arXiv subjects

Xenia de la Ossa

Publications and source records attributed to Xenia de la Ossa.

At least 19 recordsLinked to original sources

arXiv2025

Local Zeta Functions of Multiparameter Calabi-Yau Threefolds from the Picard-Fuchs Equations

The deformation approach of arXiv:2104.07816 for computing zeta functions of one-parameter Calabi-Yau threefolds is generalised to cover also multiparameter manifolds. Consideration of the multiparameter case requires the development of an improved formalism. This allows us, among other things, to make progress on some issues left open in previous work, such as the treatment of apparent and conifold singularities and changes of coordinates. We also discuss the efficient numerical computation of the zeta functions. As examples, we compute the zeta functions of the two-parameter mirror octic, a non-symmetric split of the quintic threefold also with two parameters, and the $S_5$ symmetric five-parameter Hulek-Verrill manifolds. These examples allow us to exhibit the several new types of geometries for which our methods make practical computations possible. They also act as consistency checks, as our results reproduce and extend those of arXiv:hep-th/0409202 and arXiv:math/0304169. To make the methods developed here more approachable, a Mathematica package "CY3Zeta" for computing the zeta functions of Calabi-Yau threefolds, which is attached to this paper, is presented.

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arXiv2025

Quantum aspects of heterotic $G_2$ systems

Compactifications of the heterotic string, to first order in the $α'$ expansion, on manifolds with integrable $G_2$ structure give rise to three-dimensional ${\cal N} = 1$ supergravity theories that admit Minkowski and AdS ground states. As shown in arXiv:1904.01027, such vacua correspond to critical loci of a real superpotential $W$. We perform a perturbative study around a supersymmetric vacuum of the theory, which confirms that the first order variation of the superpotential, $δW$, reproduces the BPS conditions for the system, and furthermore shows that $δ^2 W=0$ gives the equations for infinitesimal moduli. This allows us to identify a nilpotent differential, and a symplectic pairing, which we use to construct a bicomplex, or a double complex, for the heterotic $G_2$ system. Using this complex, we determine infinitesimal moduli and their obstructions in terms of related cohomology groups. Finally, by interpreting $δ^2 W$ as an action, we compute the one-loop partition function of the heterotic $G_2$ system and show it can be decomposed into a product of one-loop partition functions of Abelian and non-Abelian instanton gauge theories.

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arXiv2025

Some Identities For Periods of Hulek-Verrill Threefolds

We study the Hulek--Verrill families of Calabi--Yau threefolds. They are birationally equivalent to fibred products of elliptic surfaces, so we expect to be able to compute periods on these threefolds by integrating products of elliptic periods over a contour on $\mathbb{P}^1$. We numerically verify this in several examples. This article was submitted to MATRIX Annals (2024) for inclusion in the proceedings of the conference "The Geometry of Moduli Spaces in String Theory", held 2--13 September 2024.

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arXiv2025

$\mathcal{SW}$-algebras and strings with torsion

We explore the connection between super $\mathcal{W}$-algebras ($\mathcal{SW}$-algebras) and $\mathrm{G}$-structures with torsion. The former are realised as symmetry algebras of strings with $\mathcal{N}=(1,0)$ supersymmetry on the worldsheet, while the latter are associated with generic string backgrounds with non-trivial Neveu-Schwarz flux $H$. In particular, we focus on manifolds featuring $\mathrm{Spin}(7)$, $\mathrm{G}_2$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$-structures. We compare the full quantum algebras with their classical limits, obtained by studying the commutators of superconformal and $\mathcal{W}$-symmetry transformations, which preserve the action of the $(1,0)$ non-linear $σ$-model. We show that, at first order in the string length scale $\ell_s$, the torsion deforms some of the OPE coefficients corresponding to special holonomy through a scalar torsion class.

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arXiv2025

Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants

We revisit the series solutions of the attractor equations of 4d N=2 supergravities obtained by Calabi--Yau compactifications previously studied in Candelas, Kuusela, and McGovern (2021). While only convergent for a restricted set of black hole charges, we find that they are summable with Padé resummation providing a suitable method. By specialising these solutions to rank-two attractors, we obtain many conjectural identities of the type discovered in CKM. These equate ratios of weight-four special L-values with an infinite series whose summands are formed out of genus-0 Gromov--Witten invariants. We also present two new rank-two attractors which belong to moduli spaces each interesting in their own right. Each moduli space possesses two points of maximal unipotent monodromy. One has already been studied by Hosono and Takagi, and we discuss issues stemming from the associated L-function having nonzero rank.

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arXiv2023

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 $\mathbb{Z}_2$ 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, and these manifolds are 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 $\mathbb{Q}$ but also in algebraic extensions of $\mathbb{Q}$, 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.

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arXiv2023

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 $h^{1,1}=5$. 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 $S_{6}$ 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 $S_6$ 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.

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arXiv2021

Families of solutions of the heterotic G$_2$ system

We construct new families of solutions of the heterotic G$_2$ system on squashed homogeneous 3-Sasakian manifolds, that is, using squashed metrics on either the 7-sphere or the Aloff-Wallach space $N_{1,1}$. We obtain AdS$_3$ solutions for all values of the squashing parameter $s$ except for the nearly-parallel G$_2$ value $s=1/\sqrt{5}$, for which we don't find any solutions. Along the process, we construct different G$_2$-instanton connections on bundles over these squashed manifolds.

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arXiv2021

Almost contact structures on manifolds with a $G_2$ structure

We review the construction of almost contact metric (three-) structures, abbreviated ACM(3)S, on manifolds with a $G_2$ structure. These are of interest for certain supersymmetric configurations in string and M-theory. We compute the torsion of the $SU(3)$ structure associated to an ACMS and apply these computations to heterotic $G_2$ systems and supersymmetry enhancement. We initiate the study of the space of ACM3Ss, which is an infinite dimensional space with a local product structure and interesting topological features. Tantalising links between ACM3Ss and associative and coassociative submanifolds are observed.

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arXiv2021

Local Zeta Functions From Calabi-Yau Differential Equations

The zeta-function of a manifold is closely related to, and sometimes can be calculated completely, in terms of its periods. We report here on a practical and computationally rapid implementation of this procedure for families of Calabi-Yau manifolds with one complex structure parameter phi. Although partly conjectural, it turns out to be possible to compute the matrix of the Frobenius map on the third cohomology group of X(phi) directly from the Picard-Fuchs differential operator of the family. To illustrate our method, we compute tables of the quartic numerators of the zeta-functions for six manifolds of increasing complexity as the parameter phi varies in Fp. For four of these manifolds, we do this for the 500 primes p=5,7,...,3583, while for two manifolds we extend the calculation to 1000 primes. The tables for 5 <= p <= 97 are part of this article while the remaining tables are attached in electronic form. Interest attaches to the cases for which the numerators factorise. Some of these factorisations can be associated with parameter values for which the underlying manifold becomes singular. For the cases we consider here, the singularities are all of conifold or hyperconifold type. In these cases the numerator degenerates to a cubic and this factorises into the product of a linear and a quadratic factor. The quadratic term contains a coefficient that is the p'th coefficient of a modular form. Some of our examples have singularities when the parameter satisfies a polynomial equation that does not factorise over Q. When this happens, the corresponding forms are modular forms with neben type or Hilbert modular forms. The numerator can also factorise into two quadrics. This happens when the Hodge structure of the manifold splits, sometimes this happens for algebraic values of the parameter and we identify, in this way, attractor points of rank two of the parameter space.

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arXiv2020

Superpotential of Three Dimensional ${\cal N}=1$ Heterotic Supergravity

We dimensionally reduce the ten dimensional heterotic action on spacetimes of the form ${\cal M}_{(2,1)}\times Y$, where ${\cal M}_{(2,1)}$ is three dimensional maximally symmetric Anti de Sitter or Minkowski space, and $Y$ is a compact seven dimensional manifold with $G_2$ structure. In doing so, we derive the real superpotential functional of the corresponding three dimensional ${\cal N}=1$ theory. We confirm that extrema of this functional precisely correspond to supersymmetric heterotic compactifications on manifolds of $G_2$ structure. We make some comments on the role of the superpotential functional with respect to the coupled moduli problem of instanton bundles over $G_2$ manifolds.

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arXiv2019

A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over Q this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over Q, and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups. Thus the critical L-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the zeta-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.

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arXiv2019

$\mathcal{G}$-structure symmetries and anomalies in $(1,0)$ non-linear $σ$-models

A new symmetry of $(1,0)$ supersymmetric non-linear $σ$-models in two dimensions with Fermi and mass sectors is introduced. It is a generalisation of the so-called special holonomy $W$-symmetry of Howe and Papadopoulos associated with structure group reductions of the target space $\mathcal{M}$. Our symmetry allows in particular non-trivial flux and instanton-like connections on vector bundles over $\mathcal{M}$. We also investigate potential anomalies and show that cohomologically non-trivial terms in the quantum effective action are invariant under a corrected version of our symmetry. Consistency with heterotic supergravity at first order in $α'$ is manifest and discussed.

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arXiv2019

The Universal Geometry of Heterotic Vacua

We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c_1 = 0, each with a gauge field solving the Hermitian Yang-Mill's equations and compatible B and H fields that satisfy the anomaly cancellation conditions. Our perspective is to consider a geometry in which these backgrounds are fibred over a parameter space. If the manifold X has coordinates x, and parameters are denoted by y, then it is natural to consider coordinate transformations x \to \tilde{x}(x,y) and y \to \tilde{y}(y). Similarly, gauge transformations of the gauge field and B field also depend on both x and y. In the process of defining deformations of the background fields that are suitably covariant under these transformations, it turns out to be natural to extend the gauge field A to a gauge field \IA on the extended (x,y)-space. Similarly, the B, H, and other fields are also extended. The total space of the fibration of the heterotic structures is the Universal Geometry of the title. The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces. String vacua furnish a richer application of these ideas. One advantage of this point of view is that previously disparate results are unified into a simple tensor formulation. In a previous paper, by three of the present authors, the metric on the moduli space of heterotic theories was derived, correct through order α', and it was shown how this was related to a simple Kahler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.

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arXiv2018

Finite deformations from a heterotic superpotential: holomorphic Chern--Simons and an $L_\infty$ algebra

We consider finite deformations of the Hull--Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer--Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira--Spencer and holomorphic Chern--Simons theory. The supersymmetric locus of this action is described by an $L_3$ algebra.

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arXiv2017

The infinitesimal moduli space of heterotic $G_2$ systems

Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tildeθ)$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${\cal Q}=T^*Y\oplus{\rm End}(V)\oplus{\rm End}(TY)$ and satisfies a nilpotency condition $\check{\cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group $\check H^1_{\check{\cal D}}(\cal Q)$. We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the $α'$ expansion.

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arXiv2017

Restrictions of Heterotic $G_2$ Structures and Instanton Connections

This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. In particular, such heterotic $G_2$ systems can be rephrased in terms of a differential $\check {\cal D}$ acting on a complex $\checkΩ^*(Y , {\cal Q})$, where ${\cal Q}=T^*Y\oplus{\rm End}(TY)\oplus{\rm End}(V)$ and $\check {\cal D}$ is an appropriate projection of an exterior covariant derivative ${\cal D}$ which satisfies an instanton condition. The infinitesimal moduli are further parametrised by the first cohomology $H^1_{\check {\cal D}}(Y,{\cal Q})$. We proceed to restrict this system to manifolds $X$ with an $SU(3)$ structure corresponding to supersymmetric compactifications to four dimensional Minkowski space, often referred to as Strominger--Hull solutions. In doing so, we derive a new result: the Strominger-Hull system is equivalent to a particular holomorphic Yang-Mills covariant derivative on ${\cal Q}\vert_X=T^*X\oplus{\rm End}(TX)\oplus{\rm End}(V)$.

math.DG↗

arXiv2017

A Metric for Heterotic Moduli

Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in alpha', in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kahler, as is required by supersymmetry. Checking that the metric is in fact Kahler is quite intricate and uses the anomaly cancellation equation for the H-field, in an essential way. The Kahler potential nevertheless takes a remarkably simple form: it is Kahler potential for special geometry with the Kahler form replaced by the alpha'-corrected hermitian form.

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