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Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3,2)1/6, ( 3 ¯ ,1)−2/3 and (1,1)1. For the family B3( ∆ 4 ° ) consisting of O (1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the O (1011) QSM geometries B3( ∆ 4 ° ). The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves. In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3( ∆ 4 ° ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens. [ABSTRACT FROM AUTHOR]

Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of a G4-flux. While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic "bottom"-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs. On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry. [ABSTRACT FROM AUTHOR]

Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in dP3, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill-Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space. [ABSTRACT FROM AUTHOR]

In this paper, we examine the emergence of conserved charges on the horizon of a particular class of extremal nondilatonic black p -branes (which reduce to extremal dilatonic black holes in D = 4 dimensions upon toroidal compactification) in the presence of a probe massless scalar field in the bulk. This result is achieved by writing the black p -brane geometry in a Gaussian null coordinate system which allows us to get a nonsingular near-horizon geometry description. We find that the near-horizon geometry is A d S p + 2 × S 2 and that the A d S p + 2 section has an internal structure which can be seen as a warped product of A d S 2 × S p in Gaussian null coordinates. We show that the bulk scalar field satisfying the field equations is expanded in terms of nonnormalizable and normalizable modes, which for certain suitable quantization conditions are well-behaved at the boundary of A d S p + 2 space. Furthermore, we show that picking the normalizable modes results in the existence of conserved quantities on the horizon. We discuss the impact of these conserved quantities in the late-time regime. [ABSTRACT FROM AUTHOR]

We show that F-theory compactifications with abelian gauge factors generally exhibit a non-trivial global gauge group structure. The geometric origin of this structure lies with the Shioda map of the Mordell-Weil generators. This results in constraints on the $$ \mathfrak{u}(1) $$ charges of non-abelian matter consistent with observations made throughout the literature. In particular, we find that F-theory models featuring the Standard Model algebra actually realise the precise gauge group [SU(3) × SU(2) × U(1)]/ ℤ . Furthermore, we explore the relationship between the gauge group structure and geometric (un-)higgsing. In an explicit class of models, we show that, depending on the global group structure, an $$ \mathfrak{s}\mathfrak{u}(2)\oplus \mathfrak{u}(1) $$ gauge theory can either unhiggs into an SU(2) × SU(2) or an SU(3) × SU(2) theory. We also study implications of the charge constraints as a criterion for the F-theory 'swampland'. [ABSTRACT FROM AUTHOR]

We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with n-sections, associated with the Tate-Shafarevich group $$ {\mathbb{Z}}_n $$ . Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group $$ {\mathbb{Z}}_n $$ . By employing a conjectured Heterotic/F-theory mirror symmetry we construct dual geometries of these original toric models, where in the stable degeneration limit we obtain a discrete gauge symmetry of order two and three, for compactifications to six dimensions. We provide explicit constructions of mirror-pairs for symmetric examples with $$ {\mathbb{Z}}_2 $$ and $$ {\mathbb{Z}}_3 $$ , in six dimensions. The Heterotic models with symmetric discrete symmetries are related in field theory to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge factors, where due to the Stückelberg mechanism only a diagonal U(1) factor remains massless, and thus after Higgsing only a diagonal discrete symmetry of order n is present in the Heterotic models and detected via Heterotic/F-theory duality. These constructions also provide further evidence for the conjectured mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional sections and those with multi-sections. [ABSTRACT FROM AUTHOR]

We construct the holographic dictionary for both running and constant dilaton solutions of the two dimensional Einstein-Maxwell-Dilaton theory that is obtained by a circle reduction from Einstein-Hilbert gravity with negative cosmological constant in three dimensions. This specific model ensures that the dual theory has a well defined ultraviolet completion in terms of a two dimensional conformal field theory, but our results apply qualitatively to a wider class of two dimensional dilaton gravity theories. For each type of solutions we perform holographic renormalization, compute the exact renormalized onepoint functions in the presence of arbitrary sources, and derive the asymptotic symmetries and the corresponding conserved charges. In both cases we find that the scalar operator dual to the dilaton plays a crucial role in the description of the dynamics. Its source gives rise to a matter conformal anomaly for the running dilaton solutions, while its expectation value is the only non trivial observable for constant dilaton solutions. The role of this operator has been largely overlooked in the literature. We further show that the only non trivial conserved charges for running dilaton solutions are the mass and the electric charge, while for constant dilaton solutions only the electric charge is non zero. However, by uplifting the solutions to three dimensions we show that constant dilaton solutions can support non trivial extended symmetry algebras, including the one found by Compère, Song and Strominger [1], in agreement with the results of Castro and Song [2]. Finally, we demonstrate that any solution of this specific dilaton gravity model can be uplifted to a family of asymptotically AdS2 × S² or conformally AdS2 × S² solutions of the STU model in four dimensions, including non extremal black holes. The four dimensional solutions obtained by uplifting the running dilaton solutions coincide with the so called 'subtracted geometries', while those obtained from the uplift of the constant dilaton ones are new. [ABSTRACT FROM AUTHOR]

String consistency conditions are stronger than anomaly cancellation and can require the addition of exotics in the visible sector. We study such exotics and demonstrate that they may account for the modest excess at 750 GeV in recent diphoton resonance searches performed by the ATLAS and CMS collaborations. In a previous analysis of type II MSSM D-brane quivers we systematically added up to five exotics for the sake of satisfying string consistency conditions. Using this dataset, we demonstrate that 89780 of the 89964 quivers have exotics, 78155 of which include singlets that may couple to MSSM or exotic multiplets with coupling structures governed by U(1) symmetries that are often anomalous. We demonstrate that certain sets of exotics are far preferred over others and study the structure of singlet couplings to heavy exotics carrying standard model charges. Typical possibilities include singlets that may decay to vector-like quarks and/or vector-like leptons and subsequently to two photons. We show that a narrow width diphoton excess can be accounted for while evading existing bounds if multiple exotics are added, with vector-like leptons of mass [ABSTRACT FROM AUTHOR]

Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi–Yau manifolds. Such geometries appear in families that are parameterized by the Tate–Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former. In this note, we focus on an element of order three in the Tate–Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in I 2 -fibers that appear at certain codimension two loci in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields. In this analysis the investigation of different resolved phases of the underlying geometry plays a crucial rôle. [ABSTRACT FROM AUTHOR]

Abstract: We construct new supersymmetric grand unified models based on orientifolds with intersecting D6-branes. Unlike constructions based on orientifolds, the orbifold images of the three-cycles wrapped by D6-branes correspond to new configurations and thus allow for models in which, in addition to the chiral sector in 10 and representations of , only, there can be new sectors with and vector-pairs. We construct an example of such a globally consistent, supersymmetric model with four-families, two Standard Model Higgs pair-candidates and the gauge symmetry . In an sector, there are and vector-pairs, while another sector contains one vector-pair of 15-plets. The vector-pairs can obtain a large mass dynamically by parallel D6-brane splitting in a particular two-torus. The 15-vector-pairs provide, after symmetry breaking to the Standard Model (via parallel D-brane splitting), triplet pair candidates which can in principle play a role in generating Majorana-type masses for left-handed neutrinos, though the necessary Yukawa couplings are absent in the specific construction. This model can also be interpreted as a flipped grand unified model where the 10-vector-pairs can play the role of Higgs fields, though again there are phenomenological difficulties for the specific construction. [Copyright &y& Elsevier]

Abstract: We systematically investigate instanton corrections from wrapped Euclidean D-branes to the matter field superpotential of various classes of supersymmetric D-brane models in four dimensions. Both gauge invariance and the counting of fermionic zero modes provide strong constraints on the allowed non-perturbative superpotential couplings. We outline how the complete instanton computation boils down to the computation of open string disc diagrams for boundary changing operators multiplied by a one-loop vacuum diagram. For concreteness we focus on E2-instanton effects in type IIA vacua with intersecting D6-branes, however the same structure emerges for type IIB and heterotic vacua. The instantons wrapping rigid cycles can potentially destabilise the vacuum or generate perturbatively absent matter couplings such as proton decay operators, μ-parameter or right-handed neutrino Majorana mass terms. The latter allow the realisation of the seesaw mechanism for MSSM like intersecting D-brane models. [Copyright &y& Elsevier]

Abstract: We analyze the proton decay via dimension six operators in supersymmetric -grand unified models based on intersecting D6-brane constructions in type IIA string theory orientifolds. We include in addition to interactions also the operators arising from interactions. We provide a detailed construction of vertex operators for any massless string excitation arising for arbitrary intersecting D-brane configurations in type IIA toroidal orientifolds. In particular, we provide explicit string vertex operators for the 10 and chiral superfields and calculate explicitly the string theory correlation functions for above operators. In the analysis we chose the most symmetric configurations in order to maximize proton decay rates for the above dimension six operators and we obtain a small enhancement relative to the field theory result. After relating the string proton decay rate to field theory computations the string contribution to the proton lifetime is , which could be up to a factor of three shorter than that predicted in field theory. [Copyright &y& Elsevier]

Abstract: We present a comprehensive classification of supersymmetric vacua of M-theory compactification on seven-dimensional manifolds with general four-form fluxes. We analyze the cases where the resulting four-dimensional vacua have supersymmetry and the internal space allows for , or G 2 structures. In particular, we find for supersymmetry, that the external space–time is Minkowski and the base manifold of the internal space is conformally Kähler for structures, while for structures the internal space has to be Einstein–Sasaki and no internal fluxes are allowed. Moreover, we provide a new vacuum with supersymmetry and structure, where all fluxes are non-zero and the first order differential equations are solved. [Copyright &y& Elsevier]

Abstract: We calculate, at the classical level, the superpotential tri-linear couplings of the only known globally consistent heterotic minimal supersymmetric Standard Model [V. Bouchard, R. Donagi, An heterotic standard model, Phys. Lett. B 633 (2006) 783–791, hep-th/0512149]. This recently constructed model is based on a compactification of the heterotic string theory on a Calabi–Yau threefold with fundamental group, coupled with a slope-stable holomorphic vector bundle. In the observable sector the massless particle content is that of the three-family supersymmetric Standard Model with massless Higgs pairs, depending on the location in the vector bundle moduli space, and no exotic particles. We obtain non-zero Yukawa couplings for the three up-sector quarks, and vanishing R-parity violating terms. In particular, the proton is stable. Another interesting feature is the existence of tri-linear couplings, on the loci with massless Higgs pairs, generating μ-mass parameters for the Higgs pairs and neutrino mass terms, with specific vector bundle moduli playing the role of right-handed neutrinos. [Copyright &y& Elsevier]

We provide a pedagogical introduction to a recently studied class of phenomenologically interesting string models known as Intersecting D-Brane Models. The gauge fields of the Standard Model are localized on D-branes wrapping certain compact cycles on an underlying geometry, whose intersections can give rise to chiral fermions. We address the basic issues and also provide an overview of the recent activity in this field. This article is intended to serve non-experts with explanations of the fundamental aspects of string phenomenology and also to provide some orientation for both experts and non-experts in this active field. [ABSTRACT FROM AUTHOR]

Abstract: A supersymmetric vacuum has to obey a set of constraints on fluxes as well as first order differential equations defined by the G-structures of the internal manifold. We solve these equations for type IIB supergravity with SU(3) structures. The 6-dimensional internal manifold has to be complex, the axion/dilaton is in general non-holomorphic and a cosmological constant is only possible if the SU(3) structures are broken to SU(2) structures. The general solution is expressed in terms of one function which is holomorphic in the three complex coordinates and if this holomorphic function is constant, we obtain a flow-type solution and near poles and zeros we find the so-called type-A and type-B vacuum. [Copyright &y& Elsevier]

Abstract: Based on the T-dual constructions of supersymmetric intersecting D6-models on orientifolds, whose electroweak sector is parallel with the orientifold planes with gauge symmetry [hep-th/0407178], we derive and classify standard model-like vacua with RR and NSNS fluxes, which stabilize toroidal complex structure moduli and the dilaton. We find consistent four-family () and two-family () models with one- and two-units of the quantized flux, respectively. Such models typically possess additional gauge group factors with negative beta functions and may lead, via gaugino condensation, to stabilization of toroidal Kähler moduli. These models have chiral exotics. [Copyright &y& Elsevier]

Abstract: We study the open-string moduli of supersymmetric D6-branes, addressing both the string and field theory aspects of D6-brane splitting on type IIA orientifolds induced by open-string moduli higgsing (i.e., their obtaining VEVs). Specifically, we focus on the orientifolds and address the symmetry breaking pattern for D6-branes parallel with the orientifold 6-planes as well as those positioned at angles. We demonstrate that the string theory results, i.e., D6-brane splitting and relocating in internal space, are in one to one correspondence with the field theory results associated with the higgsing of moduli in the antisymmetric representation of gauge symmetry (for branes parallel with orientifold planes) or adjoint representation of (for branes at general angles). In particular, the moduli higgsing in the open-string sector results in the change of the gauge structure of D6-branes and thus changes the chiral spectrum and family number as well. As a by-product, we provide the new examples of the supersymmetric standard-like models with the electroweak sector arising from gauge symmetry; and one four-family example is free of chiral Standard Model exotics. [Copyright &y& Elsevier]

Abstract: We study conditions on general fluxes of massive type IIA supergravity that lead to four-dimensional backgrounds with supersymmetry. We derive these conditions in the case of - as well as -structures. -structures imply that the internal space is constrained to be a nearly Kähler manifold with all the turned on fluxes, and the negative cosmological constant proportional to the mass parameter, and the dilaton fixed by the quantized ratio of the three-form and four-form fluxes. We further discuss the implications of such flux vacua with added intersecting D6-branes, leading to the chiral non-Abelian gauge sectors (without orientifold projections). Examples that break -structures to -ones allow for the internal space conformally flat (up to orbifold and orientifold projections), for which we give an explicit example. These results provide a starting point for further study of the four-dimensional (chiral) supersymmetric solutions of massive type IIA supergravity with D-branes and fluxes, compactified on orientifolds. [Copyright &y& Elsevier]

Abstract: We provide a systematic construction of three-family supersymmetric Pati–Salam models from Type IIA orientifolds on with intersecting D6-branes. All the gauge symmetry factors arise from the stacks of D6-branes with gauge symmetries, while the “hidden sector” is specified by branes, parallel with the orientifold planes or their images. The Pati–Salam gauge symmetry can be broken down to the via D6-brane splittings, and further down to the Standard Model via D- and F-flatness preserving Higgs mechanism from massless open string states in a subsector. The models also possess at least two confining hidden gauge sectors, where gaugino condensation can in turn trigger supersymmetry breaking and (some) moduli stabilization. The systematic search yields 11 inequivalent models: 8 models with less than 9 Standard Model Higgs doublet-pairs and 1 model with only 2 Standard Model Higgs doublet-pairs, 2 models possess at the string scale the gauge coupling unification of and , and all the models possess additional exotic matters. We also make preliminary comments on phenomenological implications of these models. [Copyright &y& Elsevier]

We study N=1 supersymmetric four-dimensional solutions of massive Type IIA supergravity with intersecting D6-branes in the presence NS–NS three-form fluxes. We derive N=1 supersymmetry conditions for the D6-brane and flux configurations in an internal manifold X6 and derive the intrinsic torsion (or SU(3)-structure) related to the fluxes. In the absence of fluxes, N=1 supersymmetry implies that D6-branes wrap supersymmetric three-cycles of X6 that intersect at angles of SU(3) rotations and the geometry is deformed by SU(3)-structures. The presence of fluxes breaks the SU(3) structures to SU(2) and the D6-branes intersect at angles of SU(2) rotations; non-zero mass parameter corresponds to D8-branes which are orthogonal to the common cycle of all D6-branes. The anomaly inflow indicates that the gauge theory on intersecting (massive) D6-branes is not chiral. [Copyright &y& Elsevier]

We construct some N=1 supersymmetric three-family SU(5) grand unified models from type IIA orientifolds on T6/(Z2×Z2) with D6-branes intersecting at general angles. These constructions are supersymmetric only for special choices of untwisted moduli. We show that within the above class of constructions there are no supersymmetric three-family models with 3 copies of 10-plets unless there are simultaneously some 15-plets. We systematically analyze the construction of such models and their spectra. The M-theory lifts of these brane constructions become purely geometrical backgrounds: they are singular G2 manifolds where the grand unified gauge symmetries and three families of chiral fermions are localized at codimension 4 and codimension 7 singularities respectively. We also study some preliminary phenomenological features of the models. [Copyright &y& Elsevier]

We discuss the hierarchy of Yukawa couplings in a supersymmetric three family standard-like string model. The model is constructed by compactifying type IIA string theory on a Z2×Z2 orientifold in which the standard model matter fields arise from intersecting D6-branes. When lifted to M-theory, the model amounts to compactification of M-theory on a G2 manifold. While the actual fermion masses depend on the vacuum expectation values of the multiple Higgs fields in the model, we calculate the leading worldsheet instanton contributions to the Yukawa couplings and examine the implications of the Yukawa hierarchy. [Copyright &y& Elsevier]

We investigate the charged Schwarzschild–anti-de Sitter (SAdS) BH thermodynamics in 5d Einstein–Gauss–Bonnet gravity with electromagnetic field. The Hawking–Page phase transitions between SAdS BH and pure AdS space are studied. The corresponding phase diagrams (with critical line defined by GB term coefficient and electric charge) are drawn. The possibility to account for higher derivative Maxwell terms is mentioned.In frames of proposed dS/CFT correspondence it is demonstrated that brane gravity maybe localized similarly to AdS/CFT. SdS BH thermodynamics in 5d Einstein and Einstein–Gauss–Bonnet gravity is considered. The corresponding (complicated) surface counterterms are found and used to get the conserved BH mass, free energy and entropy. The interesting feature of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. We speculate that the appearance of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur. [Copyright &y& Elsevier]

We present a consistency condition for 8D N=1 supergravity theories with nontrivial global structure G/Z for the non-Abelian gauge group, based on an anomaly involving the Z 1-form center symmetry. The interplay with other swampland criteria identifies the majority of 8D theories with gauge group G/Z, which have no string theory realization, as inconsistent quantum theories when coupled to gravity. While this condition is equivalent to geometric properties of elliptic K3 surfaces in F-theory compactifications, it constrains the unexplored landscape of gauge groups in other 8D string models. [ABSTRACT FROM AUTHOR]

We present O(1015) string compactifications with the exact chiral spectrum of the standard model of particle physics. This ensemble of globally consistent F-theory compactifications automatically realizes gauge coupling unification. Utilizing the power of algebraic geometry, all global consistency conditions can be reduced to a single criterion on the base of the underlying elliptically fibered Calabi-Yau fourfolds. For toric bases, this criterion only depends on an associated polytope and is satisfied for at least O(1015) bases, each of which defines a distinct compactification. [ABSTRACT FROM AUTHOR]

We analytically compute the thermoelectric conductivities at zero frequency (DC) in the holographic dual of a four dimensional Einstein-Maxwell-Axion-Dilaton theory that admits a class of asymptotically hyperscaling violating Lifshitz backgrounds with a dynamical exponent z and hyperscaling violating parameter θ. We show that the heat current in the dual Lifshitz theory involves the energy flux, which is an irrelevant operator for z > 1. The linearized fluctuations relevant for computing the thermoelectric conductivities turn on a source for this irrelevant operator, leading to several novel and non-trivial aspects in the holographic renormalization procedure and the identification of the physical observables in the dual theory. Moreover, imposing Dirichlet or Neumann boundary conditions on the spatial components of one of the two Maxwell fields present leads to different thermoelectric conductivities. Dirichlet boundary conditions reproduce the thermoelectric DC conductivities obtained from the near horizon analysis of Donos and Gauntlett, while Neumann boundary conditions result in a new set of DC conductivities. We make preliminary analytical estimates for the temperature behavior of the thermoelectric matrix in appropriate regions of parameter space. In particular, at large temperatures we find that the only case which could lead to a linear resistivity ρ ∼ T corresponds to z = 4/3. [ABSTRACT FROM AUTHOR]