Inflation with a graceful exit in a random landscape Cosmological inflation is the main contender for the description of the very early universe prior to the conventional hot big bang epoch. It has strong empirical support from various cosmological probes such as e.g. the cosmic microwave background (CMB) precision data, type Ia supernovae, and baryon acoustic oscillations (BAO). However, inflation in general is more of a paradigm, as the detailed microscopic model of inflation is unknown, with the available cosmological data allowing for large classes of inflationary scalar potentials. Moreover, at the theory level, inflation is sensitive to quantum gravity effects, and hence to the short distance (UV) completion of quantum mechanics and general relativity [1]. It is this property which motivates a study of UV completions of inflation in string theory as one of our best candidates for a theory of quantum gravity. String theory is described at the worldsheet level by a two-dimensional conformal field theory (CFT). A large class of consistent effectively four-dimensional solutions to string theory, called ‘string vacua’, arises by compactifying the six extra space dimensions arising from the excess central charge of the worldsheet CFT. These string compactifications typically come with a plethora of moduli scalar fields, parametrizing deformations of extra-dimensional manifold, and axionic pseudo-scalar fields from higher-dimensional gauge potentials. Lacking observational evidence for their existence, these moduli are in need of stabilization in order to acquire large masses. A combination of tree-level sources such as quantized p-form field strengths (fluxes), perturbative string quantum effects, and non-perturbative effects such as D-branes and instantons serve to stabilize the moduli in a large discretum of meta-stable 4D string vacua, some of which can be cosmologically viable de Sitter (dS) vacua [1]. Among the many fields of this high-dimensional scalar potential ‘landscape’, inflation can arise either by the ‘accident’ via occurrence of a narrow slow-roll flat region in the scalar potential, or as a long large-field valley due to underlying structures and/or symmetries of a subsector of landscape. Examples for the latter large-field high-scale inflation models arise e.g. from the approximate shift symmetry of axion inflation models, or the asymptotic exponential series of certain volume moduli inflation models, for a recent review see [1]. In this paper we study the part of the landscape without long-range structures. In this case we can approximate inflation as arising at random by local cancellations among terms in a random potential thereby producing a narrow slow-roll flat region supporting small-field inflation. One example for such sectors of the landscape is the scalar potential for the h 2,1 1 complex structure moduli of a generic non-trivial Calabi-Yau compactification of string theory (at least, away from the limit of large complex structure [2]). Previous work [3, 4] has studied the probability of viable local dS minima in this context using the fact that the statistics of critical points and the eigenvalue distribution of their mass matrices (Hessians) in a random potential are well described by the statistics of sets of large Gaussian random matrices. Moreover, random matrix theory (RMT) has been applied in [5] in a reconstruction of a random potential along the path of steepest decent starting from a local critical point with a given mass matrix. This local reconstruction of the random scalar potential along the inflationary path rests on the description of the eigenvalue distribution of Hessian of critical points in Gaussian random potentials and its stochastic variation along random paths in field space by Dyson Brownian motion (DBM) [6, 7]. The eigenvalues of an ensemble of Gaussian random matrices describing the critical point Hessians behave like a 1D gas of electrically charged particles with logarithmic mutually repulsive potential in a common quadratic confining potential. This picture allows for an intuitive understanding of the behaviour of the eigenvalue spectrum of the Hessians along trajectories in field space, including the effect of ‘eigenvalue repulsion’. Since eigenvalues tend to repel each other, moving along such a path in field space tends to rapidly generate strongly tachyonic directions in the Hessian. This is the reason why both local dS minima and inflationary small-field critical points are exponentially rare in such structure-less sectors of the string landscape. In this paper we apply the description of the evolution of the eigenvalue distribution of critical point Hessians in a Gaussian random landscape via DBM to the question of finding a ’graceful exit’ from the inflationary regime of random inflationary small-field critical points. A ’graceful exit’ from inflation describes the requirement of finding a local, cosmologically viable dS minimum after rolling away from some inflationary critical point in the landscape. We compute the corresponding probability of a graceful exit using the DBM process over multiple correlation lengths of the underlying random potential. We do this first by numerically integrating the discretized Dyson Brownian motion equations [5, 7]. Then we apply the description of the eigenvalue distribution via the 1D Dyson gas by means of a path integral. For static eigenvalue configurations this was done by Dean and Majumdar in [8]. We generalize and extend their derivation to include the relaxation dynamics of DBM which adds a set of N linear potentials to the Hamiltonian describing the evolving Dyson gas. Then, we perform an analytical saddle point evaluation of the path integral, which allows us to derive the time-dependent average eigenvalue distribution, given an initial fluctuated Hessian. ‘Time’ here denotes the field displacement along the path in field space. Given this time-dependent eigenvalue distribution, we compute the saddle point action which gives us the transition probability as a function of distance in field space. These results are general for DBM in Gaussian random matrix theory, which itself has widespread applications beyond inflationary cosmology, including in recent years in areas like image analysis, genomics, epidemiology, engineering, economics and finance, for reviews see e.g. [9, 10]. Then, we specify our results to an ensemble of Hessians with an eigenvalue distribution describing inflationary critical points (the lightest mass eigenvalues are very slightly tachyonic to describe slow-roll). Computing the probability of a graceful exit from such a random inflationary critical, we find our central result that the exit probability for small- field inflation in the landscape is exponentially small. The suppression exponent increases quadratically with number of light fields N. We then compare this behaviour of small-field inflation in the landscape with large-field models, whose underlying structure and/or symmetry usually guarantees the existence of viable post-inflationary minimum. Taken at face value, this implies a strong exponential bias against small-field inflation being the dominant regime in the landscape. Finally, we discuss the influence of the exp(−cN2 ) suppression of small-field inflation on the probability of observing negative spatial curvature in a landscape where the various dS vacua and inflationary critical points are populated via Coleman-De Luccia (CDL) tunneling transitions. Following the methodology of [11], the exponentially strong dependence on the number N of light fields participating in a small-field inflationary critical point leads exponentially strong posterior probability distribution function for N derived from the nonobservation of spatial negative curvature. This severely limits the effective number such light fields to N 10.

On the equivalence of the 11D pure spinor and Brink-Schwarz-like superparticle cohomologies Abstract: The D = 11 pure spinor formulation of the superparticle allows a simple realization of covariant quantization, unlike the D = 11 Brink-Schwarz-like superparticle. We explicitly show the equivalence between the cohomologies of these two models in the context of two different group decompositions: SO(10, 1) → SO(1, 1) × SO(9) and SO(10, 1) → SO(3, 1)×SO(7). We also carry out a light-cone analysis of the pure spinor cohomology, and show that it correctly reproduces the SO(9) equations of motion for D = 11 linearized supergravity.

Covariant field equations in supergravity Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations that many authors often assumed, though there is no proof in the literature. We prove that, under reasonable conditions, field equations of supergravity are covariant modulo other field equations. While doing so, it is also found that for any supergravity model there exist covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. In practice, these covariant field equations are easily found by simply covariantizing the ordinary field equations.

Classical Spacetimes as Amplified Information in Holographic Quantum Theories We argue that classical spacetimes represent amplified information in the holographic theory of quantum gravity. In general, classicalization of a quantum system involves amplification of information at the cost of exponentially reducing the number of observables. In quantum gravity, the geometry of spacetime must be the analogously amplified information. Bulk local semiclassical operators probe this information without disturbing it; these correspond to logical operators acting on code subspaces of the holographic theory. From this viewpoint, we study how bulk local operators may be realized in a holographic theory of general spacetimes, which includes AdS/CFT as a special case, and deduce its consequences. In the first half of the paper, we ask what description of the bulk physics is provided by a holographic state dual to a semiclassical spacetime. In particular, we analyze what portion of the bulk can be reconstructed as spacetime in the holographic theory. The analysis indicates that when a spacetime contains a quasi-static black hole inside a holographic screen, the theory provides a description of physics as viewed from the exterior (though the interior information is not absent). In the second half, we study how and when a semiclassical description emerges in the holographic theory. We find that states representing semiclassical spacetimes are non-generic in the holographic Hilbert space; in particular, microstates for a semiclassical spacetime do not form a Hilbert space. When there are a significant number of independent microstates, semiclassical operators must be given state-dependently, irrespective of the space of the microstates. We elucidate this point using the stabilizer formalism and tensor network models. We also argue that semiclassical states, albeit exponentially rare in the Hilbert space, can be dynamically selected under time evolution. Finally, we discuss implications of the present picture for the black hole interior.

Unified field theory of forces and particles with gravitational origin of gauge symmetry in hyper-spacetime A unified field theory of all known basic forces and elementary particles is built based on a postulate of gauge invariance and coordinate independence along with a conformal scaling gauge symmetry. The hyper-spin charge of a unified hyper-spinor field is conjectured to correlate to the dimension of a hyper-spacetime with Dh = 19 via a maximal symmetry. A unified fundamental interaction is postulated to be governed by a hyper-spin gauge symmetry SP(1,Dh-1). The gravitational origin of gauge symmetry characterized by a gauge-type hyper-gravifield enables us to study gauge gravity and gravity geometry correspondences and to reveal a gauge geometry duality.

A proof of the weak gravity conjecture The weak gravity conjecture suggests that, in a self-consistent theory of quantum gravity, the strength of gravity is bounded from above by the strengths of the various gauge forces in the theory. In particular, this intriguing conjecture asserts that in a theory describing a U(1) gauge field coupled consistently to gravity, there must exist a particle whose proper mass is bounded (in Planck units) by its charge: m/mP < q. This beautiful and remarkably compact conjecture has attracted the attention of physicists and mathematicians over the last decade. It should be emphasized, however, that despite the fact that there are numerous examples from field theory and string theory that support the conjecture, we still lack a general proof of its validity. In the present Letter we prove that the weak gravity conjecture (and, in particular, the mass-charge upper bound m/mP < q) can be inferred directly from Bekenstein’s generalized second law of thermodynamics, a law which is widely believed to reflect a fundamental aspect of the elusive theory of quantum gravity.

What hadron collider is required to discover or falsify natural supersymmetry? Weak scale supersymmetry (SUSY) remains a compelling extension of the Standard Model because it stabilizes the quantum corrections to the Higgs and W, Z boson masses. In natural SUSY models these corrections are, by definition, never much larger than the corresponding masses. Natural SUSY models all have an upper limit on the gluino mass, too high to lead to observable signals even at the high luminosity LHC. However, in models with gaugino mass unification, the wino is sufficiently light that supersymmetry discovery is possible in other channels over the entire natural SUSY parameter space with no worse than 3% fine-tuning. Here, we examine the SUSY reach in more general models with and without gaugino mass unification (specifically, natural generalized mirage mediation), and show that the high energy LHC (HE-LHC), a pp collider with √ s = 33 TeV, will be able to detect the gluino signal over the entire allowed mass range. Thus, HE-LHC would either discover or conclusively falsify natural SUSY.
In summary, supersymmetric models with weak scale naturalness are well-motivated SM extensions with impressive indirect support from measurements of gauge couplings and the top-quark and Higgs boson mass. While the HL-LHC appears sufficient to completely probe natural SUSY models with gaugino mass unification, we have shown that HE-LHC with √ s = 33 TeV is required to either discover or falsify natural SUSY even in very general– but equally natural– SUSY scenarios such as nGMM with a compressed gaugino spectrum. Alternatively, an e +e − collider with √ s ∼ 0.5 − 0.7 TeV would be sufficient to either discover or falsify natural SUSY via pair production of the required light higgsinos[35]. Discovery of natural SUSY via either of these machines would then provide impetus for the construction of an even higher energy machine which could then access many of the remaining superpartners.

Infinite violation of Bell inequalities in inflation We study the violation of Bell-Mermin-Klyshko (BMK) inequalities in initial quantum states of scalar fields in inflation. We show that the Bell inequality is maximally violated by the Bunch-Davies vacuum which is a two-mode squeezed state of a scalar field. However, we find that the violation of the BMK inequalities does not increase with the number of modes to measure. We then consider a non-Bunch-Davies vacuum expressed by a four-mode squeezed state of two scalar fields. Remarkably, we find that the violation of the BMK inequalities increases exponentially with the number of modes to measure. This indicates that some evidence that our universe has a quantum mechanical origin may survive in CMB data even if quantum entanglement decays exponentially afterward due to decoherence.

Dimension and Dimensional Reduction in Quantum Gravity A number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of “dimension” and concluding with some speculative ideas of what dimensional reduction might mean for physics. What is the dimension of spacetime? For most of physics, the answer is straightforward and uncontroversial: we know from everyday experience that we live in a universe with three dimensions of space and one of time. For a condensed matter physicist, say, or an astronomer, this is simply a given. There are a few exceptions—surface states in condensed matter that act two-dimensional, string theory in ten dimensions—but for the most part dimension is simply a fixed, and known, external parameter. Over the past few years, though, hints have emerged from quantum gravity suggesting that the dimension of spacetime is dynamical and scale-dependent, and shrinks to d ∼ 2 at very small distances or high energies. The purpose of this review is to summarize this evidence and to discuss some possible implications for physics. As early as 1916, Einstein pointed out that it would probably be necessary to combine the newly formulated general theory of relativity with the emerging ideas of quantum mechanics [1]. In the century since, efforts to quantize gravity have led to many breakthroughs in fundamental physics, from gauge-fixing and ghosts to the background field method to Dirac’s analysis of constrained Hamiltonian systems. But the fundamental goal of a complete, consistent quantum theory of gravity still seems distant. In its place, we have a number of interesting but incomplete research programs: most famously string theory and loop quantum gravity, but also group field theory, causal set theory, asymptotic safety, lattice approaches such as causal dynamical triangulations, research based on noncommutative geometry, and various ideas for “emergent” gravity. In a situation like this, we need to explore many complementary lines of research. One particular strategy is to look for fundamental features that are shared by different quantization programs. There is no guarantee that such features will persist in the “correct” quantum theory of gravity, but such a pattern of recurrence at least makes it more plausible. We currently have one outstanding example of such a commonality, the predictions of black hole thermodynamics. We have not directly observed Hawking radiation or black hole entropy, but the thermodynamics properties of black holes can be derived in so many different ways, with such a variety of assumptions and approximations [2], that a claim that black holes do not radiate would now seem perverse. Dimensional reduction of spacetime near the Planck scale is a candidate for second such commonality, albeit one that is much less firmly established. In most of physics the dimension of space, or spacetime, is taken as a fixed external parameter. While the notion of dimension is ancient—see [3–5] for discussion of the history and philosophy— the mathematical formalism for a space of arbitrary dimension is fairly recent, often attributed to Sch¨afli’s work in the early 1850s [6]. The question of why our universe should have the number of dimensions it does was famously discussed in 1917 by Ehrenfest [7], who pointed out that such features as the stability of Newtonian orbits and the duality between electric and magnetic field are unique to three spatial dimensions. But Ehrenfest also warned that “the questions [of what determines the number of dimensions] have perhaps no sense.” The idea that spacetime might really have more than four dimensions was introduced into physics by Nordstr¨om [8], and became more widely known with the work of Kaluza [9] and Klein [10]. The extension beyond five dimensions first appeared, I believe, as an exercise in lecture notes by DeWitt [11]. Kaluza-Klein theory provided a useful illustration of the notion that dimension might be scale-dependent. At small enough distances, the spacetime of Kaluza-Klein theory is an n-dimensional manifold with n > 4. At larger scales, though, the compact dimensions can no longer be resolved, and the spacetime becomes effectively four-dimensional, with excitations in the compact directions appearing as towers of massive four-dimensional modes. The converse process of dimensional deconstruction [12], in which appropriate four-dimensional modes effectively “create” extra dimensions at large distances, has more recently become popular in high energy theory. Here we are interested in the opposite phenomenon, in which the number of effective dimensions decreases at short distances or high energies. The first models I know of that exhibited this behavior were introduced by Jourjine [13] , Kaplunovsky and Weinstein [14], Zeilinger and Svozil [15], and Crane and Smolin [16], all in 1985. A year later, Hu and O’Connor observed scale dependence of the effective dimension in anisotropic cosmologies [17], but the wider significance was not fully appreciated. In quantum gravity, the phenomenon of dimensional reduction first appeared in string theory, where a dimension characterizing thermodynamic behavior was found to unexpectedly drop to d = 2 at high temperatures, leading Atick and Witten to postulate “a lattice theory with a (1+1)- dimensional field theory on each lattice site” [18]. But it was only with the computation of the flow of spectral dimension in causal dynamical triangulations [19] that the idea really took hold. To proceed further, though, we will have to address a basic question: how, precisely, do we define “dimension” as a physical observable?

The Real No-Boundary Wave Function in Lorentzian Quantum Cosmology It is shown that the standard no-boundary wave function has a natural expression in terms of a Lorentzian path integral with its contour defined by Picard-Lefschetz theory. The wave function is real, satisfies the Wheeler-DeWitt equation and predicts an ensemble of asymptotically classical, inflationary universes with nearly-Gaussian fluctuations and with a smooth semiclassical origin.

M-theoretic Lichnerowicz formula and supersymmetry A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric operators to the effective action, the Bianchi identities for fluxes, and some equations of motion. Recently, such formulae have also been shown to underlie the (generalised) geometry of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that describes eleven-dimensional supergravity together with its higher-derivative couplings. The first corrections to the action appear at eight-derivative level, and the construction yields two different supersymmetric invariants, each with a free coefficient. We discuss the restriction of our construction to seven-dimensional internal spaces, and implications for compactifications on manifolds of G2 holonomy. Inclusion of fluxes and computation of contributions with higher than eight derivatives are also discussed.

Gravitino/Axino as Decaying Dark Matter and Cosmological Tensions In supersymmetric axion models, if the gravitino or axino is the lightest SUSY particle (LSP), the other is often the next-to-LSP (NLSP). We investigate the cosmology of such a scenario and point out that the lifetime of the NLSP naturally becomes comparable to the present age of the universe in a viable parameter region. This is a well-motivated example of the so-called decaying dark matter model, which is recently considered as an extension of the ΛCDM model to relax some cosmological tensions.

Supersymmetry Breakdown at Distant Branes: The Super–Higgs Mechanism A compactification of 11-dimensional supergravity with two (or more) walls is considered. The whole tower of massive Kaluza-Klein modes along the fifth dimension is taken into account. With the sources on the walls, an explicit composition in terms of Kaluza-Klein modes of massless gravitino (in the supersymmetry preserving case) and massive gravitino (in the supersymmetry breaking case) is obtained. The super–Higgs effect is discussed in detail.

Natural Supersymmetry and Implications for Higgs physics We re-analyze the LHC bounds on light third generation squarks in Natural Supersymmetry, where the sparticles have masses inversely proportional to their leading-log contributions to the electroweak symmetry breaking scale. Higgsinos are the lightest supersymmetric particles; top and bottom squarks are the next-to-lightest sparticles that decay into both neutral and charged Higgsinos with well-defined branching ratios determined by Yukawa couplings and kinematics. The Higgsinos are nearly degenerate in mass, once the bino and wino masses are taken to their natural (heavy) values. We consider three scenarios for the stop and sbottom masses: (I) t˜R is light, (II) t˜L and ˜bL are light, and (III) t˜R, t˜L, and ˜bL are light. Dedicated stop searches are currently sensitive to Scenarios II and III, but not Scenario I. Sbottom-motivated searches (2b + MET) impact both squark flavors due to t˜→ bχ˜ + 1 as well as ˜b → bχ˜ 0 1,2, constraining Scenarios I and III with somewhat weaker constraints on Scenario II. The totality of these searches yield relatively strong constraints on Natural Supersymmetry. Two regions that remain are: (1) the “compressed wedge”, where (mq˜−|µ|)/mq˜ 1, and (2) the “kinematic limit” region, where mq˜ >∼ 600-750 GeV, at the kinematic limit of the LHC searches. We calculate the correlated predictions for Higgs physics, demonstrating that these regions lead to distinct predictions for the lightest Higgs couplings that are separable with ' 10% measurements. We show that these conclusions remain largely unchanged once the MSSM is extended to the NMSSM in order to naturally obtain a large enough mass for the lightest Higgs boson consistent with LHC data.

Antilinearity Rather than Hermiticity as a Guiding Principle for Quantum Theory Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear P T symmetry. Here we seek to put such P T symmetric theories into as general a context as possible. After providing a brief overview of the P T symmetry program, we show that having an antilinear symmetry is the most general condition that one can impose on a quantum theory for which one can have an inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CP T . We thus extend the CP T theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, P T symmetry then follows, with the case for the physical relevance of the P T -symmetry program thus being advanced. While CP T symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CP T symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CP T -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.

A Supersymmetry Primer I provide a pedagogical introduction to supersymmetry. The level of discussion is aimed at readers who are familiar with the Standard Model and quantum field theory, but who have had little or no prior exposure to supersymmetry. Topics covered include: motivations for supersymmetry, the construction of supersymmetric Lagrangians, superspace and superfields, soft supersymmetry-breaking interactions, the Minimal Supersymmetric Standard Model (MSSM), R-parity and its consequences, the origins of supersymmetry breaking, the mass spectrum of the MSSM, decays of supersymmetric particles, experimental signals for supersymmetry, and some extensions of the minimal framework.

Eternal Inflation, Collapse of the Wave-Function and the Quantum Birth of Cosmic Structure We consider the eternal inflation scenario with the additional element of an objective collapse of the wave function. The incorporation of this new agent to the traditional inflationary setting responds to the necessity of addressing the lack of an explanation for the generation of the primordial anisotropies and inhomogeneities, starting from a perfectly symmetric background and invoking symmetric dynamics. We adopt the continuous spontaneous localization model, in the context of inflation, as the dynamical reduction mechanism that generates the primordial inhomogeneities. Furthermore, when enforcing the objective reduction mechanism, the condition for eternal inflation can be bypassed. In particular, the collapse mechanism incites the wave function, corresponding to the inflaton, to localize itself around the zero mode of the field. Then, the zero mode will evolve essentially unperturbed, driving inflation to an end in any region of the Universe where inflation occurred. Also, our approach achieves a primordial spectrum with an amplitude and shape consistent with the one that best fits the observational data.

Covariant c-flation Cosmology We study how a minimal generalization of Einstein’s equations, where the speed of light (c), gravitational constant (G) and the cosmological constant (Λ) are allowed to vary, might generate a dynamical mechanism to explain the special initial condition necessary to obtain the homogeneous and flat universe we observe today. Our construction preserves general covariance of the theory, which yields a general dynamical constraint in c, G and Λ. We re-write the conditions necessary in order to solve the horizon and flatness problems in this framework. This is given by the shrinking of the comoving particle horizon of this theory which leads to ω < −1/3, but not necessarily to accelerated expansion like in inflation, allowing also a decelerated expansion, contraction and a phase transition in c, in the case of null Λ. We are able to construct the action of this theory, that describes the dynamics of a scalar field that represents c or G (and Λ). This action is general and can be applied to describe different cosmological solutions. We present here how the dynamics of the field can be used to solve the problems of the early universe cosmology by means of different ways to c-inflate the horizon in the early universe, solving the old puzzles of the cosmological standard model. Without a cosmological constant, we show that we can describe the dynamics of the scalar field representing c given a potential, and derive the slow-roll conditions that this potential should obey. In this setup we do not have to introduce an extra unknown scalar field, since the degree of freedom associated to the varying constants plays this role, naturally being the field that is going to be responsible for inflating the horizon in the early universe.

Covariant Open String Field Theory on Multiple Dp-Branes We study covariant open bosonic string field theories on multiple Dp-branes by using the deformed cubic string field theory which is equivalent to the string field theory in the proper-time gauge. Constructing the Fock space representations of the three-string vertex and the four-string vertex on multiple Dp-branes, we obtain the field theoretical effective action in the zero-slope limit. On the multiple D0-branes, the effective action reduces to the Banks-Fishler-Shenker-Susskind (BFSS) matrix model. We also discuss the relation between the open string field theory on multiple Dinstantons in the zero-slope limit and the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model. The covariant open string field theory on multiple Dp-branes would be useful to study the nonperturbative properties of quantum field theories in (p+1)-dimensions in the framework of the string theory. The non-zero-slope corrections may be evaluated systematically by using the covariant string field theory.

The Power of Perturbation Theory It is known since ref. [1] that perturbative expansions in quantum field theories (QFT), as well as in quantum mechanics (QM), are generically asymptotic with zero radius of convergence. In special cases, such as the anharmonic oscillator in QM and φ 4 theories up to d = 3 space-time dimensions, the perturbative expansion turns out to be Borel resummable [2–5]. For the anharmonic oscillator it has been verified that the Borel resummed perturbative series converges to the exact result, available by other numerical methods. Perturbative series associated to more general systems and/or in higher space-time dimensions are typically non-Borel resummable, because of singularities in the domain of integration. These can be avoided by deforming the contour at the cost of introducing an ambiguity that is non-perturbative in the expansion param- 1 eter λ. The ambiguity is expected to be removed by including contributions from semiclassical instanton-like configurations (and all their corresponding series expansion), resulting in what is called transseries. There has been considerable progress in recent years on these issues in the context of the theory of resurgence [6] (see e.g. ref. [7], and refs. [8,9] for reviews and further references). A systematic implementation to generic QFT and QM is however not straightforward. A resurgent analysis requires a detailed knowledge of the asymptotic form of the perturbative coefficients, while typically only the leading large-order behaviour of the perturbative expansion might be accessed in generic QFT and QM [10–13]. Besides, the knowledge of the coefficients of the perturbative series alone is not enough to guarantee that the reconstructed transseries reproduces the full answer. Some non-perturbative information is required, such as the knowledge of some analytic properties of the observable as a function of the expansion parameter. Most importantly, the practicality of transseries beyond the weak coupling regime is hindered by the need to resum the series expansion of all the semi-classical configurations that contribute, in general infinite in number. Perturbation theory within a path integral formulation is an infinite dimensional generalization of the usual steepest-descent method to evaluate ordinary integrals. For sufficiently regular functions Picard-Lefschetz theory teaches us how to decompose the initial contour of integration into a sum of steepest-descent trajectories (called Lefschetz thimbles, or simply thimbles). A geometric approach to the path integral from this perspective, as well as an excellent introduction for physicists to these ideas, has been given by Witten [14] (see also refs. [15,16]). The theory of Lefschetz thimbles allows us to rigorously classify which saddle-point configurations contribute to a given physical observable. An interesting question to ask is under what conditions no non-trivial saddle point contributes, so that the whole result is given by the single perturbative series around the trivial saddle-point. In terms of Lefschetz thimbles, this corresponds to the simple situation in which the domain of integration of the path integral does not need any deformation being already a single Lefschetz thimble on its own. This is what should happen for instance in the anharmonic oscillator in which, as we mentioned, the perturbative series is Borel resummable and converges to the exact result. It has recently been shown in ref. [17] that several one-dimensional quantum mechanical models with a discrete spectrum admit an “exact perturbation theory” (EPT) that is able to capture the full result including non-perturbative effects, even in cases which are known to receive instanton corrections, such as the (supersymmetric) double well. In this paper we explain the reasons behind the results of ref. [17], using the path integral formulation and a Lefschetz thimble perspective. For pedagogical purposes, in sec. 2 we start by reviewing the concepts of Borel summability and Lefschetz-thimble decomposition for a class of one-dimensional integrals Z(λ), viewed as 0-dimensional path integrals, with the parameter λ playing the role of ~. In fact, all the properties of perturbation theory, the role of non-perturbative saddles as well as the definition of EPT can easily be understood in this context. The Lefschetz 2 thimble decomposition reduces Z(λ) into a sum of integrals over thimbles—steepest descent paths with a single saddle point. We prove that their saddle-point expansion is always Borel resummable to the exact answer. In contrast to previous works in the literature, there is no need to study the analytic properties of the integral as a function of λ. Indeed, thanks to a suitable change of coordinates, we are able to rewrite the integral over thimbles directly in terms of a well-defined Borel transform. This result implies the following important consequences. When the decomposition of Z(λ) involves trivially only one thimble, its ordinary perturbation theory is also Borel resummable to the whole result. On the contrary, when the decomposition involves more than one thimble, or it requires an analytic continuation in λ, the naive series expansion of Z(λ) is not Borel resummable to the exact answer. Independently of the thimble decomposition of Z(λ), we show how to introduce a second integral Zˆ(λ, λ0) which has a trivial thimble decomposition for any fixed λ0 and coincides with Z(λ) at λ0 = λ. Therefore the expansion of Zˆ(λ, λ0) in λ is Borel resummable to the exact result even when this is not the case for Z(λ). Such result is possible considering that Z(λ) and Zˆ(λ, λ0), at fixed λ0, have different analytical properties in λ. The expansion of Zˆ(λ, λ0) is the simplest implementation of EPT. In sec. 3 the Borel summability of thimbles is readily extended to multi-dimensional integrals and we discuss in some detail the non trivial generalization to path integrals in QM. In this way we are able to show that QM systems with a bound-state potential and a single non-degenerate crtitical point—the anharmonic oscillator being the prototypical example—are entirely reconstructable from their perturbative expansion. Namely, for any observable (energy eigenvalues, eigenfunctions, etc.) the asymptotic perturbation theory is Borel resummable to the exact result.1 At least for the ground state energy, this remains true also for potentials with multiple critical points as long as the absolute minimum is unique. Potentials V (x; λ) with more than one critical point are more problematic because not all observables are Borel resummable to the exact result and in general instantons are well-known to contribute. Unfortunately in most situations it is a challenging task to explicitly classify all saddle-points and evaluate the corresponding contributions (see e.g. ref. [18] for a recent attempt). In analogy to the one-dimensional integral we show how to bypass this problem by considering an alternative potential Vˆ (x; λ, λ0) admitting always a Borel resummable perturbation theory in λ and coinciding to the original one for λ0 = λ. The idea is to choose Vˆ as the sum of a tree-level and a quantum potential, with the former having only a single critical point. Since the thimble decomposition is controlled only by the saddle point of the tree-level part, the perturbative expansion of Vˆ (EPT) is guaranteed to be Borel resummable to the exact result. For any value of the coupling constant λ, EPT captures the full result. In contrast, the expansion from V requires in general also the inclusion of instanton contributions, we denote such expansion Standard Perturbation Theory (SPT). As noticed also in ref. [17], EPT works 1As far as we know, the Borel resummability of observables other than the energy levels has not received much attention in the literature. 3 surprisingly well at strong coupling, where SPT becomes impractical. In the spirit of resurgence the coefficients of the perturbative series encode the exact answer, with the crucial difference that no transseries are needed. Using this method, we can relax the requirement of having a single critical point in the original potential V , and arrive to the following statement: In one-dimensional QM systems with a bound-state potential V that admits the Vˆ defined above, any observable can be exactly computed from a single perturbative series. We illustrate our results in sec. 4 by a numerical study of the following quantum mechanical examples: the (tilted) anharmonic potential, the symmetric double well, its supersymmetric version, the perturbative expansion around a false vacuum, and pure anharmonic oscillators. In all these systems we will show that the exact ground state energy, computed by solving the Schrodinger equation, is recovered without the need of advocating non-perturbative effects, such as real (or complex) instantons. We will also show that the same applies for higher energy levels and the eigenfunctions. We conclude in sec. 5, where we also briefly report the future perspectives to extend our results in QFT. Some technical details associated to sec. 3 are reported in an appendix.

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