Mirror Partners Research Articles

On a B-field transform of generalized complex structures over complex tori

Let (Xn,Xˇn) be a mirror pair of an n-dimensional complex torus Xn and its mirror partner Xˇn. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of Xˇn→Rn/Zn and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category DGXn. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on Xn, and interpret it as the deformation XGn of Xn by a flat gerbe G. Moreover, we construct the deformation of DGXn associated to the deformation from Xn to XGn, and also discuss the homological mirror symmetry between XGn and its mirror partner on the object level.

Correlation between the charge radii difference in mirror partner nuclei and thesymmetry energy slope* *Supported by the National Natural Science Foundation of China (12275025, 11975096) and the Fundamental Research Funds for the Central Universities (2020NTST06)

A correlation between the charge radii difference of mirror partner nuclei and the slope parameter L of symmetry energy has been built to ascertain the equation of state of isospin asymmetric nuclear matter. In this work, the influences of pairing correlations and isoscalar compression modulus on the are systematically investigated based on the Skyrme energy density functional theory. The calculated results suggest that the linear correlation between and L is decreased by the surface pairing correlations. The slope parameter deduced from the difference of charge radii of mirror-pair nuclei 32Ar-32Si, 36Ca-36S, 38Ca-38Ar, and 54Ni-54Fe falls into the range of − MeV, that is, the rather soft equation of state of asymmetric nuclear matter. Besides, the range of the slope parameter can also be influenced by the effective forces classified by various isoscalar incompressibility coefficients.

Chiral Gauge Theory at the Boundary between Topological Phases.

I demonstrate how chiral fermions with an exact gauge symmetry can appear on the d-dimensional boundary of a finite volume (d+1)-dimensional manifold, without any light mirror partners. The condition for the d-dimensional boundary theory to be local is that gauge anomalies cancel and that the volume be large. This can likely be achieved on a lattice and provides a new paradigm for the lattice regularization of chiral gauge theories.

Framed duality and mirror symmetry for toric complete intersections

This paper is devoted to systematically extend f-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in [17], to families of complete intersections subvarieties. Namely, f-mirror symmetry is induced by framed duality of framed toric varieties extending Batyrev-Borisov polar duality between Fano toric varieties. Framed duality has been defined and essentially well described for families of hypersurfaces in toric varieties in the previous [17]. Here it is developed for families of complete intersections, allowing us to strengthen some previous results on hypersurfaces. In particular, the class of projective complete intersections and their mirror partners are studied in detail. Moreover, a (generalized) Landau-Ginzburg/Complete-Intersection correspondence is discussed, extending to the complete intersection setup the LG/CY correspondence firstly studied Chiodo-Ruan and Krawitz.

Binding energies of proton-rich nuclei determined from their mirror pairs* *Supported by the National Natural Science Foundation of China (12075121 and 11605089), and by the Natural Science Foundation of Jiangsu Province (BK20190067 and BK20150762)

In addition to the Coulomb displacement energy, the residual differences between the binding energies of mirror nuclei (a pair of nuclei with the same mass number plus interchanged proton and neutron numbers) contribute to the shell effect via the valence scheme in this study. To this end, one linear combining type of valence nucleon number, namely, , is chosen to tackle this shell correction, in which and are the valence proton and neutron numbers with respect to the nearest shell closure, respectively. The mass differences of mirror nuclei, as the sum of the empirical Coulomb displacement energy and shell effect correction, are then used to obtain the binding energies of proton-rich nuclei through the available data of their mirror partners to explore the proton dripline of the nuclear chart.

The bijectivity of mirror functors on tori

By the SYZ construction, a mirror pair (X,Xˇ) of a complex torus X and a mirror partner Xˇ of the complex torus X is described as the special Lagrangian torus fibrations X→B and Xˇ→B on the same base space B. Then, by the SYZ transform, we can construct a simple projectively flat bundle on X from each affine Lagrangian multisection of Xˇ→B with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.

Symmetry breaking of Gamow-Teller and magnetic-dipole transitions and its restoration in calcium isotopes

Nuclear magnetic-dipole (M1) and Gamow-Teller (GT) transitions provide insight into the spin-isospin properties of atomic nuclei. By considering them as unified spin-isospin transitions, the M1/GT transition strengths and excitation energies are subject to isospin symmetry. The excitation properties associated to the M1/GT symmetry need to be clarified within consistent theoretical approach. In this work, the relationship between the M1 and GT transitions in Ca isotopes is investigated in a unified framework based on the relativistic energy-density functional (REDF) with point-coupling interactions, using the relativistic quasi-particle random-phase approximation (RQRPA). It is shown that the isovector-pseudovector (IV-PV) residual interaction affects both transitions, and the symmetry of M1 and giant-GT transitions is disrupted by this interaction in closed-shell nuclei. In open-shell Ca isotopes, the proton-neutron pairing in the residual RQRPA interaction also plays a role in GT transitions. Due to the interplay between these interactions, the M1/GT symmetry can be restored especially in the $^{42}$Ca nucleus, i.e., the giant-GT strength can become comparable to that of the M1 mode in terms of the unified spin-isospin transitions by adjusting the PN-pairing strength to reproduce the experimental low-lying GT-excitation energies. The mirror symmetry of both M1 and GT transitions is also demonstrated for open-shell mirror partners, $^{42}$Ca and $^{42}$Ti. Further improvements are required to achieve simultaneous reproduction of M1 and GT-transition energies in the REDF framework.

Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model.

Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively (1+1)D. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.

Deformations of Dimer Models

The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon $P$ there exists a dimer model having $P$ as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.

Theoretical Constraints on Neutron-Mirror-Neutron Oscillation

Mirror models lead to the possibility that neutron (n) can oscillate into its mirror partner (n′), inspiring several experimental searches for this phenomenon. The condition for observability of this oscillation is a high degree of degeneracy between the n and n′ masses, which can be guaranteed if there is exact parity symmetry taking all particles to their mirror partners. However, consistency of these models with big-bang nucleosynthesis requires that this parity symmetry be broken in the early universe in a scenario called asymmetric inflation. In this paper, we study the consistency of an observable n−n′ oscillations signal with asymmetric inflation and derive various theoretical constraints. In particular, we find that the reheat temperature after inflation should lie below 2.5 TeV, and we predict a singlet fermion with a mass below 100 GeV. In simple models, where the right-handed neutrino is a mediator of baryon-number-violating interactions, we find that the light neutrinos are Dirac fermions with their masses arising radiatively through one-loop diagrams.

The mu rightarrow egamma decay in an EW-scale non-sterile RH neutrino model

We investigate herein the phenomenology of mu rightarrow egamma decay in a scenario of the class of extended models with non-sterile right-handed (RH) neutrinos at electroweak (EW) scale proposed by P. Q. Hung. The field content of the standard model (SM) is increased by introducing for each SM fermion a corresponding mirror partner with the same quantum numbers along with opposite chirality. Light neutrino masses are generated via the type I seesaw mechanism, and it is also proved to be relevant to low energy within the EW scale of RH neutrino masses. We introduce the model and derive the branching ratio of the mu rightarrow egamma decay at one-loop approximation with the participation of W gauge boson and neutral and singly charged Higgs scalars. We then set constraints on relevant parameters and predict the sensitivity of the decay channel under the present and future experiments.

Mass relations of mirror nuclei for both bound and unbound systems

In this paper we study nuclear masses of proton-rich systems close to (both inside and outside) the proton drip line, in terms of mass relations for mirror nuclei. We show that mass relations of mirror nuclei are remarkably accurate for all nuclei (including very light nuclei) within the proton drip line, with root-mean-square deviations (RMSD) between 70--147 keV. A very simple formula is proposed to extend the mass relations to systems beyond the proton drip line, with the RMSD value of 183 keV (or 164 keV if the $^{11}\mathrm{O}\ensuremath{-}^{11}\mathrm{Li}$ mirror partners are excluded). Based on these mass relations, we predict 162 mass excesses of proton-rich systems (136 systems outside and 26 systems inside the proton drip line) close to the proton drip line, and tabulate these results in the Supplemental Material of this paper.

An extension of polar duality of toric varieties and its consequences in Mirror Symmetry

The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called \emph{framed} duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.

Mirror energy differences above the 0f7/2 shell: First γ-ray spectroscopy of the Tz = −2 nucleus 56Zn

Excited states in 56Zn were populated following one-neutron removal from a 57Zn beam impinging on a Be target at intermediate energies in an experiment conducted at the Radioactive Isotope Beam Factory at RIKEN. Three γ rays were observed and tentatively assigned to the 6+→4+→2+→0+ yrast sequence. This turns 56Zn into the heaviest Tz=−2 nucleus in which excited states are known. The excitation-energy differences between these levels and the isobaric analogue states in the Tz=+2 mirror partner, 56Fe, are compared with large-scale shell-model calculations considering the full pf valence space and various isospin-breaking contributions. This comparison, together with an analysis of the mirror energy differences in the A=58, Tz=±1 pair 58Zn and 58Ni, provides valuable information with respect to the size of the monopole radial and the isovector multipole isospin-breaking terms in the region above doubly-magic 56Ni.

Remarks on the homological mirror symmetry for tori

Let us consider an n-dimensional complex torus TJ=T2n≔ℂn∕2π(Zn⊕TZn). Here, T is a complex matrix of order n whose imaginary part is positive definite. In particular, when we consider the case n=1, the complexified symplectic form of a mirror partner of TJ=T2 is defined by using −1T or T. However, if we assume n≥2 and that T is a singular matrix, we cannot define a mirror partner of TJ=T2n as a natural generalization of the case n=1 to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.

Isobaric-multiplet mass equation in a macroscopic-microscopic approach

We study the a, b and c coefficients of the isobaric-multiplet mass equation using a macroscopic-microscopic approach developed by P. Moeller and his collaborators in ADNDT 59, 185 (1995) and ADNDT 109-110, 1 (2016). We show that already the macroscopic part of the finite-range liquid-drop model (FRLDM) describes the general trend of the a and b coefficients relatively well, while the staggering behavior of b coefficients for doublets and quartets can be understood in terms of the difference of average proton and neutron pairing energies. The sets of isobaric masses, predicted by the full macroscopic-microscopic approaches, are used to explore the general trends of IMME coefficients up to A=100. We conclude that while the agreement for a coefficients is quite satisfactory, the global approaches have less sensitivity to predict the staggering pattern observed for b coefficients of doublets and quartets. The best set of theoretical b coefficients for multiplets up to about A=100 is used to predict masses of proton-rich nuclei based on the known experimental masses of neutron-rich mirror partners, and subsequently to investigate their one- and two-proton separation energies. The estimated position of the proton-drip line is in fair agreement with known experimental data. These masses are important for simulations of the astrophysical rp-process.

HΛΛ5 and HeΛΛ5 hypernuclei reexamined in halo/cluster effective theory

The J=1/2 iso-doublet double-$\Lambda$-hypernuclei, namely, ${}_{\Lambda\Lambda}^{5}$H and ${}_{\Lambda\Lambda}^{5}$He, are examined as the three-body cluster states, $\Lambda\Lambda t$ ($t\equiv {}^3$H or triton) and $\Lambda\Lambda h$ ($h\equiv {}^3$He or helion), respectively, in a model independent framework utilizing pionless halo effective theory. Both singlet and triplet states of the constituent $\Lambda T$ ($T\equiv t,h$) sub-system are used in the elastic channel for the study of $_\Lambda^4$H$-\Lambda$ and $_\Lambda^4$He$-\Lambda$ scattering processes. A prototypical leading order investigation using a sharp momentum cut-off regulator, irrespective of the type of the elastic channel chosen, yields almost identical cut-off dependence of the three-body binding energy or the double-$\Lambda$-separation energy ($B_{\Lambda\Lambda}$) is obtained for the mirror partners, evidently suggesting good isospin symmetry in these three-body systems. Subsequently, upon normalization of our solutions to the integral equation with respect to a single pair of input data from an {\it ab initio} potential model analysis for each mirror hypernuclei, yields $B_{\Lambda\Lambda}$ which agrees fairly well with various erstwhile regulator independent potential models for our choice of the cut-off regulator, $\sim 200$ MeV. This is either consistent with pionless effective theory or with its slightly augmented version with a hard scale around $2m_\pi$, where low-energy $\Lambda$-$\Lambda$ interactions dominated by $\pi\pi$ or $\sigma$-meson exchange. Finally, to demonstrate the predictability of our effective theory, we present preliminary estimates of the S-wave $\Lambda\Lambda T$ three-body scattering lengths and the $\Lambda$-separation energies.

Mirror Symmetry for Perverse Schobers from Birational Geometry

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers.

On the equivalence of Batyrev and BHK mirror symmetry constructions

We consider the connection between two constructions of the mirror partner for the Calabi-Yau orbifold. This orbifold is defined as a quotient by some suitable subgroup G of the phase symmetries of the hypersurface XM in the weighted projective space, cut out by a quasi-homogeneous polynomial WM. The first, Berglund–Hübsch–Krawitz (BHK) construction, uses another weighted projective space and the quotient of a new hypersurface XMT inside it by some dual group GT. In the second, Batyrev construction, the mirror partner is constructed as a hypersurface in the toric variety defined by the reflexive polytope dual to the polytope associated with the original Calabi-Yau orbifold. We give a simple evidence of the equivalence of these two constructions.

Proton and neutron skins and symmetry energy of mirror nuclei

The neutron skin of nuclei is an important fundamental property, but its accurate measurement faces many challenges. Inspired by charge symmetry of nuclear forces, the neutron skin of a neutron-rich nucleus is related to the difference between the charge radii of the corresponding mirror nuclei. We investigate this relation within the framework of the Hartree-Fock-Bogoliubov method with Skyrme interactions. Predictions for proton skins are also made for several mirror pairs in the middle mass range. For the first time the correlation between the thickness of the neutron skin and the characteristics related with the density dependence of the nuclear symmetry energy is investigated simultaneously for nuclei and their corresponding mirror partners. As an example, the Ni isotopic chain with mass number A=48−60 is considered. These quantities are calculated within the coherent density fluctuation model using Brueckner and Skyrme energy-density functionals for isospin asymmetric nuclear matter with two Skyrme-type effective interactions, SkM* and SLy4. Results are also presented for the symmetry energy as a function of A for a family of mirror pairs from selected chains of nuclei with Z=20, N=14, and N=50. The evolution curves show a similar behavior crossing at the N=Z nucleus in each chain and a smooth growing deviation when N≠Z starts. Comparison of our results for the radii and skins with those from the calculations based on high-precision chiral forces is made.