The M-theory origin of global properties of gauge theories We show that global properties of gauge groups can be understood as geometric properties in M-theory. Different wrappings of a system of N M5-branes on a torus reduce to four-dimensional theories with AN−1 gauge algebra and different unitary groups. The classical properties of the wrappings determine the global properties of the gauge theories without the need to impose any quantum conditions. We count the inequivalent wrappings as they fall into orbits of the modular group of the torus, which correspond to the S-duality orbits of the gauge theories.

2018 Workshop on the Physics and Geometry of F-Theory: Links to Talks and Slides -- C. Vafa: String Landscape and The Swampland -- F. Appruzzi: 6D SCFTs and the Predictivity of F-theory via Massive IIA -- A. Braun: From F-Theory to G2 manifolds -- A. Collinucci: Flops of length two, or the conifold -- M. Cvetic: F-Theory Global Constraints on Matter Representations -- R. Donagi: Bundles on elliptic fibrations -- I. García-Etxebarria: Global anomalies in 8d and universality of F-theory -- T. Grimm: Field distances, F-theory, and the Swampland -- J. Halverson: A Large Ensemble of F-Theory Geometries: the Weak, the Strong, and the Non-Higgsable -- A. Hanany: 3d N=4 Coulomb branch and Higgs branch in 5 and 6 dimensions -- H. Hayashi: 5-brane webs and 5d N=1 rank 2 theories -- J. Heckman: 4D Gauge Theories with Conformal Matter -- H. Jockers: Moduli spaces of string compactifications from gauge theory correlators -- C. Lawrie: F-theory and AdS3/CFT2 -- L. Martucci: Effective field theory of 3d N=2 CFT’s from holography -- P. Oehlmann: F-theory on Quotient Threefolds with (2,0) Discrete Superconformal Matter -- T. Rudelius: All 6D F-theory SCFTs from Group Theory -- F. Ruehle: NS5-branes and line bundles in Heterotic/M-/F-theory duality -- W. Taylor: Anomaly constraints and an infinite swampland for charged matter in 6D supergravity + U(1) theories -- A. Tomasiello: The frozen phase of F-theory -- R. Valandro: Abelian gauge symmetries and higher charge states from Matrix Factorization -- C. Vafa: F-Theory, Black Holes and Topological Strings -- T. Weigand: Singularities, Matter, and Gauge Backgrounds -- G. Zoccarato: T-branes and black holes

The Kähler Quotient Resolution of C3/Γ singularities, the McKay Correspondence and D=3 N=2 Chern-Simons gauge theories We advocate that the generalized Kronheimer construction of the Kahler quotient crepant resolution ¨ Mζ −→ C3/Γ of an orbifold singularity where Γ ⊂ SU(3) is a finite subgroup naturally defines the field content and the interaction structure of a superconformal Chern-Simons Gauge Theory. This latter is supposedly the dual of an M2-brane solution of D = 11 supergravity with C×Mζ as transverse space. We illustrate and discuss many aspects of this type of constructions emphasizing that the equation p∧p = 0 which provides the Kahler analogue of the holomorphic sector in the hyperK ¨ ahler moment map equations canonically defines ¨ the structure of a universal superpotential in the CS theory. Furthermore the kernel DΓ of the above equation can be described as the orbit with respect to a quiver Lie group GΓ of a special locus LΓ ⊂ HomΓ(Q ⊗R,R) that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold GΓ/FΓ, the gauge group FΓ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of Γ. These first Chern classes are represented by (1,1)-forms on Mζ and provide a basis for the cohomology group H2 (Mζ). We also discuss the relation with conjugacy classes of Γ and we provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of C2/Γ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.

Non-simply-laced Symmetry Algebras in F-theory on Singular Spaces We demonstrate how non-simply-laced gauge and flavor symmetries arise in F-theory on spaces with non-isolated singularities. The breaking from a simply-laced symmetry to one that is non-simply-laced is induced by Calabi-Yau complex structure deformation. In all examples the deformation maintains non-isolated singularities but is accompanied by a splitting of an I1 sevenbrane that opens new loops in the geometry near a non-abelian seven-brane. The splitting also arises in the moduli space of a probe D3-brane, which upon traversing the new loop experiences a monodromy that acts on 3-7 string junctions on the singular space. The monodromy reduces the symmetry algebra, which is the flavor symmetry of the D3-brane and the gauge symmetry of the seven-brane, to one that is non-simply-laced. A collision of the D3-brane with the seven-brane gives rise to a 4d N = 1 SCFTs with a non-simply-laced flavor symmetry.

Top Down Approach to 6D SCFTs Six-dimensional superconformal field theories (6D SCFTs) occupy a central place in the study of quantum field theories encountered in high energy theory. This article reviews the top down construction and study of this rich class of quantum field theories, in particular, how they are realized by suitable backgrounds in string / M- / F-theory. We review the recent F-theoretic classification of 6D SCFTs, explain how to calculate physical quantities of interest such as the anomaly polynomial of 6D SCFTs, and also explain recent progress in understanding renormalization group flows for deformations of such theories. Additional topics covered by this review include some discussion on the (weighted and signed) counting of states in these theories via superconformal indices. We also include several previously unpublished results as well as a new variant on the swampland conjecture for general quantum field theories decoupled from gravity. The aim of the article is to provide a point of entry into this growing literature rather than an exhaustive overview.

Real ADE-equivariant (co)homotopy and Super M-branes A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by Ktheory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf’s theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant S4-valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. At each stage the full Green-Schwarz action functional for the given fundamental brane species appears, as the datum associated to the morphisms in the orbit category

McKay Correspondence and new Calabi-Yau Threefolds Abstract. In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs (h 1,1, h2,1) of Hodge numbers of Calabi-Yau threefolds.

The McKay correspondence via Floer theory Abstract. We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity Cn/G for a finite subgroup G ⊂ SL(n, C) and any crepant resolution Y , we prove that the rank of positive symplectic cohomology SH∗+(Y ) is the number |Conj(G)| of conjugacy classes of G, and that twice the age grading on conjugacy classes is the Z-grading on SH∗−1+ (Y ) by the Conley-Zehnder index. The generalized McKay correspondence follows as SH∗−1+ (Y ) is naturally isomorphic to ordinary cohomology H∗(Y ), due to a vanishing result for full symplectic cohomogy. In the Appendix we construct a novel filtration on the symplectic chain complex for any nonexact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology.

D-Branes And Mirror Symmetry We study (2, 2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in nonlinear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K¨ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.

Supersymmetric Theory and Models In these introductory lectures, we review the theoretical tools used in constructing supersymmetric field theories and their application to physical models. We first introduce the technology of two-component spinors, which is convenient for describing spin- 1/2 fermions. After motivating why a theory of nature may be supersymmetric at the TeV energy scale, we show how supersymmetry (SUSY) arises as an extension of the Poincare algebra of spacetime symmetries. We then obtain the representations of the SUSY algebra and discuss its simplest realization in the Wess-Zumino model. In order to have a systematic approach for obtaining supersymmetric Lagrangians, we introduce the formalism of superspace and superfields and recover the Wess-Zumino Lagrangian. These methods are then extended to encompass supersymmetric abelian and nonabelian gauge theories coupled to supermatter. Since supersymmetry is not an exact symmetry of nature, it must ultimately be broken. We discuss several mechanisms of SUSY-breaking (both spontaneous and explicit) and briefly survey various proposals for realizing SUSY-breaking in nature. Finally, we construct the the Minimal Supersymmetric extension of the Standard Model (MSSM), and consider the implications for the future of SUSY in particle physics.

Supersymmetric Gauge Theories from String Theory The subject of this thesis are various ways to construct four-dimensional quantum field theories from string theory. In a first part we study the generation of a supersymmetric Yang-Mills theory, coupled to an adjoint chiral superfield, from type IIB string theory on non-compact Calabi-Yau manifolds, with D-branes wrapping certain subcycles. Properties of the gauge theory are then mapped to the geometric structure of the Calabi-Yau space. In particular, the low energy effective superpotential, governing the vacuum structure of the gauge theory, can in principle be calculated from the open (topological) string theory. Unfortunately, in practice this is not feasible. Quite interestingly, however, it turns out that the low energy dynamics of the gauge theory is captured by the geometry of another non-compact Calabi-Yau manifold, which is related to the original Calabi-Yau by a geometric transition. Type IIB string theory on this second CalabiYau manifold, with additional background fluxes switched on, then generates a fourdimensional gauge theory, which is nothing but the low energy effective theory of the original gauge theory. As to derive the low energy effective superpotential one then only has to evaluate certain integrals on the second Calabi-Yau geometry. This can be done, at least perturbatively, and we find that the notoriously difficult task of studying the low energy dynamics of a non-Abelian gauge theory has been mapped to calculating integrals in a well-known geometry. It turns out, that these integrals are intimately related to quantities in holomorphic matrix models, and therefore the effective superpotential can be rewritten in terms of matrix model expressions. Even if the Calabi-Yau geometry is too complicated to evaluate the geometric integrals explicitly, one can then always use matrix model perturbation theory to calculate the effective superpotential. This intriguing picture has been worked out by a number of authors over the last years. The original results of this thesis comprise the precise form of the special geometry relations on local Calabi-Yau manifolds. We analyse in detail the cut-off dependence of these geometric integrals, as well as their relation to the matrix model free energy. In particular, on local Calabi-Yau manifolds we propose a pairing between forms and cycles, which removes all divergences apart from the logarithmic one. The detailed analysis of the holomorphic matrix model leads to a clarification of several points related to its saddle point expansion. In particular, we show that requiring the planar spectral density to be real leads to a restriction of the shape of Riemann surfaces, that appears in the planar limit of the matrix model. This in turns constrains the form of the contour along which the eigenvalues have to be integrated. All these results are used to exactly calculate the planar free energy of a matrix model with cubic potential. The second part of this work covers the generation of four-dimensional supersymmetric gauge theories, carrying several important characteristic features of the standard model, from compactifications of eleven-dimensional supergravity on G2- manifolds. If the latter contain conical singularities, chiral fermions are present in the four-dimensional gauge theory, which potentially lead to anomalies. We show that, locally at each singularity, these anomalies are cancelled by the non-invariance of the classical action through a mechanism called “anomaly inflow”. Unfortunately, no explicit metric of a compact G2-manifold is known. Here we construct families of metrics on compact weak G2-manifolds, which contain two conical singularities. Weak G2-manifolds have properties that are similar to the ones of proper G2-manifolds, and hence the explicit examples might be useful to better understand the generic situation. Finally, we reconsider the relation between eleven-dimensional supergravity and the E8 ×E8-heterotic string. This is done by carefully studying the anomalies that appear if the supergravity theory is formulated on a ten-manifold times the interval. Again we find that the anomalies cancel locally at the boundaries of the interval through anomaly inflow, provided one suitably modifies the classical action.

M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture We study various aspects of the M-theory uplift of the AN−1 series of (2, 0) CFTs in 6d, which describe the worldvolume theory of N M5 branes in flat space. We show how knowledge of OPE coefficients and scaling dimensions for this CFT can be directly translated into features of the momentum expansion of M-theory. In particular, we develop the expansion of the four-graviton S-matrix in M-theory via the flat space limit of four-point Mellin amplitudes. This includes correctly reproducing the known contribution of the R4 term from 6d CFT data. Central to the calculation are the OPE coefficients for half-BPS operators not in the stress tensor multiplet, which we obtain for finite N via the previously conjectured relation [1] between the quantum W∞[N] algebra and the AN−1 (2, 0) CFT. We further explain how the 1/N expansion of W∞[N] structure constants exhibits the structure of protected vertices in the M-theory action. Conversely, our results provide strong evidence for the chiral algebra conjecture.

String Sigma Models on Curved Supermanifolds We use the techniques of integral forms to analyse the easiest example of two dimensional sigma models on a supermanifold. We write the action as an integral of a top integral form over a D=2 supermanifold and we show how to interpolate between different superspace actions. Then, we consider curved supermanifolds and we show that the definitions used for flat supermanifold can also be used for curved supermanifolds. We prove it by first considering the case of a curved rigid supermanifold and then the case of a generic curved supermanifold described by a single superfield E.

Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality We study aspects of heterotic/F-theory duality for compactifications with Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds with a rank one MordellWeil group of rational sections. By rigorously performing the stable degeneration limit in a class of toric models, we derive both the Calabi-Yau geometry as well as the spectral cover describing the vector bundle in the heterotic dual theory. We carefully investigate the spectral cover employing the group law on the elliptic curve in the heterotic theory. We find in explicit examples that there are three different classes of heterotic duals that have U(1) factors in their low energy effective theories: split spectral covers describing bundles with S(U(m)×U(1)) structure group, spectral covers containing torsional sections that seem to give rise to bundles with SU(m) × Zk structure group and bundles with purely non-Abelian structure groups having a centralizer in E8 containing a U(1) factor. In the former two cases, it is required that the elliptic fibration on the heterotic side has a non-trivial MordellWeil group. While the number of geometrically massless U(1)’s is determined entirely by geometry on the F-theory side, on the heterotic side the correct number of U(1)’s is found by taking into account a Stückelberg mechanism in the lower-dimensional effective theory. In geometry, this corresponds to the condition that sections in the two half K3 surfaces that arise in the stable degeneration limit of F-theory can be glued together globally.

Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1)×U(1)×U(1) Gauge Symmetry We analyze general F-theory compactifications with U(1)xU(1)xU(1) Abelian gauge symmetry by constructing the general elliptically fibered Calabi-Yau manifolds with a rank three Mordell-Weil group of rational sections. The general elliptic fiber is shown to be a complete intersection of two non-generic quadrics in P3 and resolved elliptic fibrations are obtained by embedding the fiber as the generic Calabi-Yau complete intersection into Bl3P3, the blow-up of P3 at three points. For a fixed base B, there are finitely many Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on these Calabi-Yau manifolds are shown to be labeled by integral points in reflexive polytopes constructed from the nef-partition of Bl3P3. We determine all 14 massless matter representations to six and four dimensions by an explicit study of the codimension two singularities of the elliptic fibration. We obtain three matter representations charged under all three U(1)-factors, most notably a tri-fundamental representation. The existence of these representations, which are not present in generic perturbative Type II compactifications, signifies an intriguing universal structure of codimension two singularities of the elliptic fibrations with higher rank Mordell-Weil groups. We also compute explicitly the corresponding 14 multiplicities of massless hypermultiplets of a six-dimensional F-theory compactification for a general base B.

F-Theory Compactifications with Multiple U(1)-Factors: Addendum The purpose of this note is to extend the results obtained in [1] in two ways. First, the six-dimensional F-theory compactifications with U(1)×U(1) gauge symmetry on elliptic Calabi-Yau threefolds, constructed as a hypersurface in dP2 fibered over the base B = P2 [1] , are generalized to Calabi-Yau threefolds elliptically fibered over an arbitrary twodimensional base B. While the representations of the matter hypermultiplets remain unchanged, their multiplicities are calculated for an arbitrary B. Second, for a specific non-generic subset of such Calabi-Yau threefolds we engineer SU(5)×U(1)×U(1) gauge symmetry. We summarize the hypermultiplet matter representations, which remain the same as for the choice of the base B = P2 [2], and determine their multiplicities for an arbitrary B. We also verify that the obtained spectra cancel anomalies both for U(1)×U(1) and SU(5)×U(1)×U(1).

F-Theory Compactifications with Multiple U(1)-Factors: Elliptic Fibrations with Rational Sections We study F-theory compactifications with U(1)×U(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B = P 2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)×U(1)- charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)×U(1) and SU(5)×U(1)×U(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl(1,0,0)P2(1, 2, 3) and Bl(0,1,0)P2 (1, 1, 2) employing birational maps and extremal transitions.

Quantized Kähler Geometry and Quantum Gravity It has been often observed that Kähler geometry is essentially a U(1) gauge theory whose field strength is identified with the Kähler form. However it has been pursued neither seriously nor deeply. We argue that this remarkable connection between the Kähler geometry and U(1) gauge theory is a missing corner in our understanding of quantum gravity. We show that the Kähler geometry can be described by a U(1) gauge theory on a symplectic manifold with a slight generalization. We derive a natural Poisson algebra associated with the Kähler geometry we have started with. The quantization of the underlying Poisson algebra leads to a noncommutative U(1) gauge theory which arguably describes a quantized Kähler geometry. The Hilbert space representation of quantized Kähler geometry eventually ends in a zero-dimensional matrix model. We then play with the zero-dimensional matrix model to examine how to recover our starting point–Kähler geometry–from the background independent formulation. The round-trip journey suggests many remarkable pictures for quantum gravity that will open a new perspective to resolve the notorious problems in theoretical physics such as the cosmological constant problem, hierarchy problem, dark energy, dark matter and cosmic inflation. We also discuss how time emerges to generate a Lorentzian spacetime in the context of emergent gravity.

TASI Lectures on Geometric Tools for String Compactifications In this work we provide a self-contained and modern introduction to some of the tools, obstacles and open questions arising in string compactifications. Techniques and current progress are illustrated in the context of smooth heterotic string compactifications to 4-dimensions. Progress is described on bounding and enumerating possible string backgrounds and their properties. We provide an overview of constructions, partial classifications, and moduli problems associated to Calabi-Yau manifolds and holomorphic bundles over them.

Brane Effective Actions, Kappa-Symmetry and Applications This is a review on brane effective actions, their symmetries and some of its applications. Its first part uncovers the Green-Schwarz formulation of single M- and D-brane effective actions focusing on kinematical aspects : the identification of their degrees of freedom, the importance of world volume diffeomorphisms and kappa symmetry, to achieve manifest spacetime covariance and supersymmetry, and the explicit construction of such actions in arbitrary on-shell supergravity backgrounds. Its second part deals with applications. First, the use of kappa symmetry to determine supersymmetric world volume solitons. This includes their explicit construction in flat and curved backgrounds, their interpretation as BPS states carrying (topological) charges in the supersymmetry algebra and the connection between supersymmetry and hamiltonian BPS bounds. When available, I emphasise the use of these solitons as constituents in microscopic models of black holes. Second, the use of probe approximations to infer about non-trivial dynamics of strongly coupled gauge theories using the AdS/CFT correspondence. This includes expectation values of Wilson loop operators, spectrum information and the general use of D-brane probes to approximate the dynamics of systems with small number of degrees of freedom interacting with larger systems allowing a dual gravitational description. Its final part briefly discusses effective actions for N D-branes and M2-branes. This includes both SYM theories, their higher order corrections and partial results in covariantising these couplings to curved backgrounds, and the more recent supersymmetric Chern-Simons matter theories describing M2-branes using field theory, brane constructions and 3-algebra considerations.

From geometry to non-geometry via T-duality Reconsideration of T-duality of the open string allows us to introduce some geometric features in non-geometric theories. First, we have found what symmetry is T-dual to the local gauge transformations. This is restricted general coordinate transformations, which includes transformations of background fields but not include transformations of the coordinates. According to this we have introduced new, up to now missing term, with additional gauge field ADi (D denotes components with Dirichlet boundary conditions). It compensate non-fulfilment of the invariance under restricted general coordinate transformation on the end-points of open string, as well as standard gauge field ANa (N denotes components with Neumann boundary conditions) compensate non-fulfilment of the gauge invariance. Using generalized procedure we will perform T-duality of vector fields linear in coordinates. We show that gauge fields ANa and ADi are T-dual to ⋆Aa D and ⋆Ai N respectively. We introduce the field strength of T-dual non-geometric theories as derivative of T-dual gauge fields along both T-dual variable yµ and its double ˜yµ. This definition allows us to obtain gauge transformation of non-geometric theories which leaves T-dual field strength invariant. Therefore, we introduce some new features of non-geometric theories where field strength has both antisymmetric and symmetric parts. This allows us to define new kind of truly non-geometric theories.

The Spectra of Supersymmetric States in String Theory The goals of the thesis, apart from for the author to become a doctor, are the following: 1. To summarise the main results of my research of the past three years. 2. To provide a compact and self-contained survey of the relevant materials for beginning graduate students or researchers in other sub-fields as a shortcut to the frontline of the current research in this area of string theory. Motivation for the First Goal: My personal motivation to pursue this line of research has two sides. First of all, in order to understand the structure of a theory, it is important to know the spectrum of the theory. Just like the spectrum of a hydrogen atom holds the key to understanding quantum mechanics, we hope that the same might be true for string theory. For a very complex theory as string theory is, the supersymmetric part of the spectrum is usually the part which is most accessible to us due to the great simplification supersymmetry offers. Nevertheless, as I hope I will convince the readers in this thesis, it is still a far from trivial task to study this part of the spectrum. In other words, we hope that the study of the spectrum of supersymmetric states of string theory will be a feasible step towards furthering our understanding of string theory. In the other direction, it has been a great challenge since the invention of Einstein gravity and quantum mechanics to understand the quantum aspects of gravity. A fundamental question since the work of Bekenstein and Hawking in the 70’s, is why black holes have entropy. Only when we can answer this question can we ever claim that we understand the nature of quantum gravity. Conversely, because of the challenging nature of the question, once we can answer this question we have a reason to believe that we are on the right track to the goal of quantising, in one way or the other, Einstein gravity. String theory, at the time of writing, still scores highest in the challenge of explaining the thermodynamical entropy of the black holes, while it is also true that most of the work done along this trajectory still focuses on black holes with supersymmetry, which are unlikely to be directly observable in nature. From this point of view, to study the supersymmetric spectrum of string theory and to use it as information about the black hole entropy, is a part of the effort towards a deeper understanding of the nature of quantum gravity. Motivation for the Second Goal: Now I will move on to explain the motivation to achieve the second goal of the thesis: providing a self-contained material serving as a shortcut to the current research on the topic In the course of development of string theory since its birth in the 70’s, it has expanded into an extremely broad and sometimes very complicated field. According Mr. Peter Woit, there might be around 30,000 papers written on the subject so far. While exactly this property makes it, in my opinion, a sufficiently fun field to be working in, it is no good news for beginners. In order to work on a topic in a specific sub-field, she or he is likely to find herself having to go through the labyrinth of a large amount of papers on various totally different but yet somehow inter-connected topics in physics and mathematics, with conflicting notations and conventions. As it could be fairly time-consuming and frustrating a process, I would like to take the chance of writing my PhD thesis to provide a service for anyone who might be able to use it, by making an attempt at a relatively compact and self-contained exposition of some of the should-know’s for performing research related to the subjects I have worked on in the past three years.

Effective action from M-theory on twisted connected sum G2-manifolds We study the four-dimensional low-energy effective N = 1 supergravity theory of the dimensional reduction of M-theory on G2-manifolds, which are constructed by Kovalev’s twisted connected sum gluing suitable pairs of asymptotically cylindrical Calabi–Yau threefolds XL/R augmented with a circle S1. In the Kovalev limit the Ricci-flat G2-metrics are approximated by the Ricci-flat metrics on XL/R and we identify the universal modulus — the Kovalevton — that parametrizes this limit. We observe that the low-energy effective theory exhibits in this limit gauge theory sectors with extended supersymmetry. We determine the universal (semi-classical) K¨ahler potential of the effective N = 1 supergravity action as a function of the Kovalevton and the volume modulus of the G2-manifold. This K¨ahler potential fulfills the noscale inequality such that no anti-de-Sitter vacua are admitted. We describe geometric degenerations in XL/R, which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of G2-manifolds.

The Standard Model in extra dimensions and its Kaluza-Klein effective Lagrangian An effective theory for the Standard Model with extra dimensions is constructed. We start from a field theory governed by the extra-dimensional Poincar´e group ISO(1, 3 + n) and by the extended gauge group GSM(M4+n) = SUC (3,M4+n) × SUL(2,M4+n) × UY (1,M4+n), which is characterized by an unknown energy scale Λ and is assumed to be valid at energies far below this scale. Assuming that the size of the extra dimensions is much larger than the distance scale at which this theory is valid, an effective theory with symmetry groups ISO(1, 3) and GSM(M4) is constructed. The transition between such theories is carried out via a canonical transformation that allows us to hide the extended symmetries {ISO(1, 3 + n), GSM(M4+n)} into the standard symmetries {ISO(1, 3), GSM(M4)}, and thus endow the Kaluza-Klein gauge fields with mass. Using a set of orthogonal functions {f(0), f(m)(¯x)}, which is generated by the Casimir invariant P¯2 associated with the translations subgroup T (n) ⊂ ISO(n), the degrees of freedom of {ISO(1, 3 + n), GSM(M4+n)} are expanded via a general Fourier series, whose coefficients are the degrees of freedom of {ISO(1, 3), G(M4)}. It is shown that these functions, which correspond to the projection on the coordinates basis {|x¯} of the discrete basis {|0, |p (m)} generated by P¯2, play a central role in defining the effective theory. It is shown that those components along the ground state f(0) = x¯|0 do not receive mass at the compactification scale, so they are identified with the Standard Model fields; but components along excited states f(m) = x¯|p (m) do receive mass at this scale, so they correspond to Kaluza-Klein excitations. In particular, it is shown that associated with any direction |p (m) 6= 0 there are a massive gauge field and a pseudo-Goldstone boson. Some resemblances of this mass-generating mechanism with the Englert-Higgs mechanism are stressed and some physical implications are discussed. We perform a comprehensive study of the couplings in all sectors of the effective theory, which includes a full catalog of Lagrangian terms that can be used to calculate Feynman rules.  

From F-theory to brane webs: Non-perturbative effects in type IIB String Theory We analyse the flavour sector of SU(5) Grand Unified Theories in F–theory. Two classes of local models are formulated, one with enhancement to E6 where the masses of the up–type quarks are generated, and one with enhancement to either E7 or E8 where the masses for all fermions of the Standard Model are generated. A full rank 3 Yukawa matrix is attained only after the inclusion of non–perturbative effects in the compactification space. By performing a scan over the parameters defining the local models we check whether realistic masses for the fermions may be attained. Secondly we present two example of the appearance of linear equivalence between cycles in D–brane models. In the first case we show how linear equivalence is tied with kinetic mixing between open and closed string massless U(1)’s and discuss potential phenomenological implications for dark matter and unification of gauge couplings. Secondly we show how taking into account the coupling with closed string moduli some of the brane moduli may acquire a mass. We clarify the microscopic origin of this effect and its connection with linear equivalence of cycles, and finally match it with the 4d supergravity description. Finally we discuss the application of topological string techniques for the computation of the Nekrasov partition function for theories in the Higgs branch. We formulate a general algorithm for the computation of the Nekrasov partition function of the 5d TN theory in a generic point of the Higgs branch. Afterwards we present a generalisation of the topological vertex applicable to a wide class of non– toric varieties. In both cases we provide some explicit examples of the application of the new rules formulated.

Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. The allowed choices of compactification data are integral points in an auxiliary polytope. In order to ensure the existence of a low-energy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. In particular, flatness of SU(5) fourfolds imposes additional linear constraints on the auxiliary polytope of compactifications, and is therefore non-generic. Abelian gauge symmetries arising in toric F-theory compactifications are studied systematically. Associated to each top, the toric Mordell-Weil group determining the minimal number of U(1) factors is computed. Furthermore, all SU(5)-tops and their splitting types are determined and used to infer the pattern of U(1) matter charges.

Four-modulus “Swiss Cheese” chiral models Abstract: We study the ‘Large Volume Scenario’ on explicit, new, compact, four-modulus Calabi-Yau manifolds. We pay special attention to the chirality problem pointed out by Blumenhagen, Moster and Plauschinn. Namely, we thoroughly analyze the possibility of generating neutral, non-perturbative superpotentials from Euclidean D3-branes in the presence of chirally intersecting D7-branes. We find that taking proper account of the Freed-Witten anomaly on non-spin cycles and of the K¨ahler cone conditions imposes severe constraints on the models. Nevertheless, we are able to create setups where the constraints are solved, and up to three moduli are stabilized.

Toric Construction of Global F-Theory GUTs We systematically construct a large number of compact Calabi-Yau fourfolds which are suitable for F-theory model building. These elliptically fibered Calabi-Yaus are complete intersections of two hypersurfaces in a six dimensional ambient space. We first construct three-dimensional base manifolds that are hypersurfaces in a toric ambient space. We search for divisors which can support an F-theory GUT. The fourfolds are obtained as elliptic fibrations over these base manifolds. We find that elementary conditions which are motivated by F-theory GUTs lead to strong constraints on the geometry, which significantly reduce the number of suitable models. The complete database of models is available at [1]. We work out several examples in more detail.

U-dualities in Type II string theories and M-theory In this thesis the recently developed duality covariant approach to string and Mtheory is investigated. In this formalism the U-duality symmetry of M-theory or Tduality symmetry of Type II string theory becomes manifest upon extending coordinates that describe a background. The effective potential of Double Field Theory is formulated only up to a boundary term and thus does not capture possible topological effects that may come from a boundary. By introducing a generalised normal we derive a manifestly duality covariant boundary term that reproduces the known Gibbons-Hawking action of General Relativity, if the section condition is imposed. It is shown that the full potential can be represented as a sum of the scalar potential of gauged supergravity and a topological term that is a full derivative. The latter is written totally in terms of the geometric f-flux and the non-geometric Q-flux integrated over the doubled torus. Next we show that the Scherk-Schwarz reduction of M-theory extended geometry successfully reproduces known structures of maximal gauged supergravities. Local symmetries of the extended space defined by a generalised Lie derivatives reduce to gauge transformations and lead to the embedding tensor written in terms of twist matrices. The scalar potential of maximal gauged supergravity that follows from the effective potential is shown to be duality invariant with no need of section condition. Instead, this condition, that assures the closure of the algebra of generalised diffeomorphisms, takes the form of the quadratic constraints on the embedding tensor.

Geometric Aspects of the Kapustin-Witten Equations This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex YangMills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow. The new results in the paper correspond to estimates on the solutions to the KW equations given an estimate on the complex part of the connection. This leaves open the problem of obtaining global estimates on the complex part of the connection.