Determinantal Quintics and Mirror Symmetry of Reye Congruences

We study a certain family of determinantal quintic hypersurfaces in \({\mathbb{P}^{4}}\) whose singularities are similar to the well-studied Barth–Nieto quintic. Smooth Calabi–Yau threefolds with Hodge numbers (h ^1,1,h ^2,1) = (52, 2) are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi–Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi–Yau threefolds in \({\mathbb{P}^{4}\times\mathbb{P}^{4}}\) which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over \({\mathbb{P}^{2}}\) and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over \({\mathbb{P}^{2}}\) completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.     

Flux vacua statistics for two‐parameter Calabi‐Yau's

We study the number of flux vacua for type IIB string theory on an orientifold of the Calabi‐Yau expressed as a hypersurface in WCP ⁴[1,1,2,2,6] by evaluating a suitable integral over the complex‐structure moduli space as per the conjecture of Douglas and Ashok. We show that away from the singular conifold locus, one gets a power law, and that the (neighborhood) of the conifold locus indeed acts as an attractor in the (complex structure) moduli space. In the process, we evaluate the periods near the conifold locus. We also study (non)supersymmetric solutions near the conifold locus, and show that supersymmetric solutions near the conifold locus do not support fluxes.     

Completing the web of ℤ3‐quotients of complete intersection Calabi‐Yau manifolds

We complete the study [1] of smooth ℤ₃‐quotients of complete intersection Calabi‐Yau threefolds by discussing the six new manifolds that admit free ℤ₃ actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ₃‐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 ℤ₃‐free quotients of complete intersection Calabi‐Yau threefolds is connected by conifold transitions.     

Gauge threshold corrections in intersecting brane world models

We calculate the one‐loop corrections to gauge couplings in N = 1 supersymmetric brane world models, which are realized in an type IIA orbifold/orientifold background with several stacks of D6 branes wrapped on 3‐cycles with non‐vanishing intersections. Contributions arise from both N = 1 and N = 2 open string subsectors. In contrast to what is known from ordinary orbifold theories, N = 1 subsectors do give rise to moduli‐dependent one‐loop corrections.     

Gauge sector statistics of intersecting D‐brane models

In this paper, which is based on the first part of the author's PhD thesis, we review the statistics of the open string sector in T⁶/(ℤ₂ × ℤ₂) orientifold compactifications of type IIA. After an introduction to the orientifold setup we discuss the two different techniques that have been developed, using either a saddle point approximation or a direct computer based method. We explain the two approaches by means of eight‐ and six‐dimensional toy models and present the results for the four‐dimensional models in detail. Special emphasis is placed on models containing phenomenologically interesting gauge groups, in particular those containing a standard model, Pati‐Salam or SU(5) part.     

D3 instantons in Calabi–Yau orientifolds with(out) fluxes

We investigate the instanton effects due to D3 branes wrapping a four-cycle in a Calabi–Yau orientifold with D7 branes. We study the condition for the nonzero superpotentials from the D3 instantons. To this aim we work out the zero mode structures of D3 branes wrapping a four-cycle both in the presence of the fluxes and in the absence of the fluxes. In the presence of the fluxes, the condition for the nonzero superpotential could be different from that without the fluxes. We explicitly work out a simple example of the orientifold of K3×T ²/Z ₂ with a suitable flux to show such behavior. The effects of the D3–D7 sectors are interesting and give further constraints for the nonzero superpotential. In a special configuration where D3 branes and D7 branes wrap the same four-cycle, multi-instanton calculus of D3 branes could be reduced to that of a suitable field theory. The structure of D5 instantons in Type I theory is briefly discussed.     

On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes

We consider certain vacua of four‐dimensional SU(N) gauge theory with the same field content as the 𝒩 = 4 supersymmetric Yang‐Mills theory, resulting from potentials which break the 𝒩 = 4 supersymmetry as well as its global SO(6) symmetry down to SO(3) × SO(3). We show that the theory behaves at intermediate scales as Yang‐Mills theory on M⁴ × S_L² × S_R², where the extra dimensions are fuzzy spheres with magnetic fluxes. We determine in particular the structure of the zero modes due to the fluxes, which leads to low‐energy mirror models.     

(MS)SM‐like models on smooth Calabi‐Yau manifolds from all three heterotic string theories

We perform model searches on smooth Calabi‐Yau compactifications for both the supersymmetric E₈ × E₈ and SO(32) as well as for the non‐supersymmetric SO(16) × SO(16) heterotic strings simultaneously. We consider line bundle backgrounds on both favorable CICYs with relatively small h₁₁ and the Schoen manifold. Using Gram matrices we systematically analyze the combined consequences of the Bianchi identities and the tree‐level Donaldson‐Uhlenbeck‐Yau equations inside the Kähler cone. In order to evaluate the model building potential of the three heterotic theories on the various geometries, we perform computer‐aided scans. We have generated a large number of GUT‐like models (up to over a few hundred thousand on the various geometries for the three heterotic theories) which become (MS)SM‐like upon using a freely acting Wilson line. For all three heterotic theories we present tables and figures summarizing the potentially phenomenologically interesting models which were obtained during our model scans.     

BPS relations from spectral problems and blowup equations

Recently, an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi–Yau threefold has been proposed. At the same time, an exact quantization condition for the cluster integrable systems associated with these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi–Yau threefold. We prove these constraints for the \(Y^{{N},m}\) geometries using the K-theoretic blowup equations for SU(N) SYM with generic Chern–Simons invariant m.

     

On the existence of non‐supersymmetric black hole attractors for two‐parameter Calabi‐Yau's and attractor equations

We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY₃(2,128) expressed as a degree‐12 hypersurface in WCP ⁴[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the existence of a non‐supersymmetric attractor along with a consistent choice of fluxes and extremum values of the complex structure moduli, could be connected to the existence of an elliptic curve fibered over C ⁸ which may also be “arithmetic” (in some cases, it is possible to interpret the extremization conditions for the black‐hole superpotential as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that existence of non‐supersymmetric black‐hole attractors corresponds to a version of A₁‐singularity in the space Image( Z ⁶→ R ²/ Z ₂ (↪ R ³)) fibered over the complex structure moduli space. The (derivatives of the) effective black hole potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP ⁵→ CP ²⁰, fibered over the complex structure moduli space. We also discuss application of Kallosh's attractor equations (which are equivalent to the extremization of the effective black‐hole potential) for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the attractor equations than the extremization of the black hole potential.     

Gauss–Manin Connection in Disguise: Calabi–Yau Threefolds

We describe a Lie Algebra on the moduli space of non-rigid compact Calabi–Yau threefolds enhanced with differential forms and its relation to the Bershadsky–Cecotti–Ooguri–Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions \({{\bf F}_{g}^{\rm alg}, g \geq 1}\), which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck’s algebraic de Rham cohomology and on the algebraic Gauss–Manin connection. In this way, we recover a result of Yamaguchi–Yau and Alim–Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi–Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.     

Kähler metrics and gauge kinetic functions for intersecting D6‐branes on toroidal orbifolds – The complete perturbative story

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kähler metrics and closed string Kähler potential on intersecting D6‐branes by matching open string one‐loop computations of gauge thresholds with field theoretical gauge couplings in 𝒩 = 1 supergravity. We consider all cases of bulk, fractional and rigid D6‐branes on T⁶/Ω ℛ and the orbifolds T⁶/(ℤ_N × Ω ℛ) and T⁶/(ℤ₂ × ℤ_2M × Ω ℛ) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kähler potential. Our analysis includes all supersymmetric configurations of vanishing and non‐vanishing angles among D6‐branes and O6‐planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kähler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one‐loop corrections: a Kähler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6‐branes, and finally a constant term from intersections with O6‐planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D‐brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kähler metrics for some T⁶/ℤ₂ × ℤ₂ examples with rigid D‐branes of [1]. As a second class of examples, the Kähler metrics and gauge kinetic functions for the fractional QCD and leptonic D6‐brane stacks of the Standard Model on T⁶/ℤ₆^′T⁶/ℤ₆^′ from [2] are given.     

Remodeling the B-Model

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi–Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A _p fibrations, where the amplitudes compute the ’t Hooft expansion of vevs of Wilson loops in Chern-Simons theory on lens spaces. We also use our formalism to predict the disk amplitude for the orbifold .     

Gravitational couplings and Z2 orientifolds

The interplay between gravitational couplings on branes and the occurrence of fractional flux in low-dimensional orientifolds is examined. It is argued that gravitational couplings need to be assigned not only to D-branes but also to orientifold planes. The fractional charges of the orientifold d-planes can be understood in terms of flux quantization of the d − 3 form potential and modified Bianchi identities. Detailed results are presented for the case of the type IIB orientifold on T⁶/Z₂, which is dual to F-theory on a complex 4-fold with terminal singularities.     

Counting curves with modular forms

We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space P⁴_(1,1,2,2,6). We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under PSL(2, Z) modular transformations. These functions are then determined by monodromy properties, by singularities at the massless monopole point of the moduli space, and by S → T exchange symmetry.     

On de Sitter vacua in type IIA orientifold compactifications

This Letter discusses the orientifold projection of the quantum corrections to type IIA strings compactified on rigid Calabi–Yau threefolds. It is shown that N=2$N=2$ membrane instanton effects give a holomorphic contribution to the superpotential, while the perturbative corrections enter into the Kähler potential. At the level of the scalar potential the corrections to the Kähler potential give rise to a positive energy contribution similar to adding anti-D3-branes in the KKLT scenario. This provides a natural mechanism to lift an AdS vacuum to a meta-stable dS vacuum.     

Mirror Symmetry on Kummer Type <Emphasis Type="Italic">K</Emphasis>3 Surfaces

We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of _N orbifold limits of K3. The resulting map gives an automorphism of order 4,8, or 12, respectively, on the smooth universal covering space of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results.