We consider the question of counting the degrees of freedom in theoretical models, with an emphasis on theories of fields and gravity. Among the possible approaches, the Hamiltonian formulation remains one of the most systematic and robust tools. However, it can easily become long and technically involved. In this work, we present a broadly applicable recipe to find the degrees of freedom directly, based on the Lagrangian formulation. We compare it to the standard approaches, highlight the challenges that may arise in the latter, and demonstrate that the proposed method leads to transparent insights about the dynamical nature of theory in a quick, simple, and straight-forward way.

Japan

Anamaria Hell

Elisa Ferreira

Germany

Dieter Lüst

South Korea

Taiwan

Misao Sasaki

The recipe for the degrees of freedom.

In this paper we take the initial steps toward a formulation of species quantum mechanics. Specifically, we consider quantum operators that correspond to the species number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>s</mi></msub></math> and the tower mass scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>t</mi></msub></math> in the context of the swampland distance conjecture. We discuss the commutation relations, a possible wave function, and symplectic duality transformations on the conjugate variables. Furthermore, we argue that the Castellano-Ruiz-Valenzuela (CRV) pattern is a consequence of the canonical commutation rules of moduli space quantum mechanics. We also connect the canonical quantization to the periods of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></math> Calabi-Yau compactifications to explore other aspects of the CRV pattern, including its possible connection to the Ooguri-Vafa-Verlinde black hole quantization procedure.

United States

Luis A. Anchordoqui

France

Severin Lüst

Species quantum mechanics

We consider a class of theories containing power-law terms in both the Ricci scalar and a scalar field, including their non-minimal couplings. As a first step, we systematically classify all non-trivial cases with a propagating scalar field that arise from the simplest general power-law formulation, which contains the minimal number of terms. We then analyze each case in detail, focusing on the structure of the degrees of freedom, by both formulating the theories in the Einstein frames and focusing on the singular points in the Jordan frame. We demonstrate that such theories can give rise to different, and sometimes unexpected structure of the modes, that can change at the leading order depending on the background.

Aspects of non-minimally coupled curvature with power laws

The quantum gravity scale within the dark dimension scenario (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mo>*</mo></mrow></msub><mo>∼</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>9</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>) roughly coincides with the energy scale of the KM3-230213A neutrino (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>∼</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>8</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>). We propose an interpretation for this intriguing coincidence in terms of Hawking evaporation of five-dimensional (5D) primordial black holes (PBHs). 5D PBHs are bigger, colder, and longer-lived than 4D PBHs of the same mass. For brane observers, PBHs residing in the higher-dimensional bulk decay essentially invisibly (only through gravitationally and sterile coupled modes). As a consequence, constraints on the density of PBHs relative to that of dark matter from null searches of Hawking evaporation can be avoided. We demonstrate that Hawking evaporation of 5D bulk PBHs can explain the KM3-230213A neutrino, evade constraints from upper limits on the gamma-ray flux, and remain consistent with IceCube upper limits on the partial decay width of superheavy dark matter particles into neutrinos.

Francis Halzen

Neutrinos from primordial black holes in theories with extra dimensions

In this work, we investigate the connection between black hole instabilities and Swampland constraints, presenting new insights into the anti–de Sitter (AdS) distance conjecture. By examining the scale at which horizon instabilities of Schwarzschild-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>AdS</mi><mi>d</mi></msub></math> black holes take place—<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mrow><mi>BH</mi></mrow></msub></math>—we uncover a universal scaling relation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mrow><mi>BH</mi></mrow></msub><mo>∼</mo><mo stretchy="false">|</mo><msub><mi mathvariant="normal">Λ</mi><mi>AdS</mi></msub><msup><mo stretchy="false">|</mo><mi>α</mi></msup></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mi>d</mi></mfrac><mo>≤</mo><mi>α</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, linking the emergence of towers of states directly to instability scales as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>AdS</mi></msub><mo stretchy="false">→</mo><mn>0</mn></math>. This approach circumvents the explicit dependence on field-space distances, offering a refined formulation of the AdS distance conjecture grounded in physical black hole scales. From a top-down perspective, we find that these instability scales correspond precisely to the Gregory-Laflamme and Horowitz-Polchinski transitions, as expected for the flat space limit, and consistently with our proposed bounds. Furthermore, revisiting explicit calculations in type IIB string theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>×</mo><msup><mi>S</mi><mn>5</mn></msup></mrow></math>, we illustrate how higher-derivative corrections may alter these bounds, potentially extending their applicability toward the interior of moduli space. Using also general results about gravitational collapse in AdS, our analysis points toward a possible breakdown of the conjecture in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>&gt;</mo><mn>10</mn></math>, suggesting an intriguing upper limit on the number of noncompact spacetime dimensions. Finally, we briefly discuss parallel considerations and implications for the dS case.

Alvaro Herráez

Carmine Montella

Black hole transitions, AdS spacetime, and the distance conjecture

We consider conformal and scale-invariant gravities in d dimensions, with a special focus on pure $\textit{R}$ gravity in the scale-invariant case. In four dimensions, the structure of these theories is well known. However, in dimensions larger than four, the behavior of the modes is so far unclear. In this work, we explore this question, studying the theories in conformally flat spacetimes as well as anisotropic backgrounds. First, we consider the pure theory in d-dimensions. We show that this theory propagates no degrees of freedom for flat space-time. Otherwise, we find the theory in the corresponding Einstein frame and show that it propagates a scalar field and two tensor modes, that arise from Einstein’s gravity. We then consider conformal gravity in d dimensions. We argue on the number of degrees of freedom for conformally flat space-times and show that for $\textit{d >}$ 4, there exists a frame in which this theory can be written as the Weyl-squared gravity with a cosmological constant, and also generalize this formulation to the $\textit{f}$($\textit{W}$) theories. Then, we consider the specific model of conformal gravity in five dimensions. We find the analytical and numerical solutions for the anisotropic Universe for this case, which admits super-Hubble and exponential expansions. Finally, we consider the perturbations around these solutions and study the number of the degrees of freedom.

Conformal and pure scale-invariant gravities in $\textit{d}$ dimensions

The latest DESI DR2 results, when combined with other independent cosmological data on the Cosmic Microwave Background and supernovas, suggest a preference for dynamical dark energy. We propose a novel cosmological scenario, which features two distinct scalar fields. One governs the magnitude of the present-day dark energy density and is related to the size of extra-dimensions. Accounting for the observed smallness of this energy density requires the scalar to reside near the boundary of field space. The second field, responsible for the time evolution of dark energy and associated with the string coupling, must instead lie in the bulk to remain consistent with the non-observation of light string states. We show that a natural candidate for such dark energy dynamics is a quintessence modular-invariant potential, in which the second scalar field rolls down a negatively curved slope, starting from a self-dual critical point. We find that this scenario is in good agreement with the latest findings by DESI.

Luis Anchordoqui

Thailand

Ignatios Antoniadis

Belgium

Niccolò Cribiori

Türkiye

Arda Hasar

Joaquin Masias

Italy

Marco Scalisi

Bulk/boundary modular quintessence and DESI

We propose a dark energy model in which a quintessence field <italic>&#981;</italic> rolls near the vicinity of a local maximum of its potential characterized by the simplest <italic>S</italic> self-dual form <math altimg='si1.svg'><mi>V</mi><mo stretchy='false'>(</mo><mi>&#981;</mi><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mi mathvariant='normal'>&#923;</mi><mspace width='0.25em' /><mrow><mi mathvariant='normal'>sech</mi></mrow><mo stretchy='false'>(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mspace width='0.2em' /><mi>&#981;</mi><mo stretchy='false'>/</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy='false'>)</mo></math>, where <math altimg='si2.svg'><msub><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow></msub></math> is the reduced Planck mass and <math altimg='si3.svg'><mi mathvariant='normal'>&#923;</mi><mo>&#8764;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>120</mn></mrow></msup><msubsup><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msubsup></math> is the cosmological constant. We confront the model with Swampland ideas and show that the <italic>S</italic>-dual potential is consistent with the distance conjecture, the de Sitter conjecture, and the trans-Planckian censorship conjecture. We also examine the compatibility of this phenomenological model with the intriguing DESI DR2 results and show that the shape of the <italic>S</italic>-dual potential is almost indistinguishable from the axion-like potential, <math altimg='si4.svg'><mi>V</mi><mo stretchy='false'>(</mo><mi>&#981;</mi><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mspace width='0.25em' /><msubsup><mrow><mi>f</mi></mrow><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mspace width='0.25em' /><mo stretchy='false'>[</mo><mn>1</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mi mathvariant='normal'>cos</mi><mo>&#8289;</mo><mo stretchy='false'>(</mo><mi>&#981;</mi><mo stretchy='false'>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo></math>, with <math altimg='si5.svg'><msub><mrow><mi>m</mi></mrow><mrow><mi>a</mi></mrow></msub></math> and <math altimg='si6.svg'><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi></mrow></msub></math> parameters fitted by the DESI Collaboration to accommodate the DR2 data. The self-dual potential has the advantage that one starts at the self-dual point and this is a theoretical motivation, because as the universe cools off the <math altimg='si7.svg'><msub><mrow><mi mathvariant='double-struck'>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> symmetry gets broken leading to a natural rolling away from the symmetric point.

<bold><italic>S</italic></bold>-dual quintessence, the Swampland, and the DESI DR2 results

In the context of the Swampland program, black hole attractors have been employed to probe infinite distances in moduli space, where the EFT cutoff goes to zero in Planck units and UV effects become significant. In this paper, we take the perspective of the two-derivative action of string theoretic effective field theories and explore various families of extremal black hole solutions that probe infinite distance limits at their horizons. While these solutions do not include higher-order corrections in the EFT expansion, we find that, in many cases, the smallest BPS black holes in these families remarkably reproduce either the species scale or some other Kaluza-Klein scale. In highly supersymmetric cases, this match with UV scales even persists in the interior of moduli space. We explore the use of the species length bound on classical black hole sizes as a criterion to rule out inconsistent EFTs, using a 9d bottom-up model arising from an $\textit{ad hoc}$ truncation of maximal supergravity. These observations suggests that the two-derivative action may encode information about relevant UV scales. We discuss the interplay of these results with emergence and UV/IR mixing in quantum gravity.

Switzerland

José Calderón-Infante

Matilda Delgado

Yixuan Li

Spain

Angel Uranga

Classical black hole probes of UV scales

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.

Israel

Saskia Demulder

Thomas Raml

Navigating string theory field space with geometric flows