We study the sub-structure of the heterotic Kahler moduli space due to the presence of non-Abelian internal gauge fields from the perspective of the four-dimensional effective theory. Internal gauge fields can be supersymmetric in some regions of the Kahler moduli space but break supersymmetry in others. In the context of the four-dimensional theory, we investigate what happens when the Kahler moduli are changed from the supersymmetric to the non-supersymmetric region. Our results provide a low-energy description of supersymmetry breaking by internal gauge fields as well as a physical picture for the mathematical notion of bundle stability. Specifically, we find that at the transition between the two regions an additional anomalous U(1) symmetry appears under which some of the states in the low-energy theory acquire charges. We compute the associated D-term contribution to the four-dimensional potential which contains a Kahler-moduli dependent Fayet-Iliopoulos term and contributions from the charged states. We show that this D-term correctly reproduces the expected physics. Several mathematical conclusions concerning vector bundle stability are drawn from our arguments. We also discuss possible physical applications of our results to heterotic model building and moduli stabilization.
We consider heterotic string solutions based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold, preserving two supercharges. The constraints on the internal manifolds with SU(3) structure are derived. They are found to be generalized half-flat manifolds with a particular pattern of torsion classes and they include half-flat manifolds and Strominger's complex non-Kahler manifolds as special cases. We also verify that previous heterotic compactifications on half-flat mirror manifolds are based on this class of solutions.
We calculate partition functions for supersymmetric heterotic theories on Melvin background with Wilson line. These functions coincide with partition functions for some of non-supersymmetric heterotic theories in appropriate limits. This suggests that non-supersymmetric heterotic theories are equivalent to supersymmetric theories on supersymmetry-breaking backgrounds, just as in the case of recently conjectured IIA-0A duality.
We give an interpretation of the non-supersymmetric heterotic theories as the supersymmetric heterotic theories on Melvin background with Wilson lines. The coincidence of the partition functions are shown for most of the non-supersymmetric theories. We also discuss the tachyonic instability in terms of the non-trivial background fields. The behavior in the strong coupling region is investigated by using the duality relations of the supersymmetric theories.
It has been conjectured that Little String Theories in six dimensions are holographic to critical string theory in a linear dilaton background. We test this conjecture for theories arising on the worldvolume of heterotic fivebranes. We compute the spectrum of chiral primaries in these theories and compare with results following from Type I-heterotic duality and the AdS/CFT correspondence. We also construct holographic duals for heterotic fivebranes near orbifold singularities. Finally we find several new Little String Theories which have Spin(32)/Z_2 or E_8 \times E_8 global symmetry but do no…
We explicitly describe, in the language of four-dimensional N=1 supersymmetric field theory, what happens when the moduli of a heterotic Calabi-Yau compactification change so as to make the internal non-Abelian gauge fields non-supersymmetric. At the edge of the region in Kahler moduli space where supersymmetry can be preserved, an additional anomalous U(1) gauge symmetry appears in the four-dimensional theory. The D-term contribution to the scalar potential associated to this U(1) attempts to force the system back into a supersymmetric configuration and provides a consistent low-energy descri…
We examine compactifications of heterotic string theory on manifolds with SU(3) structure. In particular, we study N = 1/2 domain wall solutions which correspond to the perturbative vacua of the 4D, N =1 supersymmetric theories associated to these compactifications. We extend work which has appeared previously in the literature in two important regards. Firstly, we include two additional fluxes which have been, heretofore, omitted in the general analysis of this situation. This allows for solutions with more general torsion classes than have previously been found. Secondly, we provide explicit…
Anomalous U(1) gauge symmetries can appear both in heterotic and type I string theories. In the heterotic case we find a single anomalous U(1), while in open string theories several such symmetries can appear. Nonetheless, there is a conjectured duality symmetry that might connect these two theories. We review the properties of anomalous gauge symmetries in various string theories as well as the status of this heterotic-type I/II duality. We also comment on the possible phenomenological applications of anomalous gauge symmetries in string theory.
In heterotic string theories consistency requires the introduction of a non-trivial vector bundle. This bundle breaks the original ten-dimensional gauge groups $\text{E}_8\times\text{E}_8$ or $\text{SO}(32)$ for the supersymmetric heterotic string theories and $\text{SO}(16) \times \text{SO}(16)$ for the non-supersymmetric tachyon-free theory to smaller subgroups. A vast number of MSSM-like models have been constructed up to now, most of which describe the vector bundle as a sum of line bundles. However, there are several different ways of describing these line bundles and their embedding in t…
Using finite abelian automorphism groups of $K3$ we construct orbifold candidates for Type IIA-heterotic dual pairs with maximal supersymmetry in six and lower dimensions. On the heterotic side, these results extend the series of known reduced rank theories with maximal supersymmetry. The corresponding Type IIA theories generalize the Schwarz and Sen proposal for the dual of the simplest of the reduced rank theories constructed as a novel Type IIA $\IZ_2$ orbifold.
We study heterotic string compactifications on nearly Kähler homogeneous spaces, including the gauge field effects which arise at order α'. Using Abelian gauge fields, we are able to solve the Bianchi identity and supersymmetry conditions to this order. The four-dimensional external space-time consists of a domain wall solution with moduli fields varying along the transverse direction. We find that the inclusion of α' corrections improves the moduli stabilization features of this solution. In this case, one of the dilaton and the volume modulus asymptotes to a constant value away from the doma…
A holomorphic vector bundle on a Calabi-Yau threefold, X, with h^{1,1}(X)>1 can have regions of its Kahler cone where it is slope-stable, that is, where the four-dimensional theory is N=1 supersymmetric, bounded by "walls of stability". On these walls the bundle becomes poly-stable, decomposing into a direct sum, and the low energy gauge group is enhanced by at least one anomalous U(1) gauge factor. In this paper, we show that these additional symmetries can strongly constrain the superpotential in the stable region, leading to non-trivial textures of Yukawa interactions and restrictions on al…
We study pair production of small BPS BH's in heterotic strings compactified on tori and in the FHSV model. After recalling the identification of small BH's in the perturbative BPS spectrum, we compute the tree-level amplitudes for processes initiated by massless vector bosons or gravitons. We then analyze the resulting cross sections in terms of energy and angular distributions. Finally, we briefly comment on scenari with large extra dimensions and on generalizations of our results to non-BPS, non-extremal and rotating BH's.
We study properties of 1/2 BPS Higher Spin states in heterotic compactifications with extended supersymmetry. We also analyze non BPS Higher Spin states and give explicit expressions for physical vertex operators of the first two massive levels. We then study on-shell tri-linear couplings of these Higher Spin states and confirm that BPS states with arbitrary spin cannot decay into lower spin states in perturbation theory. Finally, we consider scattering of vector bosons off higher spin BPS states and extract form factors and polarization effects in various limits.
We examine how to construct explicit heterotic string models dual to F-theory in eight dimensions. In doing so we learn about where the moduli spaces of the two theories overlap, and how non-perturbative features leave traces on a purely perturbative level. We also propose a modified T-duality linking M and F-theories based on non-perturbative structures of string theories in eight dimensions.
We use Pauli-Villars regularization to evaluate the conformal and chiral anomalies in the effective field theories from Z3 and Z7 compactifications of the heterotic string without Wilson lines. We show that parameters for Pauli-Villars chiral multiplets can be chosen in such a way that the anomaly is universal in the sense that its coefficient depends only on a single holomorphic function of the three diagonal moduli. It is therefore possible to cancel the anomaly by a generalization of the four-dimensional Green-Schwarz mechanism. In particular we are able to reproduce the results of a string…
We discuss solutions of the heterotic string theory which are analogous to bosonic and superstring backgrounds related to coset conformal field theories. A class of exact `left-right symmetric' solutions is obtained by supplementing the metric, antisymmetric tensor and dilaton of the superstring solutions by the gauge field background equal to the generalised Lorentz connection with torsion. As in the superstring case, these backgrounds are $\a'$-independent, i.e. have a `semiclassical' form. The corresponding heterotic string sigma model is obtained from the combination of the (1,0) supersymm…
The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T^6 is known to correctly produce the degeneracy of dyons of torsion 1, i.e. dyons for which gcd(Q\wedge P)=1. We propose a generalization of this formula for dyons of arbitrary torsion. Our proposal satisfies the constraints coming from S-duality invariance, wall crossing formula, black hole entropy and the gauge theory limit. Furthermore using our proposal we derive a general wall crossing formula that is valid even when both the decay products are non-primitive…
We investigate the coset structures which appear in the dimensional reduction of supergravity theories. Especially we investigate how to recognize the global symmetry groups if the coset is non-split. As an example we apply our analysis to the theories emanating from the dimensional reduction of Heterotic supergravity.
We construct heterotic string theories on spacetimes of the form R^{d-1,1} times N=2 linear dilaton, where d=6,4,2,0. There are two lines of supersymmetric theories descending from the two supersymmetric ten-dimensional heterotic theories. These have gauge groups which are lower rank subgroups of E_{8} times E_{8} and SO(32). On turning on a (2,2) deformation which makes the two dimensional part a smooth SL_{2}(R)/U(1) supercoset, the gauge groups get broken further. In the deformed theories, there are non-trivial moduli which are charged under the surviving gauge group in the case of d=6. We …
The effective action for five-dimensional heterotic M-theory in the presence of five-branes is systematically derived from Horava-Witten theory coupled to an M5-brane world-volume theory. This leads to a five-dimensional N=1 gauged supergravity theory on S^1/Z_2 coupled to four-dimensional N=1 theories residing on the two orbifold fixed planes and an additional bulk three-brane. We analyse the properties of this action, particularly the four-dimensional effective theory associated with the domain-wall vacuum state. The moduli Kahler potential and the gauge-kinetic functions are determined alon…
The heterotic and type II superstring actions are identified in different anomaly-free decompositions of a single topological sigma-model action depending on bosonic and fermionic coordinates, $X^μ$ and $\r^A$ respectively, and of their topological ghosts. This model results from gauge-fixing the topological gauge symmetry $δX^μ= ε^μ(z,\bar z)$ ($μ=1,2,\dots, 10$) and $δ\r^α= ε^α(z,\bar z)$. ($α=1,2.\dots, 16$). From another viewpoint the heterotic and type II superstring actions emerge as two different gauge-fixings of the same closed two-form. Comments are also made concerning the possibilit…
We study a class of 6d $\mathcal{N}=(1,0)$ non-geometric vacua of the $\text{Spin}(32)/\mathbb Z_2$ heterotic string which can be understood as fibrations of genus-two curves over a complex one-dimensional base. The 6d $\mathcal{N}=(1,0)$ theories living on the defects that arise when the genus-two fiber degenerates at a point of the base are analyzed by dualizing to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We consider all possible degenerations of genus-two curves and systematically attempt to resolve the singularities of the dual threefolds. As in the analogous non-…
We address the construction of non-supersymmetric vacua in heterotic compactifications with intrinsic torsion and background fluxes. In particular, we implement the approach of domain-wall supersymmetry breaking (DWSB) previously developed in the context of type II flux compactifications. This approach is based on considering backgrounds where probe NS5-branes wrapping internal three-cycles and showing up as four-dimensional domain-walls do not develop a BPS bound, while all the other BPS bounds characterizing the N=1 supersymmetric compactifications are preserved at tree-level. Via a scalar p…
We consider type II and heterotic string compactifications on an isolated singularity in the noncompact Gepner model approach. The conifold-type ADE noncompact Calabi-Yau threefolds, as well as the ALE twofolds, are modeled by a tensor product of the SL(2,R)/U(1) Kazama-Suzuki model and an N=2 minimal model. Based on the string partition functions on these internal Calabi-Yaus previously obtained by Eguchi and Sugawara, we construct new modular invariant, space-time supersymmetric partition functions for both type II and heterotic string theories, where the GSO projection is performed before t…
We review here some aspects of our recent works about the geometric engineering of heterotic little string theories using F-theory. Building on the seminal work by Aspinwall and Morrison as well as Intrilligator and Blum, we solve some longstanding open questions thanks to recent progress in our understanding of 6D (1,0) theories and their generalized symmetries. On the geometry side, these systems correspond to non-compact elliptically fibered Calabi-Yau varieties that must admit the structure of an elliptic K3 fibration. From fiberwise F-theory/Heterotic duality the K3 plays a central role -…
The compactification from the 11-dimensional Horava-Witten orbifold to 5-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is reviewed, as is the specific $SU(4)$ vector bundle leading to the "heterotic standard model" in the observable sector. A generic formalism for a consistent hidden sector gauge bundle, within the context of strongly coupled heterotic M-theory, is presented. Anomaly cancellation and the associated bulk space 5-branes are discussed in this context. The further compactification to a 4-dimensional effective field theory on a linearized BPS double domain wall is…
Thus far in the search for, and classification of, `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ the attention has been focused on the {\it symmetric} case where the holomorphic and anti-holomorphic sectors, and hence the characters $\c_L$ and $\c_R$, are associated with the same Kac-Moody algebras $\g_L=\g_R$ and levels $k_L=k_R$. In this paper we consider the more general possibility where $(\g_L,k_L)$ may not equal $(\g_R,k_R)$. We discuss which choices of algebras and levels may correspond to well-defined conformal field theories, we find the `smallest' such {\…
We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional $\mathcal{N}=(0,1)$ supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the $B$-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQF…
We study heterotic vacua with four supercharges in three and four space-time dimensions and their duals obtained as M/F-theory compactified on Calabi-Yau fourfolds. We focus on their respective moduli spaces and derive the Kahler potential for heterotic vacua obtained as circle compactifications of four-dimensional N=1 heterotic theories. The Kahler potential of the dual theory is computed by compactifying 11-dimensional supergravity on Calabi-Yau fourfolds. The duality between these theories is checked for K3-fibred fourfolds and an appropriate F-theory limit is discussed.
We analyze transitions between heterotic vacua with distinct gauge bundles using two complementary methods - the effective four-dimensional field theory and the corresponding geometry. From the viewpoint of effective field theory, such transitions occur between flat directions of the potential energy associated with heterotic stability walls. Geometrically, this branch structure corresponds to smooth deformations of the gauge bundle coupled to the chamber structure of Kähler moduli space. We demonstrate how such transitions can change important properties of the effective theory, including the…
We suggest that compactifications on Anti-de-Sitter (AdS) spaces of type IIA, IIB, heterotic strings and eleven dimensional vacuua of M-theory are related by a combination of $T$ and strong/weak dualities. Maldacena conjecture relates then all these vacuua to a conformal theory on the boundaries. Furthermore acting with discrete groups on part of the internal spaces of these theories will lead to further dual theories with less or no supersymmetry.
In these lectures, we present cosmological vacuum solutions of Horava-Witten theory and discuss their physical properties. We begin by deriving the five-dimensional effective action of strongly coupled heterotic string theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S1/Z2. The effective theory is shown to be a gauged version of five-dimensional N=1 supergravity coupled, for simplicity, to the universal hypermultiplet and four-dimensional boundary theories with gauge and universal gauge matter fields. The static vacuum of the theory is a pair of BPS three-brane doma…
We discuss heterotic string theories in two dimensions with gauge groups Spin(24) and Spin(8) x E_8. After compactification the theories exhibit a rich spectrum of states with both winding and momentum. At special points some of these stringy states become massless, leading to new first order phase transitions. For example, the thermal theories exhibit standard thermodynamics below the phase transition, but novel and peculiar behavior above it. In particular, when the radius of the Euclidean circle is smaller than the phase transition point the torus partition function is not given by the ther…
According to a recent conjecture, the moduli space of the heterotic conformal field theory on a $G\subset$ ADE singularity of an ALE space is equivalent to the moduli space of a pure $\cx N=4$ supersymmetric three-dimensional gauge theory with gauge group G. We establish this relation using geometric engineering of heterotic strings and generalize it to theories with non-trivial matter content. A similar equivalence is found between the moduli of heterotic CFT on isolated Calabi--Yau 3-fold singularities and two-dimensional Kazama-Suzuki coset theories.
Global geometry of $K3$-fibration Calabi-Yau threefolds, with Hodge number $h_{2,1}=r+1$, is used to define $N=4$ softly broken $SU(r+1)$ gauge theories, with the bare coupling constant given by the dual heterotic dilaton, and the mass of the adjoint hypermultiplet given by the heterotic string tension. The $U(r+1)$ Donagi-Witten integrable model is also derived from the $K3$-fibration structure, with the extra $U(1)$ associated to the heterotic dilaton. The case of $SU(2)$ gauge group is analyzed in detail. String physics beyond the heterotic point particle limit is partially described by the…
We study soft SUSY breaking terms in heterotic M-theory. We show that both weakly and strongly coupled heterotic string models lead to the same relations of soft SUSY breaking terms, $A=-M$ and $m^2 = M^2/3$, up to $O((αT/S)^2)$.
D=11 supergravity possesses D=5 Calabi-Yau compactified solutions that may be identified with the vacua of the Horava-Witten orbifold construction for M--theory/heterotic duality. The simplest of these solutions naturally involves two 3-brane domain walls, which may be identified with the orbifold boundary planes; this solution also possesses an unbroken $\Z_2$ symmetry. Consideration of nearby excited solutions, truncated to the zero-mode and $\Z_2$ invariant sector, yields an effective D=4 heterotic theory displaying chirality and N=1, D=4 supersymmetry.
We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analyzing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactific…
We establish a relation, conjectured recently by E. Witten, between the hypermultiplet moduli space in compactifications of the heterotic string on an A-D-E singularities, and the moduli spaces of three dimensional pure gauge theories with the corresponding A-D-E gauge groups. It is possible to add a bounded number of heterotic fivebranes sitting in the singularity, while (in leading order in $α'$) keeping the heterotic string perturbative. The corresponding hypermultiplet moduli space is given by the moduli space of a three dimensional gauge theory with matter.
We study the problem of string propagation in a general instanton background for the case of the complete heterotic superstring. We define the concept of generalized HyperK\" ahler manifolds and we relate it to (4,4) superconformal theories. We propose a generalized h-map construction that predicts a universal SU(6) symmetry for the modes of the string excitations moving in an instanton background. We also discuss the role of abstract $N$=4 moduli and, applying it to the particular limit case of the solvable SU(2) X R instanton found by Callan et al. we show that it admits deformations and cor…
We analyse the vector bundle moduli arising from generic heterotic compactifications from the point of view of quiver representations. Phenomena such as stability walls, crossing between chambers of supersymmetry, splitting of non-Abelian bundles and dynamic generation of D-terms are succinctly encoded into finite quivers. By studying the Poincaré polynomial of the quiver moduli space using the Reineke formula, we can learn about such useful concepts as Donaldson-Thomas invariants, instanton transitions and supersymmetry breaking.
Progress towards the classification of the meromorphic $c=24$ conformal field theories is reported. It is shown if such a theory has any spin-1 currents, it is either the Leech lattice CFT, or it can be written as a tensor product of Kac-Moody algebras with total central charge 24. The total number of combinations of Kac-Moody algebras for which meromorphic $c=24$ theories may exist is 221. The next step towards classification is to obtain all modular invariant combinations of Kac-Moody characters. The presently available results are sufficient to obtain a complete list of all ten-dimensional …
In this paper, we apply the methods developed in recent work for constructing A-twisted (2,2) Landau-Ginzburg models to analogous (0,2) models. In particular, we study (0,2) Landau-Ginzburg models on topologically non-trivial spaces away from large-radius limits, where one expects to find correlation function contributions akin to (2,2) curve corrections. Such heterotic theories admit A- and B-model twists, and exhibit a duality that simultaneously exchanges the twists and dualizes the gauge bundle. We explore how this duality operates in heterotic Landau-Ginzburg models, as well as other prop…
Following a recent conjecture by Lapan, Simons and Strominger, we revisit and discuss an intrinsically heterotic class of conformal field theories, emphasizing their Lagrangian construction as asymmetrically gauged WZW models, which may be useful in several applications to the study of supersymmetric strings and black holes in heterotic and type II string theory compactified on T^6 and K3 X T^2 respectively. In these cases, the leading supergravity geometry is singular, but higher order corrections remove this singularity in a way that is consistent with, for example the non-zero entropy for t…
The scattering amplitudes of gauge bosons in heterotic and open superstring theories are related by the single-valued projection which yields heterotic amplitudes by selecting a subset of multiple zeta value coefficients in the $α'$ (string tension parameter) expansion of open string amplitudes. In the present work, we argue that this relation holds also at the level of low-energy expansions (or individual Feynman diagrams) of the respective effective actions, by investigating the beta functions of two-dimensional sigma models describing world-sheets of open and heterotic strings. We analyze t…
Compactifications of the heterotic string with NS flux normally require non Calabi-Yau internal spaces which are complex but no longer Kähler. We point out that this conclusion rests on the assumption of a maximally symmetric four-dimensional space-time and can be avoided if this assumption is relaxed. Specifically, it is shown that an internal Calabi-Yau manifold is consistent with the presence of NS flux provided four-dimensional space-time is taken to be a domain wall. These Calabi-Yau domain wall solutions can still be associated with a covariant four-dimensional N=1 supergravity. In this …
A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the …
We study ${\cal N}=2$ compactifications of heterotic string theory on the CHL orbifold $(K3\times T^2)/\mathbb{Z}_N$ with $N= 2, 3, 5, 7$. $\mathbb{Z}_N$ acts as an involution on $K3$ together with a shift of $1/N$ along one of the circles of $T^2$. These compactifications generalize the example of the heterotic string on $K3\times T^2$ studied in the context of dualities in ${\cal N}=2$ string theories. We evaluate the new supersymmetric index for these theories and show that their expansion can written in terms of the McKay-Thompson series associated with the $\mathbb{Z}_N$ involution embedd…
It is shown that $D=4$ $N=1$ SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on compact Kähler manifold. The conditions of cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a non-trivial set of physical operators defined as classes of the cohomology of this BRST \op . We prove that the physical correlators are independent on external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical \op s tur…
Gravity duals for little string theories --- which give rise to four-dimensional theories that undergo permanent confinement in the infrared --- have not been studied in great detail. We address this question in the framework of heterotic SO(32) and E_8 x E_8 string theory, constructing these backgrounds by wrapping heterotic five-branes on calibrated two-cycles of non-Kahler resolved conifolds. Related to deformations of the underlying little string theories, we find numerous analytic solutions preserving N = 1 supersymmetry in four-dimensions. These theories all have non-abelian global symme…
New heterotic string theories in four dimensions are constructed by tensoring a nonstandard SCFT along with some minimal SCFT's. All such theories are identified and their particle generation number is found. We prove that from the infinite number of new heterotic string theories only the {6} theory predicts three generations as seen in nature which makes it an interesting candidate for further study.
We establish a symmetry map which relates two low-energy heterotic string theories with different numbers of the Abelian gauge fields compactified from the diverse to three dimensions on a torus. We discuss two applications of the established symmetry: a generation of the heterotic string theory solutions from the stationary Einstein-Maxwell fields and one non-trivial submersion of the heterotic string theory into the bosonic one.
We briefly review attempts to construct string theories that yield the standard model, concentrating on models with a geometric interpretation. Calabi-Yau compactifications are discussed in the context of both the weakly coupled heterotic string and the strongly coupled version, heterotic M-theory. Similarly, we consider orbifold compactifications of weakly coupled heterotic theory and of Type II theories in the presence of D-branes. The latter allows a "bottom-up" construction that so far seems the most natural and direct route to the standard model.
