683 research outputs found

    On the Global Structure of Some Natural Fibrations of Joyce Manifolds

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    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 M7M^7, the fact that M7M^7 is constructed via a generalized Kummer construction on a 7-torus {\smallBbb T}^7 with a torsion-free G2G_2-structure ϕ\phi suggests that there are natural fibrations of M7M^7 by {\smallBbb T}^3, {\smallBbb T}^4, and K3 surfaces in a way governed by ϕ\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

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    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=4d=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

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    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 ε\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

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    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

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    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 YY 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 ZZ-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold

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    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 ZZ-semistable morphisms from general Azumaya nodal curves, of genus 2\ge 2, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such ZZ-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/CC^{\infty}-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

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

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    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 CC^\infty-algebrogeometric foundations of d=4d=4, N=1N=1 supersymmetry and d=4d=4, N=1N=1 superspace in physics, with emphasis on the partial CC^\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 D^\widehat{D}-chiral maps φ^\widehat{\varphi} from a d=4d=4 N=1N=1 Azumaya/matrix superspace with a fundamental module with a SUSY-rep compatible hybrid connection ^\widehat{\nabla} to a complex manifold YY 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 (^,φ^)(\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

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    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

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    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 WW 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 WW in the ambient space of Wick rotations, the resolution of the generic metric singularities of a Lorentzian surface Σ\Sigma by local Wick rotations, and some related S1S^1-bundles over Σ\Sigma.Comment: 24 pages, 9 Postscript figure
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