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In this work, we aim to characterize the structure of higher-derivative corrections within low-energy Effective Field Theories (EFTs) arising from a UV-complete theory of quantum gravity. To this end, we use string theory as a laboratory and argue that such EFTs should exhibit a $\textit{double EFT expansion}$ involving higher-curvature operators. The $\textit{field-theoretic}$ expansion is governed by the mass of the lightest (tower of) new degrees of freedom, as expected from standard field theory considerations. Conversely, the $\textit{quantum-gravitational}$ expansion is suppressed relative to the Einstein-Hilbert term by the quantum gravity cutoff, $\Lambda_{\rm QG}$, above which no local gravitational EFT description remains valid. This structure becomes manifest in the so-called $\textit{asymptotic regime}$, where a hierarchy between the Planck scale and $\Lambda_{\rm QG}$ emerges, the latter identified herein as the species scale. Most notably, we demonstrate the features of the double EFT expansion through an amplitudes-based approach in (toroidal compactifications of) ten-dimensional Type IIA string theory, and via a detailed analysis of the supersymmetric black hole entropy in 4d $\mathcal{N}=2$ supergravities derived from Type II Calabi-Yau compactifications. We provide further evidence for our proposal across various string theory setups, including Calabi-Yau compactifications of M/F-theory and Type II string theory. Finally, we explore the implications of this framework for the Wilson coefficients of the aforementioned higher-curvature operators, revealing potentially significant constraints in the asymptotic regime and highlighting a remarkable interplay with recent results from the S-matrix bootstrap program., Comment: 57 pages + Appendices, 3 figures

Species thermodynamics has been proposed in analogy to black hole thermodynamics. The entropy scales like an area and is given by the mere counting of the number of the species. In this work, we $\textit{derive}$ the constitutive relations of species thermodynamics and explain how those $\textit{originate}$ from standard thermodynamics. We consider configurations of species in thermal equilibrium inside a box of size $L$, and show that the temperature $T$ of the system, which plays a crucial role, is always upper bounded above by the species scale $\Lambda_{\rm sp}$. We highlight three relevant regimes: (i) when $L^{-1}< T<\Lambda_{\rm sp}$, and gravitational collapse is avoided, the system exhibits standard thermodynamics features, for example, with the entropy scaling like the volume of the box; (ii) in the limit $L^{-1}\simeq T\rightarrow \Lambda_{\rm sp}$ we recover the rules of species thermodynamics with the entropy scaling like the area of the box; (iii) an intermediate regime with $ L^{-1}\simeq T< \Lambda_{\rm sp}$ that avoids gravitational collapse and fulfills the Covariant Entropy Bound; this interpolates between the previous two regimes and its entropy is given simply in terms of the counting of the species contributing to the thermodynamic ensemble. This study also allows us to find a novel and independent bottom-up rationale for the Emergent String Conjecture. Finally, we present the $\textit{Black Hole - Tower Correspondence}$ as a generalization of the celebrated Black Hole - String Correspondence. This provides us with a robust framework to interpret the results of our thermodynamic investigation. Moreover, it allows us to qualitatively account for the entropy of black holes in terms of the degrees of freedom of the weakly coupled species in the tower., Comment: 61 pages, 5 figures; typos corrected; details, references, footnotes and subsection 3.4 added

The concept of the species scale as the quantum gravity cut-off has been recently emphasised in the context of the Swampland program. Along these lines, we continue the quest for a precise understanding of its role within effective field theories of gravity as well as a global definition of the latter in case there is enough supersymmetry preserved. To do so, we exploit duality symmetries, the familiar asymptotic dependence imposed by the presence of infinite towers of light states and the known behaviour of higher-curvature corrections to the Einstein-Hilbert action in various String Theory setups. In those cases, we obtain a self-consistent result for the identification of the species scale as the quantum gravity cut-off, but also present some puzzles related to the suppression of certain higher-dimensional operators as well as minor ambiguities that may arise in the deep interior of moduli space., Comment: Minor changes, references added and Section 4 significantly extended. Matches the published version

The Swampland Distance Conjecture (SDC) states that, as we move towards an infinite distance point in moduli space, a tower of states becomes exponentially light with the geodesic distance in any consistent theory of Quantum Gravity. Although this fact has been tested in large sets of examples, it is fair to say that a bottom-up justification that explains both the geodesic requirement and the exponential behavior has been missing so far. In the present paper we address this issue by making use of the Covariant Entropy Bound as applied to the EFT. When applied to backgrounds of the Dynamical Cobordism type in theories with a moduli space, we are able to recover these main features of the SDC. Moreover, this naturally leads to universal lower and upper bounds on the 'decay rate' parameter $\lambda_{\text{sp}}$ of the species scale, that we propose as a convex hull condition under the name of Species Scale Distance Conjecture (SSDC). This is in contrast to already proposed universal bounds, that apply to the SDC parameter of the lightest tower. We also extend the analysis to the case in which asymptotically exponential potentials are present, finding a nice interplay with the asymptotic de Sitter conjecture. To test the SSDC, we study the convex hull that encodes the (asymptotic) moduli dependence of the species scale. In this way, we show that the SSDC is the strongest bound on the species scale exponential rate which is preserved under dimensional reduction and we verify it in M-theory toroidal compactifications., Comment: v2: minor changes and typos corrected. 54 pages + Appendix, 17 figures, 2 tables

Based on Quantum Gravity arguments, it has been suggested that all kinetic terms of light particles below the UV cut-off could arise in the IR via quantum (loop) corrections. These loop corrections involve infinite towers of states becoming light (e.g. Kaluza-Klein or string towers). We study implications of this Emergence Proposal for fundamental scales in the Standard Model (SM). In this scheme all Yukawa couplings are of order one in the UV and small Yukawas for lighter generations appear via large anomalous dimensions induced by the towers of states. Thus, the observed hierarchies of quark and lepton masses are a reflection of the structure of towers of states that lie below the Quantum Gravity scale, $\Lambda_{\text{QG}}$. Small Dirac neutrino masses consistent with experimental observation appear due to the existence of a tower of SM singlet states of mass $m_{0}\simeq Y_{\nu_3}M_p\simeq 7\times 10^5$ GeV, opening up a new extra dimension, while the UV cut-off occurs at $\Lambda_{\text{QG}}\lesssim 10^{14}$ GeV. Additional constraints relating the Electro-Weak (EW) and cosmological constant (c.c.) scales (denoted $M_{\text{EW}}$ and $V_0$) appear if the Swampland condition $m_{\nu_1}\lesssim V_0^{1/4}$ is imposed (with $\nu_1$ denoting the lightest neutrino), which itself arises upon applying the AdS non-SUSY Conjecture or the AdS/dS Distance Conjecture to the 3d vacua from circle compactifications of the SM. In particular, the EW scale and that of the extra dimension fulfill ${m_0 \, M_{\text{EW}}\lesssim 10^{2}\, V_0^{1/4}M_p}$, thus relating the EW hierarchy problem to that of the c.c. Hence, all fundamental scales may be written as powers of the c.c., i.e. $m_{\bullet}\sim V_0^\delta M_p^{1-4\delta}$. The scale of SUSY breaking is $m_{3/2}\lesssim 7\times 10^5$ GeV, which favours a Mini-Split scenario that could be possibly tested at LHC and/or FCC., Comment: 20 pages, 5 figures, 1 table; v2: references added and typos corrected

In the Emergence Proposal in QG it is conjectured that all light-particle kinetic terms are absent in the fundamental UV theory and are generated by quantum corrections in the IR. It has been argued that this may provide for some microscopic understanding of the WGC and SDC. In the present paper we take the first steps towards a systematic study of Emergence in the context of string theory. We emphasize the crucial role of the species scale in any EFT coupled to gravity, and discuss its computation in string theory and general systems with light towers of states. We then introduce the notion of Emergence and show how kinetic terms for moduli, gauge bosons and fermions may be generated. One-loop computations play an important role in Emergence, so we present detailed calculations in $d$ spacetime dimensions for the wave-function renormalization of scalars, vectors and fermions. We extend and check the Emergence Proposal in a number string vacua, including 4d $\mathcal{N}=2$ theories arising from type IIA on a CY$_3$, where the towers at strong coupling are comprised by D0 and (wrapped) D2-branes, and also elaborate on how instanton corrections would fit within the emergence picture. Higher dimensional examples are also discussed, including 6d and 7d models arising from F-/M-theory on an elliptic CY$_3$ or a $K3$ surface. We also consider 10d string theories and study in some detail the emergence mechanism in type IIA. We show as well how the flux potential in 4d may be obtained from the emergence prescription, by analyzing the corresponding decompactification limits to M-theory. We find that the required kinetic terms for the dual 3-form fields can arise upon integrating out towers of massive gravitini (and bosonic superpartners). Our analysis renders support to the Emergence Proposal, and to the idea that infinite distance singularities may arise in QG as an intrinsic IR phenomenon., Comment: v2: minor changes and typos corrected. 79 pages + Appendices, 11 figures, 8 tables

We analyze the boundaries of the moduli spaces of compactifications of the heterotic string on $T^d$, making particular emphasis on $d=2$ and its F-theory dual. We compute the OPE algebras as we approach all the infinite distance limits that correspond to (possibly partial) decompactification limits in some dual frame. When decompactifying $k$ directions, we find infinite towers of states becoming light that enhance the algebra arising at a given point in the moduli space of the $T^{d-k}$ compactification to its $k$-loop version, where the central extensions are given by the $k$ KK vectors. For $T^2$ compactifications, we reproduce all the affine algebras that arise in the F-theory dual, and show all the towers explicitly, including some that are not manifest in the F-theory counterparts. Furthermore, we construct the affine $SO(32)$ algebra arising in the full decompactification limit, both in the heterotic and in the F-theory sides, showing that not only affine algebras of exceptional type arise in the latter., Comment: 51 pages, 10 figures

The tadpole conjecture suggests that the complete stabilization of complex structure deformations in Type IIB and F-theory flux compactifications is severely obstructed by the tadpole bound on the fluxes. More precisely, it states that the stabilization of a large number of moduli requires a flux background with a tadpole that scales linearly in the number of stabilized fields. Restricting to the asymptotic regions of the complex structure moduli space, we give the first conceptual argument that explains this linear scaling setting and clarifies why it sets in only for a large number of stabilized moduli. Our approach relies on the use of asymptotic Hodge theory. In particular, we use the fact that in each asymptotic regime an orthogonal sl(2)-block structure emerges that allows us to group fluxes into sl(2)-representations and decouple complex structure directions. We show that the number of stabilized moduli scales with the number of sl(2)-representations supported by fluxes, and that each representation fixes a single modulus. Furthermore, we find that for Calabi-Yau four-folds all but one representation can be identified with representations occurring on two-folds. This allows us to discuss moduli stabilization explicitly and establish the relevant scaling constraints for the tadpole., Comment: 31 pages + appendices; v2: minor changes, clarifications added, published version

We show that compactifications of the heterotic string on a circle exhibit at the boundary of moduli space ($R\to 0$, or equivalently the decompactification limit $R \to \infty$) a tower of winding or momentum modes that enhance the $E_8 \times E_8$ or $SO(32)$ gauge symmetry to the affine algebras $(E_9 \oplus E_9)/\sim$ (the identification means that the two copies of $E_9$ share the same central extension) and $\hat{D}_{16}$, respectively. We also prove that these towers of modes satisfy the lattice Weak Gravity and Repulsive Force Conjectures., Comment: 27 pages, 3 tables, 1 figure

By applying the Covariant Entropy Bound (CEB) to an EFT in a box of size $1/\Lambda_{\text{IR}}$ one obtains that the UV and IR cut-offs of the EFT are necessarily correlated. We argue that in a theory of Quantum Gravity (QG) one should identify the UV cutoff with the `species scale', and give a general algorithm to calculate it in the case of multiple towers becoming light. One then obtains an upper bound on the characteristic mass scale of the tower in terms of the IR cut-off, given by $M_{\text{tower}}\lesssim (\Lambda_{\text{IR}})^{2\alpha_D}$ in Planck units, with $\alpha_D=(D-2+p)/2p(D-1)$, where $p$ depends on the density of states. Identifying the IR cut-off with a (non-vanishing) curvature in AdS one reproduces the statement of the AdS Distance Conjecture (ADC), also giving an explicit lower bound for the $\alpha$ exponent. In particular, we find that the CEB implies $\alpha \geq1/2$ in any dimension if there is a single KK tower, both in AdS and dS vacua. However values $\alpha <1/2$ are allowed if the particle tower is multiple or has a string component. We also consider the CKN constraint coming from avoiding gravitational collapse which further requires in general $\alpha \geq 1/D$ for the lightest tower. We analyse the case of the DGKT-CFI class of Type IIA orientifold models and show it has both particle and string towers below the species scale, so that a careful analysis of how the ADC is defined is needed. We find that this class of models obey but do not saturate the CEB. The UV/IR constraints found apply to both AdS and dS vacua. We comment on possible applications of these ideas to the dS Swampland conjecture as well as to the observed dS phase of the universe., Comment: 19 pages, 4 figures. v2: minor corrections, references added. v3: minor corrections and comments added

The swampland is the set of seemingly consistent low-energy effective field theories that cannot be consistently coupled to quantum gravity. In this review we cover some of the conjectural properties that effective theories should possess in order not to fall in the swampland, and we give an overview of their main applications to particle physics. The latter include predictions on neutrino masses, bounds on the cosmological constant, the electroweak and QCD scales, the photon mass, the Higgs potential and some insights about supersymmetry., Comment: 36 pages. Contribution to the Open Access Special Issue "Women Physicists in Astrophysics, Cosmology and Particle Physics'', Universe, ISSN 2218-1997, IF 1.752. v2: minor changes, references added

We conjecture that in a consistent supergravity theory with non-vanishing gravitino mass, the limit $m_{3/2}\rightarrow 0$ is at infinite distance. In particular one can write $M_{\mathrm{tower}} \sim m_{3/2}^\delta$ so that as the gravitino mass goes to zero, a tower of KK states as well as emergent strings becomes tensionless. This conjecture may be motivated from the Weak Gravity Conjecture as applied to strings and membranes and implies in turn the AdS Distance Conjecture. We test this proposal in classical 4d type IIA orientifold vacua in which one obtains a range of values $\tfrac13 \le \delta \le 1$. The parameter $\delta$ is related to the scale decoupling exponent in AdS vacua and to the $\alpha$ exponent in the Swampland Distance Conjecture for the type IIA complex structure. We present a general analysis of the gravitino mass in the limits of moduli space in terms of limiting Mixed Hodge Structures and study in some detail the case of two-moduli F-theory settings. Moreover, we obtain general lower bounds $\delta\, \geq \, \frac{1}{3}, \, \frac{1}{4}$ for Calabi--Yau threefolds and fourfolds, respectively. The conjecture has important phenomenological implications. In particular we argue that low-energy supersymmetry of order 1 TeV is only obtained if there is a tower of KK states at an intermediate scale, of order $10^8$ GeV. One also has an upper bound for the Hubble constant upon inflation $H \lesssim m_{3/2}^\delta M^{(1-\delta)}_{\text{P}}$., Comment: v2: 60 pages, 3 figures. Minor changes and references added

We compute the tree-level potential between two parallel $p$-branes due to the exchange of scalars, gravitons and $(p+1)$-forms. In the case of BPS membranes in 4d $\mathcal{N}=1$ supergravity, this provides an interesting reinterpretation of the classical Cremmer et al. formula for the F-term scalar potential in terms of scalar, graviton and 3-form exchange. In this way, we present a correspondence between the scalar potential at every point in scalar field space and a system of two interacting BPS membranes. This could potentially lead to interesting implications for the swampland program by providing a concrete way to relate conjectures about the form of scalar potentials with conjectures regarding the spectrum of charged objects., Comment: 13 pages. Some typos corrected and references added, published version

We study the separation of AdS and Kaluza-Klein (KK) scales in type II 4d AdS orientifold vacua. We first address this problem in toroidal/orbifold type IIA vacua with metric fluxes, corresponding to compactifications in twisted tori, both from the 4d and 10d points of view. We show how the naive application of the effective 4d theory leads to results which violate the AdS distance conjecture, in a class of $\mathcal{N}=1$ supersymmetric models which have a 10d lifting to a compactification on $S^3\times S^3$. We show how using KK scales properly modified by the compact metric leads to no separation of scales with $M^2_{\text{KK}} = \mathfrak{c} |\Lambda|$, with $\mathfrak{c}$ a numerical constant independent of fluxes. This applies with no need to keep non-leading fluxes fixed. We also consider a class of IIB models with non-geometric fluxes in which the effective field theory analysisseems to lead to a naive separation of scales and a violation of the AdS distance conjecture. It has a T-dual which again may be understood as a 10d type IIA theory compactified on $S^3\times S^3$. In this geometric dual one again observes that the strong AdS distance conjecture is obeyed with $M^2_{\text{KK}} = \mathfrak{c}' |\Lambda|$, if one takes into account the curvature in the internal space. These findings seem to suggest that all toroidal/orbifold models with fluxes in this class obey $M^2_{\text{KK}} = \mathfrak{c} |\Lambda|$ with $\mathfrak{c}$ a flux-independent numerical constant., Comment: 16 pages. References added and minor changes in notation. Published version

The Swampland Distance Conjecture states that at infinite distance in the scalar moduli space an infinite tower of particles become exponentially massless. We study this issue in the context of 4d type IIA and type IIB Calabi-Yau compactifications. We find that for large moduli not only towers of particles but also domain walls and strings become tensionless. We study in detail the case of type IIA and IIB ${\cal N}=1$ CY orientifolds and show how for infinite K\"ahler and/or complex structure moduli towers of domain walls and strings become tensionless, depending on the particular direction in moduli space. For the type IIA case we construct the monodromy orbits of domain walls in detail. We study the structure of mass scales in these limits and find that these towers may occur at the same scale as the fundamental string scale or the KK scale making sometimes difficult an effective field theory description. The structure of IIA and IIB towers are consistent with mirror symmetry, as long as towers of exotic domain walls associated to non-geometric fluxes also appear. We briefly discuss the issue of emergence within this context and the possible implications for 4d vacua., Comment: 69 pages, 5 figures, 8 tables

It has been recently argued that an embedding of the SM into a consistent theory of quantum gravity may imply important constraints on the mass of the lightest neutrino and the cosmological constant $\Lambda_{4}$. The constraints come from imposing the absence of any non-SUSY AdS stable vacua obtained from any consistent compactification of the SM to 3 or 2 dimensions. This condition comes as a corollary of a recent extension of the Weak Gravity Conjecture (WGC) by Ooguri and Vafa. In this paper we study $T^2/Z_N$ compactifications of the SM to two dimensions in which SM Wilson lines are projected out, leading to a considerable simplification. We analyze in detail a $T^ 2/Z_4$ compactification of the SM in which both complex structure and Wilson line scalars are fixed and the potential is only a function of the area of the torus $a^2$. We find that the SM is not robust against the appearance of AdS vacua in 2D and hence would be by itself inconsistent with quantum gravity. On the contrary, if the SM is embedded at some scale $M_{SS}$ into a SUSY version like the MSSM, the AdS vacua present in the non-SUSY case disappear or become unstable. This means that WGC arguments favor a SUSY version of the SM, independently of the usual hierarchy problem arguments. In a $T^2/Z_4$ compactification in which the orbifold action is embedded into the $B-L$ symmetry the bounds on neutrino masses and the cosmological constant are recovered. This suggests that the MSSM should be extended with a $U(1)_{B-L}$ gauge group. In other families of vacua the spectrum of SUSY particles is further constrained in order to avoid the appearance of new AdS vacua or instabilities. We discuss a possible understanding of the little hierarchy problem in this context., Comment: Minor corrections, references added, 2 figures added. 58 pages, 15 figures, 3 tables

We compute the full classical 4d scalar potential of type IIA Calabi-Yau orientifolds in the presence of fluxes and D6-branes. We show that it can be written as a bilinear form $V = Z^{AB} \rho_A\rho_B$, where the $\rho_A$ are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The $\rho_A$ only depend on the internal fluxes, the axions and the topological data of the compactification, and are fully determined by the Freed-Witten anomalies of branes that appear as 4d string defects. The quadratic form $Z^{AB}$ only depends on the saxionic partners of these axions. In general, the $\rho_A$ can be seen as the basic invariants under the discrete shift symmetries of the 4d effective theory, and therefore the building blocks of any flux-dependent quantity. All these polynomials may be obtained by derivation from one of them, associated to a universal 4-form. The standard N=1 supergravity flux superpotential is uniquely determined from this {\it master polynomial}, and vice versa., Comment: 75 pages, one figure; v2 minor corrections

We construct models in which the SM Higgs mass scans in a landscape. This is achieved by coupling the SM to a monodromy axion field through Minkowski 3-forms. The Higgs mass scans with steps given by delta(m_H^2)= eta mu f, where mu and f are the axion mass and periodicity respectively, and eta measures the coupling of the Higgs to the associated 3-form. The observed Higgs mass scale could then be selected on anthropic grounds. The monodromy axion may have a mass mu in a very wide range depending on the value of eta, and the axion periodity f. For eta=1 and f = 10^10 GeV, one has 10^{-3}eV < mu < 10^3 eV, but ultralight axions with e.g. mu = 10^{-17} eV are also possible. In a different realization we consider landscape models coupled to the MSSM. In the context of SUSY, 4-forms appear as being part of the auxiliary fields of SUSY multiplets. The scanning in the 4-forms thus translate into a landscape of vevs for the N=1 auxiliary fields and hence as a landscape for the soft terms. This could provide a rationale for the MSSM fine-tuning suggested by LHC data. In all these models there are 3-forms coupling to membranes which induce transitions between different vacua through bubble nucleation. The Weak Gravity Conjecture (WGC) set limits on the tension of these membranes and implies new physics thresholds well below the Planck scale. More generaly, we argue that in the case of string SUSY vacua in which the Goldstino multiplet contains a monodromy axion the WGC suggests a lower bound on the SUSY breaking scale m_{3/2} > M_s^2/M_p., Comment: 28 pages, 1 pdf figure