To understand in detail the contribution of a world-sheet instanton to the superpotential in a heterotic string compactification, one has to understand the moduli dependence (bundle and complex structure moduli) of the one-loop determinants from the fluctuations, which accompany the classical exponential contribution (involving Kähler moduli) when evaluating the world-volume partition function. Here we use techniques to describe geometrically these Pfaffians for spectral bundles over rational base curves in elliptically fibered Calabi-Yau threefolds, and provide a (partially exhaustive) list of cases involving {\em factorising} (or vanishing) superpotential. This gives a conceptual explanation and generalisation of the few previously known cases which were obtained just experimentally by a numerical computation.
A formalism for calculating the open supermembrane contribution to the non-perturbative superpotential of moduli in heterotic M-theory is presented. This is explicitly applied to the Calabi-Yau (1,1)-moduli and the separation modulus of the end-of-the-world BPS three-branes, whose non-perturbative superpotential is computed. The role of gauge bundles on the boundaries of the open supermembranes is discussed in detail, and a topological criterion presented for the associated superpotential to be non-vanishing.
The open supermembrane contribution to the non-perturbative superpotential of bulk space five-branes in heterotic M-theory is presented. We explicitly compute the superpotential for the modulus associated with the separation of a bulk five-brane from an end-of-the-world three-brane. The gauge and kappa-invariant boundary strings of such open supermembranes are given and the role of the holomorphic vector bundle on the orbifold fixed plane boundary is discussed in detail.
The aim of this talk is to derive two constraints imposed by supersymmetry for a class of heterotic Landau-Ginzburg models with nonholomorphic superpotentials. One of these constraints relates the nonholomorphic parameters of the superpotential to the Hermitian curvature. Various special cases of this constraint have been used to establish properties of Mathai-Quillen form analogues which arise in the corresponding heterotic Landau-Ginzburg models. The other constraint was not anticipated from studies of Mathai-Quillen form analogues.
We study the superpotential for the heterotic string compactified on non-Kahler complex manifolds. We show that many of the geometrical properties of these manifolds can be understood from the proposed superpotential. In particular we give an estimate of the radial modulus of these manifolds. We also show, how the torsional constraints can be obtained from this superpotential.
The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technica…
We present models of heterotic compactification on Calabi-Yau threefolds and compute the non-perturbative superpotential for vector bundle moduli. The key feature of these models is that the threefolds, which are elliptically fibered over del Pezzo surfaces, have homology classes with a unique holomorphic, isolated genus-zero curve. Using the spectral cover construction, we present vector bundles for which we can explicitly calculate the Pfaffians associated with string instantons on these curves. These are shown to be non-zero, thus leading to a non-vanishing superpotential in the 4D effectiv…
Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi-Yau threefold with a specific F-theory Calabi-Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi-Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing hetero…
We consider N =1 compactifications to four dimensions of heterotic string theory in the presence of fluxes. We show that up to order O(α'^2) the associated action can be written as a sum of squares of BPS-like quantities. In this way we prove that the equations of motion are solved by backgrounds which fulfill the supersymmetry conditions and the Bianchi identities. We also argue for the expression of the related superpotential and discuss the radial modulus stabilization for a class of examples.
We compare the N=1 F-theory compactification of Donagi, Grassi and Witten with modular superpotential - and some closely related models - to dual heterotic models. We read of the F-theory spectrum from the cohomology of the fourfold and discuss on the heterotic side the gauge bundle moduli sector (including the spectral surface) and the necessary fivebranes. Then we consider the N=1 superpotential and show how a heterotic superpotential matching the F-theory computation is built up by worldsheet instantons. Finally we discuss how the original modular superpotential should be corrected by an ad…
We extend the classical heterotic instanton solutions to all orders in $α'$ using the equations of anomaly-free supergravity, and discuss the relation between these equations and the string theory $β$-functions.
An explicit orbifold example of the non-zero correlation functions related to the additional contribution to the induced mass term for Higgs particles at low energies is given. We verify that they form finite dimensional representations of the target space modular transformation $SL_2(Z)$. This action of the modular group is shown to be consistent with its action on the fixed points set defining the twisted fields.
We study the non-perturbative superpotential in E_8 x E_8 heterotic string theory on a non-simply connected Calabi-Yau manifold X, as well as on its simply connected covering space \tilde{X}. The superpotential is induced by the string wrapping holomorphic, isolated, genus 0 curves. According to the residue theorem of Beasley and Witten, the non-perturbative superpotential must vanish in a large class of heterotic vacua because the contributions from curves in the same homology class cancel each other. We point out, however, that in certain cases the curves treated in the residue theorem as ly…
We study non-perturbative effects due to a heterotic M-theory five-brane wrapped on Calabi-Yau threefold. We show that such instantons contribute to derivative F-terms described recently by Beasley and Witten rather than to the superpotential.
We explore a relation between four-dimensional N=2 heterotic vacua induced by Mirror Symmetry via Heterotic/Type II duality. It allows us to compute the α' corrections to the hypermultiplet moduli space of heterotic compactifications on K3xT^2 in the limit of large base of the elliptic K3. We concentrate on the case of point-like instantons on orbifold singularities leading to low-dimensional hypermultiplet moduli spaces.
We derive the most general flux-induced superpotential for N=1 M-theory compactifications on seven-dimensional manifolds with SU(3) structure. Imposing the appropriate boundary conditions, this result applies for heterotic M-theory. It is crucial for the latter to consider SU(3) and not G_2 group structure on the seven-dimensional internal space. For a particular background that differs from CY(3) x S^1/Z_2 only by warp factors, we investigate the flux-generated scalar potential as a function of the orbifold length. We find a positive cosmological constant minimum, however at an undesirably la…
We discuss instanton fermionic zero modes in the heterotic N=(0,2) modification of the CP(1) sigma model in two dimensions. By calculating its chiral anomaly we prove that the number of fermionic zero modes is same as in the standard N=(2,2) CP(1) case, and determine their explicit form.
The Affleck-Dine-Seiberg instanton generated superpotential for SQCD with Nf=Nc-1 flavours is explicitly derived from a local model of engineered intersecting D6-branes with a single E2-instanton. This computation extends also to symplectic gauge groups with Nf=Nc flavours.
We use string field theory to fix the normalization of the D-instanton corrections to the superpotential involving the moduli fields of type II string theory compactified on an orientifold of a Calabi-Yau threefold in the absence of fluxes. We focus on $O(1)$ instantons whose only zero modes are the four bosonic modes associated with translation of the instanton in non-compact directions and a pair of fermionic zero modes associated with the two supercharges broken by the instanton. We work with a generic superconformal field theory and express our answer in terms of the spectrum of open strin…
We discuss the behaviour of non-perturbative superpotentials in 4d N=1 type II compactifications (and orientifolds thereof) near lines of marginal stability, where the spectrum of contributing BPS D-brane instantons changes discontinuously. The superpotential is nevertheless continuous, in agreement with its holomorphic dependence on the closed string moduli. The microscopic mechanism ensuring this continuity involves novel contributions to the superpotential: As an instanton becomes unstable against decay to several instantons, the latter provide a multi-instanton contribution which reconstru…
This is a transcript of lectures given at the Sixth Jorge Andre Swieca Summer School in Theoretical Physics. The subject of these lectures is soliton solutions of string theory. We construct a class of exact conformal field theories possessing a spacetime soliton or instanton interpretation and present a preliminary discussion of their physical properties.
We continue our study of heterotic/type-I duality in D<10 dimensions. We consider the heterotic and type-I theories compactified on tori to lower dimensions. We calculate the special (``BPS saturated'') F^4 and R^4 terms in the effective one-loop heterotic action. These terms are expected to be non-perturbatively exact for D>4. The heterotic result is compared with the associated type-I result. In D<9 dimensions, the type-I theory has instanton corrections due to D1 instantons. In D=8 we use heterotic-type I duality to give a simple prescription of the D-instanton calculation on the type I sid…
We review the duality between heterotic and F--theory string vacua with N=1 space-time supersymmetry in eight, six and four dimensions. In particular, we discuss two chains of four-dimensional F--theory/heterotic dual string pairs, where F--theory is compactified on certain elliptic Calabi-Yau fourfolds, and the dual heterotic vacua are given by compactifications on elliptic Calabi-Yau threefolds plus the specification of the $E_8\times E_8$ gauge bundles. We show that the massless spectra of the dual pairs agree by using, for one chain of models, an index formula to count the heterotic bundle…
We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all $\mathbb{P}^1$ curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a com…
We study different aspects of the non-perturbative superpotentials induced by Euclidean E3-branes on systems of D3/D7-branes located at Abelian orbifold singularities. We discuss in detail how the induced couplings are consistent with the U(1) symmetries carried by the D3/D7-branes. We construct different compact and non-compact examples, and show phenomenologically relevant couplings like $μ$-terms or certain Yukawa couplings generated by these E3 instantons. Some other novel effects are described. We show an example where E3-instantons combine with standard gauge instantons to yield new mult…
We study the F-terms in N=1 supersymmetric, d=4 gauge theories arising from D(p+3)-branes wrapping supersymmetric p-cycles in a Calabi-Yau threefold. If p is even the spectrum and superpotential for a single brane are determined by purely classical ($α^\prime \to 0$) considerations. If p=3, superpotentials for massless modes are forbidden to all orders in $α^\prime$ and may only be generated by open string instantons. For this latter case we find that such instanton effects are generically present. Mirror symmetry relates even and odd p and thus perturbative and nonperturbative superpotentials…
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.
We consider non-perturbative superpotentials from world-sheet instantons wrapped on holomorphic genus zero curves in heterotic string theory. These superpotential contributions feature prominently in moduli stabilization and large field axion inflation, which makes their presence or absence, as well as their functional dependence on moduli, an important issue. We develop geometric methods to compute the instanton superpotentials for heterotic string theory with monad and extension bundles. Using our methods, we find a variety of examples with a non-vanishing superpotential. In view of standard…
We study heterotic/type I duality in d=8,9 uncompactified dimensions. We consider the special (``BPS saturated'') F^4 and R^4 terms in the effective one-loop heterotic action, which are expected to be non-perturbatively exact. Under the standard duality map these translate to tree-level, perturbative and non-perturbative contributions on the type I side. We check agreement with the one-loop open string calculation, and discuss the higher-order perturbative contributions, which arise because of the mild non-holomorphicities of the heterotic elliptic genus. We put the heterotic world-sheet insta…
In this paper we show that it is possible to derive non-perturbative superpotential terms from a stringy instanton without introducing orientifold planes. The instanton is realized by a Euclidean D brane wrapping a non-trivial cycle upon which we also wrap a single space-filling D brane. The standard problem of unwanted neutral fermionic zero modes is evaded by the appearance of couplings to charged bosonic zero modes in the instanton moduli action. Since the Euclidean D brane wraps a cycle which is not associated to any low energy gauge dynamics, it can not be interpreted as an ordinary gauge…
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…
We consider two simple examples of multi-instanton configurations in type II 4d N=1 superstring compactifications. The first one involves O(1) and U(1) D2-instantons embedded in T^6/(Z_2 x Z_2') geometry with SU(4) gauge symmetry coming from D6-branes and the second is related to quiver gauge theory of orientifolded orbifold of conifold containing fractional D(-1)-instantons and D3-branes. The additional zero modes of instantons which appear at the intersection points are lifted through interactions stemming from dimensionally reduced F- and D-terms. It is shown that multi-instanton configurat…
In this paper we compute gaugino and scalar condensates in N=1 supersymmetric gauge theories with and without massive adjoint matter, using localization formulae over the multi--instanton moduli space. Furthermore we compute the chiral ring relations among the correlators of the $N=1^*$ theory and check this result against the multi-instanton computation finding agreement.
The static Yang-Mills-Higgs dyonic instanton is shown to have a non-vanishing, but anti-self-dual, angular momentum 2-form with skew eigenvalues equal to the electric charge; for large charge the angular momentum causes the instanton to expand into a hyper-spherical shell. A class of exact multi dyonic instantons is then found and then generalized to a new class of 1/4 supersymmetric, non-singular, stationary, exact solutions of the ten-dimensional supergravity/Yang-Mills theory. These self-gravitating dyonic instantons yield new heterotic string solitons, to leading order in the inverse strin…
We derive the precise relation between level matching condition and fractional instanton numbers in six dimensional, abelian and supersymmetric orbifolds of E8 x E8 heterotic string theory. The fractional part of the two E8 instanton numbers is explicitly calculated in terms of the gauge twist. This relation is then used to show that the classification of these orbifolds can be given in terms of flat bundles away from the orbifold singularities under the only constraint that the sum of the fractional parts of the gauge instanton numbers match the fractional part of the gravitational instanton …
We compute the partition function for the exotic instanton system corresponding to D-instantons on D7 branes in Type I' theory. We exploit the BRST structure of the moduli action and its deformation by RR background to fully localize the integration. The resulting prepotential describes non-perturbative corrections to the quartic couplings of the gauge field F living on the D7's. The results match perfectly those obtained in the dual heterotic theory from a protected 1-loop computation, thus providing a non-trivial test of the duality itself.
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 …
Hypermultiplet couplings in type IIA string theory on a Calabi-Yau space can be quantum corrected by D2-brane instantons wrapping special Lagrangian cycles. On the other hand, hypermultiplet couplings in the heterotic string on a K3 surface are corrected by world-sheet instantons wrapping curves. In a class of examples, we relate these two sets of instanton corrections. We first present an analogue of the c-map for the heterotic string via a dual flux compactification of M-theory. Using this duality, we propose two ways of capturing quantum corrections to hypermultiplets. We then use the orien…
As a first step towards computing instanton-generated superpotentials in heterotic standard model vacua, we determine the Gromov-Witten invariants for a Calabi-Yau threefold with fundamental group pi_1(X)=Z_3 x Z_3. We find that the curves fall into homology classes in H_2(X,Z)=Z^3+(Z_3+Z_3). The unexpected appearance of the finite torsion subgroup in the homology group complicates our analysis. However, we succeed in computing the complete genus-0 prepotential. Expanding it as a power series, the number of instantons in any integral homology class can be read off. This is the first explicit c…
D-instanton effects are studied for the IIB orientifold T^2/IΩ(-1)^{F_L} of Sen using type I/heterotic duality. An exact one loop threshold calculation of t_8 \tr F^4 and t_8(\tr F^2)^2 terms for the heterotic string on T^2 with Wilson lines breaking SO(32) to SO(8)^4 is related to D-instanton induced terms in the worldvolume of D7 branes in the orientifold. Introducing D3 branes and using the AdS/CFT correspondence in this case, these terms are used to calculate Yang-Mills instanton contributions to four point functions of the large N_c limit of N=2 USp(2N_c) SYM with four fundamental and one…
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 study superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications for vector bundles constructed from line bundle sums, monads and extensions. Within a certain class of manifolds and for certain second homology classes, we derive simple necessary conditions for a non-vanishing instanton superpotential. These show that non-vanishing instanton superpotentials are rare and require a specific pattern for the bundle construction. For the class of monad and extension bundles with this pattern, we derive a sufficient criterion for non-vanishing instanton superpotentials ba…
In this note we explore the stringy interpretation of non-perturbative effects in N=1^* deformations of the A_{k-1} quiver models. For certain types of deformations we argue that the massive vacua are described by Nk fractional D3-branes at the orbifold polarizing into k concentric 5-brane spheres each carrying fractional brane charge. The polarization of the D3-branes induces a polarization of D-instantons into string world-sheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these world-sheet instantons. We point out that for certain p…
We investigate nonperturbative effects in M-theory compactifications arising from wrapped membranes. In particular, we show that in $d=4, \mathcal{N}=1$ compactifications along manifolds of $G_2$ holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize many M-theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invarian…
We discuss non-perturbative phase transitions, within the context of heterotic M-theory, which occur when all, or part, of the wrapped five-branes in the five-dimensional bulk space come into direct contact with a boundary brane. These transitions involve the transformation of the five-brane into a ``small instanton'' on the Calabi-Yau space at the boundary brane, followed by the ``smoothing out'' of the small instanton into a holomorphic vector bundle. Small instanton phase transitions change the number of families, the gauge group or both on the boundary brane, depending upon whether a base …
We show how the level matching condition in six dimensional, abelian and supersymmetric orbifolds of the E_8 x E_8 heterotic string can be given equivalently in terms of fractional gauge and gravitational instanton numbers. This relation is used to restate the classification of the orbifolds in terms of flat bundles away from the orbifold singularities under the constraint of the level matching condition. In an outlook these results are applied to Kaluza-Klein monopoles of the heterotic string on S^1 in Wilson line backgrounds.
The non-perturbative superpotential can be effectively calculated in $M$-theory compactification to three dimensions on a Calabi-Yau four-fold $X$. For certain $X$, the superpotential is identically zero, while for other $X$, a non-perturbative superpotential is generated. Using $F$-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with $N=1$ supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come …
We study type IIA string theory compactified on Calabi-Yau fourfolds and heterotic string theory compactified on Calabi-Yau threefolds times a two-torus. We derive the resulting effective theories which have two space-time dimensions and preserve four supercharges. The duality between such vacua is established at the level of the effective theory. For type IIA vacua with non-trivial Ramond-Ramond background fluxes a superpotential is generated. We show that for a specific choice of background fluxes and a fourfold which has the structure of a threefold fibred over a sphere the superpotential c…
Two classes of stringy instanton effects, stronger than standard field theory instantons, are identified in the heterotic string theory. These contributions are established using type IIA/heterotic and type I/heterotic dualities. They provide examples for the heterotic case of the effects predicted by Shenker based on the large-order behavior of perturbation theory. The corrections vanish as the radius of the compactification goes to infinity. For appropriate amplitudes, they are computable worldsheet or worldline instanton effects on the dual side. Some potential applications are discussed.
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 consider heterotic string theories compactified on a K3 surface which lead to an unbroken perturbative gauge group of Spin(32)/Z2. All solutions obtained are combinations of two types of point-like instanton --- one ``simple type'' as discovered by Witten and a new type associated to the ``generalized second Stiefel-Whitney class'' as introduced by Berkooz et al. The new type of instanton is associated to an enhancement of the gauge symmetry by Sp(4) and the addition of a massless tensor supermultiplet. It is shown that if four simple instantons coalesce at an orbifold point in the K3 surfa…
In this paper we explore contributions to non-perturbative superpotentials arising from instantons wrapping effective divisors in smooth Calabi-Yau four-folds. We concentrate on the case of manifolds constructed as complete intersections in products of projective spaces (CICYs) or generalizations thereof (gCICYs). We systematically investigate the structure of the cone of effective (algebraic) divisors in the four-fold geometries and employ the same tools recently developed in arXiv:1507.03235 to construct more general instanton geometries than have previously been considered in the literature…
We construct an octonionic instanton solution to the seven dimensional Yang-Mills theory based on the exceptional gauge group $G_2$ which is the automorphism group of the division algebra of octonions. This octonionic instanton has an extension to a solitonic two-brane solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to $N=1$ space-time supersymmetry in the (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.
We review some exact solitonic solutions of string theory with higher-membrane structure. These include an axionic instanton solution of bosonic string theory as well as multi-instanton and multimonopole solutions of heterotic string theory. The heterotic solutions reveal some interesting aspects of string theory as a theory of quantum gravity.
Following the work of Lima et al on the exact evaluation of the nonperturbative contribution to the superpotential from open-membrane instanton in Heterotic M-Theory, we evaluate systematically the contribution to the superpotential of a membrane instanton obtained by wrapping of a single M2 brane, once, on an isolated supersymmetric 3-cycle in a G_2-holonomy manifold. We then try to relate the results obtained to those sketched out in Harvey and Moore. We also do a heat-kernel asymptotics analysis to see whether one gets similar UV-divergent terms for (one or both of) the bosonic and fermioni…
We give a detailed microscopic derivation of gauge and stringy instanton generated superpotentials for gauge theories living on D3-branes at Z_3-orientifold singularities. Gauge instantons are generated by D(-1)-branes and lead to Affleck, Dine and Seiberg (ADS) like superpotentials in the effective N=1 gauge theories with three generations of bifundamental and anti/symmetric matter. Stringy instanton effects are generated by Euclidean ED3-branes wrapping four-cycles on T^6/\Z_3. They give rise to Majorana masses in one case and non-renormalizable superpotentials for the other cases. Finally w…
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.
