Global Torelli Theorem for Teichmuller Spaces of Polarized Calabi-Yau manifolds
The result of this paper is proved in arXiv:1112.1163
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arXiv subjects
Shing-Tung Yau
Publications and source records attributed to Shing-Tung Yau.
Generalized Cohomologies and Supersymmetry
We show that the complex cohomologies of Bott, Chern, and Aeppli and the symplectic cohomologies of Tseng and Yau arise in the context of type II string theory. Specifically, they can be used to count a subset of scalar moduli fields in Minkowski compactification with RR fluxes in the presence of either O5/D5 or O6/D6 brane sources, respectively. Further, we introduce a new set of cohomologies within the generalized complex geometry framework which interpolate between these known complex and symplectic cohomologies. The generalized complex cohomologies play the analogous role for counting massless fields for a general supersymmetric Minkowski type II compactification with Ramond-Ramond flux.
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D0-brane realizations of the resolution of a reduced singular curve
Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space $Y$ via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution $Y^{\prime}\rightarrow Y$ of a variety $Y$ over ${\Bbb C}$ can be factored through an embedding of $Y^{\prime}$ into the stack ${\frak M}^{0^{A z^f}_{\;p}}_r (Y)$ of punctual D0-branes of rank $r$ on $Y$ for $r\ge r_0$ in ${\Bbb N}$, where $r_0$ depends on the germ of singularities of $Y$. We prove that this conjecture holds for the resolution $ρ: C^{\prime}\rightarrow C$ of a reduced singular curve $C$ over ${\Bbb C}$. In string-theoretical language, this says that the resolution $C^{\prime}$ of a singular curve $C$ always arises from an appropriate D0-brane aggregation on $C$ and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.
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Yukawa Couplings on Quintic Threefolds
We compute the particle spectrum and some of the Yukawa couplings for a family of heterotic compactifications on quintic threefolds X involving bundles that are deformations of TX+O_X. These are then related to the compactifications with torsion found recently by Li and Yau. We compute the spectrum and the Yukawa couplings for generic bundles on generic quintics, as well as for certain stable non-generic bundles on the special Dwork quintics. In all our computations we keep the dependence on the vector bundle moduli explicit. We also show that on any smooth quintic there exists a deformation of the bundle TX+O_X whose Kodaira-Spencer class obeys the Li-Yau non-degeneracy conditions and admits a non-vanishing triple pairing.
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D5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory
A D5 elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in P3. They provide simple examples of elliptic fibrations admitting a rich spectrum of singular fibers (not all on the list of Kodaira) without introducing singularities in the total space of the fibration and therefore avoiding a discussion of their resolutions. We study systematically the fiber geometry of such fibrations using Segre symbols and compute several topological invariants. We present for the first time Sen's (orientifold) limits for D5 elliptic fibrations. These orientifolds limit describe different weak coupling limits of F-theory to type IIB string theory giving a system of three brane-image-brane pairs in presence of a Z_2 orientifold. The orientifold theory is mathematically described by the double cover the base of the elliptic fibration. Such orientifold theories are characterized by a transition from a semi-stable singular fiber to an unstable one. In this paper, we describe the first example of a weak coupling limit in F-theory characterized by a transition to a non-Kodaira (and non-ADE) fiber. Inspired by string dualities, we obtain non-trivial topological relations connecting the elliptic fibration and the different loci that appear in its weak coupling limit. Mathematically, these are surprising relations relating the total Chern class of the D5 elliptic fibration and those of different loci that naturally appear in the weak coupling limit. We work in arbitrary dimension and our results don't assume the Calabi-Yau condition.
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Immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce's criteria: Potential image-support rigidity of A-branes that wrap around a sL $S^3$
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which resembles a similar phenomenon for holomorphic curves that wrap around a rigid smooth rational curve in a Calabi-Yau 3-fold in Gromov-Witten theory.
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The Electromagnetic Christodoulou Memory Effect in Neutron Star Binary Mergers
Gravitational waves are predicted by the general theory of relativity. In [6] D. Christodoulou showed that gravitational waves have a nonlinear memory. We proved in [3] that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In the present paper, we study this electromagnetic Christodoulou memory effect and compute it for binary neutron star mergers. These are typical sources of gravitational radiation. During these processes, not only mass and momenta are radiated away in form of gravitational waves, but also very strong magnetic fields are produced and radiated away. Thus the observed effect on test masses of a laser interferometer gravitational wave detector will be enlarged by the contribution of the electromagnetic field. Therefore, the present results are important for the planned experiments. Looking at the null asymptotics of spacetimes, which are solutions of the Einstein-Maxwell (EM) equations, we derived in [3] the electromagnetic Christodoulou memory effect. Moreover, our results allow to answer astro- physical questions, as the knowledge about the amount of energy radiated away in a neutron star binary merger enables us to gain information about the source of the gravitational waves.
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A note on small deformations of balanced manifolds
In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.
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Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations -- direct sum, tensor product, dual, and Hom -- on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher algebraic cobordisms of vector bundles on varieties is posed in the end.
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Quantum tunneling on graphs
We explore the tunneling behavior of a quantum particle on a finite graph, in the presence of an asymptotically large potential. Surprisingly the behavior is governed by the local symmetry of the graph around the wells.
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Null Asymptotics of Solutions of the Einstein-Maxwell Equations in General Relativity and Gravitational Radiation
We prove that for spacetimes solving the Einstein-Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In [5] D. Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Chapter 3, we show that to leading order, the presence of the electromagnetic field does not change the instantaneous displacement of the test masses. Following the method introduced by D. Christodoulou in [5] and asymptotics derived by N. Zipser in [8] and [9], we investigate gravitational radiation at null infinity in spacetimes solving the EM equations. We study the Bondi mass loss formula at null infinity derived in [9]. We show that the mass loss formula from [9] is compatible with the one in Bondi coordinates obtained in [4]. And we observe that the presence of the electromagnetic field increases the total energy radiated to infinity up to leading order. Moreover, we compute the limit of the area radius at null infinity in Theorem 7.
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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 $C^{\infty}$-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 from such objects with a unitary, minimally flat connection-with-singularities. This merges Donaldson's picture of special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz for D-branes on a Calabi-Yau space. Basic phenomena of D-branes such as Higgsing/un-Higgsing and large- vs. small-brane wrapping can be realized via deformations of such morphisms. Classical results of Alexander, Hilden, Lozano, Montesinos, and Thurston suggest then a genus-like expansion of the path-integral of D3-branes. Similarly for D2-branes and M2-branes. In the last section, we use the technical results of Joyce on desingularizations of special Lagrangian submanifolds with conical singularities to explain how A-branes thus defined can be driven and re-assemble under a split attractor flow, as studied in an earlier work of Denef. This section is to be read with arXiv:hep-th/0107152 of Denef and arXiv:math.DG/0303272 of Joyce.
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Finiteness of Subfamilies of Calabi-Yau n-Folds over Curves with Maximal Length of Yukawa-Coupling
We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.
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Evaluating quasilocal energy and solving optimal embedding equation at null infinity
We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.
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D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck
We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with Gómez-Sharpe (arXiv:hep-th/0008150), Donagi-Katz-Sharpe (arXiv:hep-th/0309270), and Denef (arXiv:hep-th/0107152). Some string-theoretical remarks are given at the end of each section.
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New Heterotic Non-Kahler Geometries
New heterotic torsional geometries are constructed as orbifolds of T^2 bundles over K3. The discrete symmetries considered can be freely-acting or have fixed points and/or fixed curves. We give explicit constructions when the base K3 is Kummer or algebraic. The orbifold geometries can preserve N=1,2 supersymmetry in four dimensions or be non-supersymmetric.
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Local Heterotic Torsional Models
We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kahler manifold with a non-trivial torus fiber. We solve the non-linear anomaly equation in this background exactly. We also define a new charge that detects the non-Kahlerity of our solutions.
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Recent Development on the Geometry of the Teichmuller and Moduli Spaces of Riemann Surfaces and Polarized Calabi-Yau Manifolds
We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.
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