Mirror Family Research Articles

SYZ mirror symmetry for toric Calabi-Yau manifolds

We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

Heterotic weight lifting

We describe a method for constructing genuinely asymmetric ( 2 , 0 ) heterotic strings out of N = 2 minimal models in the fermionic sector, whereas the bosonic sector is only partly build out of N = 2 minimal models. This is achieved by replacing one minimal model plus the superfluous E 8 factor by a non-supersymmetric CFT with identical modular properties. This CFT generically lifts the weights in the bosonic sector, giving rise to a spectrum with fewer massless states. We identify more than 30 such lifts, and we expect many more to exist. This yields more than 450 different combinations. Remarkably, despite the lifting of all Ramond states, it is still possible to get chiral spectra. Even more surprisingly, these chiral spectra include examples with a certain number of chiral families of SO ( 10 ) , SU ( 5 ) or other subgroups, including just SU ( 3 ) × SU ( 2 ) × U ( 1 ) . The number of families and mirror families is typically smaller than in standard Gepner models. Furthermore, in a large number of different cases, spectra with three chiral families can be obtained. Based on a first scan of about 10% of the lifted Gepner models we can construct, we have collected more than 10,000 distinct spectra with three families, including examples without mirror fermions. We present an example where the GUT group is completely broken to the standard model, but the resulting and inevitable fractionally charged particles are confined by an additional gauge group factor.

Generic Torelli theorem for one-parameter mirror families to weighted hypersurfaces

One-parameter mirror families to weighted hypersurfaces are already constructed and well studied. A generic Torelli theorem for quintic-mirror family is proved by Sampei Usui. In this article, we give the proof of a generic Torelli theorem for the other families after Usui's proof for quintic-mirror family.

Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.

A mirror symmetric construction of [formula omitted

Let G be a simple simply connected complex algebraic group. We give a Lie-theoretic construction of a conjectural mirror family associated to a general flag variety G / P , and show that it recovers the Peterson variety presentation for the T-equivariant quantum cohomology rings q H T ∗ ( G / P ) ( q ) with quantum parameters inverted. For SL n / B we relate our construction to the mirror family defined by Givental and its T-equivariant analogue due to Joe and Kim.

On solutions to Walcher’s extended holomorphic anomaly equation

Let X be a non-singular quintic hypersurface in CP4. The case of the X and its mirror is the most well-studied example of the mirror symmetry. After the construction of the mirror family of Calabi–Yau 3-folds [10], the genus 0 Gromov–Witten (GW) potential of X were computed via the Yukawa coupling of the mirror family [4]. The predicted mirror formula was proved first by Givental [7]. For higher genera, Bershadsky–Cecotti–Ooguri–Vafa (BCOV) [2] has predicted that the GW potential at genus g is obtained as a certain limit of the B-model closed topological string amplitude F (g) of genus g.1 They have also proposed a partial differential equation (PDE) for F (g), called the BCOV holomorphic anomaly equation, which determines F (g) up to a holomorphic function. The prediction of BCOV for the genus 1 GW potential was proved by Zinger [21]. Recently the open string analogue of the mirror symmetry has been developed by Walcher [18] for the pair (X,L) of the quintic 3-fold X defined over R (called a real quintic) and the set of real points L = X(R) which is a Lagrangian submanifold of X. Open mirror symmetry gave the prediction for the generating function for the disk GW invariants of X with boundary in L and it was proved by Pandharipande–Solomon–Walcher [16]. Then, Walcher [19] further proposed the open string analogue of BCOV, the extended holomorphic anomaly equation, which is a PDE for the B-model topological string amplitude F (g,h) for world-sheets with g handles and h boundaries.2

Moment zeta functions for toric Calabi–Yau hypersurfaces

Moment zeta functions provide a diophantine formulation for the distribution of rational points on a family of algebraic varieties over finite fields. They also form algebraic approximations to Dwork's p-adic unit root zeta functions. In this paper, we use l-adic cohomology to calculate all the higher moment zeta functions for the mirror family of the Calabi-Yau family of smooth projective hypersurfaces over finite fields. Our main result is a complete determination of the purity decomposition and the trivial factors for the moment zeta functions.

Family unification in five and six dimensions

In family unification models, all three families of quarks and leptons are grouped together into an irreducible representation of a simple gauge group, thus unifying the Standard Model gauge symmetries and a gauged family symmetry. Large orthogonal groups, and the exceptional groups $E_7$ and $E_8$ have been much studied for family unification. The main theoretical difficulty of family unification is the existence of mirror families at the weak scale. It is shown here that family unification without mirror families can be realized in simple five-dimensional and six-dimensional orbifold models similar to those recently proposed for SU(5) and SO(10) grand unification. It is noted that a family unification group that survived to near the weak scale and whose coupling extrapolated to high scales unified with those of the Standard model would be evidence accessible in principle at low energy of the existence of small (Planckian or GUT-scale) extra dimensions.

The transcendental part of the regulator map forK1 on a mirror family of K3-surfaces

ULLER-STACH Abstract We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K1 of a mirror family of quartic K 3 surfaces. The re- sulting multivalued function does not satisfy the hypergeometric differential equation of the periods, and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equations are related to Painlev´ e VI-type differential equations. 1. The regulator map and Picard-Fuchs equations In this paper we study the first nonclassical higher K -group K 1.X/ for a smooth com- plex projective surface X. It was conjectured by H. Esnault around 1995 that certain elements in this group can be detected in the transcendental part of the Deligne coho- mology group H 3

PICARD LATTICES OF FAMILIES OF K3 SURFACES

ABSTRACT Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.

WHAT HEAVY QUANTA BOUNDS COULD BE INFERRED FROM A HIGGS DISCOVERY?

The Higgs-gauge bosons couplings ghVV(V = g, γ, W, Z) receive non-decoupling corrections due to heavy quanta, and deviations from the SM predictions can be used to test its presence. The possible Higgs signal recently reported at LEP, with mh = 115 GeV , severely constrains the presence of heavy quanta, such as a heavy fourth family. At Tevatron, the Higgs production by gluon fusion, followed by the decay h → WW*, can also be used to probe the existence of heavy colored particles, including additional standard or mirror families, chiral colored sextets and octet quarks. Within the MSSM, we also find that gluon fusion is a sensitive probe for the spectrum of squark masses.

Mirror families in electro-weak symmetry breaking

We study symmetry breaking in a left-right symmetric extension of the Standard Model with mirror fermions, one for each Standard-Model fermion. The new particles assist a top-quark condensate in breaking electro-weak symmetry. Half of the fermions acquire electro-weak-invariant masses at around 500 GeV and would be probably accessible at future high-energy experiments like LHC or NLC. The contributions to the S and T parameters are small and negative in accordance with electro-weak precision data.

Radiative quark mass and mixing hierarchies from supersymmetric models with a fourth mirror family.

A class of supersymmetric models is discussed in which radiative quark mass as well as Kobayashi-Maskawa (KM) and neutral-current mixing hierarchies are obtained with all Yukawa couplings of the same order and no horizontal symmetries. The presence of a single mirror family is crucial. The respective KM mixing angles of the top quark and mirror top quark (${t}^{\ensuremath{'}}$) are of the same order (in particular, ${V}_{\mathrm{tb}}\ensuremath{\sim}{V}_{{t}^{\ensuremath{'}}b}\ensuremath{\sim}1$). The lighter of the two top quarks typically has a mass \ensuremath{\lesssim}100 GeV while the mirror bottom quark, considerably heavier than either top quark, typically has a mass \ensuremath{\lesssim}300 GeV. The main experimental consequence of the neutral-current mixing matrix is $B(Z\ensuremath{\rightarrow}\overline{c}t+c\overline{t})\ensuremath{\sim}{10}^{\ensuremath{-}4}\ensuremath{-}{10}^{\ensuremath{-}5}$ (if ${m}_{t}<{m}_{Z}\ensuremath{-}{m}_{c}$), with similar branching ratio for charm+mirror top quark. These decays should be observable at CERN LEP given ${10}^{7}$ $Z'\mathrm{s}$. The models provide a promising framework for nonperturbative Maiani-Parisi-Petronzio unification. The field content can, in principle, be obtained from the superstring and a search for two-generation string vacua is advocated.

Matter-parity constraints on particle spectrum in three-generation Calabi-Yau manifolds.

An analysis of the particle spectrum after intermediate scale breaking in the three-generation Calabi-Yau manifold CP/sup 3//times/CP/sup 2//Z/sub 3/ is given. The spectrum is investigated within the framework of intermediate scale breaking which preserves matter parity and breaks the rank-six gauge group SU(3)/sub /ital C///times/SU(3)/sub /ital L///times/SU(3)/sub /ital R// to the standard-model gauge group SU(3)/sub /ital C///times/SU(2)/sub /ital L///times/U(1)/sub /ital Y//. It is shown that a large number of particles and mirror particles associated with the extra families and mirror families on CP/sup 3//times/CP/sup 3//Z/sub 3/ can all be made superheavy through mass generation induced by nonrenormalizable interactions that are present in the superpotential below the compactification scale. The final spectrum of the compactified superstring below the intermediate-mass scale breaking is shown to be the spectrum of the standard /ital N/=1 supersymmetry theory and a few additional exotic leptons. Some of these exotic leptons are hybrid structures containing equal mixture of leptons and mirror leptons.

Flavour-changing Z 0 decays

We compute neutral Higgs and Z 0-boson couplings to quarks in a model with three quark-lepton families and one mirror family. Mixing between ordinary and mirrorquarks leads to flavour-changing couplings which are negligible for the known quarks. However, t-quark couplings are sizeable and the branching ratio for the decay Z 0→( tc ) + ( t c ) can be as large as 10 −4.

Matter parity, intermediate scale breaking, and sin2 theta W in Calabi-Yau superstring models.

Consequences of matter-parity invariance on intermediate scale breaking in three-generation Calabi-Yau superstring models are discussed. It is shown that values of ${\mathrm{sin}}^{2}$${\mathrm{\ensuremath{\theta}}}_{\mathrm{W}}$ in conformity with the current experimental limits can arise in the Calabi-Yau models when the extra families and mirror families gain superheavy masses through nonrenormalizable interactions. A small number of exotic leptons with electroweak-scale masses remain and may be accessible at accelerator energies.

Quark and lepton masses in superstring-type models with mirror families.

We investigate ordinary-mirror fermion mass splitting, quark isospin breaking, and both Majorana and Dirac neutrino mass seesaw mechanisms in the framework of ${\mathrm{E}}_{6}$-based superstring-type models with mirror families. Our results suggest that actual realizations containing vectorlike generations should not be overlooked in the search for a realistic low-energy phenomenology.

Mirror families and radiative SU2 x U1 breaking.

We examine the possibility that a heavy fourth family with mirror quantum numbers is the driving force for radiative ${\mathrm{SU}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathrm{U}}_{1}$ breaking in the framework of supersymmetric grand unified theories coupled to N=1 supergravity. We compare the results to the case of a sequential fourth family.

E8 family unification, mirror fermions, and new low-energy physics.

The group E/sub 8/ is investigated from the point of view of family unification. A Peccei-Quinn symmetry is used to protect the light fermions from acquiring a superlarge mass. It is found that if this protection is to be maintained without destroying perturbative unification three families are uniquely picked out with a family group SU(3)/sub fam/is contained inE/sub 6/ x SU(3)/sub fam/..-->..E/sub 8/. A relation between the (V-A)/(V+A) hierarchy and the hierarchy of M/sub PQ//M/sub GUT/ is found. It is also found that, in addition to mirror families, several exotic fermions characteristically remain light.

Cancellation of anomalies in supersymmetric sigma models based on exceptional groups

We show that there is no global obstruction in the supersymmetric E 8/SO(10)×U(1) 3 sigma model which accomodates three families of ordinary quarks and leptons and one mirror family as quasi-Nambu-Goldstone fermions. However, the other interesting sigma models based on E 7 or E 8, such as E 7/SU(5)×SU(3)×U(1) or E 8/SO(10)×SU(3)×U(1), are anomalous and hence cannot be quantized. We also point out that these anomalies are removed by introducing matter chiral-multiplets. A possible explanation of the fermion mass hierarchy is suggested in the E 7/SU(5)×U(1) 3 sigma model.