Higgs Bundles for M-theory on G2-Manifolds M-theory compactified on G2-holonomy manifolds results in 4d N = 1 supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle M3. We derive the BPS equations and find the massless spectrum for both abelian and non-abelian gauge groups in 4d. The mathematical tool that allows us to determine the spectrum is Morse theory, and more generally Morse-Bott theory. The latter generalization allows us to make contact with twisted connected sum (TCS) G2-manifolds, which form the largest class of examples of compact G2-manifolds. M-theory on TCS G2-manifolds is known to result in a non-chiral 4d spectrum. We determine the Higgs bundle for this class of G2-manifolds and provide a prescription for how to engineer singular transitions to models that have chiral matter in 4d.

GUT Scale Unification in Heterotic Strings We present a class of heterotic compactifications where it is possible to lower the string unification scale down to the GUT scale, while preserving the validity of the perturbative analysis. We illustrate this approach with an explicit example of a four-dimensional chiral heterotic vacuum with N = 1 supersymmetry.

Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory 1 Introduction 5 2 A Lightning Introduction to String Theory and Some Related Topics 8 2.1 Quantum black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Schwarzschild black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Hawking temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Page curve and unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.4 Information loss problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Some string theory and conformal field theory . . . . . . . . . . . . . . . . . . . 14 2.2.1 The conformal anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 The operator product expansion . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 The bc CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 The super-conformal field theory . . . . . . . . . . . . . . . . . . . . . . 21 2.2.5 Vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.6 Background fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.7 Beta function: finiteness and Weyl invariance . . . . . . . . . . . . . . . 24 2.2.8 String perturbation expansions . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.9 Spectrum of type II string theory . . . . . . . . . . . . . . . . . . . . . . 27 2.3 On Dp-branes and T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Coupling to abelian gauge fields . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Symmetry under the exchange of momentum and winding . . . . . . . . 32 2.3.4 Symmetry under the exchange of Neumann and Dirichlet . . . . . . . . . 34 2.3.5 Chan-Paton factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.6 Electromagnetism on a circle and Wislon lines . . . . . . . . . . . . . . . 36 2.3.7 The D-branes on the dual circle . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 Quantum gravity in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.1 Dynamical triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.2 Matrix models of D = 0 string theory . . . . . . . . . . . . . . . . . . . . 42 2.4.3 Matrix models of D = 1 string theory . . . . . . . . . . . . . . . . . . . . 44 2.4.4 Preliminary synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 M-(atrix) Theory and Matrix String Theory 46 3.1 The quantized membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The IKKT model or type IIB matrix model . . . . . . . . . . . . . . . . . . . . 50 3.3 The BFSS model from dimensional reduction . . . . . . . . . . . . . . . . . . . . 53 3.4 Introducing gauge/gravity duality . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.1 Dimensional reduction to p + 1 dimensions . . . . . . . . . . . . . . . . . 56 3.4.2 Dp-branes revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.3 The corresponding gravity dual . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 Black hole unitarity from M-theory . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.1 The black 0-brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.2 Supergravity in 11 dimensions and M-wave solution . . . . . . . . . . . . 64 3.5.3 Type IIA string theory at strong couplings is M-Theory . . . . . . . . . 69 3.6 M-theory prediction for quantum black holes . . . . . . . . . . . . . . . . . . . . 70 3.6.1 Quantum gravity corrections . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6.2 Non-perturbative tests of the gauge/gravity duality . . . . . . . . . . . . 74 3.7 Matrix string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.8 Black-hole/black-string transition as the confinement/deconfinement transition . 80 3.8.1 The black-hole/black-string phase transition . . . . . . . . . . . . . . . . 80 3.8.2 The confinement/deconfinement phase transition . . . . . . . . . . . . . 83 3.8.3 The mass gap and the Gaussian structure . . . . . . . . . . . . . . . . . 88 3.8.4 The large d approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.8.5 High temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.9 The discrete light-cone quantization (DLCQ) and infinite momentum frame (IMF) 92 3.9.1 Light-cone quantization and discrete light-cone quantization . . . . . . . 92 3.9.2 Infinite momentum frame and BFSS conjecture . . . . . . . . . . . . . . 95 3.9.3 More on light-like versus space-like compactifications . . . . . . . . . . . 96 3.10 M-(atrix) theory in pp-wave spacetimes . . . . . . . . . . . . . . . . . . . . . . . 99 3.10.1 The pp-wave spacetimes and Penrose limit . . . . . . . . . . . . . . . . . 99 3.10.2 The BMN matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.10.3 Construction of the BMN matrix model . . . . . . . . . . . . . . . . . . 104 3.10.4 Compactification on R × S3 . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.10.5 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.11 Other matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4 Type IIB Matrix Model 124 4.1 The IKKT model in the Gaussian expansion method . . . . . . . . . . . . . . . 124 4.2 Yang-Mills matrix cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.1 Lorentzian type IIB matrix model . . . . . . . . . . . . . . . . . . . . . . 128 4.2.2 Spontaneous symmetry breaking and continuum and infinite volume limits . . . . . . . . . . . . . . . . . . . . . 130 4.2.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.2.4 Role of noncommutativity . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.5 Other related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.3 Emergent gravity: introductory remarks . . . . . . . . . . . . . . . . . . . . . . 140 4.3.1 Noncommutative electromagnetism is a gravity theory . . . . . . . . . . 140 4.3.2 Seiberg-Witten map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.4 Fuzzy spheres and fuzzy CPn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.4.1 Co-adjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.4.2 Fuzzy projective space CP2 . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.4.3 Tangent projective module on fuzzy CP2 . . . . . . . . . . . . . . . . . . 152 4.4.4 Yang-Mills matrix models for fuzzy CPk . . . . . . . . . . . . . . . . . . 153 4.4.5 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4.6 Star product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4.7 Fuzzy derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.5 Fuzzy S 4 N : symplectic and Poisson structures . . . . . . . . . . . . . . . . . . . . 160 4.5.1 The spectral triple and fuzzy CPk N : another look . . . . . . . . . . . . . 160 4.5.2 Fuzzy S 4 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.5.3 Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.5.4 Poisson structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.5.5 Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.5.6 Local Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.5.7 Noncommutativity scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.5.8 Matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.6 Emergent matrix gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.6.1 Fluctuations on fuzzy S 4 N . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.6.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.6.3 Emergent geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.6.4 Emergent gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.6.5 Emergent gravity: Einstein equations . . . . . . . . . . . . . . . . . . . . 189 4.7 Emergent quantum gravity from multitrace matrix models . . . . . . . . . . . . 194

Numerical Observation of Emergent Spacetime Supersymmetry at Quantum Criticality

Spacetime supersymmetry (SUSY) was originally proposed as a fundamental symmetry of nature more than four decades ago, but no experimental evidence of SUSY in particle physics has been confirmed. Recently, it has been theoretically argued that SUSY can also spontaneously emerge in certain condensed matter systems (1–14), e.g., near the superconducting (SC) quantum critical point (QCP) of an interacting single-flavor Dirac fermions in 2 + 1–dimensional (2 + 1D) systems (5, 6). However, verification of this fascinating  SUSY of a single Dirac fermion in microscopic lattice models in 2 + 1D by nonperturbative and unbiased approaches is still lacking and is thus highly desired.

Dirac fermions are essential ingredients of modern physics that can appear as either elementary particles such as electrons and positrons or emergent quasiparticles, e.g., massless Dirac fermions in graphene and on the surface of 3D topological insulators (15, 16). For a single flavor of massless interacting Dirac fermion in 2 + 1 dimensions, there are numerous interesting phenomena and theoretical predictions, from emergent spacetime SUSY at the SC QCP (5, 6) to the surface topological order (17–20), as well as fermion dualities (21). Although a single Dirac cone can occur on the surface of 3D topological insulators, studying such interacting problems in 2 + 1D microscopic models has been highly challenging due to the notorious no-go theorem of fermion doubling. According to this theorem, it is impossible to realize a single Dirac fermion in local lattice models in two spatial dimensions while maintaining time-reversal and chiral symmetries. Usual lattice regularization of a single-flavor Dirac fermion violates some of those symmetry requirements such that existing approaches cannot reveal many fascinating features associated with a single Dirac fermion.

In this study, we investigate a novel 2D lattice model of spin-1/2 fermions that features a single Dirac point at Γ, with perfectly linear dispersion and quantized π Berry phase around Γ, and preserves both time-reversal and chiral symmetries. Fermions in this model can hop along either the x or y direction, with hopping amplitudes that decay in power law at long distances. At half-filling, namely, when the Fermi level is exactly at the neutral point of single Dirac cone, sufficiently strong attractions between fermions should induce superconductivity in the system. If the lattice regularization can capture low-energy physics of a single Dirac cone, spacetime SUSY could emerge at the SC QCP. Consequently, it is highly desired to investigate universal properties of this putative SC quantum phase transition by a reliable and nonperturbative method like quantum Monte Carlo (QMC) (22) without encountering the fermion sign problem (23, 24). However, QMC methods are sign problem free only for limited classes of interacting models (25–32).

Our lattice model of a single Dirac cone with onsite Hubbard attractive interaction U is sign problem free, which allows us to study the emergent behaviors of the SC quantum phase transition in a numerically reliable way. From the state-of-the-art QMC simulations, we provide convincing evidence that the  spacetime SUSY emerges at the SC QCP, as shown schematically in Fig. 1. First, the fermions and order parameter bosons have identical anomalous dimensions that are consistent with the exact value of 1/3 (33) associated with the  SUSY. Moreover, we obtain the correlation-length exponent ν = 0.87 ± 0.05, which is consistent with the nearly exact result of 0.917 obtained from conformal bootstrap calculations (34) of the  SUSY in 2 + 1 dimensions. Moreover, our QMC calculations show that the local electronic density of states ρ(ω) at ω ≪ 1 behaves like ρ(ω) ∝ ω^a with the exponent a = 1.37 ± 0.07, close to the exact value of 4/3 associated with the  SUSY, which can be measured in experiments such as scanning tunneling microscopy (STM) to test the predicted SUSY.

F-theory and Dark Energy Motivated by its potential use as a starting point for solving various cosmological constant problems, we study F-theory compactified on the warped product Rtime ×S 3×Y8 where Y8 is a Spin(7) manifold, and the S3 factor is the target space of an SU(2) Wess–Zumino–Witten (WZW) model at level N. Reduction to M-theory exploits the abelian duality of this WZW model to an S3/ZN orbifold. In the large N limit, the untwisted sector is captured by 11D supergravity. The local dynamics of intersecting 7-branes in the Spin(7) geometry is controlled by a Donaldson–Witten twisted gauge theory coupled to defects. At late times, the system is governed by a 1D supersymmetric quantum mechanics system with two real supercharges, which in four dimensions would appear as “N = 1/2 supersymmetry” on a curved background. This leads to a cancellation of zero point energies in the 4D field theory but a split mass spectrum for superpartners of order ∆m4D ∼√ MIRMUV specified by the IR and UV cutoffs of the model. This is suggestively close to the TeV scale in some scenarios. The classical 4D geometry has an intrinsic instability which can produce either a collapsing or expanding Universe, the latter providing a promising starting point for a number of cosmological scenarios. The resulting 1D quantum mechanics in the time direction also provides an appealing starting point for a more detailed study of quantum cosmology.

G2-Manifolds and M-Theory Compactifications The mathematical features of a string theory compactification determine the physics of the effective four-dimensional theory. For this reason, understanding the mathematical structure of the possible compactification spaces is of profound importance. It is well established that the compactification space for M-Theory must be a seven-manifold with holonomy G2, but much else remains to be understood regarding how to achieve a physically-realistic effective theory from such a compactification. Much also remains unknown about the mathematics of these G2-Manifolds, as they are quite difficult to construct. This review discusses progress with regards to both the mathematical and physical considerations surrounding spaces of holonomy G2. Special attention is given to the known constructions of G2-Manifolds and the physics of their corresponding M-Theory compactifications.

Covariant Formulation of M-Theory Part I: We propose the bosonic part of an action that defines M-theory. It possesses manifest SO(1, 10) symmetry and constructed based on the Lorentzian 3-algebra associated with U(N) Lie algebra. From our action, we derive the bosonic sector of BFSS matrix theory and IIB matrix model in the naive large N limit by taking appropriate vacua. We also discuss an interaction with fermions. Covariant Formulation of M-Theory, Part II We propose a supersymmetric model that defines M-theory. It possesses SO(1, 10) super Poincare symmetry and is constructed based on the Lorentzian 3-algebra associated with U(N) Lie algebra. This model is a supersymmetric generalization of the model in arXiv:0902.1333. From our model, we derive BFSS matrix theory and IIB matrix model in the naive large N limit by taking appropriate BPS vacua.

String Geometry and Non-perturbative Formulation of String Theory We define string geometry: spaces of superstrings including the interactions, their topologies, charts, and metrics. Trajectories in asymptotic processes on a space of strings reproduce the right moduli space of the super Riemann surfaces in a target manifold. Based on the string geometry, we define Einstein-Hilbert action coupled with gauge fields, and formulate superstring theory non-perturbatively by summing over metrics and the gauge fields on the spaces of strings. This theory does not depend on backgrounds. The theory has a supersymmetry as a part of the diffeomorphisms symmetry on the superstring manifolds. We derive the all-order perturbative scattering amplitudes that possess the super moduli in type IIA, type IIB and SO(32) type I superstring theories from the single theory, by considering fluctuations around fixed backgrounds representing type IIA, type IIB and SO(32) type I perturbative vacua, respectively. The theory predicts that we can see a string if we microscopically observe not only a particle but also a point in the space-time. That is, this theory unifies particles and the space-time.

SL(2)×R+ Exceptional Field Theory: An Action for F-Theory Exceptional Field Theory employs an extended spacetime to make supergravity fully covariant under the U-duality groups of M-theory. The 12-dimensional EFT associated to the group SL(2)×R+ together with its action is presented. Demanding the closure of the algebra of local symmetries leads to a constraint, known as the section condition, that must be imposed on all fields. This constraint has two inequivalent solutions, one giving rise to 11-dimensional supergravity and the other leading to Type IIB supergravity and F-theory. Thus SL(2)×R+ Exceptional Field Theory contains both F-theory and M-theory in a single 12-dimensional formalism.

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups Eight-dimensional nongeometric heterotic strings were constructed as duals of Ftheory on Λ1,1 ⊕E8 ⊕E7 lattice polarized K3 surfaces by Malmendier and Morrison. We study the structure of the moduli space of this construction. There are special points in this space at which the ranks of the non-Abelian gauge groups on the 7-branes in F-theory are enhanced to 18. We demonstrate that the enhanced rank-18 non-Abelian gauge groups arise as a consequence of the coincident 7-branes, which deform stable degenerations on the F-theory side. This observation suggests that the non-geometric heterotic strings include nonperturbative effects of the coincident 7-branes on the Ftheory side. The gauge groups that arise at these special points in the moduli space do not allow for perturbative descriptions on the heterotic side. We also construct a family of elliptically fibered Calabi–Yau 3-folds by fibering K3 surfaces with enhanced singularities over P1. Highly enhanced gauge groups arise in F-theory compactifications on the resulting Calabi–Yau 3-folds.

Symmetries in Quantum Field Theory and Quantum Gravity Abstract: In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any “long-range” bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the “convex hull condition” of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and ’t Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman-Mandula theorem to discrete symmetries, and an improved explanation of the decay π 0 → γγ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in d + 1 dimensions with horizon topology T^p × S^d−p−1 .

String Dualities and Gaugings of Supergravity This thesis is devoted to various questions connected with duality. It is composed of two parts. The first part discusses some aspects of timelike T-duality. We explore the possibility of compactification of supergravity theories with various signatures (low energy limit of M-theories which are dual under timelike T-dualities) on parallelizable internal seven dimensional (pseudo-)spheres. We show that, beside the standard theory, only one of the dual theories known as M0-theory can admit such a solution. The effective four dimensional theory is non-supersymmetric and due to the presence of torsion the symmetry of seven dimensional (pseudo-)sphere breaks down to Spin(3, 4). In the second part, in an attempt to have a systematic discussion of gaugings in supergravity, we show the isomorphism between the space of local deformations of the appropriate zero coupling limit of the embedding tensor Lagrangian and that of the second-order scalar-vector Lagrangian, describing the bosonic sector of supergravity ignoring gravity, in a chosen duality frame determined by embedding tensors. We analyze the BV-BRST deformation of a class of scalar-vector coupled Lagrangians, which contains supergravity Lagrangians as examples, and find a set of constraints that guarantee the consistency of the deformations of the Lagrangians. We show in principle that for a large class of theories considered in this thesis, the only deformations are those of the Yang-Mills type associated with a subgroup of the rigid symmetries.

Lectures on Extended Supergravities and Gaugings In an ungauged supergravity theory, the presence of a scalar potential is allowed only for the minimal N = 1 case. In extended supergravities, a non-trivial scalar potential can be introduced without explicitly breaking supersymmetry only through the so-called gauging procedure. The latter consists in promoting a suitable global symmetry group to local symmetry to be gauged by the vector fields of the theory. Gauged supergravities provide a valuable approach to the study of superstring flux-compactifications and the construction of phenomenologically viable, string-inspired models. The aim of these lectures is to give a pedagogical introduction to the subject of gauged supergravities, covering just selected issues and discussing some of their applications.

SO(32) heterotic standard model vacua in general Calabi-Yau compactifications We study a direct flux breaking scenario in SO(32) heterotic string theory on general CalabiYau threefolds. The direct flux breaking, corresponding to hypercharge flux breaking in the F-theory context, allows us to derive the Standard Model in general Calabi-Yau compactifications. We present a general formula leading to the three generations of quarks and leptons and no chiral exotics in a background-independent way. As a concrete example, we show the three-generation model on a complete intersection Calabi-Yau threefold.

TASI Lectures on Abelian and Discrete Symmetries in F-theory In F-theory compactifications, the abelian gauge sector is encoded in global structures of the internal geometry. These structures lie at the intersection of algebraic and arithmetic description of elliptic fibrations: While the Mordell–Weil lattice is related to the continuous abelian sector, the Tate–Shafarevich group is conjectured to encode discrete abelian symmetries in F-theory. In these notes we review both subjects with a focus on recent findings such as the global gauge group and gauge enhancements. We then highlight the application to F-theory model building.

Quantum Corrections and the de Sitter Swampland Conjecture Abstract: Recently a swampland criterion has been proposed that rules out de Sitter vacua in string theory. Such a criterion should hold at all points in the field space and especially at points where the system is on-shell. However there has not been any attempt to examine the swampland criterion against explicit equations of motion. In this paper we study four-dimensional de Sitter and quasi-de Sitter solutions using dimensionally reduced M-theory. While on one hand all classical sources that could allow for solutions with de Sitter isometries are ruled out, the quantum corrections, on the other hand, are found to allow for de Sitter solutions provided certain constraints are satisfied. A careful study however shows that generically such a constrained system does not allow for an effective field theory description in four-dimensions. Nevertheless, if some hierarchies between the various quantum pieces could be found, certain solutions with an effective field theory description might exist. Such hierarchies appear once some mild time dependence is switched on, in which case certain quasi-de Sitter solutions may be found without a violation of the swampland criterion.  

Towards de Sitter vacua and non-abelian gauge symmetry in M-theory via compactifications on 7-manifolds with G2 holonomy We study the physics of globally consistent four-dimensional N = 1 supersymmetric M-theory compactifications on G2 manifolds constructed via twisted connected sum. We study a rich example that exhibits U(1)3 gauge symmetry and a spectrum of massive charged particles that includes a trifundamental. Applying recent mathematical results to this example, we compute the form of membrane instanton corrections to the superpotential and spacetime topology change in a compact model; the latter include both the (non-isolated) G2 flop and conifold transitions. The conifold transition spontaneously breaks the gauge symmetry to U(1)2, and associated field theoretic computations of particle charges make correct predictions for the topology of the deformed G2 manifold. We discuss physical aspects of the abelian G2 landscape broadly, including aspects of Higgs and Coulomb branches, membrane instanton corrections, and some general aspects of topology change.

Classification, geometry and applications of supersymmetric backgrounds to string theory, M-theory, AdS/CFT and black holes We review the remarkable progress that has been made the last 15 years towards the classification of supersymmetric solutions with emphasis on the description of the bilinears and spinorial geometry methods. We describe in detail the geometry of backgrounds of key supergravity theories, which have applications the context of black holes, string theory, M-theory and the AdS/CFT correspondence unveiling a plethora of existence and uniqueness theorems. Some other aspects of supersymmetric solutions like the Killing superalgebras and the homogeneity theorem are also presented, and the non-existence theorem for certain smooth supergravity flux compactifications is outlined. Amongst the applications described is the proof of the emergence of conformal symmetry near black hole horizons and the classification of warped AdS backgrounds that preserve more than 16 supersymmetries.

Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications The Mordell-Weil lattices (MW lattices) associated to rational elliptic surfaces are classified into 74 types. Among them, there are cases in which the MW lattice is none of the weight lattices of simple Lie algebras or direct sums thereof. We study how such “non-Cartan MW lattices” are realized in the six-dimensional heterotic/Ftheory compactifications. In this paper, we focus on non-Cartan MW lattices that are torsion free and whose associated singularity lattices are sublattices of A7. For the heterotic string compactification, a non-Cartan MW lattice yields an instanton gauge group H with one or more U(1) group(s). We give a method for computing massless spectra via the index theorem and show that the U(1) instanton number is limited to be a multiple of some particular non-one integer. On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over P1. Except for some cases, we obtain such geometries for specific distributions of instantons. All the spectrum derived from those geometries completely match with the heterotic results.

Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.

Introduction to the AdS/CFT correspondence This is a pedagogical introduction to the AdS/CFT correspondence, based on lectures delivered by the author at the third IDPASC school. Starting with the conceptual basis of the holographic dualities, the subject is developed emphasizing some concrete topics, which are discussed in detail. A very brief introduction to string theory is provided, containing the minimal ingredients to understand the origin of the AdS/CFT duality. Other topics covered are the holographic calculation of correlation functions, quarkantiquark potentials and transport coefficients.

Negative Branes, Supergroups and the Signature of Spacetime We study the realization of supergroup gauge theories using negative branes in string theory. We show that negative branes are intimately connected with the possibility of timelike compactification and exotic spacetime signatures previously studied by Hull. Isolated negative branes dynamically generate a change in spacetime signature near their worldvolumes, and are related by string dualities to a smooth Mtheory geometry with closed timelike curves. Using negative D3 branes, we show that SU(0|N) supergroup theories are holographically dual to an exotic variant of type IIB string theory on dS3,2 × S¯5, for which the emergent dimensions are timelike. Using branes, mirror symmetry and Nekrasov’s instanton calculus, all of which agree, we derive the Seiberg-Witten curve for N = 2 SU(N|M) gauge theories. Together with our exploration of holography and string dualities for negative branes, this suggests that supergroup gauge theories may be non-perturbatively well-defined objects, though several puzzles remain.

Monodromifold Planes: Refining (p,q)-Branes and Orientifold Planes in String Theory Abstract: We consider new brane-like objects in F-theory consisting of the zero loci of the coefficient functions f and g of the Weierstrass equation, which we referred to as an “flocus plane” and a “g-locus plane”, collectively as “monodromifold planes”. If there are some monodromifold planes, the base of the fibration is divided into several cell regions, each of which corresponds to a (half of a) fundamental region in the pre-image of the J function. A cell region is bounded by several domain walls extending from these monodromifold planes and D-branes, on which the imaginary part of the J function vanishes. If one crosses through a special kind of domain walls called “S-walls”, the type IIB complex string coupling is locally S-dualized, leading to the simultaneous coexistence of a theory in the perturbative regime and its nonperturbative S-dual. Consequently, the monodromy of a 7-brane gets SL(2, Z) conjugated, and a D-brane is transmuted into a non-D (p, q)-7-brane. In the orientifold limit, the “locally S-dualized regions” shrink to infinitely small and not seen from outside. We find that in a wide range of parameters arises a bound-state-like cluster made of two D-branes and one g-locus plane. We also observe that the substructure of an orientifold plane is a “3-1-3” system consisting of two clusters, made of an f- and a g-locus planes and a D-brane, and a single g-locus plane. We also show that all the types of the Kodaira singularities are achieved by D-branes and monodromifold planes with a characteristic pattern of their configuration.

Lectures on Mirror Symmetry and Topological String Theory These are notes of a series of lectures on mirror symmetry and topological string theory given at the Mathematical Sciences Center at Tsinghua University. The N = 2 superconformal algebra, its deformations and its chiral ring are reviewed. A topological field theory can be constructed whose observables are only the elements of the chiral ring. When coupled to gravity, this leads to topological string theory. The identification of the topological string A- and Bmodels by mirror symmetry leads to surprising connections in mathematics and provides tools for exact computations as well as new insights in physics. A recursive construction of the higher genus amplitudes of topological string theory expressed as polynomials is reviewed.

Non-Lorentzian Field Theories with Maximal Supersymmetry and Moduli Space Dynamics We present gauge theories in 2+1 and 4+1 dimensions with 16 supersymmetries which are invariant under rotations and translations but not boosts. The on-shell conditions reduce the dynamics to motion on a moduli space of BPS states graded by a topologically conserved quantity. On each component of the moduli space only half the supersymmetry is realised. We argue that these theories describe M2-branes and M5-branes which have been infinitely boosted so that their worldvolume ‘time’ has become null.

A 3d Gauge Theory/Quantum K-Theory Correspondence Abstract: The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a K¨ahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N = 2 supersymmetric 3d gauge theory and the IR limit is given by Givental’s permutation equivariant quantum K-theory on X. This gives a oneparameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.

Refined SU(3) Vafa-Witten Invariants and Modularity Abstract. We conjecture a formula for the refined SU(3) Vafa-Witten invariants of any smooth surface S satisfying H1(S, Z) = 0 and pg(S) > 0. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined S-duality modularity transformation. We provide evidence for our formula by calculating virtual χy-genera of moduli spaces of rank 3 stable sheaves on S in examples using Mochizuki’s formula. Further evidence is based on the recent definition of refined SU(r) Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3).

The (2, 0) Superalgebra, Null M-branes and Hitchin’s System We present an interacting system of equations with sixteen supersymmetries and an SO(2) × SO(6) R-symmetry where the fields depend on two space and one null dimensions that is derived from a representation of the six-dimensional (2, 0) superalgebra. The system can be viewed as two M5-branes compactified on S1− × T2 or equivalently as M2-branes on R+ ×R2, where ± refer to null directions. We show that for a particular choice of fields the dynamics can be reduced to motion on the moduli space of solutions to the Hitchin system. We argue that this provides a description of intersecting null M2-branes and is also related by U-duality to a DLCQ description of four-dimensional maximally supersymmetric Yang-Mills.

Study finds serious flaw in Emergent Gravity Theory: it cannot reproduce General Relativity In recent years, some physicists have been investigating the possibility that gravity is not actually a fundamental force, but rather an emergent phenomenon that arises from the collective motion of small bits of information encoded on spacetime surfaces called holographic screens. The theory, called emergent gravity, hinges on the existence of a close connection between gravity and thermodynamics. Emergent gravity has received its share of criticism, however, and a new paper adds to this by showing that the holographic screen surfaces described by the theory do not actually behave thermodynamically, undermining a key assumption of the theory. Zhi-Wei Wang, a physicist at Jilin University in Changchun, China, and Samuel L. Braunstein, a professor of quantum computational science at the University of York in the UK, have published their paper on non-thermodynamic surfaces in a recent issue of Nature Communications. "Emergent gravity has very strong claims: that it can explain things like dark matter and dark energy, but also reproduce the decades of work coming out of regular general relativity," Wang told Phys.org. "That last claim is now knocked on its head by our work, so emergent gravity proponents will have their work cut out for themselves in showing consistency with the huge canon of observational results. We've set them back, not necessarily knocked them out." In the cosmological context, surfaces refer generally to any two-dimensional area in spacetime. Some of these surfaces, such as the horizons of black holes and other objects, are confirmed to be thermodynamic. For black hole horizons, this has been known since the 1970s, since the very laws that define black hole mechanics are directly analogous to the laws of thermodynamics. This means that black hole horizons obey thermodynamic principles such as energy conservation and having a positive temperature and entropy. More recently, surfaces that are not horizons have been conjectured to obey the laws of thermodynamics, with the holographic screens in the emergent gravity theory being one example. However, so far these conjectures have not been fully justified. In the new paper, the scientists tested whether different kinds of surfaces obey an analogue of the first law of thermodynamics, which is a special form of energy conservation. Their results reveal that, while surfaces near black holes (called stretched horizons) do obey the first law, ordinary surfaces—including holographic screens—generally do not. The only exception is that ordinary surfaces that are spherically symmetric do obey the first law. As the scientists explain, the finding that stretched horizons obey the first law is not surprising, since these surfaces inherit much of their behavior from the nearby horizons. Still, the scientists caution that the results do not necessarily imply that stretched horizons obey all of the laws of thermodynamics. On the other hand, the finding that ordinary surfaces do not obey the first law is more unexpected, especially as it is one of the key assumptions of emergent gravity. Going forward, researchers will work to understand what this means for the future of emergent gravity, as well as explore other possible implications. "We spent a large amount of time working out how to reproduce the original results for black holes from the 1970s," Braunstein said. "Although the methods from the 1970s were extremely tedious to replicate in detail, we found them very powerful and are thinking now about whether there is any way to generalize these results to other scenarios. Also, we think that our formula for the deviation away from the first law as one moves away from horizons will have important implications for quantum gravity."

TASI Lectures on the Emergence of Bulk Physics in AdS/CFT These lectures review recent developments in our understanding of the emergence of local bulk physics in AdS/CFT. The primary topics are sufficient conditions for a conformal field theory to have a semiclassical dual, bulk reconstruction, the quantum error correction interpretation of the correspondence, tensor network models of holography, and the quantum Ryu-Takayanagi formula.