Calabi-Yau Compactifications Research Articles

Emergence in string theory and Fermi gases

The Emergence Proposal suggests that some Swampland criteria, in particular on large field distances, are a consequence of the emergent nature of dynamics for fields in the infrared. In the context of type II string theory compactified on Calabi-Yau manifolds, it proposes that the cubic tree-level piece of the genus-zero prepotential is emergent from integrating out massive non-perturbative states. For a certain special non-compact Calabi-Yau, the blown-up conifold, it is known that the full all-genus prepotential can be matched onto the Grand Canonical potential of a two-dimensional Fermi gas. We propose here that this should be understood in the context of emergence: the prepotential is induced by integrating out the Fermi gas degrees of freedom. To make contact with the Swampland we need dynamical gravity, so compact Calabi-Yau manifolds. We show that for specifically the cubic term, an integrating out calculation also works for compact cases. In particular, the exact cubic term coefficient can be recovered from integrating out a Fermi gas for any compact Calabi-Yau that is an elliptic fibration over a reflexive toric base. We also propose a general map, for any one-parameter Calabi-Yau, between the Grand Canonical potential of the ultraviolet non-perturbative system and the period. In particular, this map leads to an emergent cubic term in the genus-zero prepotential for any such one-parameter model.

On the string landscape without hypermultiplets

In this work we study interesting corners of the quantum gravity landscape with 8 supercharges pushing the boundaries of our current understanding. Calabi-Yau threefolds compactifications of F/M/type II theories to 6, 5 and 4 dimensions are the most prominent examples of this class, and these always lead to a universal hypermultiplet coming from the volume/string coupling constant. We find new examples of asymmetric orbifold constructions which have no hypermultiplets in 4 or 5 dimensions and no neutral hypers in 6d. We argue that these theories can also be obtained by going to strong coupling/small volume regions of geometric constructions where a new Coulomb branch opens up and moving in this direction freezes the volume/string coupling constant. Interestingly we find that the Kodaira condition encountered in geometric limits of F-theory compactifications to 6 dimensions is violated in these corners of the landscape due to strong quantum corrections. We also construct a theory in 3 dimensions which if it were to arise by toroidal compactifications from 5d, it would have to come from pure N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with no massless scalar fields.

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

We obtain analytic and numerical results for the non-perturbative amplitudes of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our approach is based on the theory of resurgence and extends previous special results to the more general case. In particular, we obtain explicit trans-series solutions of the holomorphic anomaly equations. Our results predict the all orders, large genus asymptotics of the topological string free energies, which we test in detail against high genus perturbative series obtained recently in the compact case. We also provide additional evidence that the Stokes constants appearing in the resurgent structure are closely related to integer BPS invariants.

S-Duality and the Universal Isometries of Instanton Corrected q-Map Spaces

Given a conical affine special Kähler manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-Kähler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi–Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an SL(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}(2,{\\mathbb {Z}})$$\\end{document} acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by SL(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}(2,{\\mathbb {Z}})$$\\end{document} or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full SL(2,Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}(2,{\\mathbb {Z}})$$\\end{document} acting by isometries and admitting a quotient of finite volume by a discrete group of isometries.

Generalized symmetries, gravity, and the swampland

Generalized global symmetries are a common feature of many quantum field theories decoupled from gravity. By contrast, in quantum gravity/the Swampland program, it is widely expected that all global symmetries are either gauged or broken, and this breaking is in turn related to the expected completeness of the spectrum of charged states in quantum gravity. We investigate the fate of such symmetries in the context of 7D and 5D vacua realized by compact Calabi-Yau spaces with localized singularities in M theory. We explicitly show how gravitational backgrounds support additional dynamical degrees of freedom which trivialize (i.e., “break”) the higher symmetries of the local geometric models. Local compatibility conditions across these different sectors lead to gluing conditions for gauging higher-form and (in the 5D case) higher-group symmetries. This also leads to a preferred global structure of the gauge group and higher-form gauge symmetries. In cases based on a genus-one fibered Calabi-Yau space, we also get an F-theory model in one higher dimension with corresponding constraints on the global form of the gauge group. Published by the American Physical Society 2024

Cosmic acceleration and turns in the Swampland

We argue that field trajectories, which lead to cosmic acceleration and feature rapid turns near the boundary of the moduli space, are in the Swampland. We obtain this result by assuming the validity of the Swampland Distance Conjecture (SDC) in the presence of a positive scalar potential and by focusing on hyperbolic spaces, as prototype geometries of infinite distance limits of Calabi-Yau compactifications. We find that, in a quasi-de Sitter space with Hubble rate H and acceleration parameter ϵ, the turning rate Ω is upper bounded such as Ω/H < 𝒪(√(ϵ)). Therefore, field trajectories consistent with the SDC can only have a negligible deviation from geodesics. This has direct implications for the realization and consistency of multi-field scenarios in string theory. Moreover, it implies a tension between asymptotic accelerating expansion, consistent with observations, and the de Sitter conjecture.

Flux vacua and modularity for $\mathbb{Z}_2$ symmetric Calabi-Yau manifolds

We find continuous families of supersymmetric flux vacua in IIB Calabi-Yau compactifications for multiparameter manifolds with an appropriate \mathbb{Z}_{2}ℤ2 symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weight-two modular forms, these manifolds being said to be weight-two modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to F-theory on elliptically fibred Calabi-Yau fourfolds. If conjectural expressions for Deligne’s periods are true, then these imply that the F-theory fibre is complex-isomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in \mathbb{Q}ℚ but also in algebraic extensions of \mathbb{Q}ℚ, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.

EFT strings and emergence

We revisit the Emergence Proposal in 4d mathcal{N} = 2 vector multiplet sectors that arise from type II string Calabi-Yau compactifications, with emphasis on the role of axionic fundamental strings, or EFT strings. We focus on large-volume type IIA compactifications, where EFT strings arise from NS5-branes wrapping internal four-cycles, and consider a set of infinite-distance moduli-space limits that can be classified in terms of a scaling weight w = 1, 2, 3. It has been shown before how one-loop threshold effects of an infinite tower of BPS particles made up of D2/D0-branes generate the asymptotic behaviour of the gauge kinetic functions along limits with w = 3. We extend this result to w = 2 limits, by taking into account D2-brane multi-wrapping numbers. In w = 1 limits the leading tower involves EFT string oscillations, and one can reproduce the behaviour of both weakly and strongly-coupled U(1)’s independently on whether the EFT string is critical or not, by assuming that charged modes dominate the light spectrum.

Special Lagrangian Cycles and Calabi-Yau Transitions

We construct special Lagrangian 3-spheres in non-Kähler compact threefolds equipped with the Fu–Li–Yau geometry. These non-Kähler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic 2-cycles for special Lagrangian 3-cycles.

Taming the landscape of effective theories

We introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, we propose a Tameness Conjecture that states that all valid effective theories are labelled by a definable parameter space and must have scalar field spaces and coupling functions that are definable using the tame geometry built from an o-minimal structure. We give a brief introduction to tame geometry and describe how it restricts sets, manifolds, and functions. We then collect evidence for the Tameness Conjecture by studying various effective theories arising from string theory compactifications by using some recent advances in tame geometry. In particular, we will exploit the fact that coset spaces and period mappings are definable in an o-minimal structure and argue for non-trivial tameness results in higher-supersymmetric theories and in Calabi-Yau compactifications. As strongest evidence for the Tameness Conjecture over a discrete parameter space, we then discuss a recent theorem stating that the locus of self-dual flux vacua of F-theory admits a tame geometry even if one allows for any flux choice satisfying the tadpole constraint. This result implies the finiteness of self-dual flux vacua in F-theory.

Flux vacua with approximate flat directions

We present a novel method to obtain type IIB flux vacua with flat directions at tree level. We perform appropriate choices of flux quanta that induce relations between the flux superpotential and its derivatives. This method is implemented in toroidal and Calabi-Yau compactifications in the large complex structure limit. Explicit solutions are obtained and classified on the basis of duality equivalences. In the toroidal case we present solutions with N = 1 and N = 2 supersymmetry and arbitrarily weak coupling. In Calabi-Yaus we find novel perturbatively flat vacua, as well as solutions with non-zero flux superpotential and an axionic flat direction which represent a promising starting point for de Sitter constructions from non-zero F-terms in the complex structure sector. The higher order (perturbative and non-perturbative) effects that can lift these flat directions are discussed. We also outline applications in a wide variety of settings involving the classical Regge growth conjecture, inflation and quintessence, supersymmetry breaking and F-term de Sitter uplifting.

Tameness, Strings, and the Distance Conjecture

The Distance Conjecture states that an infinite tower of modes becomes exponentially light when approaching an infinite distance point in field space. We argue that the inherent path-dependence of this statement can be addressed when combining the Distance Conjecture with the recent Tameness Conjecture. The latter asserts that effective theories are described by tame geometry and implements strong finiteness constraints on coupling functions and field spaces. By exploiting these tameness constraints we argue that the region near the infinite distance point admits a decomposition into finitely many sectors in which path-independent statements for the associated towers of states can be established. We then introduce a more constrained class of tame functions with at most polynomial asymptotic growth and argue that they suffice to describe the known string theory effective actions. Remarkably, the multi-field dependence of such functions can be reconstructed by one-dimensional linear test paths in each sector near the boundary. In four-dimensional effective theories, these test paths are traced out as a discrete set of cosmic string solutions. This indicates that such cosmic string solutions can serve as powerful tool to study the near-boundary field space region of any four-dimensional effective field theory. To illustrate these general observations we discuss the central role of tameness and cosmic string solutions in Calabi-Yau compactifications of Type IIB string theory.

Time reversal and CP invariance in Calabi-Yau compactifications

We revisit the question of time reversal and CP invariance in Calabi-Yau compactifications. We show that time reversal invariance is respected by quantum corrections to the prepotential. In particular, field independent θ angles whose presence is dictated by requiring integrality of relevant monodromy transformations can take precisely the quantized values compatible with time reversal invariance. Furthermore, monodromy symmetry enlarges the region on moduli space on which time reversal is not spontaneously broken. We define the action of the CP transformation for multi-parameter models and argue that on the slice of moduli space where it is defined, CP is trivially a symmetry of the theory. For supersymmetric vacua that lie in this slice, we derive a condition on the third cohomology of the compactification manifold which determines whether CP preserving fluxes exist that stabilize the moduli to such points. In the case of one-parameter models, the condition is always satisfied.

Instanton solutions in open superstring field theory

Open superstring field theory admits a “hybrid” formulation where N = 1 D = 4 supersymmetry is manifest for Calabi-Yau compactifications to four dimensions. Using this formulation, the half-BPS instanton solution of four-dimensional super-Yang-Mills can be easily generalized to the full open superstring field theory. In this paper, we compute the first stringy correction to the super-Yang-Mills instanton solution which involves turning on certain fields at the first massive level.

S-Duality and the Universal Isometries of q-Map Spaces

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension n-1ge 0, a quaternionic Kähler (QK) manifold of dimension 4n+4. It is known that the resulting QK manifold admits a (3n+5)-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi–Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an mathrm {SL}(2,{mathbb {R}}) group of isometries related to the mathrm {SL}(2,{mathbb {Z}}) S-duality symmetry of the full 10d theory. We present a purely mathematical proof that any tree-level q-map space admits such an mathrm {SL}(2,{mathbb {R}}) action by isometries, enlarging the previous universal group of isometries to a (3n+6)-dimensional group G. As part of this analysis, we describe how the (3n+5)-dimensional subgroup interacts with the mathrm {SL}(2,{mathbb {R}})-action, and find a codimension one normal subgroup of G that is unimodular. By taking a quotient with respect to a lattice in the unimodular group, we obtain a quaternionic Kähler manifold fibering over a projective special real manifold with fibers of finite volume, and compute the volume as a function of the base. We furthermore provide a mathematical treatment of results from the physics literature concerning the twistor space of the tree-level q-map space and the holomorphic lift of the (3n+6)-dimensional group of universal isometries to the twistor space.

Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature

We obtain the bosonic Lagrangians of vector and hypermultiplets coupled to four-dimensional mathcal{N} = 2 supergravity in signatures (0, 4), (1, 3) and (2,2) by compactification of type-II string theories in signatures (0,10), (1,9) and (2,8) on a Calabi-Yau threefold. Depending on the signature and the distinctions between type-IIA/IIA*/IIB/IIB*/IIB’ the resulting scalar geometries are special Kähler or special para-Kähler for vector multiplets and quaternion-Kähler or para-quaternion Kähler for hypermultiplets. By spacelike and timelike reductions we obtain three-dimensional mathcal{N} = 4 supergravity theories coupled to two sets of hypermultiplets. We determine the c-maps relating vector to hypermultiplets, and show how the four-dimensional theories are related by spacelike, timelike and mixed, signature-changing T-dualities.

On singular log Calabi-Yau compactifications of Landau-Ginzburg models

We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index . For coverings of degree greater than the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration, the conjecture that the number of components of the fibre over infinity is equal to the dimension of an anticanonical system of the Fano variety. Bibliography: 46 titles.

Spontaneous CP violation and symplectic modular symmetry in Calabi-Yau compactifications

We explore the geometrical origin of CP and the spontaneous CP violation in Calabi-Yau compactifications. We find that the CP symmetry is identified with an outer automorphism of the symplectic modular group in the large complex structure regime of Calabi-Yau threefolds, thereby enlarging the symplectic modular group to their semidirect product group. The spontaneous CP violation is realized by the introduction of fluxes, whose effective action is invariant under CP as well as the discrete Z2 symmetry or Z4 R-symmetry. We explicitly demonstrate the spontaneous CP violation on a specific Calabi-Yau threefold.

The Standard Model quiver in de Sitter string compactifications

We argue that the Standard Model quiver can be embedded into compact Calabi-Yau geometries through orientifolded D3-branes at del Pezzo singularities dPn with n ≥ 5 in a framework including moduli stabilisation. To illustrate our approach, we explicitly construct a local dP5 model via a combination of Higgsing and orientifolding. This procedure reduces the original dP5 quiver gauge theory to the Left-Right symmetric model with three families of quarks and leptons as well as a Higgs sector to further break the symmetries to the Standard Model gauge group. We embed this local model in a globally consistent Calabi-Yau flux compactification with tadpole and Freed-Witten anomaly cancellations. The model features closed string moduli stabilisation with a de Sitter minimum from T-branes, supersymmetry broken by the Kähler moduli, and the MSSM as the low energy spectrum. We further discuss phenomenological and cosmological implications of this construction.

Constructing local models for Lagrangian torus fibrations

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.