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A brief review on geometry and spectrum of graphsApr 14 2012This is a survey paper. We study the Ricci curvature and spectrum of graphs, as well as the exterior forms and deRahm cohomology on graphs.

A Construction of stable bundles and reflexive sheaves on Calabi-Yau threefoldsMay 22 2014We use Serre construction and deformation to construct stable bundles and reflexive sheaves on Calabi-Yau threefolds.

A remark on our paper "Negative Holomorphic curvature and positive canonical bundle"Sep 06 2016This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof ... More

D0-brane realizations of the resolution of a reduced singular curveNov 21 2011Based 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, ... More

Perfect state transfer on graphs with a potentialNov 07 2016Nov 09 2016In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential ... More

More on the admissible condition on differentiable maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(\varphi,\nabla)$Nov 29 2016In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $\varphi: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with the fundamental module $E$) with a connection $\nabla$ ... More

Davies-Gaffney-Grigor'yan Lemma on GraphsFeb 14 2014Nov 26 2015We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality. ... More

Sharp Davies-Gaffney-Grigor'yan Lemma on GraphsApr 07 2016In this note, we prove the sharp Davies-Gaffney-Grigor'yan lemma for minimal heat kernels on graphs.

A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-upsAug 11 2004In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form $Y_1\cup_D Y_2$ that arises from a degeneration $W/{\Bbb A}^1$ and a theory of relative Gromov-Witten ... More

Extracting Gromov-Witten invariants of a conifold from semi-stable reduction and relative GW invariants of pairsNov 02 2004Nov 28 2004The study of open/closed string duality and large $N$ duality suggests a Gromov-Witten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an open Gromov-Witten theory. In this work we employ the result of Jun Li on ... More

Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-GrothendieckJul 02 2009In 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 ... More

Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpointSep 11 2007We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition of D-branes (of B-type) as an Azumaya-type noncommutative ... More

D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theoryDec 02 2014In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 ... More

Quantum tunneling on graphsJan 13 2011We 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.

Heterotic String Compactification and New Vector BundlesDec 26 2014We propose a construction of K\"ahler and non-K\"ahler Calabi-Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected ... More

D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifoldsApr 08 2015We 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, ... More

D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifoldsJun 04 2014We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous ... More

Some Recent Progress in Classical General RelativityJan 21 2000In this short survey paper, we discuss certain recent results in classical gravity. Our main attention is restricted to two topics: the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and ... More

Non-Existence of Time-Periodic Solutions of the Dirac Equation in a Reissner-Nordstrom Black Hole BackgroundMay 13 1998Jan 20 2000It is shown analytically that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordstrom black hole background; in particular, there are no static solutions of the Dirac equation in such a background field. The physical interpretation ... More

Particle-Like Solutions of the Einstein-Dirac EquationsJan 22 1998Jan 25 1999The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of soliton-like solutions of these equations. The stability ... More

Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino termJun 28 2016In earlier works, D(1) (arXiv:0709.1515 [math.AG]), D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we have explained why a D-brane in string theory, when treated as a fundamental dynamical ... More

Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curvesSep 12 2008This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson's theory-space where "branes" are still branes. In this work, we focus on D-strings and construct the moduli stack ... More

Positivity of the Bondi mass in Bondi's radiating spacetimesApr 07 2006Sep 22 2006We find two conditions related to the {\it news functions} of the Bondi's radiating vacuum spacetimes. We provide a complete proof of the positivity of the Bondi mass by using Schoen-Yau's method under one condition and by using Witten's method under ... More

On A-twisted moduli stack for curves from Witten's gauged linear sigma modelsDec 23 2002Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [L-L-Y] that relates Gromov-Witten invariants ... More

Graph invariants from ideas in physics and number theorySep 20 2014Jun 17 2015We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman ... More

Global Torelli Theorem for Teichmuller Spaces of Polarized Calabi-Yau manifoldsDec 29 2009Dec 08 2011The result of this paper is proved in arXiv:1112.1163

Non-Existence of Time-Periodic Solutions of the Dirac Equation in an Axisymmetric Black Hole GeometryMay 14 1999May 16 2000We prove that, in the non-extreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chandrasekhar's separation of the Dirac equation in this geometry. A similar non-existence theorem is established ... More

Linear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole PhysicsJan 09 2008Jul 25 2009This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for ... More

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr GeometryOct 04 2003Mar 06 2008We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial ... More

The Long-Time Dynamics of Dirac Particles in the Kerr-Newman Black Hole GeometryMay 19 2000Dec 24 2003We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's ... More

Li-Yau inequality on graphsJun 11 2013Sep 17 2013We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature ... More

Smooth static solutions of the Einstein-Yang/Mills equationOct 01 1992We consider the Einstein/Yang-Mills equations in $3+1$ space time dimensions with $\SU(2)$ gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat ... More

Adiabatic Isometric Mapping Algorithm for Embedding 2-Surfaces in Euclidean 3-SpaceMar 25 2015Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly non-negative Gaussian curvature at each vertex can be isometrically embedded uniquely in $\mathbb{R}^3$ as a convex polyhedron. Due to the nonconstructive nature of his ... More

Ricci-flat cubic graphs with girth fiveFeb 08 2018We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph ... More

Gap of the First Two Eigenvalues of the Schrödinger Operator with Nonconvex PotentialFeb 13 2009We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator with a nonconvex potential in terms of a distance associated with the potential. The results here can be applied to the double well potential.

Structures of Three-ManifildsJul 31 2006Aug 21 2006This is lecture notes of a talk I gave at the Morningside Center of Mathematics on June 20, 2006. In this talk, I survey on Poincare and geometrization conjecture.

An Estimate of the Gap of the First Two Eigenvalues in the Schrödinger OperatorFeb 13 2009We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a ... More

Perspectives on geometric analysisFeb 16 2006Feb 16 2006This is a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis. It grew from a talk I gave on the occasion of seventieth anniversary of Chinese Mathematical Society. I ... More

Geometry of three manifolds and existence of Black Hole due to boundary effectSep 07 2001Feb 28 2002In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive ... More

4d N=2 SCFT and singularity theory Part I: ClassificationOct 05 2015This is the first of a series of papers in which we systematically use singularity theory to study four dimensional N=2 superconformal field theories. Our main focus in this paper is to identify what kind of singularity is needed to define a SCFT. The ... More

Semicontinuity of 4d N=2 spectrum under renormalization group flowOct 20 2015We study renormalization group flow of four dimensional N=2 SCFTs defined by isolated hypersurface three-fold singularities. We define the spectrum of N=2 theory as the set of scaling dimensions of the parameters on the Coulomb branch, which include Coulomb ... More

On exotic sphere fibrations, topological phases, and edge states in physical systemsMar 29 2013May 15 2013We suggest that exotic sphere fibrations can be mapped to band topologies in condensed matter systems. These fibrations can correspond to geometric phases of two double bands or state vector bases with second Chern numbers m+n and -n respectively. They ... More

Invariant Solutions to the Strominger System on Complex Lie Groups and Their QuotientsJul 29 2014Jun 21 2015Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on complex Lie groups. Both flat and non-flat cases are discussed in detail.

On cohomology theory of (di)graphsSep 22 2014To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW complex turns ... More

Local geometry of the G2 moduli spaceFeb 05 2008We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $\phi$ and compute the expansion of the Hodge star of $\phi$ to fourth order in the deformations of $\phi$. By considering M-theory compactified on a G2 manifold, ... More

Absence of Zero Energy States in the Simplest d=3 (d=5?) Matrix ModelsJun 18 1998The method introduced in [hep-th/9805020] is simplified, and used to calculate the asymptotic form of all SU(2) \times SO(d=3, resp. 5) invariant wave functions satisfying $Q_{\hat{\beta}} \Psi = 0, \hat{\beta} = 1 ... 4$ resp. 8, where $Q_{\hat{\beta}}$ ... More

New N = 2 dualitiesFeb 10 2016We consider N = 2 superconformal field theory with following properties: a) Coulomb branch operators have fractional scaling dimensions, b) there are exact marginal deformations . The weakly coupled gauge theory descriptions are found by decomposing 3d ... More

Computing Conformal Structure of SurfacesDec 13 2002This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct ... More

The Geometry on Smooth Toroidal Compactifications of Siegel varietiesJan 18 2012May 18 2014This is a part of our joint program. The purpose of this paper is to study smooth toroidal compactifications of Siegel varieties and their applications, we also try to understand the K\"ahler-Einstein metrics on Siegel varieties through the compactifications. ... More

Small resolutions of SU(5)-models in F-theoryJul 04 2011Sep 10 2013We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered fourfolds defined by Weierstrass models admitting a split A_4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically ... More

Absence of Zero Energy States in Reduced SU(N) 3d Supersymmetric Yang Mills TheoryNov 21 1997May 31 1999For the SU(N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions we argue that <Psi,chi> = 0 for the previously obtained solution of Q chi = 0, Q^{dagger} Psi = 0.

Hodge Bundles on Smooth Compactifications of Siegel Varieties and ApplicationsJan 18 2012May 18 2014We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be embedded into the ... More

The Existence of Supersymmetric String Theory with TorsionNov 12 2004Apr 23 2006We derived an existence criterion to the Supersymmetric String Theory with Torsion proposed by Strominger and proved the existence of such theory for a class of Calabi-Yau threefolds.

BPS States, String Duality, and Nodal Curves on K3Dec 17 1995We describe the counting of BPS states of Type II strings on K3 by relating the supersymmetric cycles of genus $g$ to the number of rational curves with $g$ double points on K3. The generating function for the number of such curves is the left-moving ... More

Topological String Partition Functions as PolynomialsJun 09 2004Jul 05 2004We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit ... More

An explicit formula of hitting times for random walks on graphsNov 30 2013We prove an explicit formula of hitting times in terms of enumerations of spanning trees for random walks on general connected graphs. We apply the formula to improve Lawler's bound of hitting times for general graphs, prove a sharp bound of hitting times ... More

A note on small deformations of balanced manifoldsApr 29 2011Jun 07 2011In 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.

A special Lagrangian type equation for holomorphic line bundlesNov 27 2014Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that ... More

Negative Holomorphic curvature and positive canonical bundleMay 21 2015In this note we show that if a projective manifold admits a K\"ahler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Elliptic Virasoro Conformal BlocksNov 02 2015We study certain six dimensional theories arising on $(p,q)$ brane webs living on $\mathbb{R}\times S^1$. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads ... More

Balanced metrics on non-Kahler Calabi-Yau threefoldsSep 27 2008Mar 14 2012We construct balanced metrics on the family of non-K\"ahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on K\"ahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex ... More

A generalization of Liu-Yau's quasi-local massFeb 15 2006In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric ... More

On the Splitting Type of an Equivariant Vector Bundle over a Toric ManifoldFeb 04 2000From 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 ... More

Canonical Metrics on the Moduli Space of Riemann Surfaces IISep 14 2004In this paper we continue our study on the canonical metrics on the Teichm\"uller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carath\'eodory metric to the K\"ahler-Einstein metric, solving another ... More

Particle-Like Solutions of the Einstein-Dirac-Maxwell EquationsFeb 03 1998Aug 23 1999We consider the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions ... More

Nonexistence for complete Kähler Einstein metrics on some noncompact manifoldsMar 30 2016Let $M$ be a compact K\"ahler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete K\"ahler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ ... More

Affine Manifolds, SYZ Geometry, and the "Y" VertexMay 04 2004Sep 09 2008We study the real Monge-Amp\`ere equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere ... More

Calabi-Yau modular forms in limit: Elliptic FibrationsNov 04 2015We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy ... More

The theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equationApr 10 2006The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-Kahler manifolds with torsion.

The Geometric Dual of a-maximisation for Toric Sasaki-Einstein ManifoldsMar 24 2005Aug 14 2006We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R^n which depends only on the toric data that defines ... More

Mirror Maps, Modular Relations and Hypergeometric Series IJul 27 1995Motivated by the recent work of Kachru-Vafa in string theory, we study in Part A of this paper, certain identities involving modular forms, hypergeometric series, and more generally series solutions to Fuchsian equations. The identity which arises in ... More

Gravitational Waves and Their Memory in General RelativityMay 20 2015General relativity explains gravitational radiation from binary black hole or neutron star mergers, from core-collapse supernovae and even from the inflation period in cosmology. These waves exhibit a unique effect called memory or Christodoulou effect, ... More

Cohomology and Hodge Theory on Symplectic Manifolds: ISep 29 2009Oct 01 2012We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on ... More

The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Bound StatesJan 21 2000Aug 02 2000We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang/Mills equations are derived. Using numerical methods, we find different types ... More

Airy Equation for the Topological String Partition Function in a Scaling LimitJun 03 2015We use the polynomial formulation of the holomorphic anomaly equations governing perturbative topological string theory to derive the free energies in a scaling limit to all orders in perturbation theory for any Calabi-Yau threefold. The partition function ... More

Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a $B$-field background à la Polchinski-Grothendieck AnsatzSep 12 2009In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,\alpha_B)$ ... More

Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth caseAug 10 2015In this follow-up of our earlier two works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) in the D-project, we study further the notion of a `differentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture ... More

D-branes and Azumaya noncommutative geometry: From Polchinski to GrothendieckMar 05 2010We 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 ... More

Perfect state transfer on graphs with a potentialNov 07 2016In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential ... More

D5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theoryOct 27 2011A 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 ... More

Quasilocal mass in general relativityApr 08 2008Dec 04 2008There have been many attempts to define the notion of quasilocal mass for a spacelike 2-surface in spacetime by the Hamilton-Jacobi analysis. The essential difficulty in this approach is to identify the right choice of the background configuration to ... More

Arithmetic Properties of Mirror Map and Quantum CouplingNov 30 1994Dec 05 1994We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror ... More

Taming symplectic forms and the Calabi-Yau equationMar 26 2007We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature ... More

Cohomology and Hodge Theory on Symplectic Manifolds: IINov 04 2010Oct 01 2012We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms ... More

Generalized Cohomologies and SupersymmetryNov 29 2011We 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 ... More

Accelerating black holes, spin-3/2 fields and C-metricApr 29 2014We consider spin-3/2 particles on the background of general accelerating black holes and C-metric. The Rarita-Schwinger equations of spin-3/2 particles are analyzed on these backgrounds. The emission and absorption probabilities of the spin-3/2 particles ... More

The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Black HolesOct 13 1999Jun 27 2000We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon ... More

The Einstein-Dirac-Maxwell Equations - Black Hole SolutionsOct 08 1999In this summary article, we review and discuss the non-existence of stationary black hole solutions for the Einstein-Dirac-Maxwell equations.

The Coupling of Gravity to Spin and ElectromagnetismJun 09 1999The coupled Einstein-Dirac-Maxwell equations are considered for a static, spherically symmetric system of two fermions in a singlet spinor state. Stable soliton-like solutions are shown to exist, and we discuss the regularizing effect of gravity from ... More

Non-Existence of Black Hole Solutions for a Spherically Symmetric, Static Einstein-Dirac-Maxwell SystemOct 14 1998Feb 04 2008We consider for j=1/2, 3/2,... a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein-Dirac-Maxwell equations ... More

Non-Existence of Black Hole Solutions to Static, Spherically Symmetric Einstein-Dirac Systems - a Critical DiscussionNov 12 2002This short note compares different methods to prove that Einstein-Dirac systems have no static, spherically symmetric solutions.

Gravitational wave memory in de Sitter spacetimeSep 03 2015We examine gravitational wave memory in the case where sources and detector are in an expanding cosmology. For simplicity, we treat the case where the cosmology is de Sitter spacetime, and discuss the possibility of generalizing our results to the case ... More

Null Asymptotics of Solutions of the Einstein-Maxwell Equations in General Relativity and Gravitational RadiationNov 10 2010Dec 14 2010We 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 ... More

Limit of quasilocal mass at spatial infinityJun 01 2009Jun 08 2009We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the ... More

Positivity of quasi-local mass IIDec 15 2004Jan 17 2005We prove the following stronger verson of the positivity of quasi-local mass stated in gr-qc/0303019: the quasi-local energy (mass) of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition ... More

Positivity of Quasilocal MassMar 05 2003Jun 19 2003Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy ... More

Mirror Symmetry is T-DualityJun 07 1996Jun 14 1996It is argued that every Calabi-Yau manifold $X$ with a mirror $Y$ admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space $Y$. The mirror transformation ... More

Isometric embeddings into the Minkowski space and new quasi-local massMay 09 2008Nov 11 2008The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should ... More

Sasaki-Einstein Manifolds and Volume MinimisationMar 03 2006Apr 23 2008We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert ... More

Duality and Fibrations on G_2 ManifoldsMar 23 2002We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and various string and ... More