>8</mn></mrow></mmultiscripts></mrow></math> and even increasing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>α</mi><mtext>−</mtext><mi>α</mi></mrow></math> separation. Notably, our analysis predicts weakly bound <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>α</mi><mtext>−</mtext><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> states in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>4</mn></mrow></mmultiscripts></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></math> channels, a result not reported in prior studies, which suggests that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow><mrow><mn>8</mn></mrow></mmultiscripts></math> may not be a Borromean nucleus. The sensitivity of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><mrow><mi>ϕ</mi></mrow><mrow><mn>8</mn></mrow></mmultiscripts></mrow><mo stretchy="false">(</mo><msubsup><mn>4</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></math> state—transitioning from bound to resonant depending on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math>-particle radius—highlights the subtle dynamics at play. These results provide a systematic theoretical comparison of how different vector mesons modify nuclear clustering, offering critical predictions for future experimental searches of such exotic hadron-nucleus systems.

China

Hao Zhou

Xiang Liu

Three-body resonances of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi><mi>α</mi><mi>M</mi></math> clusters (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mo>=</mo><mi>ϕ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math>) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Be</mi></mrow><mprescripts></mprescripts><mrow><mi>M</mi></mrow><mrow><mn>8</mn></mrow></mmultiscripts></mrow></math> nuclei

Scalar fields with masses protected by global shift symmetries, commonly referred to as axions, are abundantly used in effective field theories in cosmology and particle physics. However, global symmetries cannot be expected to be protected at the fundamental level. Finding consistent ultra-violet completions for axions is therefore a necessity. In this work, we identify the axion with the position mode of a charged 3-brane in (4+1)-dimensions. The shift symmetry of the axion is then a residual diffeomorphism in the fifth dimension orthogonal to the brane. Meanwhile, the brane is coupled to a flux in the fifth dimension. From the (3+1)-dimensional perspective, this construction generates (perturbatively) a mass for the axion and matches previously known proposals in the literature based on the coupling between the axion and a three-form gauge field. In a second step, we uplift this (4+1)-dimensional model to M-theory, where the same three-form is found to couple to the membrane with a (2+1)-dimensional worldvolume. In particular, our proposal also elucidates the duality between the axion and a two-form gauge field in the literature. We show that this dual two-form couples to the boundary of an open membrane in M-theory. Finally, we comment on the relations to and differences from other closed and open string axion monodromy models.

Sweden

Florian Niedermann

Denmark

Ziqi Yan

Axions, three-forms, and M-theory

We study the viability of light thermal dark matter (DM) in the sub-GeV mass range in view of the stringent new DAMIC-M limits on DM-electron scattering. Considering a Dirac fermion singlet DM charged under a new Abelian gauge symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math>, we outline two possibilities: (i) family nonuniversal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> gauge coupling with resonantly enhanced DM annihilation into Standard Model (SM) fermions, and (ii) family universal dark <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> gauge symmetry where relic density is set by DM annihilation into light gauge bosons. As an illustrative example of the first class of models, we consider a gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> extension of the SM having interesting detection prospects at several experiments. While both of these classes of models lead to the observed DM relic and consistency with DAMIC-M together with other experimental limits, the second class of models also lead to strong DM self-interactions, potentially solving the small-scale structure issues of cold dark matter. While a vast part of the parameter space in both the models is already ruled out, the current allowed region of parameter space can be further probed at ongoing or future experiments, keeping the models testable.

India

Debasish Borah

Satyabrata Mahapatra

Narendra Sahu

Vicky Singh Thounaojam

Light thermal dark matter models in the light of DAMIC-M 2025 constraints

Flavor hierarchies emerge from a single hierarchical parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> in a one-flavon Froggatt–Nielsen scheme. Fixing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>=</mo><mn>5.357</mn></math> from charged-lepton ratios (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>:</mo><mtext> </mtext><mtext> </mtext><msub><mrow><mi>m</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>:</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>∝</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>:</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mn>1</mn></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ε</mi><mo>=</mo><mn>1</mn><mo>/</mo><mi>B</mi></math>), we reproduce quark masses and Cabibbo-Kobayashi-Maskawa targets at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>Z</mi></msub></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> coefficients. The same <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ε</mi></math> gives viable lepton textures and benchmarks for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>U</mi><mrow><mi>PMNS</mi></mrow></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∑</mo><msub><mi>m</mi><mi>ν</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mrow><mi>β</mi><mi>β</mi></mrow></msub></math>. Compact correlations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo><mo>,</mo><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>,</mo><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>,</mo><msubsup><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow><mrow><mi>PMNS</mi></mrow></msubsup></mrow></math>) follow from powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ε</mi></math>.

United States

Vernon Barger

One-flavon flavor: A single hierarchical parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> organizes quarks and leptons at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>Z</mi></msub></math>

In this work, we derive a new class of analytic black hole solutions within the framework of <math altimg='si8.svg'><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>,</mo><mi>T</mi><mo>)</mo></mrow></math> gravity, where the black hole is surrounded by an anisotropic fluid acting as the matter source. We consider both linear and nonlinear forms of the function <math altimg='si8.svg'><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>,</mo><mi>T</mi><mo>)</mo></mrow></math>, enabling a detailed exploration of how anisotropic pressures and different <math altimg='si8.svg'><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>,</mo><mi>T</mi><mo>)</mo></mrow></math> influence the spacetime structure. Furthermore, we derive the conditions on the coupling parameters ( <italic>&#946;</italic> <inf>1</inf>, <italic>&#946;</italic> <inf>2</inf>, <italic>&#946;</italic> <inf>3</inf>) under which the energy condition is satisfied. Utilising these constraints, we then investigate the thermodynamic behaviour of the resulting black hole solutions in the presence of various matter fields. An important outcome of this study is that the results obtained deviate from those predicted by standard General Relativity. It is also observed that these deviations depend explicitly on the interaction between the matter Lagrangian <math altimg='si9.svg'><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub></math> and the trace <italic>T</italic> of the energy-momentum tensor.

Aniruddha Ghosh

Ujjal Debnath

The new black hole solution with anisotropic fluid in <math altimg='si8.svg'><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>,</mo><mi>T</mi><mo>)</mo></mrow></math> gravity: Thermodynamics

We initiate the study of flux compactifications on non-supersymmetric Riemann-flat manifolds (RFM’s) with Casimir energy. While curvature and other corrections are suppressed in RFM’s, the inclusion of Casimir energies allows one to evade standard dS no-go theorems, and the absence of orientifolds or other singular sources means that the construction is completely captured by ten or eleven-dimensional supergravity. We obtain a fully explicit formula for the Casimir stress-energy in a general RFM, including its ten or eleven-dimensional profile. The Casimir energy localizes in particular loci of the RFM, which we call “Casimir branes”. The tension of Casimir branes sometimes cancels exactly, due to a spacetime analog of worldsheet Atkin-Lehner symmetry. We use Casimir energies to construct an explicit $\textit{dS}$ maximum solution of a flux compactification of M-theory on a specific 6-dimensional RFM. The resulting solution is scale-separated, has a vacuum energy of 10 in five-dimensional Planck units, the Hubble radius is 10 Planck lengths, and the light fields have masses of order $\textit{H}$. This is a fully explicit, top-down de Sitter maximum in M-theory, with precisely computable vacuum energy. While the solution is not parametric, it is under extraordinary control: higher derivative and loop corrections to the vacuum energy are suppressed in powers of a small parameter $\textit{δV}$/$\textit{V}$ ~ 10 , and M2 and M5-brane instantons are negligible. In short, the solution survives all known corrections. Nevertheless, it might be sensitive to more exotic ones, such as e.g. loops of 11d Planckian virtual black holes if there were a large enough number of them. We also extend the Ewald numerical method for lattice sums to arbitrary dimensions and develop an efficient numerical implementation.

Spain

Bruno Bento

Miguel Montero

An M-theory dS maximum from Casimir energies on Riemann-flat manifolds

The observed matter–antimatter asymmetry of the Universe remains a fundamental challenge in modern physics. In this work, we explore gravitational baryogenesis within the framework of <italic>f</italic>( <italic>T, L<inf>m</inf></italic>) gravity, where the gravitational Lagrangian depends on both the torsion scalar <italic>T</italic> and the matter Lagrangian <italic>L<inf>m</inf></italic>. We consider three representative models and examine their ability to generate the observed baryon-to-entropy ratio. Our analysis shows that couplings involving both torsion and the matter Lagrangian, <math altimg='si35.svg'><mrow><msub><mi>&#8706;</mi><mi>&#956;</mi></msub><mrow><mo>(</mo><mo>&#8722;</mo><mi>T</mi><mo linebreak='badbreak'>&#8722;</mo><mfrac><msub><mi>L</mi><mi>m</mi></msub><msub><mi>L</mi><mn>0</mn></msub></mfrac><mo>)</mo></mrow></mrow></math>, can successfully account for the baryon asymmetry for decoupling temperatures in the range 10 12&#8211;10 14&#8201;GeV, while remaining consistent with small deviations from General Relativity. These results highlight the capacity of <italic>f</italic>( <italic>T, L<inf>m</inf></italic>) gravity to provide novel mechanisms for baryogenesis, demonstrating that the interplay between torsion and matter-sector contributions can naturally generate the observed asymmetry. The framework also remains compatible with late-time cosmological evolution, offering a unified setting for early- and late-time dynamics.

Portugal

Daniel F P Cruz

David S. Pereira

Francisco S N Lobo

Gravitational baryogenesis in <italic>f</italic>( <italic>T, L<inf>m</inf></italic>) gravity

In the context of the left-right symmetric model, we study the interplay of neutrinoless double beta (0 <italic>&#957;&#946;&#946;</italic>) decay, parity-violating M&#248;ller scattering, and high-energy colliders, resulting from the Yukawa interaction of the right-handed doubly-charged scalar to electrons, which could evade the severe constraints from charged lepton flavor violation. The 0 <italic>&#957;&#946;&#946;</italic> decay amplitude receives additional contributions from right-handed sterile neutrinos. The half-life, calculated in the effective field theory framework, allows for an improved description of the contributions involving non-zero mixing between left- and right-handed <italic>W</italic> bosons and those arising from exchanging a light right-handed neutrino. We find that the relative sensitivities between the low-energy (or high-precision) and high-energy experiments are affected by the left-right mixing. On the other hand, our results show how the interplay of collider and low-energy searches provides a manner to explore regions that are inaccessible to 0 <italic>&#957;&#946;&#946;</italic> decay experiments.

Gang Li

Michael J. Ramsey-Musolf

Sebastián Urrutia Quiroga

Italy

Juan Carlos Vasquez

Dissecting lepton number violating interactions in the left-right symmetric model: 0 <italic>&#957;&#946;&#946;</italic> decay, M&#248;ller scattering, and collider searches

This study rigorously examines the influence of dark matter within the framework of <math altimg='si10.svg'><mrow><mi>f</mi><mo>(</mo><mi mathvariant='script'>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity, with a particular emphasis on its critical role in facilitating traversable wormhole solutions. By employing anisotropic matter sources modeled through Bose-Einstein condensates and Einasto density profiles, we derive a novel class of wormhole solutions. The construction of the shape function relies on the density profile equation in conjunction with modified field equations, thereby establishing a theoretical foundation for these solutions. Careful selection of parameters ensures that the proposed solutions satisfy all requisite conditions for traversability. Moreover, we conduct a systematic evaluation of energy conditions across two distinct scenarios, further validating the physical viability of the models. The investigation also extends to analyzing anisotropic effects, exoticity parameters, and active gravitational mass. This work presents a comprehensive exploration of the physical feasibility and stability of wormhole structures within the context of <math altimg='si10.svg'><mrow><mi>f</mi><mo>(</mo><mi mathvariant='script'>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> modified gravity theories, offering significant contributions to our understanding of the interplay between dark matter and advanced gravitational phenomena.

H Aruna Kumara

Chaitra Chooda Chalavadi

V. Venkatesha

Investigating wormhole structures in <math altimg='si10.svg'><mrow><mi>f</mi><mo>(</mo><mi mathvariant='script'>R</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity: Implications for dark matter

We present, to our knowledge, the first systematic study of early-late cosmic evolution and acceleration in the framework of <math altimg='si15.svg'><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity, an extension of teleparallel theories coupling torsion with the matter Lagrangian. By incorporating the Big-Bang Nucleosynthesis (BBN) bound on the freeze-out temperature, we obtain a tight constraint on the inverse-torsion parameter, ensuring consistency with early-time physics. Employing Markov Chain Monte Carlo analyses with progressively richer observational datasets, CC, Union3, and SN22 supernovae, we constrain a well-motivated model and reconstruct key cosmological functions. The reconstructed Hubble and distance modulus functions show excellent agreement with the observations, confirming the observational viability of the model. The model successfully reproduces the observed late-time expansion history, yielding a transition from deceleration to acceleration through the deceleration parameter. The effective equation of state is found to remain negative throughout, with present values <math altimg='si16.svg'><mrow><msub><mi>w</mi><mn>0</mn></msub><mo linebreak='goodbreak'>&gt;</mo><mo>&#8722;</mo><mn>1</mn></mrow></math>, indicating a quintessence-like behavior rather than a cosmological constant or phantom regime. These results highlight the ability of <math altimg='si15.svg'><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity to mimic the concordance scenario while allowing controlled deviations in the expansion history.

Sai Swagat Mishra

Suchita Patel

P.K. Sahoo

BBN to late-time acceleration in <math altimg='si15.svg'><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity