Calabi-Yau Research Articles

The tadpole conjecture in the interior of moduli space

We revisit moduli stabilization on Calabi-Yau manifolds with a discrete symmetry. Invariant fluxes allow for a truncation to a symmetric locus in complex structure moduli space and hence drastically reduce the moduli stabilization problem in its dimensionality. This makes them an ideal testing ground for the tadpole conjecture. For a large class of fourfolds, we show that an invariant flux with non-zero on-shell superpotential on the symmetric locus necessarily stabilizes at least 60% of the complex structure moduli. In case this invariant flux induces a relatively small tadpole, it is thus possible to bypass the bound predicted by the tadpole conjecture at these special loci. As an example, we discuss a Calabi-Yau hypersurface with h3,1 = 3878 and show that we can stabilize at least 4932 real moduli with a flux that induces M2-charge Nflux = 3.

SYZ Geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa Type Metrics

We construct a family of Calabi-Yau metrics on C 3 \mathbb {C}^3 with properties analogous to the Taub-NUT metric on C 2 \mathbb {C}^2 , and construct a family of Calabi-Yau 3-fold metric models on the positive and negative vertices of SYZ fibrations with properties analogous to the Ooguri-Vafa metric.

Probing M-theory with tetrahedron instantons

The duality between type IIA superstring theory and M-theory enables us to lift bound states of D0-branes and n parallel D6-branes to M-theory compactified on an n-centered multi-Taub-NUT space TN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{TN} $$\\end{document}n. Accordingly, the rank n K-theoretic Donaldson-Thomas invariants of ℂ3 are connected with the index of M-theory on ℂ3 × TN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{TN} $$\\end{document}n. In this paper, we extend this connection by considering intersecting D6-branes. In the presence of a suitable Neveu-Schwarz B-field, the system preserves two supercharges. This system is T-dual to the configuration of tetrahedron instantons which we introduced in [1]. We conjecture a closed-form expression for the K-theoretic tetrahedron instanton partition function, which is the generating function of the D0-D6 partition functions. We find that the tetrahedron instanton partition function coincides with the partition function of the magnificent four model for special values of the parameters, leading us to conjecture that our system of intersecting D6-branes can be obtained from the annihilation of D8-branes and anti-D8-branes. Remarkably, the K-theoretic tetrahedron instanton partition function allows an interpretation in terms of the index of M-theory on a noncompact Calabi-Yau fivefold which is related to a superposition of Kaluza-Klein monopoles. The dimensional reduction of the system allows us to express the cohomological tetrahedron instanton partition function in terms of the MacMahon function, generalizing the correspondence between Gromov-Witten invariants and Donaldson-Thomas invariants for Calabi-Yau threefolds.

The movable cone of Calabi–Yau threefolds in ruled Fano manifolds

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi–Yau threefold in a non-split (n+4)-dimensional Pn-ruled Fano manifold of index n+1 and Picard number two. Moreover, all birational minimal models of such Calabi–Yau threefolds are found whose number is finite.

Chiral Matter Multiplicities and Resolution-Independent Structure in 4D F-Theory Models

Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic Calabi–Yau fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {U}}(1)$$\\end{document} gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document} over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the independence of the intersection pairing of the vertical part of the middle cohomology for the resolutions we study. In each model, we find that vertical flux backgrounds can produce nonzero multiplicities for a spanning set of anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints on allowed matter representations beyond those implied by 4D anomaly cancellation.

Review on Special Geometry and Mirror Symmetry for Calabi–Yau Manifolds (Brief Review)

Review on Special Geometry and Mirror Symmetry for Calabi–Yau Manifolds (Brief Review)

Exact multi-instantons in topological string theory

Topological string theory has multi-instanton sectors which lead to non-perturbative effects in the string coupling constant and control the large order behavior of the perturbative genus expansion. As proposed by Couso, Edelstein, Schiappa and Vonk, these sectors can be described by a trans-series extension of the BCOV holomorphic anomaly equations. In this paper we find exact, closed form solutions for these multi-instanton trans-series in the case of local Calabi-Yau manifolds. The resulting multi-instanton amplitudes turn out to be very similar to the eigenvalue tunneling amplitudes of matrix models. Their form suggests that the flat coordinates of the Calabi-Yau manifold are naturally quantized in units of the string coupling constant, as postulated in large NN dualities. Based on our results we obtain a general picture for the resurgent structure of the topological string in the local case, which we illustrate with explicit calculations for local \mathbb{P}^2ℙ2.

Eigenvalues and eigenforms on Calabi–Yau threefolds

We present a numerical algorithm for computing the spectrum of the Laplace–de Rham operator on Calabi–Yau manifolds, extending previous work on the scalar Laplace operator. Using an approximate Calabi–Yau metric as input, we compute the eigenvalues and eigenforms of the Laplace operator acting on (p,q)-forms for the example of the Fermat quintic threefold. We provide a check of our algorithm by computing the spectrum of (p,q)-eigenforms on P3.

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.

Mirror symmetry for five-parameter Hulek-Verrill manifolds

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the Hulek-Verrill manifolds and their monodromies. On the mirror, we compute the genus-zero and -one instanton numbers, which are labelled by 5 indices, as h^{1,1}\!=5h1,1=5. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an S_{6}S6 symmetry made manifest by certain constructions of the complex structure moduli space of the Hulek-Verrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than S_6S6 and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a “web” structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term “positive webs”. This structure has consequences for instanton numbers at all genera.

On the Rational Parametric Solution of Diagonal Quartic Varieties

ABSTRACT In this paper, we exhibit a rational parametric solution for the Diophantine equations of diagonal quartic varieties. Our approach is based on utilizing the Calabi-Yau varieties including elliptic curves and diagonal quartic surfaces.

Coefficients of the monodromy matrices of one-parameter families of double octic Calabi-Yau threefolds at a half-conifold point

Doran and Morgan introduced in [12] a rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values L(f,1) and L(f,2) of the corresponding modular form f and one constant. We also present related results concerning the rank of the group of period integrals generated by the action of the monodromy group on the conifold period.

Modular differential equations of the elliptic genus of Calabi–Yau fourfolds

We study modular differential equations (MDEs) of the elliptic genus of four-dimensional complex varieties with trivial first Chern class. We construct modular differential equations of orders 3, 4, 5 and 6 with respect to the heat operator for every weak Jacobi form of weight 0 and index 2. We prove that the elliptic genus of a Calabi–Yau fourfold satisfies a MDE of the minimal possible order 3 if and only if its Euler number is equal to 48 or −18. We construct MDEs of order 5 of the elliptic genus of hyperkähler fourfolds of types Hilb2(K3) and Kum2(A).

Exponential Networks, WKB and Topological String

We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $\mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Symmetric fluxes and small tadpoles

The analysis of type IIB flux vacua on warped Calabi-Yau orientifolds becomes considerably involved for a large number of complex structure fields. We however show that, for a quadratic flux superpotential, one can devise simplifying schemes which effectively reduce the large number of equations down to a few. This can be achieved by imposing the vanishing of certain flux quanta in the large complex structure regime, and then choosing the remaining quanta to respect the symmetries of the underlying prepotential. One can then implement an algorithm to find large families of flux vacua with a fixed flux tadpole, independently of the number of fields. We illustrate this approach in a Calabi-Yau manifold with 51 complex structure moduli, where several reduction schemes can be implemented in order to explicitly solve the vacuum equations for that sector. Our findings display a flux-tadpole-to-stabilized-moduli ratio that is marginally above the bound proposed by the Tadpole Conjecture, and we discuss several effects that would take us below such a bound.

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

Relationship between two Calabi–Yau orbifolds arising as hyper–surfaces in a quotient of the same weighted projective space

In this article we consider a question: what is the relation between two Calabi-Yau manifolds of two different Berglund–Hubsch types if they appear as hyper–surfaces in the quotient of the same weighted projective space. We show that these manifolds are connected by a special change of coordinates, which we call the resonance transformation.

Affine characters at negative level and elliptic genera of non-critical strings

We study the elliptic genera of the non-critical strings of six dimensional superconformal field theories from the point of view of the strings’ worldsheet theory. We formulate a general ansatz for these in terms of characters of the affine Lie algebra associated to the 6d gauge group at negative level, and provide ample evidence for the validity of this ansatz for 6d theories obtained via F-theory compactification on elliptically fibered Calabi-Yau manifolds over a Hirzebruch base. We obtain novel closed form results for many elliptic genera in terms of our ansatz, and show that our results specialize consistently when moving along Higgsing trees.