18184003
On the splitting type of an equivariant vector bundle over a toric manifold
From the work of Lian, Liu, and Yau on ”Mirror Principle”, in the explicit computation of the Euler data Q = { Q0, Q1, � � � } for an equivariant concavex bundle E over a toric manifold, there are two places the structure of the bundle comes into play: (1) the multiplicative characteric class Q0 of V one starts with, and (2) the splitting type of E. Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of E to the classifying data of Kaneyama. From these relations, we compute the splitting type of a couple of nonsplittable equivariant vector bundles over toric manifolds that may be of interest to string theory and mirror symmetry. A code in Mathematica that carries out the computation of some of these examples is attached.
14489670
Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field background a la Polchinski-Grothendieck Ansatz
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,\alpha_B)$ along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of $B$-field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to $(Y,\alpha_B)$, are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of $B$-field to both the D-brane world-volume and the superstring target-space(-time) $Y$. This brings the notion of twisted ${\cal D}$-modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled "Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3] (arXiv:hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately "Dijkgraaf-Holland-Su{\l}kowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [D-H-S-V] (arXiv:0709.4446 [hep-th]) and [D-H-S] (arXiv:0810.4157 [hep-th]) of Dijkgraaf, Hollands, Su{\l}kowski, and Vafa.
341730
Target-space duality between simple compact Lie groups and Lie algebras under the Hamiltonian formalism: I. Remnants of duality at the classical level
AbstractIt has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function Γ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation Φ generated by Γ together with an Ad-invariant metric and a B-field on the associated Lie algebra
$$\mathfrak{g}$$
 ofG so thatG and
$$\mathfrak{g}$$
 form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation Φ including a careful analysis of its domain and image. The geometry of the T-dual structure on
$$\mathfrak{g}$$
 is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper.
116949875
A Degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y1 ∪D Y2 that arises from a degeneration W/A 1 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair (Y,D). As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Wt of W/A 1 to the Gromov-Witten invariants of the singular fiber W0 = Y1 ∪D Y2 via gluing the relative pairs (Y1,D) and (Y2,D). The finite sum mentioned above depends on a relative ample line bundle H on W/A 1 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li’s theory, one needs a refined degeneration formula that depends
40162216
Azumaya noncommutative geometry and D-branes: An Origin of the master nature of D-branes
In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of Douglas and Moore in 1996. A conjectural relation of our setting with `D-geometry' in the sense of Douglas is also given. The lecture is based on a series of works on D-branes with Shing-Tung Yau, and in part with Si Li and Ruifang Song.
15607702
Extracting Gromov-Witten invariants of a conifold from semi-stable reduction and relative GW invariants of pairs
The study of open/closed string duality and large N duality suggests a GromovWitten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an open Gromov-Witten theory. The work of Jun Li on Gromov-Witten theory for a projective singular variety of the gluing form Y1 ∪D Y2, where D is a smooth divisor on smooth Y1 and Y2, suggests two methods to study Gromov-Witten invariants for a projective conifold: one by a direct generalization of his construction to the conifold singularity and the other by an appropriate semi-stable reduction of a degeneration to a conifold and then apply his results on this new degeneration to extract Gromov-Witten invariants of the original conifold. In this work we carry out the second method. Suggested by the semi-stable reduction, we associate to a conifold
117463737
A Supersymmetric model with the gauge symmetry SU(3)(1) x SU(2)(1) x U(1)(1) x SU(3)(2) x SU(2)(2) x U(1)(2)
A supersymmetric model with two copies of the Standard Model gauge groups is constructed in the gauge mediated supersymmetry breaking scenario. The supersymmetry breaking messengers axe in a simple form. The Standard Model is obtained after first step gauge symmetry breaking. In the case of one copy of the gauge interactions being strong, a scenario of electroweak symmetry breaking is discussed, and the gauginos are generally predicted to be heavier than the sfermions.
2644316
Four quark operators relevant to B meson lifetimes from QCD sum rules
At the order of 1/m_b^3, the B meson lifetimes are controlled by the hadronic matrix elements of some four-quark operators. The nonfactorizable magnitudes of these four-quark operator matrix elements are analyzed by QCD sum rules in the framework of heavy quark effective theory. The vacuum saturation for color-singlet four-quark operators is justified at hadronic scale, and the nonfactorizable effect is at a few percent level. However for color-octet four-quark operators, the vacuum saturation is violated sizably that the nonfactorizable effect cannot be neglected for the B meson lifetimes. The implication to the extraction of some of the parameters from B decays is discussed. The B meson lifetime ratio is predicted as \tau(B^-)/\tau(B^0)=1.09\pm 0.02. However, the experimental result of the lifetime ratio \tau(\Lambda_b)/\tau(B^0) still cannot be explained.
118542460
Coherent Neutron Scattering in Polycrystalline Deuterium and its Implications for Ultracold Neutron Production
This paper presents a calculation of the neutron cross-sections in solid materials (used in practical neutron sources) with a large coherent scattering contribution. In particular, the dynamic structure function S(Q, !) of polycrystalline ortho-D2 is evaluated using a Monte-Carlo calculation that performs an average over scattering angles relative to crystal axes in random orientations. This method uses an analytical dispersion function with force constants derived from neutron scattering data of single crystal D2 in the framework of an axially symmetric force tensor. The resulting two dimensional map of S(Q, !) captures details of the phonon branches as well as the molecular rotations, that can be compared directly to data from inelastic neutron scattering on polycrystalline D2. This high resolution information is used to calculate the absolute cross-sections of production and upscattering loss of ultracold neutron (UCN). The resulting scattering cross-sections are significantly different, especially for UCN upscattering, from the previous predictions using the approach centered on the incoherent approximation.
36290076
Embedding flipped SU(5) into SO(10)
We embed the flipped SU(5) models into the SO(10) models. After the SO(10) gauge symmetry is broken down to the flipped SU(5) x U(1)(X) gauge symmetry, we can split the five/one-plets and ten-plets in the spinor 16 and 16 Higgs fields via the stable sliding singlet mechanism. As in the flipped SU(5) models, these ten-plet Higgs fields can break the flipped SU(5) gauge symmetry down to the Standard Model gauge symmetry. The doublet-triplet splitting problem can be solved naturally by the missing partner mechanism, and the Higgsino-exchange mediated proton decay can be suppressed elegantly. Moreover, we show that there exists one pair of the light Higgs doublets for the electroweak gauge symmetry breaking. Because there exist two pairs of additional vector-like particles with similar intermediate-scale masses, the SU(5) and U(1)(X) gauge couplings can be unified at the GUT scale which is reasonably (about one or two orders) higher than the SU(2)(L) x SU(3)(C) unification scale. Furthermore, we briefly discuss the simplest SO(10) model with flipped SU(5) embedding, and point out that it can not work without fine-tuning.
119447810
Note on the slope parameter of the baryonic Lambda(b) ---> Lambda(c) Isgur-Wise function
Abstract Using the framework of the Heavy Quark Effective Theory we have re-analyzed the Isgur–Wise function describing semileptonic Λ b → Λ c decays in the QCD sum rule approach. The slope parameter of the Isgur–Wise function is found to be ρ 2 = 1.35 ± 0.13 , which is consistent with an experimental measurement and a lattice calculation. To O ( 1 / m b , 1 / m c ) of the heavy quark expansion the integrated Λ b decay width is used to extract the CKM matrix element V c b for which we obtain a value of | V c b | = 0.041 ± 0.004 ± 0.001 in excellent agreement with the value of | V c b | determined from semileptonic B → D ∗ decays.