Heterotic backgrounds with torsion preserving minimal supersymmetry in four dimensions can be obtained as orbifolds of principal $T^{2}$ bundles over $K3$. We consider a worldsheet description of these backgrounds as gauged linear sigma-models (GLSMs) with $(0,2)$ supersymmetry. Such a formulation provides a useful framework in order to address the resolution of singularities of the orbifold geometries. We investigate the constraints imposed by discrete symmetries on the corresponding torsional GLSMs. In particular, the principal $T^{2}$ connection over $K3$ is inherited from $(0,2)$ vector multiplets. As these vectors gauge global scaling symmetries of products of projective spaces, the corresponding $K3$ geometry is naturally realized as an algebraic hypersurface in such a product (or as a branched cover of it). We outline the general construction for describing such orbifolds. We give explicit constructions for automorphisms of order two and three.
We determine the two--dimensional Weyl, Lorentz and $κ$--anomalies in the $D=10$ Green--Schwarz heterotic string sigma--model, in an $SO(1,9)$-Lorentz covariant background gauge, and prove their cancellation.
We report on some recent progress in understanding heterotic flux compactifications, from a worldsheet perspective mainly. We consider local models consisting in torus fibration over warped Eguchi-Hanson space and non-Kähler resolved conifold geometries. We analyze the supergravity solutions and define a double-scaling limit of the resolved singularities, defined such that the geometry is smooth and weakly coupled. We show that, remarkably, the heterotic solutions admit solvable worldsheet CFT descriptions in this limit. This allows in particular to understand the important role of worldsheet …
We propose a new formulation of the $D=3$ type II superstring which is manifestly invariant under both target-space $N=2$ supersymmetry and worldsheet $N=(1,1)$ super reparametrizations. This gives rise to a set of twistor (commuting spinor) variables, which provide a solution to the two Virasoro constraints. The worldsheet supergravity fields are shown to play the rôle of auxiliary fields.
We review recent results on string coupling selection rules for heterotic orbifolds, derived using conformal field theory. Such rules are the first step towards understanding the viability of the recently obtained compactifications with potentially realistic particle spectra. They arise from the properties of the worldsheet instantons that mediate the couplings, and include stringy effects that would seem 'miraculous' to an effective field theory observer.
By defining the heterotic Green-Schwarz superstring action on an N=(2,0) super-worldsheet, rather than on an ordinary worldsheet, many problems with the interacting Green-Schwarz superstring formalism can be solved. In the light-cone approach, superconformally transforming the light-cone super-worldsheet onto an N=(2,0) super-Riemann surface allows the elimination of the non-trivial interaction-point operators that complicate the evaluation of scattering amplitudes. In the Polyakov approach, the ten-dimensional heterotic Green-Schwarz covariant action defined on an N=(2,0) super-worldsheet can…
We compute the spectrum of massless gauge singlets in some heterotic string compactifications using Landau-Ginzburg, orbifold and non-linear sigma-model methods. This probes the worldsheet instanton corrections to the quadratic terms in the spacetime superpotential. Previous results predict that some of these states remain massless when instanton effects are included. We find vanishing masses in many cases not covered by these predictions. However, we discover that in the case of the Z-manifold the corrections do not vanish. Despite this, in all the examples studied, we find that the massless …
In this paper we consider the role that worldsheet instantons in the heterotic string could play in spin-statistics violations. Such violations are nonperturbative in the string tension and so would not appear in the spacetime effective action, producing a unique signature of string theory and the details of compactification. By performing a Bogomol'nyi transformation it is shown that there are no instanton solutions in the simplest model proposed by Harvey and Liu, but it is conjectured that more sophisticated models may yield solutions. If such instantons do exist, their effect might be meas…
It is well-known that the beta functions on a string worldsheet correspond to the target space equations of motion, e.g. the Einstein equations. We show that the BPS equations, i.e. the conditions of vanishing supersymmetry variations of the space-time fermions, can be directly derived from the worldsheet. To this end we consider the RNS-formulation of the heterotic string with (2,0) supersymmetry, which describes a complex torsion target space that supports a holomorphic vector bundle. After a detailed account of its quantization and renormalization, we establish that the cancellation of the …
We study the stability of heterotic compactifications described by (0,2) gauged linear sigma models with respect to worldsheet instanton corrections to the space-time superpotential following the work of Beasley and Witten. We show that generic models elude the vanishing theorem proved there, and may not determine supersymmetric heterotic vacua. We then construct a subclass of linear models for which a vanishing theorem holds, generating an extensive list of consistent heterotic backgrounds.
Asymmetric heterotic orbifolds are discussed from the worldsheet perspective. Starting from Buscher's gauging of a theory of D compact bosons the duality covariant description of Tseytlin is obtained after a non-Lorentz invariant gauge fixing. A left-over of the gauge symmetry can be used to removed the doubled constant zero modes so that D physical target space coordinate remain. This can be thought of as the worldsheet realization of the strong constraint of double field theory. The extension of this description to the heterotic theory is straightforward as all results are written in terms o…
We propose a new formulation of the heterotic $D=10$ Green-Schwarz superstring whose worldsheet is a superspace with two even and eight odd coordinates. The action is manifestly invariant under both target-space supersymmetry and a worldsheet reparametrisation supergroup. It contains only a finite set of auxiliary fields. The key ingredient are the commuting spinor (twistor) variables, which naturally arise as worldsheet superpartners of the target space Grassmann coordinates. These spinors parametrise the sphere $S^8$ regarded as a coset space of the $D=10$ Lorentz group. The sphere is associ…
We study the quantum dynamics of vortex strings in N=1 SQCD with U(N_c) gauge group and N_f=N_c quarks. The classical worldsheet of the string has N=(0,2) supersymmetry, but this is broken by quantum effects. We show how the pattern of supersymmetry breaking and restoration on the worldsheet captures the quantum dynamics of the underlying 4d theory. We also find qualitative matching of the meson spectrum in 4d and the spectrum on the worldsheet.
We construct a (1,2) heterotic string with gauge symmetry and determine its particle spectrum. This theory has a local N=1 worldsheet supersymmetry for left movers and a local N=2 worldsheet supersymmetry for right movers and describes particles in either two or three space-time dimensions. We show that fermionizing the bosons of the compactified N=1 space leads to a particle spectrum which has nonabelian gauge symmetry. The fermionic formulation of the theory corresponds to a dimensional reduction of self dual Yang Mills. We also give a worldsheet action for the theory and calculate the one-l…
We analyze the behavior of the heterotic string near an A-D-E singularity without small instantons. This problem is governed by a strongly coupled worldsheet conformal field theory, which, by a combination of O(alpha') corrections and worldsheet instantons, smooths out the singularities present in the classical geometry.
We use a D-instanton or physical gauge approach to re-derive the heterotic string worldsheet instanton contribution to the superpotential in Calabi-Yau compactification. We derive an analogous formula for worldsheet instanton corrections to the moduli space metric in heterotic string or Type I compactification on a K3 surface. In addition, we give a global analysis of the phase of the worldsheet path integral of the heterotic string, showing precisely how the B-field must be interpreted.
We study the spacetime decay to nothing in string theory and M-theory. First we recall a nonsupersymmetric version of heterotic M-theory, in which bubbles of nothing -- connecting the two E_8 boundaries by a throat -- are expected to be nucleated. We argue that the fate of this system should be addressed at weak string coupling, where the nonperturbative instanton instability is expected to turn into a perturbative tachyonic one. We identify the unique string theory that could describe this process: The heterotic model with one E_8 gauge group and a singlet tachyon. We then use worldsheet meth…
We study the constraints of spacetime supersymmetry for perturbative three- and two-dimensional Minkowski vacua of the critical heterotic string. Assuming a standard RNS construction of the spacetime supersymmetry generators and a compact unitary internal superconformal worldsheet theory, we describe the worldsheet structures associated to various spacetime supersymmetries. In three dimensions we show that there are no CFT surprises: each allowed spacetime supersymmetry is realized by a supergravity compactification. As a recent orbifold construction shows, in two dimensions there are more exo…
Heterotic String Sigma Models: Discrete Light Cone Quantization and Its Current-Current Deformation
University of Groningen（格罗宁根大学）；National Institute of Physics, University of the Philippines（菲律宾大学国家物理研究所）；Universidad de Murcia（穆尔西亚大学）；KTH Royal Institute of Technology and Stockholm University（瑞典皇家理工学院和斯德哥尔摩大学）
We propose an action describing chiral fermions with an arbitrary gauge group and with manifest $(8,0)$ worldsheet supersymmetry. The form of the action is inspired by and adapted for completing the twistor-like formulation of the $D=10$ heterotic superstring.
Heterotic string compactifications can be conveniently described in the language of (2,0) gauged linear sigma models (GLSMs). Such models allow for Fayet-Iliopoulos (FI)-terms, which can be interpreted as Kahler parameters and axions on the target space geometry. We show that field dependent non-gauge invariant FI-terms lead to a Green-Schwarz-like mechanism on the worldsheet which can be used to cancel worldsheet anomalies. However, given that these FI-terms are constrained by quantization conditions due to worldsheet gauge instantons, the anomaly conditions turn out to be still rather constr…
It is argued that recent developments point to the existence of an AdS_3 x S^2 x T^5 holographic dual for the 2D CFT living on the worldsheet of N coincident heterotic strings in a T^5 compactification, which can in turn be described by an exact worldsheet CFT. A supergravity analysis is shown to imply that the global supergroup is Osp(4^*|4), with 16 supercharges and an affine extension given, surprisingly, by a nonlinear N=8 2D superconformal algebra. Possible supergroups with 16 supercharges are also found to match the expected symmetries for T^n compactification with 0<=n<=7.
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.
Compactifications of the heterotic string are a viable route to phenomenologically realistic vacua and interesting new mathematics. While supergravity aspects of heterotic compactifications are largely well-understood their worldsheet description remains largely unexplored. We review recent work in developing linear sigma model techniques aimed at elucidating the underlying worldsheet description.
We discuss the AdS_3/CFT_2 duality of a heterotic three-charge model with (0,4) target space supersymmetry. The worldsheet theory for heterotic strings on the AdS_3 x S^3/Z_N x T^4 near-horizon geometry was constructed by Kutasov, Larsen and Leigh in [hep-th/9812027]. We propose that the dual conformal field theory is given by a two-dimensional (0,4) sigma model arising on the Higgs branch of an orbifolded ADHM model. As a non-trivial consistency check of the correspondence, we find that the left- and right-moving central charges of the infrared conformal field theory agree with those predicte…
Worldsheet string theory on $\mathrm{AdS}_3 \times \mathbf{S}^3 \times \mathrm{K3}$
University of the Chinese Academy of Sciences（中国科学院大学）；Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Institut für Theoretische Physik, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；Department of Mathematics, King’s College London（伦敦国王学院数学系）；Center for Quantum Mathematics and Physics (QMAP) Department of Physics & Astronomy, University of California, Davis（加州大学戴维斯分校物理学与天文学系量子数学与物理中心）
We examine to what extent heterotic string worldsheets can describe arbitrary E8xE8 gauge fields. The traditional construction of heterotic strings builds each E8 via a Spin(16)/Z2 subgroup, typically realized as a current algebra by left-moving fermions, and as a result, only E8 gauge fields reducible to Spin(16)/Z2 gauge fields are directly realizable in standard constructions. However, there exist perturbatively consistent E8 gauge fields which can not be reduced to Spin(16)/Z2, and so cannot be described within standard heterotic worldsheet constructions. A natural question to then ask is …
In this work we attempt to bridge the gap between heterotic orbifold models and Calabi-Yau compactifications using gauged linear sigma models (GLSMs) with (2,0) worldsheet supersymmetry. We associate a specific GLSM to a heterotic orbifold model with twisted states that have non-vanishing vacuum expectation values (VEVs): The charges of the GLSM superfields are essentially determined by the shifted momenta of these states. When a twisted state contains an oscillator excitation, a fermionic gauging is introduced on the worldsheet inducing a non-Abelian gauge bundle, e.g. the standard embedding.…
In $D=3,4,6$ and 10 space--time dimensions considered is a string model invariant under transformations of $N=1$ space--time supersymmetry and $n=D-2$ local worldsheet supersymmetry with the both Virasoro constraints solved in the twistor form. The twistor solution survives in a modified form even in the presence of the heterotic fermions.
Alternative gauge choices for worldsheet supersymmetry can elucidate dynamical phenomena obscured in the usual superconformal gauge. In the particular example of the tachyonic $E_8$ heterotic string, we use a judicious gauge choice to show that the process of closed-string tachyon condensation can be understood in terms of a worldsheet super-Higgs effect. The worldsheet gravitino assimilates the goldstino and becomes a dynamical propagating field. Conformal, but not superconformal, invariance is maintained throughout.
We study four-dimensional heterotic flux vacua with N=2 spacetime supersymmetry. A worldsheet perspective is used to clarify quantization conditions associated to the fluxes and the constraints these place on the moduli spaces of resulting compactifications. We propose that these vacua fit naturally in the context of heterotic/IIA duality as heterotic duals to compactifications on K3-fibered but not elliptically fibered Calabi-Yau three-folds. We present some examples of such potential dual pairs.
We compute the one-loop threshold corrections to the gauge and gravitational couplings for a large class of N=2 non-Kähler heterotic compactifications with three-form flux, consisting in principal two-torus bundles over K3 surfaces. We obtain the results as sums of BPS-states contributions, depending on the topological data of the bundle. We analyse also the worldsheet non-perturbative corrections coming from instantons wrapping the torus fiber, that are mapped under S-duality to D-instanton corrections in type I flux compactifications.
In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only…
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.
The recently established B-RNS-GSS formalism is extended for the description of the heterotic superstring in curved backgrounds. We propose a generalized action and BRST charge defined in the small Hilbert space with the standard form of an $\mathcal{N}=(1,0)$ worldsheet superconformal theory with superconformal generator $G$ and stress tensor $T$. We show that $\big\{G,G\big\}=-2T$ implies the D=10 N=1 supergravity and super-Yang-Mills equations of motion, as well as holomorphicity of the BRST charge.
We consider non-Abelian BPS-saturated flux tubes (strings) in N=2 supersymmetric QCD deformed by superpotential terms of a special type breaking N=2 supersymmetry down to N=1. Previously it was believed that worldsheet supersymmetry is "accidentally" enhanced due to the facts that N =(1,1) SUSY is automatically elevated up to N=(2,2) on CP(N-1) and, at the same time, there are no N=(0,2) generalizations of the bosonic CP(N-1) model. Edalati and Tong noted that the target space is in fact CP(N-1) x C rather than CP(N-1). This allowed them to suggest a "heterotic" N=(0,2) sigma model, with the C…
$\mathcal{G}$-structures, where $\mathcal{G}$ is a Lie group, are a uniform characterisation of many differential geometric structures of interest in supersymmetric compactifications of string theories. Calabi-Yau $n$-folds are instances of torsion-free $SU(n)$-structures, while more general structures with non-zero torsion are required for heterotic flux compactifications. Exceptional geometries in dimensions $7$ and $8$ with $\mathcal{G}=G_2$ and $Spin(7)$ also feature prominently in this thesis. We discuss multiple connections between such geometries and the worldsheet theory describing str…
We continue explorations of non-Abelian strings, focusing on the solution of a heterotic deformation of the CP(N-1) model with an extra right-handed fermion field and N=(0,2) supersymmetry. This model emerges as a low-energy theory on the worldsheet of the BPS-saturated flux tubes (strings) in N=2 supersymmetric QCD deformed by a superpotential of a special type breaking N=2 supersymmetry down to N=1. Using large-N expansion we solve this model to the leading order in 1/N. Our solution exhibits spontaneous supersymmetry breaking for all values of the deformation parameter. We identify the Gold…
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-…
The sigma model action described in this paper differs in four important features from the usual sigma model action for the four-dimensional Green-Schwarz heterotic superstring in a massless background. Firstly, the action is constructed on an N=(2,0) super-worldsheet using a Kahler potential and an Ogievetsky-Sokatchev constraint; secondly, the target-space background fields are unconstrained; thirdly, the target-space dilaton couples to the two-dimensional curvature; and fourthly, the action reduces in a flat background to a free-field action. A conjecture is made for generalizing this N=(2,…
I argue that the ten dimensional non--supersymmetric tachyonic superstrings may serve as good starting points for the construction of viable phenomenological vacua. Thus, enlarging the space of possible solutions that may address some of the outstanding problems in string phenomenology. A tachyon free six generation Standard--like Model is presented, which can be regarded as an orbifold of the $SO(16)\times E_8$ heterotic--string in ten dimensions. I propose that any $(2,0)$ heterotic--string in four dimensions can be connected to a $(2,2)$ one via an orbifold or by interpolations and provide …
We calculate the beta function of non-linear sigma models with S^{D+1} and AdS_{D+1} target spaces in a 1/D expansion up to order 1/D^2 and to all orders in α'. This beta function encodes partial information about the spacetime effective action for the heterotic string to all orders in α'. We argue that a zero of the beta function, corresponding to a worldsheet CFT with AdS_{D+1} target space, arises from competition between the one-loop and higher-loop terms, similarly to the bosonic and supersymmetric cases studied previously in hep-th/0512355. Various critical exponents of the non-linear si…
A new two dimensional $\mathcal{N}=(0,2)$ Supersymmetric Non-Linear Sigma Model describes the dynamics of internal moduli of the BPS semi-local vortex string supported in four dimensional $\mathcal{N}=2$ SQED. While the core of these strings is very similar to Abrikosov-Nielsen-Olesen vortices, they are defined with a characteristic size modulus, much like the instanton lump size. This entails that the constituting fields of the vortex do not decay exponentially, as one goes far away from the core of the string, but as a rational function. The appearance of an extra scale in the problem also a…
It is shown that the O(d,d;R) deformations of the superstring vacua and the O(d,d+16;R) deformations of the heterotic string vacua preserve extended worldsheet supersymmetry and, hence, generate superconformal deformations. The transformations of the complex structures are given explicitly and the action of the discrete duality subgroup is discussed. The results are valid when the complex structures are independent of the d coordinates which appear in the transformations. It is shown that generic deformations do not preserve the known superfield formulations of (2,2) extended supersymmetry. Th…
The $Z_2\times Z_2$ heterotic string orbifold gives rise to a large space of phenomenological three generation models that serves as a testing ground to explore how the Standard Model of particle physics may be incorporated in a theory of quantum gravity. Recently, we demonstrated the existence of type 0 $Z_2\times Z_2$ heterotic string orbifolds in which there are no massless fermionic states. In this paper we demonstrate the existence of non--supersymmetric tachyon--free $Z_2\times Z_2$ heterotic string orbifolds that do not contain any massless bosonic states from the twisted sectors. We du…
We study heterotic supergravity on the conifold and its Z2 orbifold with Abelian gauge fields and three-form flux. At large distances, these solutions are locally Ricci-flat, have a magnetic flux through the two-sphere at infinity as well as non-zero five-brane charge. For a given flux, our family of solutions has three real parameters, the size of the pair of two spheres in the IR and the dilaton zero mode. We present an explicit analytic solution for the decoupled near horizon region where for a given flux, the size of the cycles is frozen and the only parameter is the dilaton zero mode. We …
We consider C/Z_N and C^2/Z_N orbifolds of heterotic string theories and Z_N orbifolds of AdS_3. We study theories with N=2 worldsheet superconformal invariance and construct RG flows. Following Harvey, Kutasov, Martinec and Moore, we compute g_cl and show that it decreases monotonically along RG flows- as conjectured by them. For the heterotic string theories, the gauge degrees of freedom do not contribute to the computation of g_cl.
We calculate couplings of arbitrary order from correlation functions among twisted strings, using conformal field theory. Twisted strings arise in heterotic string compactified on orbifolds yielding matter fields in the low energy limit. We calculate completely the classical and the quantum amplitude including normalization, up to a contribution from Kahler potential. The classical action has saddle points which are interpreted as worldsheet instantons described by metastable untwisted strings, formed by twisted strings distributed at certain fixed points. This understanding generalizes the ar…
We study to what extent Wilson lines in heterotic Calabi-Yau compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson lines are basic ingredients for Standard Model constructions but their induced H-flux may affect the consistency of the leading order background geometry and of the two-dimensional worldsheet theory. Moreover H-flux in heterotic compactifications would play an important role for moduli stabilization and could strongly constrain the supersymmetry breaking scale. We show how to compute H-flux and the corresponding superpotential, given an explicit complete inte…
We propose a twistor--like formulation of N=1, D=3,4,6 and 10 null superstrings. The model possesses N=1 target space supersymmetry and n=D-2 local worldsheet supersymmetry, the latter replaces the kappa-symmetry of the conventional approach to the strings. Adding a Wess--Zumino term to a null superstring action we observe a string tension generation mechanism: the induced worldsheet metric becomes non-degenerate and the resulting model turns out to be classically equivalent to the heterotic string.
M(atrix) theory on an orbifold and classical two-branes therein are studied with particular emphasis to heterotic M(atrix) theory on $S_1/Z_2$ relevant to strongly coupled heterotic and dual Type IA string theories. By analyzing orbifold condition on Chan-Paton factors, we show that three choice of gauge group are possible for heterotic M(atrix) theory: SO(2N), SO(2N+1) or USp(2N). By examining area-preserving diffeomorphism that underlies the M(atrix) theory, we find that each choices of gauge group restricts possible topologies of two-branes. The result suggests that only the choice of SO(2N…
We begin with an $E_{8}\times E_{8}$ Heterotic model broken to an $SU(5)_{gauge}$ and a mirror $SU(5)_{gauge}$, where one $SU(5)$ and its spectrum is identified as the visible sector while the other can be identified as a hidden mirror world. In both cases we obtain the minimal supersymmetric standard model spectrum after Wilson-line symmetry-breaking enhanced by a low energy R-parity enforced by a local (or global) $U(1)_{X}$-symmetry. Using Heterotic/$F$-theory duality, we show how to eliminate the vector-like exotics which were obtained in previous constructions. In these constructions, the…
We construct worldsheet descriptions of heterotic flux vacua as the IR limits of N=2 gauge theories. Spacetime torsion is incorporated via a 2d Green-Schwarz mechanism in which a doublet of axions cancels a one-loop gauge anomaly. Manifest (0,2) supersymmetry and the compactness of the gauge theory instanton moduli space suggest that these models, which include Fu-Yau models, are stable against worldsheet instanton effects, implying that they, like Calabi-Yaus, may be smoothly extended to solutions of the exact beta functions. Since Fu-Yau compactifications are dual to KST-type flux compactifi…
