arXiv subjects

Chien-Hao Liu

Publications and source records attributed to Chien-Hao Liu.

45 records · Page 3Linked to original sources

arXiv2000

A reconstruction of Euler data

We apply the mirror principle of [L-L-Y] to reconstruct the Euler data $Q=\{Q_d\}_{d\in{\tinyBbb N}\cup\{0\}}$ associated to a vector bundle $V$ on ${\smallBbb C}{\rm P}^n$ and a multiplicative class $b$. This gives a direct way to compute the intersection number $K_d$ without referring to any other Euler data linked to $Q$. Here $K_d$ is the integral of the cohomology class $b(V_d)$ of the induced bundle $V_d$ on a stable map moduli space. A package '{\tt \verb+EulerData_MP.m+}' in Maple V that carries out the actual computation is provided. For $b$ the Chern polynomial, the computation of $K_1$ for the bundle $V=T_{\ast}{\smallBbb C}{\rm P}^2$, and $K_d$, $d=1,2,3$, for the bundles ${\cal O}_{{\tinyBbb C}{\rm P}^4}(l)$ with $6\le l\le 10$ done using the code are also included.

math.AG↗

arXiv2000

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 $Q=\{Q_0, Q_1, ... \}$ for an equivariant concavex bundle ${\cal E}$ over a toric manifold, there are two places the structure of the bundle comes into play: (1) the multiplicative characteric class $Q_0$ of $V$ one starts with, and (2) the splitting type of ${\cal E}$. 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 ${\cal E}$ 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.

math.AG↗

arXiv1999

On K3-Thurston 7-manifolds and their deformation space: A case study with remarks on general K3T and M-theory compactification

M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this article, we construct a special class of such 7-manifolds, named as {\it K3-Thurston} (K3T) 7-manifolds. The factor from the K3 part of the deformation space of these K3T 7-manifolds admits a Kähler structure, while the factor of the deformation space from the Thurston part admits a special Kähler structure. The latter rings with the nature of the scalar manifold of a vector multiplet in an N=2 $d=4$ supersymmetric gauge theory. Remarks and examples on more general K3T 7-manifolds and issues to possible interfaces of K3T to M-theory are also discussed.

hep-th↗

arXiv1998

On extremal transitions of Calabi-Yau threefolds and the singularity of the associated 7-space from rolling

M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone. After providing some background in Sec. 1, the simplest case of conifold transitions is studied in Sec. 2. In Sec. 3 we employ topological methods, Smale's classification theorem of smooth simply-connected spin closed 5-manifolds, and a computer code in the Appendix to understand the 5-manifolds that appear as the link of the singularity of a singuler Calabi-Yau threefolds from a Type II primitive contraction of a smooth one. From this we obtain many locally admissible extremal transition pairs of Calabi-Yau threefolds, listed in Sec. 4. Their global realization will require further study. As a mathematical byproduct in the pursuit of the subject, we obtain a formula to compute the topology of the boundary of the tubular neighborhood of a Gorenstein rational singular del Pezzo surface embedded in a smooth Calabi-Yau threefold as a divisor.

hep-th↗

arXiv1998

On the Global Structure of Some Natural Fibrations of Joyce Manifolds

The study of fibrations of the target manifolds of string/M/F-theories has provided many insights to the dualities among these theories or even as a tool to build up dualities since the work of Strominger, Yau, and Zaslow on the Calabi-Yau case. For M-theory compactified on a Joyce manifold $M^7$, the fact that $M^7$ is constructed via a generalized Kummer construction on a 7-torus ${\smallBbb T}^7$ with a torsion-free $G_2$-structure $ϕ$ suggests that there are natural fibrations of $M^7$ by ${\smallBbb T}^3$, ${\smallBbb T}^4$, and K3 surfaces in a way governed by $ϕ$. The local picture of some of these fibrations and their roles in dualities between string/M-theory have been studied intensively in the work of Acharya. In this present work, we explain how one can understand their global and topological details in terms of bundles over orbifolds. After the essential background is provided in Sec. 1, we give general discussions in Sec. 2 about these fibrations, their generic and exceptional fibers, their monodromy, and the base orbifolds. Based on these, one obtains a 5-step-routine to understand the fibrations, which we illustrate by examples in Sec. 3. In Sec. 4, we turn to another kind of fibrations for Joyce manifolds, namely the fibrations by the Calabi-Yau threefolds constructed by Borcea and Voisin. All these fibrations arise freely and naturally from the work of Joyce. Understanding how the global structure of these fibrations may play roles in string/M-theory duality is one of the major issues for further pursuit.

hep-th↗

arXiv1997

Remarks on the Geometry of Wick Rotation in QFT and its Localization on Manifolds

The geometric aspect of Wick rotation in quantum field theory and its localization on manifolds are explored. After the explanation of the notion and its related geometric objects, we study the topology of the set of landing $W$ for Wick rotations and its natural stratification. These structures in two, three, and four dimensions are computed explicitly. We then focus on more details in two dimensions. In particular, we study the embedding of $W$ in the ambient space of Wick rotations, the resolution of the generic metric singularities of a Lorentzian surface $Σ$ by local Wick rotations, and some related $S^1$-bundles over $Σ$.

hep-th↗

arXiv1996

Topological Membrane Solitons and Loop Orders of Membrane Scatterings in M-theory

Two topological issues on membranes in M-theory are studied: (1) Soliton is an important subject in M-theory. Under the framework of obstruction theory with the help from framed links in $S^3$, we give a complete enumeration of topological membrane solitons in a string-admissible target-space of the form a product of Minkowskian space-times, tori, and K3-surfaces. Patching of these solitons and their topological charges are also defined and discussed. (2) Loop order of membrane scatterings is the basis for a perturbative M-theory. We explore this concept with emphases on its distinct features from pointlike and stringlike particles. For completeness, a light exposition on homologies of compact oriented 3-manifolds is given in the Appendix.

hep-th↗

arXiv1996

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.

hep-th↗

arXiv1995

Target Space Duality between Simple Compact Lie Groups and Lie Algebras under the Hamiltonian Formalism: I. Remnants of Duality at the Classical Level

It 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 group $G$ 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 $\frak g$ of $G$ so that $G$ and $\frak g$ 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 $\frak g$ is lightly touched.

hep-th↗