We consider the compactification on a circle of the Heterotic Superstring effective action to first order in the Regge slope parameter $α'$ and re-derive the $α'$-corrected Buscher rules first found in arXiv:hep-th/9506156 , proving the T duality invariance of the dimensionally-reduced action to that order in $α'$. We use Iyer and Wald's prescription to derive an entropy formula that can be applied to black-hole solutions which can be obtained by a single non-trivial compactification on a circle and discuss its invariance under the $α'$-corrected T duality transformations. This formula has been successfully applied to $α'$-corrected 4-dimensional non-extremal Reissner-Nordström black holes in arXiv:1910.14324 and we apply it here to a heterotic version of the Strominger-Vafa 5-dimensional extremal black hole.
We perform the toroidal compactification of the full Bergshoeff-de Roo version of the Heterotic Superstring effective action to first orderin $α'$. The dimensionally-reduced action is given in a manifestly-O(n,n)-invariant form which we use to derive a manifestly-O(n,n)-invariant Wald entropy formula which we then use to compute the entropy of $α'$-corrected, 4-dimensional, 4-charge, static, extremal, supersymmetric black holes.
We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in alpha prime using Wald's formalism. We carefully take into account all the symmetries of the theory and, as a result, we obtain a manifestly gauge- and Lorentz-invariant entropy formula in which all the terms can be computed explicitly. An entropy formula with these properties allows unambiguous calculations of macroscopic black-hole entropies to first order in alpha prime that can be reliably used in a comparison with the microscopic ones. Such a formula was still lac…
We compute the first-order $α'$ corrections of the non-extremal Strominger-Vafa black hole and its non-supersymmetric counterparts in the framework of the Bergshoeff-de Roo formulation of the heterotic superstring effective action. The solution passes several tests: its extremal limit is the one found in an earlier publication and the effect of a T duality transformation on it is another solution of the same form with T dual charges. We compute the Hawking temperature and Wald entropy showing that they are related by the first law and Smarr formula. On the other hand, these two contain additio…
A simplified pure spinor superstring $b$ ghost in a curved heterotic background was constructed recently. The $b$ ghost is a composite operator and it is not holomorphic. However, it satisfies $\bar\partial b = [ Q , Ω]$, where $Q$ is the BRST charge. In this paper, we find a possible $Ω$.
Using the RNS-like fermionic vector variables introduced in arXiv:1305.0693, the pure spinor $b$ ghost in a curved heterotic superstring background is easily constructed. This construction simplifies and completes the $b$ ghost construction in a curved background of arXiv:1311.7012.
In the framework of the four-dimensional effective theory of heterotic superstrings at low energies, which is considered as a generalised theory of gravity, we search for new black hole solutions. The analytical expressions of all the scalar fields of the effective theory is determined in a Kerr-Newman black hole background and the evasion of the existing "no-hair" theorems in the presence of higher curvature gravitational terms is demonstrated. New black hole solutions, such as the Dilatonic and the Coloured black holes, are determined by use of numerical integration and their properties are …
Superstrings have been postulated based on parafermionic partition functions which permit spacetime supersymmetry by generalized Jacobi identities. A comprehensive search finds new such identities. Quadrilateral anomaly cancellation gives constraints on allowed chiral fermions. Bosonic left-movers and $Z_4$ parafermionic right-movers combine in a new heterotic superstring, more constrained than the old one, yet equally applicable to physics.
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 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 consider the effective action of the Heterotic Superstring to first order in alpha prime and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method and making minimal coordinate and frame choices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the relations existing between the left-h…
We review the derivation and the basic properties of the perturbative prepotential in N=2 compactifications of the heterotic superstring. We discuss the structure of the perturbative monodromy group and the embedding of rigidly supersymmetric monodromies associated with enhanced gauge groups, at both perturbative and non-perturbative level.
In this paper we begin the study of compactifications of the pure spinor formalism for superstrings. As a first example of such a process we study the case of the heterotic string in a Calabi-Yau background. We explicitly construct a BRST operator imposing N=1 four-dimensional supersymmetry and show that nilpotence implies Kahler and Ricci-flatness conditions. The massless spectrum is computed using this BRST operator and it agrees with the expected result.
We study quantum effects in five dimensions in heterotic superstring theory compactified on K_3 x S_1 and analyze the conjecture that its dual effective theory is eleven-dimensional supergravity compactified on a Calabi-Yau threefold. This theory is also equivalent to type II superstring theory compactified on the same Calabi-Yau manifold, in an appropriate large volume limit. In this limit the conifold singularity disappears and is replaced by a singularity associated to enhanced gauge symmetries, as naively expected from the heterotic description. Furthermore, we exhibit the existence of add…
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 derive the low energy effective action of the heterotic superstring in superspace. This is achieved by coupling the covariantly quantized Green-Schwarz superstring of Berkovits to a curved background and requiring that the sigma model has superconformal invariance at tree level and at one loop in $\a'$. Tree level superconformal invariance yields the complete supergravity algebra, and one-loop superconformal invariance the equations of motion of the low energy theory. The resulting low energy theory is old-minimal supergravity coupled to a tensor multiplet. The dilaton is part of the compen…
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,…
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 describe an $N=2$ heterotic superstring model of rank-3 which is dual to the type-II string compactified on a Calabi-Yau manifold with Betti numbers $b_{1,1}=2$ and $b_{1,2}=86$. We show that the exact duality symmetry found from the type II realization contains the perturbative duality group of the heterotic model, as well as the exact quantum monodromies of the rigid $SU(2)$ super-Yang-Mills theory. Moreover, it contains a non-perturbative monodromy which is stringy in origin and corresponds roughly to an exchange of the string coupling with the compactification radius.
The main aspects of a recently developed method for classification of heterotic superstring vacua in the Free Fermionic Formulation are presented. Using these techniques we classify a big number of approximately $10^{15}$ heterotic string vacua with Pati--Salam, SU(4)x{SU(2)}_Lx{SU(2)}_R gauge symmetry with respect to their main phenomenological features as the number of families, Pati--Salam breaking Higgs, Standard Model Higgs doublets, additional triplets and exotic charge states. We identify an interesting subclass of these vacua, approximately one to one million, whose massless spectrum i…
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 consider how to normalize the scattering amplitudes of 4D heterotic superstrings in a Minkowski background. We fix the normalization of the vacuum amplitude (the string partition function) at each genus, and of every vertex operator describing a physical external string state in a way consistent with unitarity of the $S$-matrix. We also provide an explicit expression for the map relating the vertex operator of an incoming physical state to the vertex operator describing the same physical state, but outgoing. This map is related to hermitean conjugation and to the hermiticity properties of t…
We provide some additional evidence in favour of the strong - weak coupling duality between the SO(32) heterotic and type I superstring theories by comparing terms quartic in the gauge field strength in their low-energy effective actions. We argue that these terms should not receive higher-loop string corrections so that duality should relate the leading-order perturbative coefficients in the two theories. In particular, we demonstrate that the coefficient of the $F^4$-term in the one-loop (torus) part of the SO(32) heterotic string action is exactly the same as the coefficient of the $F^4$-te…
We compare order $R^4$ terms in the 10-dimensional effective actions of SO(32) heterotic and type I superstrings from the point of view of duality between the two theories. Some of these terms do not receive higher-loop corrections being related by supersymmetry to `anomaly-cancelling' terms which depend on the antisymmetric 2-tensor. At the same time, the consistency of duality relation implies that the `tree-level' $R^4$ super-invariant (the one which has $ζ(3)$-coefficient in the sphere part of the action) should appear also at higher orders of loop expansion, i.e. should be multiplied by a…
We present the technical tools needed to compute any one-loop amplitude involving external spacetime fermions in a four-dimensional heterotic string model a` la Kawai-Lewellen-Tye. As an example, we compute the one-loop three-point amplitude with one "photon" and two external massive fermions ("electrons"). As a check of our computation, we verify that the one-loop contribution to the Anomalous Magnetic Moment vanishes if the model has spacetime supersymmetry, as required by the supersymmetric sum rules.
Two-dimensional sigma models are defined for the new manifestly spacetime supersymmetric description of four-dimensional compactified superstrings. The resulting target-superspace effective action is constrained by the way the spacetime dilaton couples to the worldsheet curvature: For the heterotic superstring, the worldsheet curvature couples to the real part of a chiral multiplet, and for Type II it couples to the real part of the sum of a vector multiplet and a tensor hypermultiplet. For the Type II superstring, this contradicts the standard folklore that only a hypermultiplet counts string…
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 discuss the prepotential describing the effective field theory of N=2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duali…
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…
We study compactifications of the heterotic pure spinor superstring to six and four dimensions focusing on two simple Calabi-Yau orbifolds. We show that the correct spectrum can be reproduced only if, in the twisted sector, there remain exactly 5 and 2 pure spinor components untwisted, respectively. This naturally defines a "small" Hilbert space of untwisted variables. We point out that the cohomology of the reduced differential on this small Hilbert space can be used to describe the states in the untwisted sector, provided certain auxiliary constraints are defined. In dimension six, the misma…
We compute explicitly the first-order in $α'$ corrections to a family of solutions of the Heterotic Superstring effective action that describes fundamental strings with momentum along themselves, parallel to solitonic 5-branes with Kaluza-Klein monopoles (Gibbons-Hawking metrics) in their transverse space. These solutions correspond to 4-charge extremal black holes in 4 dimensions upon dimensional reduction on $\mathrm{T}^{6}$. We show that some of the $α'$ corrections can be cancelled by introducing solitonic $\mathrm{SU}(2)\times \mathrm{SU}(2)$ Yang-Mills fields, and that this family of $α'…
We derive the conditions for the one-loop contribution to the cosmological constant to be exponentially suppressed in a class of heterotic string compactifications with three generations of chiral matter, where supersymmetry is spontaneously broken à la Scherk-Schwarz. Using techniques of partial unfolding based on the Hecke congruence subgroup $Γ_0(2)$ of the modular group to extract the leading asymptotics, we show that the super no-scale condition $n_B=n_F$ between the degeneracies of massless bosons $n_B$ and fermions $n_F$ in the full string spectrum is necessary but not sufficient, and n…
In a previous paper, we introduced a heterotic standard model and discussed its basic properties. This vacuum has the spectrum of the MSSM with one additional pair of Higgs-Higgs conjugate fields and a small number of uncharged moduli. In this paper, the requisite vector bundles are formulated; specifically, stable, holomorphic bundles with structure group SU(N) on smooth Calabi-Yau threefolds with Z_3 x Z_3 fundamental group. A method for computing bundle cohomology is presented and used to evaluate the cohomology groups of the standard model bundles. It is shown how to determine the Z_3 x Z_…
The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, …
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 a family of unstable heterotic string theories in more than ten dimensions which are connected via tachyon condensation to the ten-dimensional supersymmetric vacuum of heterotic string theory with gauge group SO(32). Calculating the spectrum of these theories, we find evidence for an S-duality which relates type I string theory in ten dimensions with n additional ninebrane-antininebrane pairs to heterotic string theory in 10+n dimensions with gauge group SO(32+n). The Kaluza-Klein modes of the supercritical dimensions are dual to non-singlet bound states of open strings.
Fractional superstrings are recently-proposed generalizations of the traditional superstrings and heterotic strings. They have critical spacetime dimensions which are less than ten, and in this paper we investigate model-building for the heterotic versions of these new theories. We concentrate on the cases with critical spacetime dimensions four and six, and find that a correspondence can be drawn between the new fractional superstring models and a special subset of the traditional heterotic string models. This allows us to generate the partition functions of the new models, and demonstrate th…
We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N=1 spacetime supersymmetric background. The genus g topological partition function $F^g$ corresponds to a term in the effective action of the form $W^{2g}$, where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the fo…
These lectures begin by reviewing the evidence for S duality of the toroidally compactified heterotic string in 4d that was obtained in the period 1992--94. Next they review recently discovered dualities that relate all five of the 10d superstring theories and a quantum extension of 11d supergravity called M theory. The study of p-branes of various dimensions (some of which are D-branes) plays a central role. The final sections survey supersymmetric string vacua in 6d and some of the dual constructions by which they can be obtained. Special emphasis is given to a class of N=1 models that exhib…
We investigate the connections between four-dimensional, N=2 M-theory vacua constructed as orbifolds of type II, heterotic, and type I strings. All these models have the same massless spectrum, which contains an equal number of vector multiplets and hypermultiplets, with a gauge group of the maximal rank allowed in a perturbative heterotic string construction. We find evidence for duality between two type I compactifications recently proposed and a new heterotic construction that we present here. This duality allows us to gain insight into the non-perturbative properties of these models. In pa…
We study supersymmetry conditions for the heterotic pure superstring preserving and N=1 supersymmetry in four dimensions directly from the curved superspace defined by the Berkovits-Howe constraints.
We construct a supersymmetric standard model in the context of the Z_{12-I} orbifold compactification of the E_8 x E_8' heterotic string theory. The gauge group is SU(3)_c x SU(2)_L x U(1)_Y x U(1)^4 x [SO(10) x U(1)^3]' with sin^2θ_W = 3/8. We obtain three families of SO(10) spinor-like chiral matter states, and Higgs doublets. All other extra states are exactly vector-like under the standard model gauge symmetry. There are numerous standard model singlets, many of which get VEVs such that only the standard model gauge symmetry survives and desired Yukawa couplings can be generated at lower e…
Superunification underwent a major paradigm shift in 1984 when eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings. This last year has witnessed a new shift of equal proportions: perturbative ten-dimensional superstrings have in their turn been superseded by a new non-perturbative theory called {\it $M$-theory}, which describes supermembranes and superfivebranes, which subsumes all five consistent string theories and whose low energy limit is, ironically, eleven-dimensional supergravity. In particular, six-dimensional string/string duality follows from …
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 have explored the possibility that the universe at very early stage was dominated by (macroscopic) heterotic strings. We have found that the dimensionless parameter $Gμ$ for the heterotic strings varies from $10^{-2}$ to $10^{-4}$ as the universe evolve from the matter dominance to radiation dominance. This led to the interesting consequence of epoch dependent gauge coupling constant. The gauge coupling constant at early times was found to be much stronger than the present strong interaction.
The lengthy discussion and derivations given in this paper have been corrected, and completed with additional results and greater simplicity. We show that the high temperature limits of the heterotic E8xE8 and Spin(32)/Z2 strings and their Type I A/B superstring duals are finite and convergent. The Hagedorn growth of the degeneracies in the string mass level expansion is suppressed by an exponential linear in the mass level number for both heterotic strings, and suppressed by the exponential of the negative square root of the mass level number for the Type IB superstring. However, in the massl…
We show that the high temperature limits of the heterotic E8xE8 and Spin(32)/Z2 strings and their Type I A/B superstring duals are finite and convergent. The Hagedorn growth of the degeneracies in the string mass level expansion is suppressed by an exponential linear in the mass level number for both heterotic strings, and suppressed by the exponential of the negative square root of the mass level number for the Type IB superstring. However, in the massless gauge field theoretic limit of the Type IB open and closed superstring, we find clear evidence for the thermal deconfinement phase transit…
This thesis is the discussion of heterotic and type I string phenomenology. The heterotic string model is based on the free--fermionic formalism. This is the first case where non--Abelian VEV's, as opposed to singlet VEV's are required for the cancellation of the Fayet--Iliopoulos term. It is noted that non--Abelian fields are the only fields that can give rise to the satisfaction of the D--flat constraints in this model. The type I models are based on T^6/Z_2x(Z_2)^s and T^6/Z_2xZ_2x(Z_2)^s compactifications. The first example has N=2 supersymmetry and includes a rank reduction of the D5 gaug…
We develop techniques to classify D- and F-flat directions for N=1 supersymmetric string vacua of the perturbative heterotic string theory, which possess an anomalous U(1) gauge group at the tree level. Genus-one corrections generate a Fayet-Iliopoulos term for the D-term of U(1)_A, which is canceled by non-zero vacuum expectation values (VEVs) of certain massless multiplets in such a way that the anomalous U(1) is broken, while maintaining the D- and F-flatness of the effective field theory. A systematic analysis of flat directions is given for non-zero VEVs of non-Abelian singlets, and the t…
We derive exact relations to all orders in the $α'$ expansion for the charges of a bound system of heterotic strings, solitonic 5-branes and, optionally, a Kaluza-Klein monopole. The expressions, which differ from those of the zeroth-order supergravity approximation, coincide with the values obtained when only the corrections of quadratic order in curvature are included. Our computation relies on the consistency of string theory as a quantum theory of gravity; the relations follow from the matching of the Wald entropy with the microscopic degeneracy. In the heterotic frame, the higher-curvatur…
It is shown that the pure spinor formulation of the heterotic superstring in a generic gravitational and super Yang-Mills background has vanishing one-loop beta functions.
The computation of $κ$-anomalies in the Green-Schwarz heterotic superstring sigma-model and the corresponding Wess-Zumino consistency condition constitute a powerful alternative approach for the derivation of manifestly supersymmetric string effective actions. With respect to the beta-function approach this technique presents the advantage that a result which is obtained with the computation of beta-functions at $n$ loops can be obtained through the calculation of $κ$-anomalies at \hbox{$n-1$} loops. In this paper we derive by a direct one-loop perturbative computation the $κ$-anomaly associat…
The quest for the Standard Model among the huge number of string vacua is usually based on a set of phenomenological criteria related to the massless spectrum of string models. In this work we study criteria associated with interactions in the effective low energy theory and in particular with the presence of the coupling that provides mass to the top quark. Working in the context of the Free Fermionic Formulation of the heterotic superstring, we demonstrate that, in a big class of phenomenologically promising $Z_2\times Z_2$ compactifications, these criteria can be expressed entirely in terms…
We explore the problem of braneworld black holes in the heterotic braneworld scenario of Lukas, Ovrut, Stelle and Waldram (LOSW). We show that black string solutions are unstable, and demonstrate some unusual asymptotics in the linearized metric. We also solve the fully coupled brane and bulk Einstein equations, finding an exact, though singular, solution which corresponds to a brane black hole in which the branes spike apart at the Schwarzschild radius.
We analyze the free ambitwistor string field theory action for the bosonic string, heterotic string and both GSO sectors of the Type II string. The spectrum contains non-unitary states and provides an interesting consistency test for one-loop ambitwistor string computations.
