Toric Self-Dual Einstein Metrics as Quotients

We use the quaternion Kähler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov–Gauduchon and Bryant.During the preparation of this work the first and third authors were supported by NSF grant DMS-0203219. The second author was supported by the Leverhulme Trust, the William Gordon Seggie Brown Trust and an EPSRC Advanced Fellowship. The fourth author was supported by the MIUR Project Proprietà Geometriche delle Varietà Reali e Complesse. The authors are also grateful for support from EDGE, Research Training Network HPRN-CT-2000-00101, funded by the European Human Potential Programme.Acknowledgement The first author thanks the Ecole Polytechnique, Palaiseau and the Università di Roma La Sapienza for hospitality and support. The third author would like to thank the Università di Roma La Sapienza, I.N.d.A.M, M.P.I-Bonn, and IHES as parts of this paper were written during his visits there. The fourth named author would like to thank University of New Mexico for hospitality and support. The authors are grateful to Paul Gauduchon, Michael Singer and Pavel Winternitz for invaluable discussions.     

Indefinite Kähler-Einstein Metrics on Compact Complex Surfaces

We completely classify those compact complex surfaces which admit indefinite Ricci-flat K?hler metrics. Slightly weaker results are also obtained for indefinite K?hler-Einstein metrics with non-zero scalar curvature. Received: 6 December 1996 / Accepted: 17 March 1997     

Uniqueness Problem for Quantum Lattice Systems with Compact Spins

Quantum lattice systems with compact spins and nearest-neighbour interactions are considered. Uniqueness of the corresponding Euclidean Gibbs states is proved uniformly with respect to the temperature, in the case where the particles have a sufficiently small mass.     

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ^d , and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.     

Crystal Melting and Toric Calabi-Yau Manifolds

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.     

Schwarzschild Metrics and Quasi-Universes

The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows us to obtain some results that otherwise are less apparent or even hidden in other coordinate systems.     

Toric complete intersections and weighted projective space

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension two more than doubles the number of spectra with h¹¹=1. Altogether we find 87 new (mirror pairs of) Hodge data, mainly with h¹¹≤4.     

Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels \({\Pi_{h^k}(z,w)}\) and Berezin kernels \({P_{h^k}(z,w)}\) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) where \({D(z,w)}\) is the diastasis. When the metric is only \({C^{\infty}}\) this asymptotic cannot hold for all \({(z,w)}\) since the diastasis is not even defined for all \({(z,w)}\) close to the diagonal. Our main result is that for general toric \({C^{\infty}}\) metrics, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) as long as w lies on the \({\mathbb{R}_+^m}\)-orbit of z, and for general \({(z,w)}\), \({{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}\) where \({D(z, w^*)}\) is the diastasis between z and the translate of w by \({(S^1)^m}\) to the \({\mathbb{R}_+^m}\) orbit of z. These results are complementary to Mike Christ’s negative results showing that \({P_{h^k}(z,w)}\) does not have off-diagonal exponential decay at “speed” k if \({(z,w)}\) lies on the same \({(S^1)^m}\)-orbit.     

Quasi-Spherical Metrics and Applications

In this paper, using the idea of Bartnik [B2] on quasi-spherical metrics we continue our study on the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Unlike the previous work [ST] of the authors and the work of Liu-Yau [LY], we only assume each boundary component has nonnegative curvature which is not identically zero. We also study the case that the boundary is embedded in the quotient of the infinity of the Euclidean space over a finite group. The regularity of the black hole boundary condition of quasi-spherical metrics is also discussed.Research partially supported by NSF of China, Projects 10001001 and 10231010.Research partially supported by Earmarked Grant of Hong Kong #CUHK4032/02P.     

Energy Properness and Sasakian-Einstein Metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition of admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.     

On Degenerate Metrics and Electromagnetism

A theory of degenerate metrics is developed and applied to the problem of unifying gravitation with electromagnetism. The approach is similar to the Kaluza-Klein approach with a fifth dimension, however no ad hoc conditions are needed to explain why the extra dimension is not directly observable under everyday conditions. Maxwell's theory is recovered with differences only at very small length scales, and a new formula is found for the Coulomb potential that is regular everywhere.     

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold ${\mathcal{X}}$ and that of its toric crepant resolution Y coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Ruan’s original CRC (Gromov-Witten theory of spin curves and orbifolds, contemp math, Amer. Math. Soc., Providence, RI, pp 117–126, 2006). We prove the open CRC for the weighted projective spaces ${\mathcal{X} = \mathbb{P}(1,\ldots,1, n)}$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.     

Linear Perturbations of Quaternionic Metrics

We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold ${\mathcal{M}}We extend the twistor methods developed in our earlier work on linear deformations of hyperk?hler manifolds [1] to the case of quaternionic-K?hler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-K?hler manifold M{\mathcal{M}} are in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperk?hler cone S{\mathcal{S}}. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space Z_S{\mathcal{Z}_\mathcal{S}}, with a suitable homogeneity condition that ensures that the hyperk?hler cone property is preserved. Equivalently, we show that the deformations of M{\mathcal{M}} can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space Z_M{\mathcal{Z}_\mathcal{M}} of M{\mathcal{M}}, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-K?hler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.     

Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

We present a proof of the mirror conjecture of Aganagic and Vafa (Mirror Symmetry, D-Branes and Counting Holomorphic Discs. http://arxiv.org/abs/hep-th/0012041v1, 2000) and Aganagic et al. (Z Naturforsch A 57(1–2):128, 2002) on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Graber-Zaslow, Contemp Math 310:107–121, 2002), (ii) an outer brane at arbitrary framing in the resolved conifold ${\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)}$ O P 1 ( - 1 ) ⊕ O P 1 ( - 1 ) (Zhou, Open string invariants and mirror curve of the resolved conifold. http://arxiv.org/abs/1001.0447v1 [math.AG], 2010), and (iii) an outer brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Brini, Open topological strings and integrable hierarchies: Remodeling the A-model. http://arxiv.org/abs/1102.0281 [hep-th], 2011).     

Symmetries of Taub-NUT Dual Metrics

In this paper we study the symmetries of thedual Taub-nut metrics. Generic and non-genericsymmetries of dual Taub-nut metrics are investigated.The existence of the Runge-Lenz type symmetry isanalyzed for dual Taub-nut metrics. We find that in somecases the symmetries of the dual metrics are the samewith the symmetries of Taub-nut metric.     

Generalization of the Gross-Perry Metrics

A class of SO(n+1) symmetric solutions of the (N+n+1)-dimensional Einstein equations is found. It contains 5-dimensional metrics of Gross and Perry and Millward.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.