We describe all supersymmetric solutions of the heterotic string which preserve 8 supersymmetries and show that are distinguished by the holonomy, ${\rm hol}(\hat\nabla)$, of the connection, $\hat\nabla$, with skew-symmetric torsion. The ${\rm hol}(\hat\nabla) \subseteq SU(2)$ solutions are principal bundles over a 4-dimensional hyper-Kähler manifold equipped with a anti-self-dual connection and fibre group $G$ which has Lie algebra, ${\mathfrak Lie} (G)=\bR^{5,1}$, $\mathfrak{sl}(2,\bR)\oplus \mathfrak{su}(2)$ or $\mathfrak{cw}_6$. Some of the solutions have the interpretation as 5-branes wrapped on $G$ with transverse space any hyper-Kähler 4-dimensional manifold. We construct new solutions for ${\mathfrak Lie} (G)=\mathfrak{sl}(2,\bR)\oplus \mathfrak{su}(2)$ and show that are characterized by 3 integers and have continuous moduli. There is also a smooth family in this class with one asymptotic region and the dilaton is bounded everywhere on the spacetime. We also demonstrate that the worldvolume theory of the backgrounds with holonomy SU(2) can be understood in terms of gauged WZW models for which the gauge fields are composite. The ${\rm hol}(\hat\nabla) \subseteq\bR^8$ solutions are superpositions of fundamental strings and pp-waves in flat space, which may also include a null rotation. The ${\rm hol}(\hat\nabla)=\{1\}$ heterotic string backgrounds which preserve 8 supersymmetries are Lorentzian group manifolds.
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 investigate asymmetric orbifold models constructed from non-supersymmetric heterotic strings. We systematically classify the asymmetric orbifold models with standard embeddings and present a list of asymmetric orbifolds which are geometrically interpreted as toroidal compactifications of non-supersymmetric heterotic strings. By studying non-standard embedding models, we also construct examples of the {\em supersymmetric} asymmetric orbifold models based on non-supersymmetric heterotic strings.
Generalizing the work by Cherkis and Schwarz [1], we carry out the double dimensional reduction of supersymmetric M5 brane on K3 to obtain the supersymmetric action of heterotic string in 7-dimensional flat space-time. Motivated by this result, we propose the supersymmetric heterotic action in 10-dimensional flat space-time where the current algebra is realized in a novel way. We explicitly verify the kappa-symmetry of the proposed action.
The previously proposed generalized action principle approach to supersymmetric extended objects is considered in some details for the case of heterotic string in $D=3, 4, 6 ~and~ 10$ space--time dimensions. The proof of the 'off--shell' superdiffeomorphism invariance of the generalized action is presented. The doubly supersymmetric geometric approach to heterotic string is constructed on the basis of generalized action principle (instead of the geometrodynamic condition, used for this previously). It is demonstrated that $D=3$ heterotic string is described by $n=(1,0)$ supersymmetric generali…
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.
Exact black hole solutions of the five dimensional heterotic $S$-$T$-$U$ model including all perturbative quantum corrections and preserving $1/2$ of $N=2$ supersymmetry are studied. It is shown that the quantum corrections yield a bound on electric charges and harmonic functions of the solutions.
A supersymmetric solution to type II supergravity is constructed by superposing two hyperkählers with torsion metrics. The solution is given by a Käler with torsion metric with $SU(3)$ holonomy. The metric is embedded into a heterotic solution obeying the Strominger system, together with a Yang-Mills instanton obtained by the standard embedding. T dualities lead to an $SO(6)$ instanton describing a symmetry breaking from $E_8$ to $SO(10)$. The compactification by taking a periodic array yields a supersymmetric domain wall solution of heterotic supergravity.
We investigate $α'$-corrections of heterotic double field theory up to quadratic order in the language of supersymmetric O(D,D+dim G) gauged double field theory. After introducing double-vielbein formalism with a parametrization which reproduces heterotic supergravity, we show that supersymmetry for heterotic double field theory up to leading order $α'$-correction is obtained from supersymmetric gauged double field theory. We discuss the necessary modifications of the symmetries defined in supersymmetric gauged double field theory. Further, we construct supersymmetric completion at quadratic o…
We simplify the classification of supersymmetric solutions with compact holonomy of the Killing spinor equations of heterotic supergravity using the field equations and the additional assumption that the 3-form flux is closed. We determine all the fractions of supersymmetry that the solutions preserve and find that there is a restriction on the number of supersymmetries which depends on the isometry group of the background. We examine the geometry of spacetime in all cases. We find that the supersymmetric solutions of heterotic supergravity are associated with a large number of geometric struc…
We present the main features of the solution of the gravitino and dilatino Killing spinor equations derived in hep-th/0510176 and hep-th/0703143 which have led to the classification of geometric types of all type I backgrounds. We then apply these results to the supersymmetric backgrounds of the heterotic string. In particular, we solve the gaugino Killing spinor equation together with the other two Killing spinor equations of the theory. We also use our results to classify all supersymmetry conditions of ten-dimensional gauge theory.
We consider the low-energy effective field theory of heterotic string theory compactified on a seven-torus, and we construct electrically charged as well as more general solitonic solutions. These solutions preserve 1/2, 1/4 and 1/8 of N=8, D=3 supersymmetry and have Killing spinors which exist due to cancellation of holonomies. The associated space-time line elements do not exhibit the conical structure that often arises in 2+1 dimensional gravity theories.
Phenomenological explorations of heterotic strings have conventionally focused primarily on the E8xE8 theory. We consider smooth compactifications of all three ten-dimensional heterotic theories to exhibit the many similarities between the non-supersymmetric SO(16)xSO(16) theory and the related supersymmetric E8xE8 and SO(32) theories. In particular, we exploit these similarities to determine the bosonic and fermionic spectra of Calabi-Yau compactifications with line bundles of the non-supersymmetric string. We use elements of four-dimensional supersymmetric effective field theory to character…
An off-shell manifestly (8,0) worldsheet supersymmetric formulation of a multiplet describing physical chiral fermions is given. The multiplet can be used to complete the doubly supersymmetric (twistor-like) action for the heterotic string. Correction of technical errors.
We construct the supersymmetric completion of quartic $R+R^4$-actions in the ten-dimensional effective action of the heterotic string. Two invariants, of which the bosonic parts are known from one-loop string amplitude calculations, are obtained. One of these invariants can be generalized to an $R+F^2+F^4$-invariant for supersymmetric Yang-Mills theory coupled to supergravity. Supersymmetry requires the presence of $B\wedge R\wedge R\wedge R\wedge R$-terms, ($B\wedge F\wedge F\wedge F\wedge F$ for Yang-Mills) which correspond to counterterms in the Green-Schwarz anomaly cancellation. Within th…
We explicitly compute the anomalous magnetic moment at one loop level for an ``electron'' in a 4d heterotic string theory. The anomalous magnetic moment vanishes if the model is spacetime supersymmetric, as required by the supersymmetric sum rules.
We compute one-loop quantum corrections to gravitational couplings in the effective action of four-dimensional heterotic strings where supersymmetry is spontaneously broken by Scherk-Schwarz fluxes. We show that in both heterotic and type II theories of this class, no moduli dependent corrections to the Planck mass are generated. We explicitly compute the one-loop corrections to the $\mathcal R^2$ coupling and find that, despite the absence of supersymmetry, its contributions may still be organised into representations of subgroups of the modular group, and admit a universal form, determined u…
N=2 supersymmetric quantum black holes in the heterotic S-T-U model are presented. In particular three classes of axion-free quantum black holes with half the N=2, D=4 supersymmetries unbroken are considered. First, these quantum black holes are investigated at generic points in moduli space. Then linearized non-abelian black holes are investigated representing a subset of non-abelian black hole solutions at critical points of perturbative gauge symmetry enhancement in moduli space. It is shown that the entropy of linearized non-abelian black holes can be obtained, starting at non-critical poi…
Within the context of the weakly coupled E8 x E8 heterotic string, we study the hidden sector of heterotic standard model compactifications to four-dimensions. Specifically, we present a class of hidden sector vector bundles - composed of the direct sum of line bundles only - that, together with an effective bulk five-brane, renders the heterotic standard model entirely N=1 supersymmetric. Two explicit hidden sectors are constructed and analyzed in this context; one with the gauge group E7 x U(1) arising from a single line bundle and a second with an SO(12) x U(1) x U(1) gauge group constructe…
In this talk we review recent investigations of the non-supersymmetric heterotic SO(16)xSO(16) string on orbifolds and smooth Calabi-Yaus. Using such supersymmetry preserving backgrounds allows one to re-employ commonly known model building techniques. We will argue that tachyons do not appear on smooth Calabi-Yaus to leading order in alpha' and g_s. Twisted tachyons may arise on singular orbifolds, where some of these approximations break down. However, they get lifted in full blow-up. Finally, we show that model searches is viable by identifying over 12,000 of SM-like models on various orbif…
We analyze the AdS_3 x M_7 type supersymmetric solutions, including non-trivial fluxes, of the Killing spinor equations in the heterotic supergravity. We classify these solutions by their G-structures and intrinsic torsions, for the cases that the number of seven-dimensional Killing spinors N are equal to 1,2,3 and 4. We find that the solutions cannot have non-trivial warp factor and the seven dimensional manifold M_7 is charactrized by G_2(SU(3))-structures for N=1 (2) case and SU(2)-structure for N=3 and 4 cases. They are further classified by their non-trivial intrinsic torsions. It is show…
We study quantum corrections for a family of 24 non-supersymmetric heterotic strings in two dimensions. We compute their genus two cosmological constant using the hyperelliptic formalism and the genus one two-point functions for the massless states. From here we get the mass corrections to the states in the massless sector and discuss the role of the infrared divergences that appear in the computation. We also study some tree-level aspects of these theories and find that they are classified not only by the corresponding Niemeier lattice but also by their {\it hidden} right-moving gauge symmetr…
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.
Maximal Enhancements in Eight-Dimensional Non-Supersymmetric Heterotic Strings
Kyushu University（九州大学）；Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Quantum and Spacetime Research Institute (QuaSR), Kyushu University（量子与时空研究所（QuaSR），九州大学）
We compute one-loop threshold corrections to non-abelian gauge couplings in four-dimensional heterotic vacua with spontaneously broken $\cal N = 2 \to \cal N = 0$ supersymmetry, obtained as Scherk-Schwarz reductions of six-dimensional K3 compactifications. As expected, the gauge thresholds are no-longer BPS protected, and receive contributions also from the excitations of the RNS sector. Remarkably, the difference of thresholds for non-abelian gauge couplings is BPS saturated and exhibits a universal behaviour independently of the orbifold realisation of K3. Moreover, the thresholds and their …
We study the effective action of the heterotic string compactified on particular half-flat manifolds which arise in the context of mirror symmetry with NS-NS flux. We explicitly derive the superpotential and Kahler potential at lowest order in alpha' by a reduction of the bosonic action. The superpotential contains new terms depending on the Kahler moduli which originate from the intrinsic geometrical flux of the half-flat manifolds. A generalized Gukov formula, valid for all manifolds with SU(3) structure, is derived from the gravitino mass term. For the half-flat manifolds it leads to a supe…
We discuss supersymmetric compactifications of heterotic strings in the presence of H-flux and general condensates using the formalism of G-structures and intrinsic torsion. We revisit the examples based on nearly-Kaehler coset spaces and show that supersymmetric solutions, where the Bianchi identity is satisfied, can be obtained when both gaugino and dilatino condensates are present.
We study one-loop quantum corrections to gauge couplings in heterotic vacua with spontaneous supersymmetry breaking. Although in non-supersymmetric constructions these corrections are not protected and are typically model dependent, we show how a universal behaviour of threshold differences, typical of supersymmetric vacua, may still persist. We formulate specific conditions on the way supersymmetry should be broken for this to occur. Our analysis implies a generalised notion of threshold universality even in the case of unbroken supersymmetry, whenever extra charged massless states appear at …
We investigate under which conditions the cosmological constant vanishes perturbatively at the one-loop level for heterotic strings on non-supersymmetric toroidal orbifolds. To obtain model-independent results, which do not rely on the gauge embedding details, we require that the right-moving fermionic partition function vanishes identically in every orbifold sector. This means that each sector preserves at least one, but not always the same Killing spinor. The existence of such Killing spinors is related to the representation theory of finite groups, i.e. of the point group that underlies the…
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…
Recent developments in string theory have reinforced the notion that the space of stable supersymmetric and non-supersymmetric string vacua fills out a ``landscape'' whose features are largely unknown. It is then hoped that progress in extracting phenomenological predictions from string theory -- such as correlations between gauge groups, matter representations, potential values of the cosmological constant, and so forth -- can be achieved through statistical studies of these vacua. To date, most of the efforts in these directions have focused on Type I vacua. In this note, we present the firs…
We study compact non-supersymmetric Z_N orbifolds in various dimensions. We compute the spectrum of several tachyonic type II and heterotic examples and partially classify tachyon-free heterotic models. We also discuss the relation to compactification on K3 and Calabi-Yau manifolds.
A description is given of how to construct (0,2) supersymmetric conformal field theories as coset models. These models may be used as non-trivial backgrounds for Heterotic String Theory. They are realised as a combination of an anomalously gauged Wess-Zumino-Witten model, right-moving supersymmetric fermions, and left-moving current algebra fermions. Requiring the sum of the gauge anomalies from the bosonic and fermionic sectors to cancel yields the final model. Applications discussed include exact models of extremal four-dimensional charged black holes and Taub-NUT solutions of string theory.…
In this paper, we search for accelerating power-law and ekpyrotic solutions in heterotic string theory with NS-NS fluxes compactified on half-flat and generalized half-flat manifolds. We restrict our searches to the STZ sector of the theory. We also considered linear order $α'$ corrections for the half-flat case. The power-law solutions that we find are neither accelerating nor ekpyrotic in any of the models.
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 explicitly construct massive (0,4) supersymmetric ADHM sigma models which have heterotic p-brane solitons as their conformal fixed points. These yield the familiar gauge 5-brane and a new 1-brane solution which preserve 1/2 and 1/4 of the spacetime supersymmetry respectively. We also discuss an analogous construction for the type II NS-NS p-branes using (4,4) supersymmetric models.
Inspired by the tachyon-free non-supersymmetric heterotic SO(16)xSO(16) string we consider a special class of non-supersymmetric field theories: Those that can be obtained from supersymmetric field theories by supersymmetry breaking twists. We argue that such theories, like their supersymmetric counter parts, may still possess some fermionic symmetries as left-overs of the super gauge transformations and have special one-loop non-renormalization properties due to holomorphicity. In addition, we extend the supergraph techniques to these theories to calculate some explicit supersymmetry-breaking…
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 showed that there is no SU(2) Witten anomaly in a large class of 4d N = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.
A description is given of how to construct $(0,2)$ supersymmetric conformal field theories as coset models. These models may be used as non--trivial backgrounds for Heterotic String Theory. They are realised as a combination of an anomalously gauged Wess--Zumino--Witten model, right--moving supersymmetric fermions, and left--moving current algebra fermions. Requiring the sum of the gauge anomalies from the bosonic and fermionic sectors to cancel yields the final model. Applications discussed include exact models of extremal four--dimensional charged black holes and Taub--NUT solutions of strin…
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…
In this thesis, we consider heterotic string vacua based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold preserving only two supercharges. Thus, they correspond to half-BPS states of heterotic supergravity. The constraints on the internal manifolds with SU(3) structure are derived. They are found to be a generalization of 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 heterotic compactifications on half-flat mir…
We study supersymmetric solutions in 7- and 8-dimensional Abelian heterotic supergravity theories. In dimension 7, the solutions are described by $G_{2}$ with torsion equations. When a $G_{2}$ manifold has principal orbits $S^{3} \times S^{3}$, the equations are reduced to ordinary differential equations with four radial functions. For these equations we obtain explicit ALC metrics with $S^{3}$-bolt and $T^{1,1}$-bolt singularities. In dimension 8, we study supersymmetric solutions to $Spin(7)$ with torsion equations associated with 3-Sasakian manifolds by using a similar method to the case $G…
In this letter we establish that the supersymmetric R^2 effective action for the heterotic string, obtained from the supersymmetrisation of the Lorentz Chern-Simons term, is to order $α'$ equivalent modulo field redefinitions to heterotic string effective actions computed by different methods.
We investigate orbifold compactifications of the heterotic string, addressing in detail their construction, classification and phenomenological potential. We present a strategy to search for models resembling the minimal supersymmetric extension of the standard model (MSSM) in Z_6-II orbifold compactifications. We find several MSSM candidates with the gauge group and the exact spectrum of the MSSM, and supersymmetric vacua below the compactification scale. They also exhibit the following realistic features: R-parity, seesaw suppressed neutrino masses, and intermediate scale of supersymmetry br…
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 …
We construct explicit compact solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic string equations in dimensions seven and eight. We present a quadratic condition on the curvature which is necessary and sufficient the heterotic supersymmetry and the anomaly cancellation to imply the heterotic equations of motion in dimensions seven and eight. We show that some of our examples are compact supersymmetric solutions of the heterotic equations of motion in dimensions seven and eight.
The constraints imposed on heterotic compactifications by global consistency and phenomenology seem to be very finely balanced. We show that weakening these constraints, as was proposed in some recent works, is likely to lead to frivolous results. In particular, we construct an infinite set of such frivolous models having precisely the massless spectrum of the MSSM and other quasi-realistic features. Only one model in this infinite collection (the one constructed in arXiv:hep-th/0512149) is globally consistent and supersymmetric. The others might be interpreted as being anomalous, or as non-su…
On The Consistent Supersymmetric Reduction Of Heterotic Supergravity With Geometrically Arising Yang-Mills Symmetries
George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University（乔治·P和辛西娅·伍兹基础物理与天文学研究所，德克萨斯农工大学）；DAMTP, Centre for Mathematical Sciences, Cambridge University（剑桥大学数学科学中心DAMTP）
Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory -- including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first is…
It is shown that four-dimensional N=1 supersymmetric QCD with massive flavors in the fundamental representation of the gauge group can be realized in the hidden sector of E8xE8 heterotic string vacua. The number of flavors can be chosen to lie in the range of validity of the free-magnetic dual, using which one can demonstrate the existence of long-lived meta-stable non-supersymmetric vacua. This is shown explicitly for the gauge group Spin(10), but the methods are applicable to Spin(Nc), SU(Nc) and Sp(Nc) for a wide range of color index Nc. Hidden sectors of this type can potentially be used a…
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…
We present a Z_6 orbifold compactification of the E_8xE_8 heterotic string which leads to the (supersymmetric) Standard Model gauge group and matter content. The quarks and leptons appear as three 16-plets of SO(10), whereas the Higgs fields do not form complete SO(10) multiplets. The model has no exotic states at low energies and is consistent with gauge coupling unification. The top quark Yukawa coupling arises from gauge interactions and is of the order of the gauge couplings, whereas the other Yukawa couplings are suppressed.
