Deformations of calibrated subbundles of Euclidean spaces via twisting by special sections

We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.Comment: 16 pages, no figures. Version 2: Only minor cosmetic and typographical revisions. To appear in "Annals of Global Analysis and Geometry.

Calibrated Sub-Bundles in Non-Compact Manifolds of Special Holonomy

This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.Comment: 20 pages; for Revised Version: Minor cosmetic changes, some paragraphs rewritten for improved clarit

M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized Spin(7) structures

The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of the form of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifold and ${\cal M}$ is three-dimensional Minkowski or AdS space, implies the existence of a nowhere-vanishing Majorana spinor $\xi$ on $X$. $\xi$ lifts to a nowhere-vanishing spinor on the auxiliary nine-manifold $Y:=X\times S^1$, where $S^1$ is a circle of constant radius, implying the reduction of the structure group of $Y$ to $Spin(7)$. In general, however, there is no reduction of the structure group of $X$ itself. This situation can be described in the language of generalized $Spin(7)$ structures, defined in terms of certain spinors of $Spin(TY\oplus T^*Y)$. We express the condition for ${\cal N}=1$ supersymmetry in terms of differential equations for these spinors. In an equivalent formulation, working locally in the vicinity of any point in $X$ in terms of a `preferred' $Spin(7)$ structure, we show that the requirement of ${\cal N}=1$ supersymmetry amounts to solving for the intrinsic torsion and all irreducible flux components, except for the one lying in the $\bf{27}$ of $Spin(7)$, in terms of the warp factor and a one-form $L$ on $X$ (not necessarily nowhere-vanishing) constructed as a $\xi$ bilinear; in addition, $L$ is constrained to satisfy a pair of differential equations. The formalism based on the group $Spin(7)$ is the most suitable language in which to describe supersymmetric compactifications on eight-manifolds of $Spin(7)$ structure, and/or small-flux perturbations around supersymmetric compactifications on manifolds of $Spin(7)$ holonomy.Comment: 24 pages. V2: introduction slightly extended, typos corrected in the text, references added. V3: the role of Spin(7) clarified, erroneous statements thereof corrected. New material on generalized Spin(7) structures in nine dimensions. To appear in JHE

Closed forms and multi-moment maps

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.Comment: 36 page

Fluxes in M-theory on 7-manifolds and G structures

We consider warp compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes and investigate the constraints imposed by supersymmetry. As long as the 7-manifold supports only one Killing spinor we infer from the Killing spinor equations that non-trivial 4-form fluxes will necessarily curve the external 4-dimensional space. On the other hand, if the 7-manifold has at least two Killing spinors, there is a non-trivial Killing vector yielding a reduction of the 7-manifold to a 6-manifold and we confirm that 4-form fluxes can be incorporated if one includes non-trivial SU(3) structures.Comment: 13 pages, Latex; minor changes & add reference

Introduction to G2\mathrm{G}_2 geometry

These notes give an informal and leisurely introduction to $\mathrm{G}_2$ geometry for beginners. A special emphasis is placed on understanding the special linear algebraic structure in $7$ dimensions that is the pointwise model for $\mathrm{G}_2$ geometry, using the octonions. The basics of $\mathrm{G}_2$-structures are introduced, from a Riemannian geometric point of view, including a discussion of the torsion and its relation to curvature for a general $\mathrm{G}_2$-structure, as well as the connection to Riemannian holonomy. The history and properties of torsion-free $\mathrm{G}_2$ manifolds are considered, and we stress the similarities and differences with Kahler and Calabi-Yau manifolds. The notes end with a brief survey of three important theorems about compact torsion-free $\mathrm{G}_2$ manifolds.Comment: 37 pages. To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics". Version 2: Corrected the references. No other change

Simultaneous endovascular repair of an iatrogenic carotid-jugular fistula and a large iliocaval fistula presenting with multiorgan failure: a case report

<p>Abstract</p> <p>Introduction</p> <p>Iliocaval fistulas can complicate an iliac artery aneurysm. The clinical presentation is classically a triad of hypotension, a pulsatile mass and heart failure. In this instance, following presentation with multiorgan failure, management included the immediate use of an endovascular stent graft on discovery of the fistula.</p> <p>Case presentation</p> <p>A 62-year-old Caucasian man presented to our tertiary hospital for management of iatrogenic trauma due to the insertion of a central venous line into his right common carotid artery, causing transient ischemic attack. Our patient presented to a peripheral hospital with fever, nausea, vomiting, acute renal failure, acute hepatic dysfunction and congestive heart failure. A provisional diagnosis of sepsis of unknown origin was made. There was a 6.5 cm×6.5 cm right iliac artery aneurysm present on a non-contrast computed tomography scan. An unexpected intra-operative diagnosis of an iliocaval fistula was made following the successful angiographic removal of the central line to his right common carotid artery. Closure of the iliocaval fistula and repair of the iliac aneurysm using a three-piece endovascular aortic stent graft was then undertaken as part of the same procedure. This was an unexpected presentation of an iliocaval fistula.</p> <p>Conclusion</p> <p>Our case demonstrates that endovascular repair of a large iliac artery aneurysm associated with a caval fistula is safe and effective and can be performed at the time of the diagnostic angiography. The presentation of an iliocaval fistula in this case was unusual which made the diagnosis difficult and unexpected at the time of surgery. The benefit of immediate repair, despite hemodynamic instability during anesthesia, is clear. Our patient had two coronary angiograms through his right femoral artery decades ago. Unusual iatrogenic causes of iliocaval fistulas secondary to previous coronary angiograms with wire and/or catheter manipulation should be considered in patients such as ours.</p

Induced differential forms on manifolds of functions

Differential forms on the Fr\'echet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a constant function) and the bar map associating p-forms on F(S,M) to p-forms on M (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians.Comment: 17 page

Chiral de Rham complex on Riemannian manifolds and special holonomy

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi-Yau threefold and conjecture similar results for G_2 manifolds. We also discuss quasi-classical limits of these algebras.Comment: 49 pages, title changed, major rewrite with no changes in the main theorems, published versio