Model building with the non-supersymmetric heterotic SO(16)xSO(16) string
Stefan Groot Nibbelink
备注：
10 pages LaTeX. Based on talks presented at the "Frontiers in String Phenomenology" workshop, Ringberg Castle, Tegernsee, Germany, the conference "The string universe", Mainz, Germany, conference "DISCRETE 2014", King's College, London, UK and Heidelberg university, Heidelberg, Germany
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 orbifold geometries.
We extend strong/weak coupling duality to string theories without spacetime supersymmetry, and focus on the case of the unique ten-dimensional, nonsupersymmetric, tachyon-free $SO(16)\times SO(16)$ heterotic string. We construct a tachyon-free heterotic string model that interpolates smoothly between this string and the ten-dimensional supersymmetric $SO(32)$ heterotic string, and we construct a dual for this interpolating model. We find that the perturbative massless states of our dual theories precisely match within a certain range of the interpolation. Further evidence for this proposed dua…
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…
Motivated by the study of black attractors in 8D supergravity with 16 supersymmetries, we use the field theory approach and 8D supersymmetry with non trivial central charges to shed light on the exact duality between heterotic string on T^2 and type IIA on real connected and compact surfaces Σ2. We investigate the two constraints that should be obeyed by Σ2 and give their solutions in terms of intersecting 2-cycles as well their classification using Dynkin diagrams of affine Kac-Moody algebras. It is shown as well that the moduli space of these dual theories is given by SO(1,1)x((SO(2,r+2))/(S…
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.
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…
Four dimensional heterotic SO(32) orbifold models are classified systematically with model building applications in mind. We obtain all Z3, Z7 and Z2N models based on vectorial gauge shifts. The resulting gauge groups are reminiscent of those of type-I model building, as they always take the form SO(2n_0)xU(n_1)x...xU(n_{N-1})xSO(2n_N). The complete twisted spectrum is determined simultaneously for all orbifold models in a parametric way depending on n_0,...,n_N, rather than on a model by model basis. This reveals interesting patterns in the twisted states: They are always built out of vectors…
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 ten dimensional string theories as well as eleven dimensional supergravity are conjectured to arise as limits of a more basic theory, traditionaly dubbed M--theory. This notion is confined to the ten dimensional supersymmetric theories. String theory, however, also contains ten dimensional nonsupersymmetric theories that have not been incorporated into this picture. In this note we explore the possibility of generating the low energy spectra of various nonsupersymmetric heterotic string vacua form the Horava--Witten model. We argue that this can be achieved by imposing on the Horava--Witte…
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.
We investigate orbifold and smooth Calabi-Yau compactifications of the non-supersymmetric heterotic SO(16)xSO(16) string. We focus on such Calabi-Yau backgrounds in order to recycle commonly employed techniques, like index theorems and cohomology theory, to determine both the fermionic and bosonic 4D spectra. We argue that the N=0 theory never leads to tachyons on smooth Calabi-Yaus in the large volume approximation. As twisted tachyons may arise on certain singular orbifolds, we conjecture that such tachyonic states are lifted in the full blow-up. We perform model searches on selected orbifol…
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 find that the gas of IIA strings undergoes a phase transition into a gas of IIB strings at the self-dual temperature. A gas of free heterotic strings undergoes a Kosterlitz-Thouless duality transition with positive free energy and positive specific heat but vanishing internal energy at criticality. We examine the consequences of requiring a tachyon-free thermal string spectrum. We show that in the absence of Ramond-Ramond fluxes the IIA and IIB string ensembles are thermodynamically ill-defined. The 10D heterotic superstrings have nonabelian gauge fields and in the presence of a temperature…
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.
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 perform model searches on smooth Calabi-Yau compactifications for both the supersymmetric E8xE8 and SO(32) as well as for the non-supersymmetric SO(16)xSO(16) heterotic strings simultaneously. We consider line bundle backgrounds on both favorable CICYs with relatively small h_11 and the Schoen manifold. Using Gram matrices we systematically analyze the combined consequences of the Bianchi identities and the tree-level Donaldson-Uhlenbeck-Yau equations inside the Kahler cone. In order to evaluate the model building potential of the three heterotic theories on the various geometries, we perfo…
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…
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…
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 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.…
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.
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 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 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 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…
A non-supersymmetric ten-dimensional open string theory is constructed as an orbifold of type I string theory, and as an orientifold of the bosonic type B theory. It is purely bosonic, and cancellation of massless tadpoles requires the gauge group to be SO(32)xSO(32). The spectrum of the theory contains a closed string tachyon, and open string tachyons in the (32,32) multiplet. The D-branes of this theory are analyzed, and it is found that the massless excitations of one of the 1-branes coincide with the world-sheet degrees of freedom of the D=26 bosonic string theory compactified on the SO(32…
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 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 wra…
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 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…
