Non-perturbative Superpotentials Research Articles

D-instanton, threshold corrections, and topological string

In this note, we prove that the one-loop pfaffian of the non-perturbative superpotential generated by Euclidean D-branes in type II compactifications on orientifolds of Calabi-Yau threefolds is determined by the moduli integral of the new supersymmetric index defined by Cecotti, Fendley, Intriligator, and Vafa. As this quantity can be computed via topological string theory, Chern-Simons theory, matrix models, or by solving the holomorphic anomaly equation, this result provides a method to directly compute the one-loop pfaffian of the non-perturbative superpotential. The relation between the one-loop pfaffian, threshold corrections to the gauge coupling, and the one-loop partition function of open topological string theory is discussed.

Superpotentials from singular divisors

We study Euclidean D3-branes wrapping divisors D in Calabi-Yau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf {mathcal{O}}_D applies when D is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf {mathcal{O}}_{overline{D}} of the normalization overline{D} of D. We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, {h}_{+}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(1,0,0right) and {h}_{-}^{bullet}left({mathcal{O}}_{overline{D}}right)=left(0,0,0right) give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups Γ. We use the action of Γ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strong-weak coupling dualities involving inversion of divisor volumes.

Topological constraints in the LARGE-volume scenario

We elaborate on recent results regarding the self-consistency of de Sitter vacua in the LARGE-volume scenario of type IIB string theory. In particular, we analyze to what extent the control over warping, curvature and gs corrections depends on the topology and the orientifold/brane data of a compactification. We compute a general bound on the magnitude of these corrections which strongly constrains the D3 tadpole. The minimally required tadpole ranges from mathcal{O} (500) to mathcal{O} (106) or more and depends strongly on other data, in particular on the Euler number of the Calabi-Yau 3-fold, the triple-self-intersection and Euler numbers of the small divisor and the coefficient as appearing in the non-perturbative superpotential. We give arguments suggesting that satisfying these constraints is very challenging and perhaps impossible.

Monodromy charge in D7-brane inflation

In axion monodromy inflation, traversing N axion periods corresponds to discharging N units of a quantized charge. In certain models with moving D7-branes, such as Higgs-otic inflation, this monodromy charge is D3-brane charge induced on the D7-branes. The stress-energy of the induced charge affects the internal space, changing the inflaton potential and potentially limiting the field range. We compute the backreaction of induced D3-brane charge in Higgs-otic inflation. The effect on the nonperturbative superpotential is dramatic even for N = 1, and may preclude large-field inflation in this model in the absence of a mechanism to control the backreaction.

Heterotic instantons for monad and extension bundles

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 vanishing theorems, we speculate that these results are likely to be attributed to the non-compactness of the instanton moduli space. We test this proposal, for the case of monad bundles, by considering gauged linear sigma models where compactness of the instanton moduli space can be explicitly checked. In all such cases, we find that the geometric results are consistent with the vanishing theorems. Surprisingly, linearly dependent Pfaffians even arise for cases with a non-compact instanton moduli space. This suggests some gauged linear sigma models with a non-compact instanton moduli space may still have a vanishing instanton superpotential.

Modular symmetries and the swampland conjectures

Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d mathcal{N}=1 compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the Kähler and complex structure moduli, in particular SL(2, Z) T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I’ or Horava-Witten theory, in which the SL(2, Z) acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. The non-perturbative examples explored point to potentials not leading to weak coupling at infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamically forbidding the access to points with global symmetries. We perform a study of general modular invariant potentials and find that there are dS maxima and saddle points but no dS minima, and that all examples explored obey the refined dS conjecture.

Non-Perturbative Superpotentials and Discrete Torsion

We discuss the non-perturbative superpotential in $${{E}_{8}} \times {{E}_{8}}$$ 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 zero curves. We show, in a specific example, that the superpotential is non-zero both on $$\tilde {X}$$ and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.

Infinitely many M2-instanton corrections to M-theory on G2-manifolds

We consider the non-perturbative superpotential for a class of four-dimensional mathcal{N}=1 vacua obtained from M-theory on seven-manifolds with holonomy G2. The class of G2-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over S3. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of [1], which relates M-theory on TCS G2-manifolds to E8 × E8 heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the G2-compactification.

One-loop Pfaffians and large-field inflation in string theory

We study the consistency of large-field inflation in low-energy effective field theories of string theory. In particular, we focus on the stability of Kähler moduli in the particularly interesting case where the non-perturbative superpotential of the Kähler sector explicitly depends on the inflaton field. This situation arises generically due to one-loop corrections to the instanton action. The field dependence of the modulus potential feeds back into the inflationary dynamics, potentially impairing slow roll. We distinguish between world-sheet instantons from Euclidean D-branes, which typically yield polynomial one-loop Pfaffians, and gaugino condensates, which can yield exponential or periodic corrections. In all scenarios successful slow-roll inflation imposes bounds on the magnitude of the one-loop correction, corresponding to constraints on possible compactifications. While we put a certain emphasis on Type IIB constructions with mobile D7-branes, our results seem to apply more generally.

Weak gravity conjecture as a razor criterium for exotic D-brane instantons

We discuss implications of weak gravity conjecture (WGC) for exotic D-brane instantons. In particular, WGC leads to indirect stringent bounds on non-perturbative superpotentials generated by exotic instantons with many implications for phenomenology: R-parity violating processes, neutrino mass, [Formula: see text]-problem, neutron–antineutron transitions and collider physics.

De Sitter vacua and supersymmetry breaking in six-dimensional flux compactifications

We consider six-dimensional supergravity with Abelian bulk flux compactified on an orbifold. The effective low-energy action can be expressed in terms of N=1 chiral moduli superfields with a gauged shift symmetry. The D-term potential contains two Fayet-Iliopoulos terms which are induced by the flux and by the Green-Schwarz term canceling the gauge anomalies, respectively. The Green-Schwarz term also leads to a correction of the gauge kinetic function which turns out to be crucial for the existence of Minkowski and de Sitter vacua. Moduli stabilization is achieved by the interplay of the D-term and a nonperturbative superpotential. Varying the gauge coupling and the superpotential parameters, the scale of the extra dimensions can range from the GUT scale down to the TeV scale. Supersymmetry is broken by F- and D-terms, and the scale of gravitino, moduli, and modulini masses is determined by the size of the compact dimensions.

Instanton superpotentials, Calabi-Yau geometry, and fibrations

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 provide examples of instanton configurations on Calabi-Yau manifolds that are elliptically and $K3$-fibered and explore their consequences in the context of string dualities. The examples discussed include manifolds containing infinite families of divisors with arithmetic genus, $\chi(D, \mathcal O_D)=1$ and superpotentials exhibiting modular symmetry.

Inflation with Fayet-Iliopoulos terms

Two of the most attractive realizations of inflation in supergravity are based upon the presence of a constant Fayet-Iliopoulos (FI) term. In D-term hybrid inflation it is the FI term itself which sets the energy scale of inflation. Alternatively, the breaking of a U(1) symmetry induced by the FI term can dynamically generate the quadratic potential of chaotic inflation. The purpose of this note is to study the possible UV embedding of these schemes in terms of the `field-dependent FI term' related to a string modulus field which is stabilized by a non-perturbative superpotential. We find that in settings where the FI term drives inflation, gauge invariance prevents a decoupling of the modulus from the inflationary dynamics. The resulting inflation models generically contain additional dynamical degrees of freedom compared to D-term hybrid inflation. However, the dynamical realization of chaotic inflation can be obtained in complete analogy to the case of a constant FI term. We present a simple string-inspired toy model of this type.

Gauge threshold corrections and field redefinitions

We review the argument for field redefinitions arising from threshold corrections to heterotic string gauge couplings, and the relation between the linear and the chiral multiplet. In the type IIB case we argue that the necessity for moduli mixing at one-loop order has not been clearly established, since this is based on extending the background field expansion way beyond its regime of validity. We also resolve some issues related to the form of non-perturbative terms resulting from gaugino condensation. This enables us to estimate the effective cutoff in the field theory by evaluating the non-perturbative superpotential by two different methods, and find that it is around the Kaluza–Klein scale, as one might have expected on general grounds of self-consistency.

Non-perturbative superpotentials in F-theory and string duality

We use open-closed string duality between F-theory on K3xK3 and type II strings on CY manifolds without branes to study non-perturbative superpotentials in generalized flux compactifications. On the F-theory side we obtain the full flux potential including D3-instanton contributions and show that it leads to an explicit and simple realization of the three ingredients of the KKLT model for stringy dS vacua. The D3-instanton contribution is highly non-trivial, can be systematically computed including the determinant factors and demonstrates that a particular flux lifts very effectively zero modes on the instanton. On the closed string side, we propose a generalization of the Gukov-Vafa-Witten superpotential for type II strings on generalized CY manifolds, depending on all moduli multiplets.

Sequestering in string compactifications

We study the mediation of supersymmetry breaking in string compactifications whose moduli are stabilized by nonperturbative effects. We begin with a critical review of arguments for sequestering in supergravity and in string theory. We then show that geometric isolation, even in a highly warped space, is insufficient to achieve sequestering: in type IIB compactifications, nonperturbative superpotentials involving the Kahler moduli introduce cross-couplings between well-separated visible and hidden sectors. The scale of the resulting soft terms depends on the moduli stabilization scenario. In the Large Volume Scenario, nonperturbative superpotential contributions to the soft trilinear $A$ terms can introduce significant flavor violation, while in KKLT compactifications their effects are negligible. In both cases, the contributions to the $\mu$ and $B\mu$ parameters cannot be ignored in general. We conclude that sequestered supersymmetry breaking is possible in nonperturbatively-stabilized compactifications only if a mechanism in addition to bulk locality suppresses superpotential cross-couplings.

Quantum corrections to heterotic moduli potentials

In a recent paper, we derived the leading alpha' corrections to the Kahler potentials for moduli in (0,2) heterotic compactifications. In the same spirit as the LARGE volume stabilization scenario for type IIB orientifolds, we examine whether these quantum corrections, together with a combination of tree-level and non-perturbative superpotentials, are sufficient to stabilize the overall volume modulus at large values. This is not a priori obvious, since the corrections we found are of a lower order than those used in the type IIB setting. Nevertheless, we find that stabilizing the volume at (exponentially) large values may be possible (under certain conditions) in these heterotic backgrounds.

Low-scale inflation and supersymmetry breaking in racetrack models

In many moduli stabilization schemes in string theory, the scale of inflation appears to be of the same order as the scale of supersymmetry breaking. For low-scale supersymmetry breaking, therefore, the scale of inflation should also be low, unless this correlation is avoided in specific models. We explore such a low-scale inflationary scenario in a racetrack model with a single modulus in type IIB string theory. Inflation occurs near a point of inflection in the K\"ahler modulus potential. Obtaining acceptable cosmological density perturbations leads to the introduction of magnetized D7-branes sourcing non-perturbative superpotentials. The gravitino mass, m_{3/2}, is chosen to be around 30 TeV, so that gravitinos that are produced in the inflaton decay do not affect big-bang nucleosynthesis. Supersymmetry is communicated to the visible sector by a mixture of anomaly and modulus mediation. We find that the two sources contribute equally to the gaugino masses, while scalar masses are decided mainly by anomaly contribution. This happens as a result of the low scale of inflation and can be probed at the LHC.

NOTES ON UNORIENTED D-BRANE INSTANTONS

We review basic aspects of worldsheet and penta-brane instantons as well as (unoriented) D-brane instantons, which is our main focus here, and threshold corrections to BPS-saturated couplings. We then consider nonperturbative superpotentials generated by "gauge" and "exotic" instantons living on D3-branes at orientifold singularities. Moreover, we discuss the interplay between worldsheet and D-string instantons on T4/Z2. We focus on a 4-Fermi amplitude, give Heterotic and perturbative Type I descriptions, and offer a multi-D-string instanton interpretation. We conclude with possible interesting developments.

Non-perturbative effects and wall-crossing from topological strings

We argue that the Gopakumar-Vafa interpretation of the topological string partition function can be used to compute and resum certain non-perturbative brane instanton effects of type II CY compactifications. In particular the topological string A-model encodes the non-perturbative corrections to the hypermultiplet moduli space metric from general D1/D(-1)-brane instantons in 4d = 2 IIB models. We also discuss the reduction to 4d = 1 by fluxes and/or orientifolds and/or D-branes, and the prospects to resum brane instanton contributions to non-perturbative superpotentials. We argue that the connection between non-perturbative effects and the topological string underlies the continuity of non-perturbative effects across lines of BPS stability. We also confirm this statement in mirror B-model matrix model examples, relating matrix model instantons to non-perturbative D-brane instantons. The computation of non-perturbative effects from the topological string requires a 3d circle compactification and T-duality, relating effects from particles and instantons, reminiscent of that involved in the physical derivation of the Kontsevich-Soibelmann wall-crossing formula.