Narrow Results

Generalized Calabi–Gray manifolds are non-Kähler complex manifolds with very explicit geometry yet not being homogeneous. In this note, we demonstrate how generalized Calabi–Gray manifolds can be used to answer some questions in non-Kähler geometry.

We study the algebraic structure of the configuration space of the Kodaira–Spencer gravity theory on a Calabi–Yau threefold. We then investigate the deformation problem of the Kodaira–Spencer gravity at the classical level using the algebraic tools obtained here.

The mathematical development of Yang–Mills theory is an extremely fruitful subject. The purpose of this paper is to give non-experts and researchers in interdisciplinary areas a quick overview of some history, key ideas and recent developments in this subject.

In this paper, we apply the technique of chamber structures of stability polarizations to construct the full moduli space of rank-2 stable sheaves with certain Chern classes on Calabi–Yau manifolds which are anti-canonical divisor of ℙ¹×ℙⁿ or a double cover of ℙ¹×ℙⁿ. These moduli spaces are isomorphic to projective spaces. As an application, we compute the holomorphic Casson invariants defined by Donaldson and Thomas.

We investigate a method of construction of Calabi–Yau manifolds, that is, by smoothing normal crossing varieties. We develop theories for calculating the Picard groups of the Calabi–Yau manifolds obtained in this method. As an application, we show that many examples of Kawamata and Namikawa are different Calabi–Yau three-folds although they have same Hodge numbers.

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi–Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov’s sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.

We present new non-Ricci-flat Kähler metrics with U(N) and O(N) isometries as target manifolds of superconformally invariant sigma models with an anomalous dimension. They are so-called Ricci solitons, special solutions to a Ricci-flow equation. These metrics explicitly contain the anomalous dimension and reduce to Ricci-flat Kähler metrics on the canonical line bundles over certain coset spaces in the limit of vanishing anomalous dimension.

This is an exposition of recent progress in the categorical approach to D-brane physics. I discuss the physical underpinnings of the appearance of homotopy categories and triangulated categories of D-branes from a string field theoretic perspective, and with a focus on applications to homological mirror symmetry.

We initiate the study of Brane Gas Cosmology (BGC) on manifolds with nontrivial holonomy. Such compactifications are required within the context of superstring theory in order to make connections with realistic particle physics. We study the dynamics of brane gases constructed from various string theories on background spaces having a K3 submanifold. The K3 compactifications provide a stepping stone for generalizing the model to the case of a full Calabi–Yau threefold. Duality symmetries are discussed within a cosmological context. Using a duality, we arrive at an N=2 theory in four dimensions compactified on a Calabi–Yau manifold with SU(3) holonomy. We argue that the Brane Gas model compactified on such spaces maintains the successes of the trivial toroidal compactification while greatly enhancing its connection to particle physics. The initial state of the universe is taken to be a small, hot and dense gas of p-branes near thermal equilibrium. The universe has no initial singularity and the dynamics of string winding modes allow three spatial dimensions to grow large, providing a possible solution to the dimensionality problem of string theory.

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa.

In this work, we reconsider the study of 5D black branes in M-theory compactifications by means of ${𝒩}=2$ supergravity formalism. Precisely, we provide a model relaying on a three-parameter Calabi–Yau (CY) manifold in the ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^2$ projective space factorization, referred to as economical model. First, we investigate the stability of 5D BPS and non-BPS black holes obtained from wrapped M2-branes on nonholomorphic two-cycles in such a CY manifold. Then, we approach the stability of 5D black strings derived from wrapped M5-branes on nonholomorphic four-cycles. Among others, we find various stable and unstable black brane solutions depending on the charge regions of the involved moduli space.

We consider the adiabatic limit of a sequence of Hermitian Yang–Mills connections on a SU(r)-bundle over a semi-flat smooth special Lagrangian torus fibration π:M→B. The restriction of these connections to the fiber tori can be viewed as a family of SU(r)-connections on the fiber tori which are parameterized by B. We show that there is a gauge equivalent subsequence such that the restriction of each connection to the fiber tori converges in the Hausdorff topology, away from some closed subset of B of codimension at least two, to a limit which defines a r-sheeted special Lagrangian cycle in the dual special Lagrangian torus fibration.

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi–Yau hypersurfaces in toric varieties near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi–Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit X₀. In this paper, we will deal with the region near the smooth top dimensional torus fibers of X₀ and its mirror dual situation — the region near the 0-dimensional fibers of X₀.

In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi–Yau varieties. Furthermore, for certain asymptotically flat Calabi–Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

In this work, we reconsider the study of black holes and black strings in the compactification of M-theory on a Calabi–Yau three-fold, considered as a complete intersection of hypersurfaces in a product of weighted projective spaces given by ${𝕎\mathbb{P}}^4(\omega ,1,1,1,1)\times {\mathbb{P}}^1$. Using the $N=2$ supergravity formalism in five dimensions, we examine the BPS and non-BPSsolutions by wrapping M-branes on appropriate cycles in such a Calabi–Yau geometry. For the black hole case, we compute certain thermodynamical quantities. In particular, we calculate the entropy taking a maximal value corresponding to the ordinary projective space ${\mathbb{P}}^4$ with $\omega =1$. Using extended black hole entropies, we evaluate the temperature involving a minimal value for ${\mathbb{P}}^4$. Then, we approach the stability of the non-BPS black holes via the recombination factor. In the allowed electric charge regions, we show that such states are unstable. For the black string solutions, we calculate the tension taking a minimal value corresponding to ${\mathbb{P}}^4$. Computing the recombination factor, we show that the associated non-BPS black string states are stable in the allowed magnetic charge regions of the moduli space.

Bipartite graphs, especially drawn on Riemann surfaces, have of late assumed an active rôle in theoretical physics, ranging from MHV scattering amplitudes to brane tilings, from dimer models and topological strings to toric AdS/CFT, from matrix models to dessins d’enfants in gauge theory. Here, we take a brief and casual promenade in the realm of brane tilings, quiver SUSY gauge theories and dessins, serving as a rapid introduction to the reader.