arXiv subjects

Philip Candelas

Publications and source records attributed to Philip Candelas.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2011

The MSSM Spectrum from (0,2)-Deformations of the Heterotic Standard Embedding

We construct supersymmetric compactifications of E_8 \times E_8 heterotic string theory which realise exactly the massless spectrum of the Minimal Supersymmetric Standard Model (MSSM) at low energies. The starting point is the standard embedding on a Calabi-Yau threefold which has Hodge numbers (h^11,h^21) = (1,4) and fundamental group Z_12, which gives an E_6 grand unified theory with three net chiral generations. The gauge symmetry is then broken to that of the standard model by a combination of discrete Wilson lines and continuous deformation of the gauge bundle. On eight distinct branches of the moduli space, we find stable bundles with appropriate cohomology groups to give exactly the massless spectrum of the MSSM.

hep-th↗

arXiv2010

Completing the Web of $Z_3$ - Quotients of Complete Intersection Calabi-Yau Manifolds

We complete the study of smooth $Z_3$-quotients of complete intersection Calabi-Yau threefolds by discussing the six new manifolds that admit free $Z_3$ actions that were discovered discovered recently by Braun. These manifolds were missed in an earlier work and complete the web of smooth $Z_3$-quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of $Z_3$-free quotients of complete intersection Calabi-Yau threefolds is connected by conifold transitions.

hep-th↗

arXiv2009

A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers

We present a complete intersection Calabi-Yau manifold Y that has Euler number -72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h^11,h^21)=(2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.

hep-th↗

arXiv2008

New Calabi-Yau Manifolds with Small Hodge Numbers

It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi-Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi-Yau manifolds where the Hodge numbers (h^{11}, h^{21}) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with χ=-6 and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross-Popescu manifolds that have Euler number zero.

hep-th↗

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.

hep-th↗

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.

hep-th↗

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.

hep-th↗

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.

hep-th↗

arXiv2000

Codimension-Three Bundle Singularities in F-Theory

We study new nonperturbative phenomena in N=1 heterotic string vacua corresponding to pointlike bundle singularities in codimension three. These degenerations result in new four-dimensional infrared physics characterized by light solitonic states whose origin is explained in the dual F-theory model. We also show that such phenomena appear generically in $E_7 \to E_6$ Higgsing and describe in detail the corresponding bundle transition.

hep-th↗

arXiv1997

F-theory, SO(32) and Toric Geometry

We show that the F-theory dual of the heterotic string with unbroken Spin(32)/Z_2 symmetry in eight dimensions can be described in terms of the same polyhedron that can also encode unbroken E_8\times E_8 symmetry. By considering particular compactifications with this K3 surface as a fiber, we can reproduce the recently found `record gauge group' in six dimensions and obtain a new `record gauge group' in four dimensions. Our observations relate to the toric diagram for the intersection of components of degenerate fibers and our definition of these objects, which we call `tops', is more general than an earlier definition by Candelas and Font.

hep-th↗

arXiv1997

Matter from Toric Geometry

We present an algorithm for obtaining the matter content of effective six-dimensional theories resulting from compactification of F-theory on elliptic Calabi-Yau threefolds which are hypersurfaces in toric varieties. The algorithm allows us to read off the matter content of the theory from the polyhedron describing the Calabi-Yau manifold. This is based on the generalized Green-Schwarz anomaly cancellation condition.

hep-th↗

arXiv1997

Toric Geometry and Enhanced Gauge Symmetry of F-Theory/Heterotic Vacua

We study F-theory compactified on elliptic Calabi-Yau threefolds that are realised as hypersurfaces in toric varieties. The enhanced gauge group as well as the number of massless tensor multiplets has a very simple description in terms of toric geometry. We find a large number of examples where the gauge group is not a subgroup of E8xE8, but rather, is much bigger (with rank as high as 296). The largest of these groups is the group recently found by Aspinwall and Gross. Our algorithm can also be applied to elliptic fourfolds, for which the groups can become extremely large indeed (with rank as high as 121328). We present the gauge content for two of the fourfolds recently studied by Klemm et al.

hep-th↗

arXiv1997

Duality Between the Webs of Heterotic and Type II Vacua

We discuss how transitions in the space of heterotic K3*T^2 compactifications are mapped by duality into transitions in the space of Type II compactifications on Calabi-Yau manifolds. We observe that perturbative symmetry restoration, as well as non-perturbative processes such as changes in the number of tensor multiplets, have at least in many cases a simple description in terms of the reflexive polyhedra of the Calabi-Yau manifolds. Our results suggest that to many, perhaps all, four-dimensional N=2 heterotic vacua there are corresponding type II vacua.

hep-th↗

arXiv1997

Comments on A,B,C Chains of Heterotic and Type II Vacua

We construct, as hypersurfaces in toric varieties, Calabi-Yau manifolds corresponding to F-theory vacua dual to E8*E8 heterotic strings compactified to six dimensions on K3 surfaces with non-semisimple gauge backgrounds. These vacua were studied in the recent work of Aldazabal, Font, Ibanez and Uranga. We extend their results by constructing many more examples, corresponding to enhanced gauge symmetries, by noting that they can be obtained from previously known Calabi-Yau manifolds corresponding to K3 compactification of heterotic strings with simple gauge backgrounds by means of extremal transitions of the conifold type.

hep-th↗

arXiv1997

F-Theory Duals of Nonperturbative Heterotic E8xE8 Vacua in Six Dimensions

We present a systematic way of generating F-theory models dual to nonperturbative vacua (i.e., vacua with extra tensor multiplets) of heterotic E8xE8 strings compactified on K3, using hypersurfaces in toric varieties. In all cases, the Calabi-Yau is an elliptic fibration over a blow up of the Hirzebruch surface F_n. We find that in most cases the fan of the base of the elliptic fibration is visible in the dual polyhedron of the Calabi-Yau, and that the extra tensor multiplets are represented as points corresponding to the blow-ups of the F_n.

hep-th↗

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.

hep-th↗

arXiv1994

Mirror Symmetry for Two Parameter Models -- II

We describe in detail the space of the two Kähler parameters of the Calabi--Yau manifold $¶_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.

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