18945028
S 1 -fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram
In the series of work [L-L-Y1, III: Sec. 5.4] on mirror principle, two of the current authors (K.L. and S.-T.Y.) with Bong H. Lian developed a method to compute the integral R X τ � e H·t ∩ 1d for a flag manifold X = Fl r1, ..., rI(C n ) via an extended mirror principle diagram. This integral determines the fundamental hypergeometric series HG[1] X (t) and is also related to the computation of the Gromov-Witten invariants (string world-sheet instanton numbers) on X. This method turns the required localization computation on the augmented moduli stack M0,0(CP 1 × X) of stable maps to a localization computation on a hyper-Quot-scheme HQuot(E n ) of inclusion sequences of subsheaves of a trivialized trivial bundle E n of rank n on CP 1 . In this article, the detail of this localization computation on HQuot(E n ) is carried out. The necessary ingredients in the computation, notably, the S 1 -fixed-point components and the distinguished ones E(A;0) in HQuot(E n ), the S 1 -equivariant Euler class of E(A;0) in HQuot(E n ), and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its base manifold are given. With these, an exact expression of R X τ � e H·t ∩ 1d is obtained. When X is a Grassmannian manifold, the same route reproduces the known exact expression for HG[1] X (t). For a general flag manifold X, our expression determines HG[1] X (t) implicitly. Remarks on what it suggests for general Hori-Vafa formula are given. Due to the technical necessity, a discussion on the general construction of restrictive flag manifold bundle, its natural embedding in a flag manifold bundle, and the Thom class of this embedding is also given. This work generalizes the result in [L-L-L-Y]. This work gives explicit formulas for mirror principle computations of Calabi-Yau manifolds in flag manifolds.
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.
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.
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.
119105228
Liapunov exponents and the reversibility of molecular dynamics algorithms
Abstract We study the phenomenon of lack of reversibility in molecular dynamics algorithms for the case of Wilson's lattice QCD. We demonstrate that the classical equations of motion that are employed in these algorithms are chaotic in nature. The leading Liapunov exponent is determined in a range of coupling parameters. We consider the consequences of the breakdown of reversibility due to round-off errors.
12038460
Two particle states and the S-matrix elements in multi-channel scattering
Using a quantum mechanical model, the exact energy eigenstates for two-particle two-channel scattering are studied in a cubic box with periodic boundary conditions in all three directions. A relation between the exact energy eigenvalue in the box and the two-channel S-matrix elements in the continuum is obtained. This result can be viewed as a generalization of the well-known Luscher's formula which establishes a similar relation in elastic scattering.
121803018
PRELIMINARY RESULTS FOR LIGHT HADRON MASSES FROM LATTICE QCD ON ASYMMETRIC LATTICES
118790054
Phenomenological aspects of R-parity violating supersymmetry with a vectorlike extra generation
Phenomenological analysis to the R-parity violating supersymmetry with a vectorlike extra generation is performed in detail. It is found that, via the trilinear couplings, the correct neutrino spectrum can be obtained. The Higgs mass rises to 125 GeV by new up-type Yukawa couplings of vectorlike quarks with no need of very heavy superpartners. Phenomena of new heavy fermions at LHC are predicted. DOI: 10.1103/PhysRevD.87.075012
119464468
Two Particle States in a Box and the $s$-Matrix in Multi-Channel Scattering
Using a quantum mechanical model, the exact energy eigenstates for two-particle two-channel scattering are studied in a cubic box with periodic boundary conditions. 
A relation between the exact energy eigenvalue in the box and the two-channel $S$-matrix elements in the continuum is obtained. This result can be viewed as a generalization of the well-known L\"uscher's formula which establishes a similar relation in elastic scattering.