117221986
D-branes of A-type, their deformations, and Morse cobordism of A-branes on Calabi-Yau 3-folds under a split attractor flow: Donaldson/Alexander-Hilden-Lozano-Montesinos-Thurston/Hurwitz/Denef-Joyce meeting Polchinski-Grothendieck
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya C 1 manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical definition of supersymmetric D-branes of A-type (i.e. A-branes) – in an appropriate region of the Wilson’s theory-space of string theory – as special Lagrangian morphisms with a unitary, minimally flat connectionwith-singularity. This merges Donaldson’s picture of special Lagrangian submanifolds (1999) and the Polchinski-Grothendieck Ansatz for D-branes in a Calabi-Yau space (Sec. 2.1). The Higgsing/un-Higgsing and the large- vs. small-brane wrapping of A-branes in string theory can be achieved via deformations of such morphisms (Sec. 2.2 and Sec. 2.3). For the case of Calabi-Yau 3-folds, classical results of Alexander (1920), Hilden (1974) and Montesinos (1976), Thurston (1982), and Hilden-Lozano-Montesinos (1983) on 3-manifolds branchedcovering S 3 implies that any embedded special Lagrangian submanifold with a complex vector bundle with a unitary flat connection on a Calabi-Yau 3-fold is the image of a special Lagrangian morphism from an Azumaya 3-sphere with a fundamental module, with a unitary minimally flat connection. This suggests a genus-like expansion of the path-integral of D3-branes in type IIB string theory compactified on Calabi-Yau 3-folds that resembles the genus expansion of the path-integral of strings (Sec. 2.4.2). Similarly, for the path-integral of D2-branes and M2-branes respectively. In Sec. 3, we use the technical results of Joyce (2002-2003) on desingularizations of special Lagrangian submanifolds with conical singularities to explain how supersymmetric D3-branes thus defined can be driven and re-assemble under a reverse split attractor flow at a point on the wall of marginal stability in Type IIB superstring theory compactified on varying Calabi-Yau 3-folds, studied by Denef (2001). This last section is to be read alongside the works [De3] (arXiv:hep-th/0107152) of Denef and [Joy3: V] (arXiv:math.DG/0303272) of Joyce. To cover the basic type of deformations of morphisms from Azumaya spaces in this note and its sequel, we discuss in Sec. 1 Morse cobordisms of manifolds and their promotion to Morse cobordisms of Azumaya manifolds with a fundamental module, and of morphisms from Azumaya manifolds to complex manifolds. The notion of cone of special Lagrangian cycles in a Calabi-Yau manifold is brought out in Sec. 2.4.1 for further study. A summary of the needed results from Joyce is given in the appendix.
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.
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.
14577943
Lorentz Surfaces and Lorentzian CFT --- with an appendix on quantization of string phase space
The interest in string Hamiltonian system has recently been rekindled due to its application to target-space duality. In this article, we explore another direction it motivates. In Sec.\ 1, conformal symmetry and some algebraic structures of the system that are related to interacting strings are discussed. These lead one naturally to the study of Lorentz surfaces in Sec.\ 2. In contrast to the case of Riemann surfaces, we show in Sec.\ 3 that there are Lorentz surfaces that cannot be conformally deformed into Mandelstam diagrams. Lastly in Sec.\ 4, we discuss speculatively the prospect of Lorentzian conformal field theory. 
Additionally, to have a view of what quantum picture a string Hamiltonian system may lead to, we discuss independently in the Appendix a formal geometric quantization of the string phase space.
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.
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
2525683
Two particle states in an asymmetric box
The exact two-particle energy eigenstates in an asymmetric rectangular box with periodic boundary conditions in all three directions are studied. Their relation with the elastic scattering phases of the two particles in the continuum are obtained. These results can be viewed as a generalization of the corresponding formulae in a cubic box obtained by Luscher before. In particular, the s-wave scattering length is related to the energy shift in the finite box. Possible applications of these formulae are also discussed.
118837551
New experimental limit on the electric dipole moment of the electron in a paramagnetic insulator
We report results of an experimental search for the intrinsic Electric Dipole Moment (EDM) of the electron using a solid-state technique. The experiment employs a paramagnetic, insulating gadolinium gallium garnet (GGG) that has a large magnetic response at low temperatures. The presence of the eEDM would lead to a small but non-zero magnetization as the GGG sample is subject to a strong electric field. We search for the resulting Stark-induced magnetization with a sensitive magnetometer. Recent progress on the suppression of several sources of background allows the experiment to run free of spurious signals at the level of the statistical uncertainties. We report our first limit on the eEDM of ( 5.57 ± 7.98 ± 0.12)×10 25 e·cm with 5 days of data averaging. The search for the electric dipole moment (EDM) of elementary particles is motivated to test the discrete symmetries assumed in the Standard Model (SM) of particle physics. An account of the different transformation properties of the EDM (a polar vector) and the spin (a pseudo-vector), the fundamental physical laws governing particles must violate both time-reversal (T) and parity-inversion (P) symmetries for a fermion to acquire an EDM [1]. While the phenomena of P violation is firmly established in numerous experiments, T violation has only been observed directly in the neutral kaon system [2], with more searches in the B system underway [3]. Measurements of EDMs of elementary particles use different experimental techniques, often on low energy systems at Q ≃ 0, to probe the physics of T violation (with no flavor-changing) at energy scales higher than tens of TeV, and could provide information complementary to that from high-energy collider experiments on the nature of symmetry breaking. The physics of T violation is often linked, via the CPT theorem, to the violation of the combined Charge conjugate (C) and P symmetry. The only confirmed source of CP violation in the SM is the complex phase of the CKM matrix (that describes the quark mixing in charged-current weak interactions). With it, the electron EDM (eEDM) manifests through high-order loop couplings that involve flavor-changing quark interactions with the exchange of W ± weak bosons. The resulting size of eEDM predicted within the framework of SM is no larger than 10 −38 e·cm, which is well beyond the reach of current experimental techniques. The current experimental upper bound is established using thallium atomic beams with a sensitivity of 1.6×10 −27 e·cm [4]. New sources of CP violation introduced by theories beyond
118572828
Glueballs at finite temperature inSU(3)Yang-Mills theory
Thermal properties of glueballs in SU(3) Yang-Mills theory are investigated in a large temperature range from $0.3T_c$ to $1.9T_c$ on anisotropic lattices. The glueball operators are optimized for the projection of the ground states by the variational method with a smearing scheme. Their thermal correlators are calculated in all 20 symmetry channels. It is found in all channels that the pole masses $M_G$ of glueballs remain almost constant when the temperature is approaching the critical temperature $T_c$ from below, and start to reduce gradually with the temperature going above $T_c$. The correlators in the $0^{++}$, $0^{-+}$, and $2^{++}$ channels are also analyzed based on the Breit-Wigner $\emph{Ansatz}$ by assuming a thermal width $\Gamma$ to the pole mass $\omega_0$ of each thermal glueball ground state. While the values of $\omega_0$ are insensitive to $T$ in the whole temperature range, the thermal widths $\Gamma$ exhibit distinct behaviors at temperatures below and above $T_c$. The widths are very small (approximately few percent of $\omega_0$ or even smaller) when $T T_c$ and reach values of roughly $\Gamma\sim \omega_0/2$ at $T\approx 1.9T_c$.
17450923
Anisotropic Lattice Calculation of Pion Scattering Using an Asymmetric Box
Pion-pion elastic scattering phase shifts in the I = 2, J = 0 channel are calculated within quenched approximation using improved gauge and Wilson fermion actions on anisotropic lattices in an asymmetric box. Using the modified Lüscher’s formula applicable in an asymmetric volume, we are able to access more low-momentum modes than a cubic box with similar volume. The calculation is performed at three lattice spacings at β = 2.080, β = 2.215 and β = 2.492. For each lattice spacing, several valence quark mass values are taken in the quenched approximation which enable us to extrapolate the results to the chiral limit. Continuum limit extrapolation is then performed and the extrapolated results are compared with the known results from both theory and experiment. We find that our lattice results for the scattering phase shifts in this channel are consistent with the experimental data when the three-momentum of the pion is below 300MeV. The results are Preprint submitted to Elsevier 2 March 2008 also consistent with previous lattice calculations and other theoretical calculations using Chiral Perturbation Theory or dispersion relations. We have also given the pion-pion scattering length in the chiral and continuum limit.
119368335
The Higgs model with a complex ghost pair
Abstract A higher derivative term is introduced in the kinetic energy of the Higgs lagrangian in the minimal Standard Model. A logically consistent and finite field theory is obtained when some excitations of the Higgs field are quantized with indefinite metric in the Hilbert space. The Landau ghost phenomenon of the ponventional triviality problem is replaced by the state vectors of a complex ghost pair at a finite mass scale with observable physical consequences. It is shown that the ghost states exhibit unusual resonance properties and correspond to a complex conjugate pair of Pauli-Villars regulator masses in the euclidean path integral formulation of the theory. An argument is given that microscopic acausality effects associated with the ghost pair remain undetectable in scattering processes with realistic wave packets, and the S -matrix should exhibit unitarity in the observable sector of the Hilbert space.