683 research outputs found

### 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 $\phi$ suggests that
there are natural fibrations of $M^7$ by {\smallBbb T}^3, {\smallBbb T}^4,
and K3 surfaces in a way governed by $\phi$. 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.Comment: 36 page

### 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 K\"{a}hler structure,
while the factor of the deformation space from the Thurston part admits a
special K\"{a}hler 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.Comment: 39 page

### Quantum fluctuations, conformal deformations, and Gromov's topology --- Wheeler, DeWitt, and Wilson meeting Gromov

The moduli space of isometry classes of Riemannian structures on a smooth
manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum
gravity. A natural question concerning it is: What is a natural topology on
such moduli space that reflects best quantum fluctuations of the geometries
within the Planck's scale? This very question has been addressed by B.DeWitt
and others. In this article we introduce Gromov's $\varepsilon$-approximation
topology on the above moduli space for a closed smooth manifold. After giving
readers some feel of this topology, we prove that each conformal class in the
moduli space is dense with respect to this topology. Implication of this
phenomenon to quantum gravity is yet to be explored. When going further to
general metric spaces, Gromov's geometries-at-large-scale based on his
topologies remind one of K.Wilson's theory of renormalization group. We discuss
some features of both and pose a question on whether both can be merged into a
single unified theory.Comment: 23+2 pages, 8 figures. Two brief notes after the first posting are
added on p.23: One on a historical account linking to the related work of
David Edwards in 1968; and the other on the proof of the Main Theorem and an
inflation scenario in cosmolog

### 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.Comment: 60 pages, 22 Postscript figure

### Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck

In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2]
(respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and
arXiv:0901.0342 [math.AG]), we study a D-brane probe on a conifold from the
viewpoint of the Azumaya structure on D-branes and toric geometry. The details
of how deformations and resolutions of the standard toric conifold $Y$ can be
obtained via morphisms from Azumaya points are given. This should be compared
with the quantum-field-theoretic/D-brany picture of deformations and
resolutions of a conifold via a D-brane probe sitting at the conifold
singularity in the work of Klebanov and Witten [K-W] (arXiv:hep-th/9807080) and
Klebanov and Strasser [K-S] (arXiv:hep-th/0007191). A comparison with
resolutions via noncommutative desingularizations is given in the end.Comment: 23+2 pages, 4 figure

### A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold

In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054
[math.AG]) as the foundation to define the notion of $Z$-semistable morphisms
from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module
to a projective Calabi-Yau 3-fold and show that the moduli stack of such
$Z$-semistable morphisms of a fixed type is compact. This gives us a counter
moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten
theory to the fundamental string. It serves and prepares for us the basis
toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string
world-sheet instanton numbers in superstring theory. This note is written
hand-in-hand with D(10.1) and is to be read side-by-side with ibidem.Comment: 47 + 2 pages, 3 figure

### D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

We lay down an elementary yet fundamental lemma concerning a finite
algebraicness property of a smooth map from an Azumaya/matrix manifold with a
fundamental module to a smooth manifold. This gives us a starting point to
build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry
and calibrated geometry that are both tailored to and guided by D-brane
phenomena in string theory and along the line of our previous works D(11.1)
(arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).Comment: 19 pages, 6 figure

### $N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace

As the necessary background to construct from the aspect of Grothendieck's
Algebraic Geometry dynamical fermionic D3-branes along the line of
Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the
building blocks are given in the current notes: (1) basic
$C^\infty$-algebrogeometric foundations of $d=4$, $N=1$ supersymmetry and
$d=4$, $N=1$ superspace in physics, with emphasis on the partial
$C^\infty$-ring structure on the function ring of the superspace, (2) the
notion of SUSY-rep compatible hybrid connections on bundles over the superspace
to address connections on the Chan-Paton bundle on the world-volume of a
fermionic D3-brane, (3) the notion of $\widehat{D}$-chiral maps
$\widehat{\varphi}$ from a $d=4$ $N=1$ Azumaya/matrix superspace with a
fundamental module with a SUSY-rep compatible hybrid connection
$\widehat{\nabla}$ to a complex manifold $Y$ as a model for a dynamical
D3-branes moving in a target space(-time). Some test computations related to
the construction of a supersymmetric action functional for SUSY-rep compatible
$(\widehat{\nabla}, \widehat{\varphi})$ are given in the end, whose further
study is the focus of a separate work. The current work is a sequel to
D(11.4.1) (arXiv:1709.08927 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th])
and is the first step in the supersymmetric generalization, in the case of
D3-branes, of the standard action functional for D-branes constructed in
D(13.3) (arXiv:1704.03237 [hep-th]).Comment: 85+2 pages, 5 figure

### Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck

In this continuation of [L-Y1] and [L-L-S-Y], we explain how the Azumaya
structure on D-branes together with a netted categorical quotient construction
produces the same resolution of ADE orbifold singularities as that arises as
the vacuum manifold/variety of the supersymmetric quantum field theory on the
D-brane probe world-volume, given by Douglas and Moore [D-M] under the
string-theory contents and constructed earlier through hyper-K\"{a}hler
quotients by Kronheimer and Nakajima. This is consistent with the moral behind
this project that Azumaya-type structure on D-branes themselves -- stated as
the Polchinski-Grothendieck Ansatz in [L-Y1] -- gives a mathematical reason for
many originally-open-string-induced properties of D-branes.Comment: 20 pages, 2 figure

### 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 $\Sigma$ by local Wick rotations, and
some related $S^1$-bundles over $\Sigma$.Comment: 24 pages, 9 Postscript figure

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