On (orientifold of) type IIA on a compact Calabi‐Yau

We study the gauged sigma model and its mirror Landau‐Ginsburg model corresponding to type IIA on the Fermat degree‐24 hypersurface in WCP ⁴[1,1,2,8,12] (whose blow‐up gives the smooth CY₃(3,243)) away from the orbifold singularities, and its orientifold by a freely‐acting antiholomorphic involution. We derive the Picard‐Fuchs equation obeyed by the period integral as defined in [1, 2], of the parent 𝒩 = 2 type IIA theory of [3]. We obtain the Meijer's basis of solutions to the equation in the large and small complex structure limits (on the mirror Landau‐Ginsburg side) of the abovementioned Calabi‐Yau, and make some remarks about the monodromy properties associated based on [4], at the same and another MATHEMATICAlly interesting point. Based on a recently shown 𝒩 = 1 four‐dimensional triality [6] between Heterotic on the self‐mirror Calabi‐Yau CY₃(11,11), M theory on and F‐theory on an elliptically fibered CY₄ with the base given by CP ¹ × Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY₃(3,243), and then explicitly verify the same using mirror symmetry formulation of [2] for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau‐Ginsburg model corresponding to the resolved Calabi‐Yau as well.     

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.     

Operators from Mirror Curves and the Quantum Dilogarithm

Mirror manifolds to toric Calabi–Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi–Yau threefolds, these operators are of trace class. In some simple geometries, like local \({\mathbb{P}^2}\), we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi–Yau threefolds.     

Topological phase transitions in Calabi‐Yau compactifications of M‐theory

In this thesis we construct five‐dimensional gauged supergravity actions which describe flop and conifold transitions in M‐theory compactified on Calabi‐Yau threefolds. While the vector multiplet sector is determined exactly, we use the Wolf spaces to model the universal hypermultiplet together with N charged hypermultiplets corresponding to winding states of M2‐branes. After specifying the hypermultiplet sector the actions are uniquely determined by M‐theory. As an application we consider five‐dimensional Kasner cosmologies. Including the dynamics of the winding modes, we find smooth cosmological solutions which undergo flop and conifold transitions. Instead of the usual runaway behavior the scalar fields of these solutions generically stabilize in the transition region where they oscillate around the transition locus. The scalar potential thereby induces short episodes of accelerated expansion in the space‐time.     

Compactifications of the heterotic string with unitary bundles

We describe a large new class of four‐dimensional supersymmetric string vacua defined as compactifications of the E₈ × E₈ and the SO(32) heterotic string on smooth Calabi‐Yau threefolds with unitary gauge bundles and heterotic five‐branes. The conventional gauge symmetry breaking via Wilson lines is replaced by the embedding of non‐flat line bundles into the ten‐dimensional gauge group, thus opening up the way for phenomenologically interesting string compactifications on simply connected manifolds. After a detailed analysis of the four‐dimensional effective theory we exemplify the general framework by means of a couple of explicit examples involving the spectral cover construction of stable holomorphic bundles. As for the SO(32) heterotic string, the resulting vacua can be viewed, in the S‐dual Type I picture, as a generalisation of magnetized D9/D5‐brane models. In the case of the E₈ × E₈ string, we find a natural way to construct realistic MSSM‐like models, either directly or via a flipped SU(5) GUT scenario.     

Localization Computation of One-Point Disk Invariants of Projective Calabi–Yau Complete Intersections

We define one-point disk invariants of a smooth projective Calabi–Yau complete intersection in the presence of an anti-holomorphic involution via localization. We show that these invariants are rational numbers and obtain a formula for them which confirms, in particular, a conjecture by Jinzenji–Shimizu [(Int J Geom Method M 11(1):1456005, 2014), Conjecture 1].     

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.     

Hyperconifold transitions,mirror symmetry,and string theory

Multiply-connected Calabi–Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi–Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrev?s construction of Calabi–Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds — the Z₃${\mathbb{Z}}_3$ and Z₅${\mathbb{Z}}_5$ cases — also have Calabi–Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi–Yau manifolds, with small Hodge numbers and fundamental group Z₃${\mathbb{Z}}_3$ or Z₅${\mathbb{Z}}_5$. Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.     

Toric complete intersections and weighted projective space

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension two more than doubles the number of spectra with h¹¹=1. Altogether we find 87 new (mirror pairs of) Hodge data, mainly with h¹¹≤4.     

(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.     

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.     

The effective action of D‐branes in Calabi‐Yau orientifold compactifications

In this review article we study type IIB superstring compactifications in the presence of space‐time filling D‐branes while preserving 𝒩=1 supersymmetry in the effective four‐dimensional theory. This amount of unbroken supersymmetry and the requirement to fulfill the consistency conditions imposed by the space‐time filling D‐branes lead to Calabi‐Yau orientifold compactifications. For a generic Calabi‐Yau orientifold theory with space‐time filling D3‐ or D7‐branes we derive the low‐energy spectrum. In a second step we compute the effective 𝒩=1 supergravity action which describes in the low‐energy regime the massless open and closed string modes of the underlying type IIB Calabi‐Yau orientifold string theory. These 𝒩=1 supergravity theories are analyzed and in particular spontaneous supersymmetry breaking induced by non‐trivial background fluxes is studied. For D3‐brane scenarios we compute soft‐supersymmetry breaking terms resulting from bulk background fluxes whereas for D7‐brane systems we investigate the structure of D‐ and F‐terms originating from worldvolume D7‐brane background fluxes. Finally we relate the geometric structure of D7‐brane Calabi‐Yau orientifold compactifications to 𝒩=1 special geometry.     

Heterotic String Compactification and New Vector Bundles

We propose a construction of Kähler and non-Kähler Calabi–Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n\({\mathbb{P}^{2}}\)s. We also construct K3-fibered Calabi–Yau manifolds from the branched double covers of the blow-ups of the twistor spaces. These manifolds can be used in heterotic string compactifications to four dimensions. We also construct stable and polystable vector bundles. Some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions.     

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.     

A geometrical construction of rational boundary states in linear sigma models

Starting from the geometrical construction of special Lagrangian submanifolds of a toric variety, we identify a certain subclass of A-type D-branes in the linear sigma model for a Calabi–Yau manifold and its mirror with the A- and B-type Recknagel–Schomerus boundary states of the Gepner model, by reproducing topological properties such as their labeling, intersection, and the relationships that exist in the homology lattice of the D-branes. In the non-linear sigma model phase these special Lagrangians reproduce an old construction of 3-cycles relevant for computing periods of the Calabi–Yau, and provide insight into other results in the literature on special Lagrangian submanifolds on compact Calabi–Yau manifolds. The geometrical construction of rational boundary states suggests several ways in which new Gepner model boundary states may be constructed.     

Decay Dynamics of 3‐methyl‐3‐pentene‐2‐one from the light absorbing S2(ππ*) state ‐ Resonance Raman Spectroscopy and CASSCF Study

The photophysics of 3‐methyl‐3‐pentene‐2‐one (3M3P2O) after excitation to the S₂(ππ*) electronic state were studied using the resonance Raman spectroscopy and complete active space self‐consistent field (CASSCF) method calculations. The A‐band resonance Raman spectra were obtained in cyclohexane, acetonitrile, and methanol with excitation wavelengths in resonance with the first intense absorption band to probe the structural dynamics of 3M3P2O. The B3LYP‐TD/6‐31++G(d, p) computation was carried out to determine the relative A‐band resonance Raman intensities of the fundamental modes, and the result was used to reproduce the corresponding fundamental band intensities of the 223.1 nm resonance Raman spectrum and thus to examine whether the vibronic‐coupling existed in Franck‐Condon region or not. CASSCF calculations were carried out to determine the minimal singlet excitation energies of S_{1, FC}, S_1,min (nπ*), S_{2, FC}, S_2,min (ππ*), the transition energies of the conical intersection points S_n/S_π, S_n/S₀, and the optimized excited state geometries as well as the geometry structures of the conical intersection points. The A‐band short‐time structural dynamics and the corresponding decay dynamics of 3M3P2O were obtained by the analysis of the resonance Raman intensity pattern and CASSCF computations. It was revealed that the initial structural dynamics of 3M3P2O was towards the simultaneous C₃=C₄ and C₂=O₇ bond elongation, with the C₃=C₄ bond length lengthening greater at the very beginning, whereas the C₂=O₇ bond length changing greater at the later evolution time before reaching the CI(S₂/S₁) conical intersection point. The decay dynamics from S₂(ππ*) to S₁(nπ*) via S₂(ππ*)/S₁(nπ*) in singlet realm and from S₁(nπ*) to T₁(nπ*) via ISC[S₁(nπ*)/T₂(ππ*)/T₁(nπ*)] in triplet realm are proposed. Copyright © 2014 John Wiley & Sons, Ltd.     

Enhanced gauge symmetry in 6D F‐theory models and tuned elliptic Calabi‐Yau threefolds

We systematically analyze the local combinations of gauge groups and matter that can arise in 6D F‐theory models over a fixed base. We compare the low‐energy constraints of anomaly cancellation to explicit F‐theory constructions using Weierstrass and Tate forms, and identify some new local structures in the “swampland” of 6D supergravity and SCFT models that appear consistent from low‐energy considerations but do not have known F‐theory realizations. In particular, we classify and carry out a local analysis of all enhancements of the irreducible gauge and matter contributions from “non‐Higgsable clusters,” and on isolated curves and pairs of intersecting rational curves of arbitrary self‐intersection. Such enhancements correspond physically to unHiggsings, and mathematically to tunings of the Weierstrass model of an elliptic CY threefold. We determine the shift in Hodge numbers of the elliptic threefold associated with each enhancement. We also consider local tunings on curves that have higher genus or intersect multiple other curves, codimension two tunings that give transitions in the F‐theory matter content, tunings of abelian factors in the gauge group, and generalizations of the “E₈” rule to include tunings and curves of self‐intersection zero. These tools can be combined into an algorithm that in principle enables a finite and systematic classification of all elliptic CY threefolds and corresponding 6D F‐theory SUGRA models over a given compact base (modulo some technical caveats in various special circumstances), and are also relevant to the classification of 6D SCFT's. To illustrate the utility of these results, we identify some large example classes of known CY threefolds in the Kreuzer‐Skarke database as Weierstrass models over complex surface bases with specific simple tunings, and we survey the range of tunings possible over one specific base.