arXiv subjects

Xenia de la Ossa

Publications and source records attributed to Xenia de la Ossa.

35 records · Page 2Linked to original sources

arXiv2016

The Heterotic Superpotential and Moduli

We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at $\mathcal{O}(α')$ to account for the Green-Schwarz anomaly cancellation mechanism, we investigate properties of four-dimensional Minkowski vacua of this theory. Considering the restrictions arising from F-terms and D-terms we identify the infinitesimal massless moduli space of the theory. We show that it agrees with the results that have recently been obtained from a ten-dimensional perspective where supersymmetric Minkowski solutions including the Bianchi identity correspond to an integrable holomorphic structure, with infinitesimal moduli calculated by its first cohomology. As has recently been noted, interplay of complex structure and bundle deformations through holomorphic and anomaly constraints can lead to fewer moduli than may have been expected. We derive a relation between the number of complex structure and bundle moduli removed from the low energy theory in this way, and give conditions for there to be no complex structure moduli or bundle moduli remaining in the low energy theory. The link between Yukawa couplings and obstruction theory is also briefly discussed.

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arXiv2016

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_θ}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_θ}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on $(Y,V)$ when $α'=0$.

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arXiv2015

Two One-Parameter Special Geometries

The special geometries of two recently discovered Calabi-Yau threefolds with $h^{11}=1$ are analyzed in detail. These correspond to the 'minimal three-generation' manifolds with $h^{21}=4$ and the `24-cell' threefolds with $h^{21}=1$. It turns out that the one-dimensional complex structure moduli spaces for these manifolds are both very similar and surprisingly complicated. Both have 6 hyperconifold points and, in addition, there are singularities of the Picard-Fuchs equation where the threefold is smooth but the Yukawa coupling vanishes. Their fundamental periods are the generating functions of lattice walks, and we use this fact to explain why the singularities are all at real values of the complex structure.

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arXiv2015

Connections, Field Redefinitions and Heterotic Supergravity

We study heterotic supergravity at $\mathcal{O}(α')$, first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at $\mathcal{O}(α')$ the Hull connection gives a consistent supergravity theory with supersymmetry transformations given in the usual way. We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We are interested in the moduli space of such field redefinitions which allow for supersymmetric solutions to the equations of motion. We show that for solutions on $M_4\times X$, where $M_4$ is Minkowski and $X$ is compact, the moduli space of infinitesimal field redefinitions is given by $H^{(0,1)}(X,\textrm{End}(TX))$. This space corresponds to infinitesimally close connections for which the equations of motion are satisfied. The setup suggests a symmetry between the gauge connection and the tangent bundle connection, as also employed by Bergshoeff and de Roo. We argue that this symmetry should be kept to higher orders in $α'$, and propose a natural choice for the corresponding tangent bundle connection used in curvature computations. In particular, the Hull connection should be corrected at second and higher orders in $α'$ from this point of view.

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arXiv2015

The Massless Spectrum of Heterotic Compactifications

We discuss the four-dimensional massless spectrum of supersymmetric Minkowski compactifications of ten-dimensional heterotic supergravity, including the anomaly cancelation condition. This can be calculated from restrictions arising from F-term conditions in a four-dimensional effective theory. The results agree with computations of the infinitesimal moduli space recently performed from a ten-dimensional perspective. The paper is based on a talk given by Eirik Eik Svanes in Leuven for the workshop on "The String Theory Universe".

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arXiv2014

Holomorphic Bundles and the Moduli Space of N=1 Supersymmetric Heterotic Compactifications

We describe the first order moduli space of heterotic string theory compactifications which preserve $N=1$ supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on $TX$ to a field, the moduli space corresponds to deformations of a holomorphic structure $\bar D$ on a bundle $\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$ with an extension class $\cal H$ which precisely enforces the anomaly cancelation condition. The deformations corresponding to the bundle $E$ are simultaneous deformations of the holomorphic structures on the poly-stable bundles $V$ and $TX$ together with those of the complex structure of $X$. We discuss the fact that the "moduli" corresponding to ${\rm End}(TX)$ cannot be physical, but are however needed in our mathematical structure to be able to enforce the anomaly cancelation condition. In the Appendix we comment on the choice of connection on $TX$ which has caused some confusion in the community before. It has been shown by Ivanov and others that this connection should also satisfy the instanton equations, and we give another proof of this fact.

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arXiv2014

Exploring SU(3) Structure Moduli Spaces with Integrable G2 Structures

We study the moduli space of SU(3) structure manifolds X that form the internal compact spaces in four-dimensional N=1/2 domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, X forms a non-compact 7-manifold Y with torsionful G2 structure. We use this G2 embedding to explore how X(t) varies along paths C(t) in the SU(3) structure moduli space. Our analysis includes the Bianchi identities which strongly constrain the flow. We show that requiring that the SU(3) structure torsion is preserved along the path leads to constraints on the G2 torsion and the embedding of X in Y. Furthermore, we study flows along which the torsion classes of X go from zero to non-zero values. In particular, we present evidence that the flow of half-flat SU(3) structures may contain Calabi-Yau loci, in the presence of non-vanishing H-flux.

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arXiv2012

Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetry of the pair of genus six curves. The subgroup A_5, of even permutations, is an automorphism of each curve, while the odd permutations interchange the two curves. The ten exceptional curves of dP_5 each intersect each of the genus six curves in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the genus six curves are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold the genus six curves develop six nodes and may be resolved to a P^1. The group A_5 acts on this P^1 and we describe this action.

math.AG↗

arXiv2007

Triadophilia: A Special Corner in the Landscape

It is well known that there are a great many apparently consistent vacua of string theory. We draw attention to the fact that there appear to be very few Calabi--Yau manifolds with the Hodge numbers h^{11} and h^{21} both small. Of these, the case (h^{11}, h^{21})=(3,3) corresponds to a manifold on which a three generation heterotic model has recently been constructed. We point out also that there is a very close relation between this manifold and several familiar manifolds including the `three-generation' manifolds with χ=-6 that were found by Tian and Yau, and by Schimmrigk, during early investigations. It is an intriguing possibility that we may live in a naturally defined corner of the landscape. The location of these three generation models with respect to a corner of the landscape is so striking that we are led to consider the possibility of transitions between heterotic vacua. The possibility of these transitions, that we here refer to as transgressions, is an old idea that goes back to Witten. Here we apply this idea to connect three generation vacua on different Calabi-Yau manifolds.

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arXiv2007

The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists

In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a one parameter family of quintic threefolds, and study the zeta-function as a function of the parameter ϕ. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(ϕ), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(ϕ) is given by a relation U(ϕ)=E^{-1}(ϕ^p)U(0)E(ϕ) with E(ϕ) a Wronskian matrix formed from the periods of the manifold. The periods are defined by series that converge for $|ϕ|_p < 1$. The values of ϕthat are of interest are those for which ϕ^p = ϕso, for nonzero ϕ, we have |\vph|_p=1. We explain how the process of p-adic analytic continuation applies to this case. The matrix U(ϕ) breaks up into submatrices of rank 4 and rank 2 and we are able from this perspective to explain some of the observations that have been made previously by numerical calculation.

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arXiv2004

Calabi-Yau Manifolds Over Finite Fields, II

We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

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arXiv2002

Supersymmetry, G-flux and Spin(7) manifolds

In this note we study warped compactifications of M-theory on manifolds of Spin(7) holonomy in the presence of background 4-form flux. The explicit form of the superpotential can be given in terms of the self -dual Cayley calibration on the Spin(7) manifold, in agreement with the general formula propsed in hep-th/9911011.

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arXiv2001

D-Branes on Noncompact Calabi-Yau Manifolds: K-Theory and Monodromy

We study D-branes on smooth noncompact toric Calabi-Yau manifolds that are resolutions of abelian orbifold singularities. Such a space has a distinguished basis {S_i} for the compactly supported K-theory. Using local mirror symmetry we demonstrate that the S_i have simple transformation properties under monodromy; in particular, they are the objects that generate monodromy around the principal component of the discriminant locus. One of our examples, the toric resolution of C^3/(Z_2 x Z_2), is a three parameter model for which we are able to give an explicit solution of the GKZ system.

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arXiv2000

Calabi-Yau Manifolds Over Finite Fields, I

We study Calabi-Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show also, for a one parameter family of quintic threefolds, that the number of rational points of the manifold is closely related to as the number of rational points of the mirror manifold. Our interest is primarily with Calabi-Yau threefolds however we consider also the interesting case of elliptic curves and even the case of a quadric in CP_1 which is a zero dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial dependence on the parameter over C but a not trivial arithmetic structure.

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arXiv1994

Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.

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arXiv1993

Mirror Symmetry for Two Parameter Models -- I

We study, by means of mirror symmetry, the quantum geometry of the Kähler-class parameters of a number of Calabi-Yau manifolds that have $b_{11}=2$. Our main interest lies in the structure of the moduli space and in the loci corresponding to singular models. This structure is considerably richer when there are two parameters than in the various one-parameter models that have been studied hitherto. We describe the intrinsic structure of the point in the (compactification of the) moduli space that corresponds to the large complex structure or classical limit. The instanton expansions are of interest owing to the fact that some of the instantons belong to families with continuous parameters. We compute the Yukawa couplings and their expansions in terms of instantons of genus zero. By making use of recent results of Bershadsky et al. we compute also the instanton numbers for instantons of genus one. For particular values of the parameters the models become birational to certain models with one parameter. The compactification divisor of the moduli space thus contains copies of the moduli spaces of one parameter models. Our discussion proceeds via the particular models $¶_4^{(1,1,2,2,2)}[8]$ and $¶_4^{(1,1,2,2,6)}[12]$. Another example, $¶_4^{(1,1,1,6,9)}[18]$, that is somewhat different is the subject of a companion paper.

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arXiv1993

Periods for Calabi--Yau and Landau--Ginzburg Vacua

The complete structure of the moduli space of \cys\ and the associated Landau-Ginzburg theories, and hence also of the corresponding low-energy effective theory that results from (2,2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. These periods are shown to be readily calculable for a great many such models. We illustrate this by computing the periods explicitly for a number of classes of \cys. We also point out that it is possible to read off from the periods certain important information relating to the mirror manifolds.

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