We consider local models of magnetised D7 branes in IIB string compactifications, focussing on cases where an explicit metric can be written for the local 4-cycle. The presence of an explicit metric allows analytic expressions for the gauge bundle and for the chiral matter wavefunctions through solving the Dirac and Laplace equations. The triple overlap of the normalised matter wavefunctions generates the physical Yukawa couplings. Our main examples are the cases of D7 branes on P1xP1 and P2. We consider both supersymmetric and non-supersymmetric gauge backgrounds and both Abelian and non-Abelian gauge bundles. We briefly outline potential phenomenological applications of our results.
Flux compactifications of $M$ theory on certain $G_2$-manifolds have vacua in which all moduli are determined.These vacua have discrete parameters - the flux quanta. Small changes in these quanta produce small changes in the moduli and these can lead to large changes in the Yukawa couplings between Higgses, quarks and leptons. We argue that a large number of vacua exist with a wide variety of Yukawa hierarchies.
We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.
In the case of F-theory compactifications on genus-one fibrations without section there are naturally appearing discrete symmetries, which we argue to be associated to geometrically massive U(1) gauge symmetries. These discrete symmetries are shown to induce non-trivial selection rules for the allowed Yukawa couplings in SU(N) gauge theories. The general discussion is exemplified using a concrete Calabi-Yau fourfold realizing an SU(5) GUT model. We observe that M2 instanton effects appear to play a key role in the generation of new superpotential terms and in the dynamics close to phase transi…
Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results in the literature used differential geometric methods to explain the origin of some of this structure. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved …
We compute the particle spectrum and some of the Yukawa couplings for a family of heterotic compactifications on quintic threefolds X involving bundles that are deformations of TX+O_X. These are then related to the compactifications with torsion found recently by Li and Yau. We compute the spectrum and the Yukawa couplings for generic bundles on generic quintics, as well as for certain stable non-generic bundles on the special Dwork quintics. In all our computations we keep the dependence on the vector bundle moduli explicit. We also show that on any smooth quintic there exists a deformation o…
We construct stringy cosmic string solutions corresponding to compactifications of F-theory on several elliptic Calabi-Yau manifolds by solving the equations of motion of low energy effective action of ten dimensional type IIB superstring theory. Existence of such solutions supports the compactifications of F-theory.
F-theory is one of the frameworks where all the Yukawa couplings of grand unified theories are generated and their computation is possible. The Yukawa couplings of charged matter multiplets are supposed to be generated around codimension-3 singularity points of a base complex 3-fold, and that has been confirmed for the simplest type of codimension-3 singularities in recent studies. However, the geometry of F-theory compactifications is much more complicated. For a generic F-theory compactification, such issues as flux configuration around the codimension-3 singularities, field-theory formulati…
Critical values of Wilson lines and general background fields for toroidal compactifications of heterotic string theories are constructed systematically using Dynkin diagrams.
In this paper we discuss the singularities in the Yukawa and gauge couplings of N=1 compactifications of the SO(32) heterotic string in four space-time dimensions. Such singularities can arise from the strong coupling dynamics of a confined non-perturbative gauge group.
We provide a qualitative review of flux compactifications of string theory, focusing on broad physical implications and statistical methods of analysis.
We perform a systematic analysis of globally consistent D-brane quivers which realize the MSSM, and analyze them with respect to their Yukawa couplings. Often, desired couplings are perturbatively forbidden due to the presence of global U(1) symmetries. We investigate the conditions under which D-brane instantons will induce these missing couplings without generating other phenomenological drawbacks, such as R-parity violating couplings or a mu-term which is too large. Furthermore, we systematically analyze which quivers allow for a mechanism that can account for the small neutrino masses and …
In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general characterization of heterotic …
We investigate special compactifications of the heterotic string for which the space of half-BPS states is, in a natural way, a representation of various subgroups of the Conway group. These compactifications provide a useful framework for analyzing the action of some of the large symmetry groups appearing in discussions of Moonshine in the physics literature. We investigate toroidal compactifications of heterotic string with sixteen supersymmetries as well as asymmetric toroidal orbifolds with $N=2$ supersymmetry in four dimensions that arise as $K3 \times T^2$ compactifications. The latter C…
We give necessary conditions for the existence of perturbative heterotic and common sector type II string warped compactifications preserving four and eight supersymmetries to four spacetime dimensions, respectively. In particular, we find that the only compactifications of heterotic string with the spin connection embedded in the gauge connection and type II strings are those on Calabi-Yau manifolds with constant dilaton. We obtain similar results for compactifications to six and to two dimensions.
Using the field theoretical approach introduced by Cremades, Ibanez and Marchesano for describing open strings attached to D9 branes having different magnetizations, we give a procedure for determining the Kaehler metrics of those open strings in toroidal compactifications.
We give a short review of a large class of warped string geometries, obtained via F-theory compactified on Calabi-Yau fourfolds, that upon reduction to 5 dimensions give consistent supersymmetric realizations of the RS compactification scenario.
Replaced with revised (uncorrupted) version We present a general scheme for generating (2,2) symmetric fermionic string models and classify the models in D=8 and D=6 space time dimensions. We discuss the relationship to other compactification schemes.
We reconsider the issue of large-volume compactifications of the heterotic string in light of the recent discoveries about strongly-coupled string theories. Our conclusion remains firmly negative with respect to classical compactifications of the ten-dimensional field theory, albeit for a new reason: When the internal sixfold becomes large in heterotic units, the theory acquires an additional threshold at energies much less then the naive Kaluza-Klein scale. It is this additional threshold that imposes the ultimate limit on the compactification scale: Any compactification must have M_{Kaluza K…
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.
We study supersymmetry conditions for the heterotic pure superstring preserving and N=1 supersymmetry in four dimensions directly from the curved superspace defined by the Berkovits-Howe constraints.
We study supersymmetric compactifications of type II strings on eightfolds to two dimensions. It is demonstrated that the type IIB string on an eightfold is free of gravitational anomalies. T-duality requires that this theory when further compactified on a circle must have a vacuum momentum; this is explicitly shown to be present and to have the right value. A subtlety in the relation of IIB compactifications and M-Theory orientifolds to two dimensions is pointed out.
We study the vacuum structure of compactifications of type II string theories on orientifolds with SU(3)xSU(3) structure. We argue that generalised geometry enables us to treat these non-geometric compactifications using a supergravity analysis in a way very similar to geometric compactifications. We find supersymmetric Minkowski vacua with all the moduli stabilised at weak string coupling and all the tadpole conditions satisfied. Generically the value of the moduli fields in the vacuum is parametrically controlled and can be taken to arbitrarily large values.
A general formula is obtained for Yukawa couplings in compactification on \LGO{s} and corresponding \CY\ spaces. Up to the kinetic term normalizations, this equates the classical Koszul ring structure with the \LGO\ chiral ring structure and the true super\CFT\ ring structure.
We study compactifications of the $N=2$ 6D tensionless string on various complex two-folds down to two-dimensions. In the IR limit they become non-trivial conformal field theories in 2D. Using results of Vafa and Witten on the partition functions of twisted Super-Yang-Mills theories, we can study the resulting CFT. We also discuss the contribution of instantons made by wrapping strings on 2-cycles of the complex two-fold.
We propose a scenario for dynamical supersymmetry breaking in string compactifications based on geometric engineering of quiver gauge theories. In particular we show that the runaway behavior of fractional branes at del Pezzo singularities can be stabilized by a flux superpotential in compact models. Our construction relies on homological mirror symmetry for orientifolds.
We study the hypermultiplet moduli space of the type II string compactified on a Calabi-Yau space $\bX$. We do this by using IIA/IIB duality in a compactification of the same theory on $\bX\times \bS ^1$ and by using recent results on three dimensional field theory.
We investigate the lifting of half-maximal four-dimensional gauged supergravities to compactifications of string theory. It is shown that a class of such supergravities can arise from compactifications of IIA string theory on manifolds of SU(2)-structure which may be thought of as K3 fibrations over T^2. Examples of these SU(2)-structure backgrounds, as smooth K3 bundles and as compactifications with H-flux, are given and we also find evidence for a class of non-geometric, Mirror-fold backgrounds. By applying the duality between IIA string theory on K3 and Heterotic string theory on T^4 fibrew…
We search for $N=2$, $d=4$ theories which can be realized both as heterotic string compactifications on $K_{3}\times T^{2}$ and as type II string compactifications on Calabi-Yau threefolds. In such cases, the exact non-perturbative superpotential of one string theory is given in terms of tree level computations in the other string theory. In particular we find concrete examples which provide the stringy realization of the results of Seiberg and Witten on N=2 Yang-Mills theory, corrected by gravitational/stringy effects. We also discuss some examples which shed light on how the moduli spaces of…
We compute the two-loop contributions to the free energy in the null compactification of perturbative string theory at finite temperature. The cases of bosonic, Type II and heterotic strings are all treated. The calculation exploits an explicit reductive parametrization of the moduli space of infinite-momentum frame string worldsheets in terms of branched cover instantons. Various arithmetic and physical properties of the instanton sums are described. Applications to symmetric product orbifold conformal field theories and to the matrix string theory conjecture are also briefly discussed.
We argue that the higher space-time derivative terms that occur in closed bosonic string field theory are reminiscent of those found in the Landau theory of liquid-crystal transitions. We examine the lowest level approximation of the closed bosonic string and find evidence for the existence of a new vacuum that spontaneously breaks Lorentz and translation symmetries. This effect can be interpreted as a spontaneous compactification of the theory.
In this short review, we outline three sets of developments in understanding heterotic string compactifications. First, we outline recent progress in heterotic analogues of quantum cohomology computations. Second, we discuss a potential swampland issue in heterotic strings, and new heterotic string constructions that can be used to fill in the naively missing theories. Third, we discuss recent developments in string compactifications on stacks and their applications, concluding with an outline of work-in-progress on heterotic string compactifications on gerbes.
We develop techniques, based on differential geometry, to compute holomorphic Yukawa couplings for heterotic line bundle models on Calabi-Yau manifolds defined as complete intersections in projective spaces. It is shown explicitly how these techniques relate to algebraic methods for computing holomorphic Yukawa couplings. We apply our methods to various examples and evaluate the holomorphic Yukawa couplings explicitly as functions of the complex structure moduli. It is shown that the rank of the Yukawa matrix can decrease at specific loci in complex structure moduli space. In particular, we co…
We review a duality procedure that relates standard matter-coupled N=1 supergravity to dual formulations in which auxiliary fields are replaced by field-strengths of gauge three-forms. As examples, we consider the dualization of the rigid Polonyi model and of effective field theories associated with Type IIA string compactifications with fluxes in supergravity.
We review the case for light U(1) gauge bosons in the hidden-sector of heterotic and type II string compactifications, present estimates of the size of their kinetic mixing with the visible-sector hypercharge U(1), and discuss their possibly very interesting phenomenological consequences in particle physics and cosmology.
The Narain lattice construction of string compactifications is generalized to include spontaneously broken supersymmetry. Consistency conditions from modular invariance and Lorentz symmetry are solved in full generality. This framework incorporates models where supersymmetry breaking is inversely proportional to the radii of compact dimensions. The enhanced lattice description, however, might allow for models with a different geometrical or even non-geometrical interpretation.
We present a pedagogical overview of flux compactifications in string theory, from the basic ideas to the most recent developments. We concentrate on closed string fluxes in type II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyze the resulting four-dimensional effective theories, as well as some of its perturbative and non-perturbative corrections, focusing on moduli stabilization. Finally, we briefly review statistical st…
For the gauge couplings, which arise after toroidal compactification of six-dimensional heterotic N=1 string theories from the T^2 torus, we calculate their one-loop corrections. This is performed by considering string amplitudes involving two gauge fields and moduli fields. We compare our results with the equations following from N=2 special geometry and the underlying prepotential of the theory. Moreover we find relations between derivatives of the N=2, d=4 prepotential and world-sheet tau-integrals which appear in various string amplitudes of any T^2-compactification.
We discuss four dimensional effective actions of string theory flux compactifications. These effective actions describe four dimensional gravity coupled to overall Kahler modulus of the compactification manifold. We demonstrate the agreement between ten dimensional equations of motion of supergravity with localized branes, and equations of motion derived from the effective action. The agreement is lost however if one evaluates the full effective action on the equations of motion for a subset of the supergravity modes, provided these modes depend on-shell on the Kahler modulus.
We study the supersymmetric index of four dimensional theories obtained by compactifications of the six dimensional E string theory on a Riemann surface. In particular we derive the difference operator introducing certain class of surface defects to the index computation. The difference operator turns out to be, up to a constant shift, an analytic difference operator discussed by van Diejen.
A general method of computing string corrections to the Kähler metric and Yukawa couplings is developed at the one-loop level for a general compactification of the heterotic superstring theory. It also provides a direct determination of the so-called Green-Schwarz term. The matter metric has an infrared divergent part which reproduces the field-theoretical anomalous dimensions, and a moduli-dependent part which gives rise to threshold corrections in the physical Yukawa couplings. Explicit expressions are derived for symmetric orbifold compactifications.
In this paper, we continue the analysis of heterotic string compactifications on half-flat mirror manifolds by including the 10-dimensional gauge fields. It is argued, that the heterotic Bianchi identity is solved by a variant of the standard embedding. Then, the resulting gauge group in four dimensions is still E6 despite the fact that the Levi-Civita connection has SO(6) holonomy. We derive the associated four-dimensional effective theories including matter field terms for such compactifications. The results are also extended to more general manifolds with SU(3) structure.
We study compactifications of type IIB supergravity on Calabi-Yau threefolds. The resulting low energy effective Lagrangian is displayed in the large volume limit and its symmetry properties - with specific emphasis on the SL(2,Z) - are discussed. The explicit map to type IIA string theory compactified on a mirror Calabi-Yau is derived. We argue that strong coupling effects on the worldsheet break the SL(2,Z).
Using heterotic/type II string duality, we obtain exact nonperturbative results for the point particle limit (alpha' -> 0) of some particular four dimensional, N=2 supersymmetric compactifications of heterotic strings. This allows us to recover recent exact nonperturbative results on N=2 gauge theory directly from tree-level type II string theory, which provides a highly non-trivial, quantitative check on the proposed string duality. We also investigate to what extent the relevant singular limits of Calabi-Yau manifolds are related to the Riemann surfaces that underlie rigid N=2 gauge theory.
We provide an overview of recent work which aims to understand patterns of vanishing Yukawa couplings that arise in models of particle physics derived from string theory. These patterns are seemingly linked to a plethora of different geometrical structures and our understanding of the subject has yet to be consolidated in a unified framework. This short note is based upon a talk that was given by one of the authors at the Nankai Symposium on Mathematical Dialogues. Therefore it is aimed at a mathematical audience of mixed academic background.
Basics of some topics on perturbative and non-perturbative string theory are reviewed. After a mathematical survey of the Standard Model of particle physics and GUTs, the bosonic string kinematics for the free case and with interaction is described. The effective action of the bosonic string and the spectrum is also discussed. Five perturbative superstring theories and their spectra is briefly outlined. Calabi-Yau three-fold compactifications of heterotic strings and their relation to some four-dimensional physics are given. T-duality in closed and open strings are surveyed. D-brane definition…
Two classes of stringy instanton effects, stronger than standard field theory instantons, are identified in the heterotic string theory. These contributions are established using type IIA/heterotic and type I/heterotic dualities. They provide examples for the heterotic case of the effects predicted by Shenker based on the large-order behavior of perturbation theory. The corrections vanish as the radius of the compactification goes to infinity. For appropriate amplitudes, they are computable worldsheet or worldline instanton effects on the dual side. Some potential applications are discussed.
In this work we use the Matrix Model of Strings in order to extract some non-perturbative information on how the Hagedorn critical temperature arises from eleven-dimensional physics. We study the thermal behavior of M and Matrix theories on the compactification backgrounds that correspond to string models. We obtain some information that allows us to state that the Hagedorn temperature is not unique for all Matrix String models and we are also able to sketch how the $S$-duality transformation works in this framework.
We investigate the low--energy properties of a Z_12 orbifold with continuous Wilson line moduli. They give rise to a (0,2) superstring compactification. Their Kaehler potentials and Yukawa couplings are calculated. We study the discrete symmetries of the model and their implications on the threshold corrections to the gauge couplings as well as for string unification.
A new framework is found for the compactification of supersymmetric string theory. It is shown that the massless spectra of Calabi--Yau manifolds of complex dimension $D_{crit}$ can be derived from noncritical manifolds of complex dimension $2k + D_{crit}$, $k\geq 1$. These higher dimensional manifolds are spaces whose nonzero Ricci curvature is quantized in a particular way. This class is more general than that of Calabi--Yau manifolds because it contains spaces which correspond to critical string vacua with no Kähler deformations, i.e. no antigenerations, thus providing mirrors of rigid Cala…
We present the Higgs mechanism in (0,2) compactifications. The existence of a vector bundle data duality (VBDD) in $(0,2)$ compactifications which is present at the Landau-Ginzburg point allows us to connect in a smooth manner theories with different gauge groups with the same base manifold and same number of effective generations. As we move along the Kahler moduli space of the theories with $E_6$ gauge group, some of the gauginos pick up masses and break the gauge group to $SO(10)$ or $SU(5)$.
Motivated by recent developments in the computation of periods for string compactifications with $c=9$, we develop a complementary method which also produces a convenient basis for related calculations. The models are realized as Calabi--Yau hypersurfaces in weighted projective spaces of dimension four or as Landau-Ginzburg vacua. The calculation reproduces known results and also allows a treatment of Landau--Ginzburg orbifolds with more than five fields.
We describe a new class of supersymmetric string compactifications to 4d Minkowski space. These solutions involve type II strings propagating on (orientifolds of) non Calabi-Yau spaces in the presence of background NS and RR fluxes. The simplest examples have descriptions as cosets, generalizing the three-dimensional nilmanifold. They can also be thought of as twisted tori. We derive a formula for the (super)potential governing the light fields, which is generated by the fluxes and certain ``twists'' in the geometry. Detailed consideration of an example also gives strong evidence that in some …
These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string theory compactifications. The emphasis is wholly on the geometry side, not on the physics. The treated topics include toroidal orbifolds, methods of toric geometry, desinglularization of toroidal orbifolds and their orientifold quotients.
F-theory is one of frameworks in string theory where supersymmetric grand unification is accommodated, and all the Yukawa couplings and Majorana masses of right-handed neutrinos are generated. Yukawa couplings of charged fermions are generated at codimension-3 singularities, and a contribution from a given singularity point is known to be approximately rank 1. Thus, the approximate rank of Yukawa matrices in low-energy effective theory of generic F-theory compactifications are minimum of either the number of generations N_gen = 3 or the number of singularity points of certain types. If there i…
In this thesis we study string compactifications on manifolds equipped with a $G$-structure, placing a special emphasis on the interplay between geometry and physics. We follow two complementary approaches. In the first part of the thesis we adopt a sigma model perspective and focus on the worldsheet superconformal field theory. We consider compactifications on 7-dimensional Extra Twisted Connected Sum (ETCS) G$_2$ manifolds as well as 8-dimensional Generalized Connected Sum (GCS) Spin(7) manifolds. We find that the geometric construction is reproduced in the worldsheet algebra via a diamond o…
We investigate quantum corrections to the classical four-dimensional low-energy effective action of type II string theory compactified on SU(3) structure geometries. Various methods previously developed for Calabi-Yau compactifications are adopted to determine - under some simple assumptions about the low-energy degrees of freedom - the leading perturbative corrections to the moduli space metrics in both alpha' and the string coupling constant. We find - in complete analogy to the Calabi-Yau case - that the corrections take a universal form dependent only on the Euler characteristic of the six…
