We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

Vesicles consisting of a bilayer membrane of amphiphilic lipid molecules are remarkably flexible surfaces that show an amazing variety of shapes of different symmetry and topology. Owing to the fluidity of the membrane, shape transitions such as budding can be induced by temperature changes or the action of optical tweezers. Thermally excited shape fluctuations are both strong and slow enough to be visible by video microscopy. Depending on the physical conditions, vesicles adhere to and unbind from each other or a substrate. This article describes the systematic physical theory developed to understand the static and dynamic aspects of membrane and vesicle configurations. The preferred shapes arise from a competition between curvature energy, which derives from the bending elasticity of the membrane, geometrical constraints such as fixed surface area and fixed enclosed volume, and a signature of the bilayer aspect. These shapes of lowest energy are arranged into phase diagrams, which separate regi...

Configurations of fluid membranes and vesicles

String theory axions appear to be promising candidates for explaining cosmological constant via quintessence. In this paper, we study conditions on the string compactifications under which axion quintessence can happen. For sufficiently large number of axions, cosmological constant can be accounted for as the potential energy of axions that have not yet relaxed to their minima. In compactifications that incorporate unified models of particle physics, the height of the axion potential can naturally fall close to the observed value of cosmological constant.

https://iopscience.iop.org/article/10.1088/1126-6708/2006/06/051/pdf

Axions In String Theory

Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The Sussmann and Ambrose-Singer theorems Calculus of the endpoint map and existence of geodesics Bibliography Index.

A Tour of Subriemannian Geometries, Their Geodesics and Applications

All purely bosonic supersymmetric solutions of minimal supergravity in five dimensions are classified. The solutions preserve either one half or all of the supersymmetry. Explicit examples of new solutions are given, including a large family of plane-fronted waves and a maximally supersymmetric analogue of the Godel universe which lifts to a solution of 11-dimensional supergravity that preserves 20 supersymmetries.

All supersymmetric solutions of minimal supergravity in five- dimensions

Basic theorems Cartan-Khler theory linear differential systems the characteristic variety prolongation theory applications of commutative algebra and algebraic geometry to the study of exterior differential systems partial differential equations linear differential operators.

Exterior Differential Systems

It is proved that there exist metrics with holonomy G2 and Spin(7), thus settling the remaining cases in Berger's list of possible holonomy groups. We first reformulate the "holonomy H" condition as a set of differential equations for an associated H-structure on a given manifold. We collect the needed algebraic facts about G2 and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kahler theorem) to prove the existence of solutions whose holonomy is G2 or Spin(7). We also provide explicit examples and some information about the "generality" of the space of

Metrics with exceptional holonomy

On construit explicitement trois metriques completes distinctes d'holonomie egale a G 2 , une metrique complete d'holonomie egale a Spin(7) et diverses metriques incompletes d'holonomie exceptionnelle

On the construction of some complete metrics with exceptional holonomy

so the two functional differ by a constant. The functional i^(X) has the advantage that its integrand is nonnegative and vanishes exactly at the umbilic points of the immersion X. Obviously iT(X) = 0 iff M = S and X is totally umbilic. Thus, the absolute minimum of #\"on the space of immersions X: S -> E 3 is 0 and the critical locus of such X is known. When M is a torus, Willmore provided an example of an immersion X: M -* E with iΓ(X) = 2ττ and showed that iΓ(X) > 2π for all smooth surfaces of revolution. He then conjectured that iΓ(X)^ 2*π for all immersions of the torus with equality only for the example he provided: the anchor ring swept out by revolving a circle of radius r about the line whose distance from the center of the circle was r]/ϊ. White then pointed out that the two-form (H K) dA had the property of being invariant under conformal transformations of E 3 plus the \"point at infinity\

/pdf/a-duality-theorem-for-willmore-surfaces-284p56jel1.pdf

A duality theorem for Willmore surfaces

/pdf/conformal-and-minimal-immersions-of-compact-surfaces-into-2kkbn8lndo.pdf

Robert L. Bryant

Papers

Exterior Differential Systems

Metrics with exceptional holonomy

On the construction of some complete metrics with exceptional holonomy

A duality theorem for Willmore surfaces

Conformal and minimal immersions of compact surfaces into the 4-sphere