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The dark dimension scenario, predicting one extra mesoscopic dimension in the micron range, has emerged by applying various Swampland principles to the dark energy. In this note we find that realizing the grand unification of gauge forces is highly constraining in this context. Without actually constructing any GUT models, we argue that the mere assumption of grand unification of forces in this scenario, together with the experimental bounds on massive replicas of the Standard Model gauge bosons, predicts an upper bound for the GUT scale, $M_{GUT}\lesssim 10^{16}\ {\rm GeV}$. Combined with the experimental bound on the proton lifetime, this predicts that the $X$ gauge boson mediating proton decay is a 5d solitonic string of Planckian tension stretched across a length scale $L\sim ({\rm 1-10\ TeV})^{-1}$ ending on gauge branes of the same diameter $\sim L$. This leads to a mass of $M_X\sim 10^{15}-10^{16}\ {\rm GeV}$. In particular assuming grand unification in the dark dimension scenario results in a tower of Kaluza-Klein excitations of Standard Model gauge bosons on the gauge branes in the 1-10 TeV range. This suggests that the diameter/separation $L$ of the gauge branes correlates with both the weak scale $\sim 1/L$ near a TeV \it and the GUT scale $\sim M_5^2 L$ at $10^{16}\ {\rm GeV}$.

In this note, we propose a generalized notion of distance between vacua in the theory of a scalar field $\phi$ with scalar potential $V(\phi)$ coupled to gravity. We propose the normalized tension of domain wall connecting different field values, with a varying normalization relative to a local energy scale, as the distance. We show this definition reproduces the usual moduli space distance for zero potential, as well as the $d\propto |\log \Lambda|$ behavior with the vacuum energy $\Lambda$ in the AdS case, previously proposed in the literature. In the case of large AdS we also obtain the expected exponent of mass versus distance in one particular case, when the mass of the light tower is $m\sim \sqrt \Lambda$ and there is a single extra dimension decompactifying. We also discuss the features and shortcomings of alternative but related proposals.

We propose new constraints for 6d (1, 0) supergravity theories based on consistency conditions on the Kahler moduli spaces of their 5d reductions. The requirement that both the metric and the BPS string tensions in the Kahler moduli space are positive imposes specific restrictions on the Chern-Simons coefficients in the 5d effective Lagrangians that are derived from the Kaluza-Klein reductions of 6d theories. Moreover, the emergence of local interacting 5d CFTs when the moduli space metric degenerates introduces additional constraints coming from the analysis of 5d SCFTs. Focusing on the moduli spaces of 6d supergravity theories without a tensor multiplet and their Higgsings, we show that these constraints require the presence of certain primary states in the 2d worldvolume CFTs on 1/2 BPS strings. We specifically analyze a class of SU(2) models and infinite families of U(1) models using these constraints, and demonstrate that the theories featuring a 1-form symmetry in their massless spectra, unless the 1-form symmetry is gauged, fail to satisfy the constraints and therefore belong to the Swampland.

In four decades of string theory research, only a handful of non-supersymmetric tachyon-free strings with only one neutral scalar at tree level were found. We construct new such non-supersymmetric tachyon-free string theories using asymmetric orbifolds that serve as the lower-dimensional counterparts to the $O(16) \times O(16)$ string in 4d, 6d, and 8d, each featuring only one neutral scalar at tree level, chiral matter and positive leading order cosmological constant. The 4d construction uses a quasicrystalline orbifold.

The dark dimension scenario, which is motivated from Swampland principles and predicts a single micron scale extra dimension, suggests a consistent framework for the dark sector of the universe. We consider the implications of this scenario for the QCD axion. We find that in the scenario in which the axion is localized on the standard model brane (which we will argue is natural), a combination of theoretical (being bounded by the 5D Planck mass) and observational constraints forces it to have decay constant in a narrow range $f \sim 10^9 - 10^{10}$ GeV. This corresponds to a mass for the QCD axion of $m_a \sim (1 - 10)$ meV. The axion mass surprisingly coincides with the mass scale for the dark energy, the dark matter tower, and the neutrinos. In this scenario axions are not expected to form a large fraction of the dark matter but nevertheless this range of axion parameters is accessible to observations in near future experiments.

Microscopic black hole entropy calculations in string theory usually proceeds through identifying them as wrapped strings in one higher dimension. For M-theory on elliptic Calabi-Yau threefolds this proceeds via its relation to F-theory in one higher dimension. Here we show how this method can be extended to M-theory on non-elliptic Calabi-Yau threefolds such as the quintic via conifold transition to elliptic threefolds. This leads to the computation of the black hole entropy through elliptic genera of the strings. However the Cardy formula for the computation of the black hole entropy of these strings fails because the relevant momentum excitations on the string are much smaller than the central charge of the strings. We show how the black hole attractor entropy formula leads to predicting corrections to the Cardy formula in this regime.

Quantum gravity (QG) has a natural cutoff given by the Planck scale $M_{\rm pl}$. However, it is known that the EFT of gravity can break down at a lower scale, the species scale $\Lambda_s\lesssim M_{\rm pl}$, if there are light species of particles. Here we point out that there is a third scale $\Lambda_{\rm BH}\lesssim \Lambda_s\lesssim M_{\rm pl}$, which marks the inverse length (or the temperature) of the smallest black hole where the EFT gives a correct description of its entropy and free energy. This latter scale is hard to detect from the viewpoint of EFT as it represents a phase transition to a state with lower free energy. We illustrate this using examples drawn from consistent QG landscape. In particular $\Lambda_{\rm BH}$ gets related to Gregory--Laflamme transition in the decompactification limits of quantum gravity and to the Horowitz--Polchinski solution in the light perturbative string limits. We propose the existence of $\Lambda_{\rm BH}$ marking the temperature at which neutral black holes undergo a phase transition, as a new Swampland condition for all consistent quantum theories of gravity. In the asymptotic regimes of field space $\Lambda_{\rm BH}$ is close to the mass scale of the lightest tower but deviates from it as we move inwards in the moduli space.

We provide a short overview of recent progress made in our understanding of the dark sector based on the Swampland program which in turn is rooted in lessons from string theory. We explain how the existence of one extra mesoscopic dimension (the ``dark dimension") in the micron range emerges and how this can lead to a unification of the dark energy and dark matter. In particular the smallness of the dark energy leads to the prediction of the existence of a tower of weakly interacting light particles which can naturally play the role of dark matter. Moreover this unifies dark matter with gravity as dark matter ends up being excitations of graviton in the dark dimension. We also explain how in combination with other Swampland principles one finds an explanation of the ``why now" and the ``cosmological coincidence" problems. This model is consistent with the cosmological bounds as well as the Newton's inverse square law, but makes predictions which differ from $\Lambda$CDM. It also gives rise to an appealing picture of hierarchy of scales in particle physics pegged to the dark energy, including a possible origin of the electroweak hierarchy and the prediction of masses of QCD axion and sterile neutrinos both in the 1-10 meV range. This review is intended for a broad audience of high energy theorists and cosmologists without prior knowledge of string theory and it explains the motivations and predictions of this program in a non-technical form.

We explore the cosmology of the Dark Dimension scenario taking into account perturbations in the linear regime. In the context of the Dark Dimension scenario, a natural candidate for dark matter in our universe is the excitations of a tower of massive spin-2 KK gravitons. These dark gravitons are produced in the early universe and decay to lighter KK gravitons during the course of cosmological evolution. The decay causes the average dark matter mass to decrease as the universe evolves. In addition, the kinetic energy liberated in each decay leads to a kick velocity for the dark matter particles leading to a suppression of structure formation. Using current CMB (Planck), BAO and cosmic shear (KiDS-1000) data, we put a bound on the dark matter kick velocity today $v_\mathrm{today} \leq 2.2 \times 10^{-4} c$ at 95\% CL. This leads to rather specific regions of parameter space for the dark dimension scenario. The combination of the experimental bounds from cosmology, astrophysics and table-top experiments lead to the range $l_5\sim 1- 10 \, \mu m$ for the size of the Dark Dimension. The Dark Dimension scenario is found to be remarkably consistent with current observations and provides signatures that are within reach of near-future experiments.

In a quantum theory of gravity, the species scale $\Lambda_s$ can be defined as the scale at which corrections to the Einstein action become important or alternatively as codifying the "number of light degrees of freedom", due to the fact that $\Lambda_s^{-1}$ is the smallest size black hole described by the EFT involving only the Einstein term. In this paper, we check the validity of this picture in diverse dimensions and with different amounts of supersymmetry and verify the expected behavior of the species scale at the boundary of the moduli space. This also leads to the evaluation of the species scale in the interior of the moduli space as well as to the computation of the diameter of the moduli space. We also find evidence that the species scale satisfies the bound $\big|{\nabla \Lambda_s \over \Lambda_s} \big|^2\leq {1\over d-2}$ all over moduli space including the interior.

In this work we study interesting corners of the quantum gravity landscape with 8 supercharges pushing the boundaries of our current understanding. Calabi-Yau threefolds compactifications of F/M/type II theories to 6, 5 and 4 dimensions are the most prominent examples of this class, and these always lead to a universal hypermultiplet coming from the volume/string coupling constant. We find that there are asymmetric orbifold constructions which have no hypermultiplets in 4 or 5 dimensions and no neutral hypers in 6d. We argue that these theories can also be obtained by going to strong coupling/small volume regions of geometric constructions where a new Coulomb branch opens up and moving in this direction freezes the volume/string coupling constant. Interestingly we find that the Kodaira condition encountered in geometric limits of F-theory compactifications to 6 dimensions is violated in these corners of the landscape due to strong quantum corrections. We also construct a theory in 3 dimensions which if it were to arise by toroidal compactifications from 5d, it would have to come from pure ${\mathcal N}=1$ supergravity with no massless scalar fields.

In the dark dimension scenario, which predicts an extra dimension of micron scale, dark gravitons (KK modes) are a natural dark matter candidate. In this paper, we study observable features of this model. In particular, their decay to standard matter fields can distort the CMB and impact other astrophysical signals. Using this we place bounds on the parameters of this model. In particular we find that the natural range of parameters in this scenario is consistent with these constraints and leads to the prediction that the mean mass of the dark matter today is close to a few hundred keV and the effective size of the extra dimension is around $1 - 30 \;\mu\mathrm{m}$.

In the context of quantum gravitational systems, we place bounds on regions in field space with slowly varying positive potentials. Using the fact that $V<\Lambda_s^2$, where $\Lambda_s(\phi)$ is the species scale, and the emergent string conjecture, we show this places a bound on the maximum diameter of such regions in field space: $\Delta \phi \leq a \log(1/V) +b$ in Planck units, where $a\leq \sqrt{(d-1)(d-2)}$, and $b$ is an $\mathcal{O}(1)$ number and expected to be negative. The coefficient of the logarithmic term has previously been derived using TCC, providing further confirmation. For type II string flux compactifications on Calabi--Yau threefolds, using the recent results on the moduli dependence of the species scale, we can check the above relation and determine the constant $b$, which we verify is $\mathcal{O}(1)$ and negative in all the examples we studied.

The species scale $\Lambda_s\leq M_{pl}$ serves as a UV cutoff in the gravitational sector of an EFT and can depend on the moduli of the theory as the spectrum of the theory varies. We argue that the dependence of the species scale $\Lambda_s (\phi)$ on massless (or light) modes $\phi^i$ satisfies $M_{pl}^{d-2} |\Lambda_s'/\Lambda_s|^2< \mathcal{O}(1)$. This bound is true at all points in moduli space including also its interior. The argument is based on the idea that the short distance contribution of massless modes to gravitational terms in the EFT cannot dramatically affect the black hole entropy. Based on string theory arguments we expect the $\mathcal{O}(1)$ constant in this bound to be equal to ${1\over {d-2}}$ as we approach the boundary of the moduli space. However, we find that the slope of the species scale can approach its asymptotic value from above as we go from interior points to the boundaries, thereby implying that the constant in the bound must be larger than ${1\over {d-2}}$. The bound on the variation of the species scale also implies that the mass of towers of light modes cannot go to zero faster than exponential in field distance in accordance with the Distance Conjecture.

The counting of the number of light modes in a gravitational theory is captured by the notion of the `species scale', which serves as an effective UV cutoff below the Planck scale. We propose to define a moduli-dependent species scale in the context of 4d, ${\cal N}=2$ theories, using the one loop topological free energy $F_1$, which we relate to a gravitational version of the $a$-function. This leads to $\Lambda_{\rm sp}\sim 1/\sqrt{F_1}$ from which we recover the expected scaling of the species scale in various corners of the moduli space. Moreover by minimizing $F_1$ we define the center of the moduli space (the `desert point') as a point where the species scale is maximal. At this point the number of light degrees of freedom is minimized.

We provide an overview of the string landscape and the Swampland program. Our review of the string landscape covers the worldsheet and spacetime perspectives, including vacua and string dualities. We then review and motivate the Swampland program from the lessons learned from the string landscape. These lecture notes are aimed to be self-contained and thus can serve as a starting point for researchers interested in exploring these ideas. These notes are an expanded version of two courses "The String Landscape and the Swampland" taught by C.~Vafa at Harvard University in 2018 with a focus on the landscape, written by M.~J.~Kang with additional material from N.~B.~Agmon, and in 2022 with a focus on the Swampland, written by A.~Bedroya.

We consider cosmological aspects of the Dark Dimension (a mesoscopic dimension of micron scale), which has recently been proposed as the unique corner of the quantum gravity landscape consistent with both the Swampland criteria and observations. In particular we show how this leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the dark dimension (the "dark gravitons") as an unavoidable dark matter candidate. Assuming a lifetime for the current de Sitter phase of our universe of order Hubble, which follows from both the dS Swampland Conjecture and TCC, we show that generic features of the dark dimenson cosmology can naturally lead to the correct dark matter density and a resolution of the cosmological coincidence problem, where the matter/radiation equality temperature ($T\sim$ 1 eV) coincides with the temperature where the dark energy begins to dominate. Thus one does not need to appeal to Weinberg's anthropic argument to explain this coincidence. The dark gravitons are produced at $T\sim$ 4 GeV, and their composition changes as they mainly decay to lighter gravitons, without losing much total mass density. The mass of dark gravitons is $m_{\text{DM}}\sim 1-100$ keV today.

Motivated by principles from the Swampland program, which characterize requirements for a consistent UV completion of quantum gravity, combined with observational data, we are led to a unique corner of the quantum gravity landscape. In particular, using the Distance/Duality conjecture and the smallness of dark energy, we predict the existence of a light tower of states and a unique extra mesoscopic dimension of length $l\sim \Lambda^{-\frac{1}{4}}\sim 10^{-6}\, m$, with extra massless fermions propagating on it. This automatically leads to a candidate for a tower of sterile neutrinos, and an associated active neutrino mass scale $m_{\nu}\sim \langle H\rangle^2\, \Lambda^{-\frac{1}{12}}M_{pl}^{-\frac{2}{3}}$. Moreover, assuming the mechanism for stabilization of this dark dimension leads to similar masses for active and sterile neutrinos we are led to the prediction of a Higgs vev $\langle H\rangle \sim \Lambda^{\frac{1}{6}}M_{pl}^{\frac{1}{3}}$. Another prediction of the scenario is a species scale ${\hat M} \sim \Lambda^ {\frac{1}{12}}M_{pl}^{\frac{2}{3}}\sim 10^{9}-10^{10} GeV$, corresponding to the higher-dimensional Planck scale. This energy scale may be related to the resolution of the instability of the Higgs effective potential present at a scale of $\sim 10^{11}\, GeV$. We also speculate about the interplay between this energy scale and the GZK limit on ultra-high energy cosmic rays.

The KKLT scenario, one of the few ideas to realize dS vacua in string theory, consists of two steps: the first involves the construction of a supersymmetric AdS vacuum with a small negative cosmological constant, and the second involves breaking supersymmetry and uplifting the energy to achieve dS. In this paper we use conventional holography to argue why it is not possible to complete the first step. We obtain this by putting a bound on the central charge of the dual theory which involves branes wrapping special Lagrangian cycles in CY 4-folds. We find that $l_{\rm AdS}^2 \lesssim \chi(CY_4)$. Since $l_{\rm species}^2\gtrsim \chi(CY_4)$ this leads to $l_{\rm AdS}/l_{\rm species}\lesssim 1$ leading at best to a highly curved AdS which is beyond the validity of the EFT.

We consider CFT's arising from branes probing singularities of internal manifolds. We focus on holographic models with internal space including arbtirary Sasaki-Einstein manifolds coming from CY as well as arbitrary sphere quotients. In all these cases we show that there is a universal upper bound (depending only on the spacetime dimension) for the conformal dimension of the first non-trivial spin 2 operator in the dual CFT and a minimal diameter (in AdS units) for the internal space of the holographic dual and conjecture it holds for all CFT's.

The most natural expectation away from asymptotic limits in moduli space of supergravity theories is the desert scenario, where there are few states between massless fields and the quantum gravity cutoff. In this paper we initiate a systematic study of these regions deep in the moduli space, and use it to place a bound on the number of massless modes by relating it to the black hole species problem. There exists a consistent sub-Planckian UV cutoff (the species scale) which resolves the black hole species problem without bounding the number of light modes. We reevaluate this in the context of supersymmetric string vacua in the desert region and show that even though heuristically the species scale is compatible with expectations, the BPS states of the actual string vacua lead to a stronger dependence of the cutoff scale on the number of massless modes. We propose that this discrepancy, which can be captured by the "BPS desert conjecture", resurrects the idea of a uniform bound on the number of light modes as a way to avoid the black hole species problem. This conjecture also implies a stronger form of the Tadpole Conjecture, which leads to an obstruction in stabilizing all moduli semi-classically for large number of moduli in flux compactifications.

We view and provide further evidence for a number of Swampland criteria, including the Weak Gravity Conjecture, Distance Conjecture and bounds on the finiteness of the quantum gravity vacua from the prism of the finiteness of black hole entropy. Furthermore we propose that at least all of these Swampland statements may be more fundamentally a consequence of the finiteness of quantum gravity amplitudes.

We demonstrate the validity of the String Lamppost Principle-that all consistent theories of quantum gravity are in the String Landscape-for supersymmetric theories in $d>6$ using compactness and connectedness of the moduli space of small instantons, as well as the classification of the associated Coulomb branch.

We consider supergravity theories with 8 supercharges in $d=6$. We show that all the proposed anomaly free theories with unbounded number of massless modes are restricted to a finite subset and thus argue that there is an upper bound on the number of massless modes, consistent with the String Lamppost Principle.

Demanding that charged Nariai black holes in (quasi-)de Sitter space decay without becoming super-extremal implies a lower bound on the masses of charged particles, known as the Festina Lente (FL) bound. In this paper we fix the $\mathcal{O}(1)$ constant in the bound and elucidate various aspects of it, as well as extensions to $d>4$ and to situations with scalar potentials and dilatonic couplings. We also discuss phenomenological implications of FL including an explanation of why the Higgs potential cannot have a local minimum at the origin, thus explaining why the weak force must be broken. For constructions of meta-stable dS involving anti-brane uplift scenarios, even though the throat region is consistent with FL, the bound implies that we cannot have any light charged matter fields coming from any far away region in the compactified geometry, contrary to the fact that they are typically expected to arise in these scenarios. This strongly suggests that introduction of warped anti-branes in the throat cannot be decoupled from the bulk dynamics as is commonly assumed. Finally, we provide some evidence that in certain situations the FL bound can have implications even with gravity decoupled and illustrate this in the context of non-compact throats.

We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi-Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon $AdS_3$ limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles).

We sharpen Swampland constraints on 8d supergravity theories by studying consistency conditions on worldvolume theory of 3-brane probes. Combined with a stronger form of the cobordism conjecture, this leads to the reconstruction of the compact internal geometry and implies strong restrictions on the gauge algebra and on some higher derivative terms (related to the level of the current algebra on the 1-brane). In particular we argue that 8d supergravity theories with $\mathfrak{g}_2$ gauge symmetry are in the Swampland. These results provide further evidence for the string lamppost principle in 8d with 16 supercharges.

We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in $d>2$ spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Stated geometrically, the diameter of a non-compact conformal manifold must diverge logarithmically in the higher-spin gap. In the holographic context our conjectures are related to the Distance Conjecture in the swampland program. Interpreted gravitationally, they imply that approaching infinite distance in moduli space at fixed AdS radius, a tower of higher-spin fields becomes massless at an exponential rate that is bounded from below in Planck units. We discuss further implications for conformal manifolds of superconformal field theories in three and four dimensions.

Motivated by string dualities we propose topological gravity as the early phase of our universe. The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology. A concrete realization of this scenario using Witten's four dimensional topological gravity is considered. This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients $a,c$. In particular the strength of the fluctuation is controlled by $1/a$ and its tilt by $c g^2$ where $g$ is the coupling constant of topological gravity. The positivity of $c$, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum. In contrast with standard inflationary models, this scenario predicts $\mathcal{O}(1)$ non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations.

We consider consequences of triviality of cobordism classes and anomaly cancellation in supergravity theories in $d>6$. We argue that this leads to the existence of certain defects which we call "I-folds" (a generalization of orientifolds). The requirement that compactifications to lower dimensions involving these defects be anomaly free leads to conditions on the higher dimensional theory. We show that in theories with 16 supercharges in $d>6$ this leads to restrictions on the rank of the allowed gauge groups and thus provides an explanation for the observed restrictions in known string theory constructions. In particular, in eight and nine dimensions the only solutions to our constraints are precisely the ones realized in string theory compactifications. We also use these techniques to place constraints on the global structure of the gauge group in eight and nine dimensions.

A number of Swampland conjectures and in particular the Trans-Planckian Censorship Conjecture (TCC) suggest that de Sitter space is highly unstable if it exists at all. In this paper we construct effective theories of scalars rolling on potentials which are dual to a chain of short-lived dS spaces decaying from one to the next through a cascade of non-perturbative nucleation of bubbles. We find constraints on the effective potential resulting from various swampland criteria, including TCC, Weak Gravity Conjecture and Distance Conjecture. Surprisingly we find that TCC essentially incorporates all the other ones, and leads to a subclass of possible dual effective potentials. These results marginally rule out emergence of eternal inflation in the dual effective theory. We discuss some cosmological implications of our observations.

We propose Swampland constraints on consistent 5-dimensional ${\cal N}=1$ supergravity theories. We focus on a special class of BPS magnetic monopole strings which arise in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the low-energy particle spectrum and the effective action of the 5d supergravity based on unitarity of the worldsheet CFT. In M-theory, where these theories are realized by compactification on Calabi-Yau 3-folds, the special monopole strings arise from wrapped M5-branes on special ("semi-ample") 4-cycles in the threefold. We identify various necessary geometric conditions for such cycles to lead to requisite BPS strings and translate these into constraints on the low-energy theories of gravity. These and other geometric conditions, some of which can be related to unitarity constraints on the monopole worldsheet, are additional candidates for Swampland constraints on 5-dimensional ${\cal N}=1$ supergravity theories.

On the basis of a number of Swampland conditions, we argue that the Hilbert space of baby universe states must be one-dimensional in a consistent theory of quantum gravity. This scenario may be interpreted as a type of "Gauss's law for entropy" in quantum gravity, and provides a clean synthesis of the tension between Euclidean wormholes and a standard interpretation of the holographic dictionary, with no need for an ensemble. Our perspective relies crucially on the recently-proposed potential for quantum-mechanical gauge redundancies between states of the universe with different topologies. By an application of the state-operator correspondence, this proposal rules out the possibility of nontrivial, strictly well-defined bulk operators supported in a compact region. We further comment on the possible exceptions in $d\leq 3$ for this hypothesis, and the role of an ensemble for holographic theories in low dimensions, such as JT gravity in $d = 2$ and possible cousins in $d=3$. We argue that these examples are incomplete physical theories that should be viewed as branes in a higher dimensional theory of quantum gravity, for which an ensemble plays no role.

For certain terms in the action, supersymmetry can forbid an infinite number of possible contributions. We study whether such protection can occur in quantum gravity even without sufficient supersymmetry. We focus on whether the superpotential can vanish exactly in four-dimensional N=1 theories, and if the prepotential can be exactly cubic in N=2 theories. We investigate these questions in string theory and find that for almost all known string constructions the corrections allowed by supersymmetry do occur. However, we do find some special settings where all the corrections can be proven to vanish. These examples all share the common feature that they are related, through a certain orbifolding by a discrete gauged R-symmetry element, to a higher supersymmetric theory. Motivated by these results, we propose a Swampland criterion that any theory which enjoys such protection beyond its realised supersymmetry must have a direct connection to a higher supersymmetric theory.

We study supersymmetry breaking deformations of the $\mathcal{N}=1$ 5d fixed point known as $E_1$, the UV completion of $SU(2)$ super-Yang-Mills. The phases of the non-supersymmetric theory can be characterized by Chern-Simons terms involving background $U(1)$ gauge fields, allowing us to identify a phase transition at strong coupling. We propose that this may signify the emergence of a non-trivial, non-supersymmetric CFT in $d=4+1$ dimensions.

We consider supergravity theories with 16 supercharges in Minkowski space with dimensions $d>3$. We argue that there is an upper bound on the number of massless modes in such theories depending on $d$. In particular we show that the rank of the gauge symmetry group $G$ in $d$ dimensions is bounded by $r_G\leq 26-d$. This in particular demonstrates that 4 dimensional ${\cal N}=4$ SYM theories with rank bigger than 22, despite being consistent and indeed finite before coupling to gravity, cannot be consistently coupled to ${\cal N}=4$ supergravity in Minkowski space and belong to the swampland. Our argument is based on the swampland conditions of completeness of spectrum of defects as well as a strong form of the distance conjecture and relies on unitarity as well as supersymmetry of the worldsheet theory of BPS strings. The results are compatible with known string constructions and provide further evidence for the string lamppost principle (SLP): that string theory lamppost seems to capture ${\it all}$ consistent quantum gravitational theories.

We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau threefolds provide hitherto unknown M-theory duals of F-theory configurations compactified on a circle with twists. For certain exceptional KK theories that do not admit a standard geometric description we propose an algebraic description that appears to retain the properties of the local Calabi-Yau threefolds necessary to determine RG flows to 5d SCFTs, along with other relevant physical data.

We study the implications of the recently proposed Trans-Planckian Censorship Conjecture (TCC) for early universe cosmology and in particular inflationary cosmology. The TCC leads to the conclusion that if we want inflationary cosmology to provide a successful scenario for cosmological structure formation, the energy scale of inflation has to be lower than $10^9$ GeV. Demanding the correct amplitude of the cosmological perturbations then forces the generalized slow-roll parameter $\epsilon$ of the model to be very small ($<10^{-31}$). This leads to the prediction of a negligible amplitude of primordial gravitational waves. For slow-roll inflation models, it also leads to severe fine tuning of initial conditions.

Using a reformulation of topological ${\cal N}=2$ QFT's in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a $G_2$ manifold constructed from the space of self-dual 2-forms over $X^4$, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes' construction of the Seiberg-Witten invariants when $X^4$ is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT's arising from ${\cal N}=2$ QFT's from all Gaiotto theories on arbitrary 4-manifolds.

Motivated by the swampland dS conjecture, we consider a rolling scalar field as the source of dark energy. Furthermore, the swampland distance conjecture suggests that the rolling field will lead at late times to an exponentially light tower of states. Identifying this tower as residing in the dark sector suggests a natural coupling of the scalar field to the dark matter, leading to a continually reducing dark matter mass as the scalar field rolls in the recent cosmological epoch. The exponent in the distance conjecture, $\tilde{c}$, is expected to be an $\mathcal{O}(1)$ number. Interestingly, when we include the local measurement of $H_0$, our model prefers a non-zero value of the coupling $\tilde{c}$ with a significance of $2.8\sigma$ and a best-fit at $\tilde{c} \sim 0.3$. Modifying the recent evolution of the universe in this way improves the fit to data at the $2\sigma$ level compared to $\Lambda$CDM. This string-inspired model automatically reduces cosmological tensions in the $H_0$ measurement as well as $\sigma_8$.

We show that the non-gravitational sectors of certain 6d and 5d supergravity theories can be decomposed into superconformal field theories (SCFTs) which are coupled together by pairwise identifying and gauging mutual global symmetries. In the case of 6d supergravity, we consider F-theory on compact elliptic Calabi-Yau 3-folds with base $B=T^4/\mathbb{Z}_n\times \mathbb{Z}_m$ and we show in many examples that the non-gravitational field theory sectors can be described as configurations of coupled 6d $(1,0)$ SCFTs. We also conjecture that the effective 2d $(0,4)$ SCFTs living on the self-dual strings of the 6d theories lead to holographically dual descriptions of type IIB string theory on $AdS_3\times S^3\times B$ and moreover that their elliptic genera can be used to compute the degeneracies of 5d spinning BPS black holes along with all-genus topological string amplitudes on the corresponding compact 3-fold. In the case of 5d supergravity, we consider M-theory on compact Calabi-Yau 3-folds and using similar ideas as in the 6d case we show the complete non-gravitational sector of 5d supergravity theories can be decomposed into coupled 5d $\mathcal{N}=1$ SCFTs. Furthermore, using this picture we propose a generalized topological vertex formalism which, excluding some curve classes, seems to capture all-genus topological string amplitudes for the mirror quintic.

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1,0) theories and a better understanding of the relation between 2d (0,1) theories and TMF spectra.

Among Swampland conditions, the distance conjecture characterizes the geometry of scalar fields and the de Sitter conjecture constrains allowed potentials on it. We point out a connection between the distance conjecture and a refined version of the de Sitter conjecture in any parametrically controlled regime of string theory by using Bousso's covariant entropy bound. The refined version turns out to evade all counter-examples at scalar potential maxima that have been raised. We comment on the relation of our result to the Dine-Seiberg problem.

In this paper we study compactifications of ADE type conformal matter, N M5 branes probing ADE singularity, on torus with flux for global symmetry. We systematically construct the four dimensional theories by first going to five dimensions and studying interfaces. We claim that certain interfaces can be associated with turning on flux in six dimensions. The interface models when compactified on a circle comprise building blocks for constructing four dimensional models associated to flux compactifications of six dimensional theories on a torus. The theories in four dimensions turn out to be quiver gauge theories and the construction implies many interesting cases of IR symmetry enhancements and dualities of such theories.

We study the open refined topological string amplitudes using the refined topological vertex. We determine the refinement of holonomies necessary to describe the boundary conditions of open amplitudes (which in particular satisfy the required integrality properties). We also derive the refined holonomies using the refined Chern-Simons theory.

We discuss the four dimensional models obtained by compactifying a single M5 brane probing $D_{N}$ singularity (minimal D-type $(1,0)$ conformal matter in six dimensions) on a torus with flux for abelian subgroups of the $SO(4N)$ flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel ${\cal N}=1$ dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.

We formulate geometric conditions necessary for engineering 5d superconformal field theories (SCFTs) via M-theory compactification on a local Calabi-Yau 3-fold. Extending the classification of the rank 1 cases, which are realized geometrically as shrinking del Pezzo surfaces embedded in a 3-fold, we propose an exhaustive classification of local 3-folds engineering rank 2 SCFTs in 5d. This systematic classification confirms that all rank 2 SCFTs predicted using gauge theoretic arguments can be realized as consistent theories, with the exception of one family which is shown to be non-perturbatively inconsistent and thereby ruled out by geometric considerations. We find that all rank 2 SCFTs descend from 6d (1,0) SCFTs compactified on a circle possibly twisted with an automorphism together with holonomies for global symmetries around the Kaluza-Klein circle. These results support our conjecture that every 5d SCFT can be obtained from the circle compactification of some parent 6d (1,0) SCFT.

We study compactifications of the 6d E-string theory, the theory of a small E_8 instanton, to four dimensions. In particular we identify N=1 field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non-abelian flat connections as well as fluxes for the abelian sub-groups of the E_8 flavor symmetry. This sheds light on emergent symmetries in a number of 4d N=1 SCFTs (including the `E7 surprise' theory) as well as leads to new predictions for a large number of 4-dimensional exceptional dualities and symmetries.

We propose a number of apparently equivalent criteria necessary for the consistency of a 5d SCFT in its Coulomb phase and use these criteria to classify 5d SCFTs arising from a gauge theory with simple gauge group. These criteria include the convergence of the 5-sphere partition function; the positivity of particle masses and monopole string tensions; and the positive definiteness of the metric in some region in the Coulomb branch. We find that for large rank classical groups simple classes of SCFTs emerge where the bounds on the matter content and the Chern-Simons level grow linearly with rank. For classical groups of rank less than or equal to 8, our classification leads to additional cases which do not fit in the large rank analysis. We also classify the allowed matter content for all exceptional groups.

We introduce a new duality for $\mathcal{N}=1$ supersymmetric gauged matrix models. This $0d$ duality is an order 4 symmetry, namely an equivalence between four different theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general $\mathcal{N}=1$ matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds.

We study the geometry of 4d N=1 SCFT's arising from compactification of 6d (1,0) SCFT's on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d N=1 SCFT's arising from M5 branes probing Z_k singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on Z_2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.

Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3d) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings may be realized in the 6 dimensional phase space, where the physical space defects play the role of topological D-branes.