Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form \omega . This generalises Freed's definition of (affine) special K\"ahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and K\"ahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n. Such a realisation induces a canonical \nabla-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special K\"ahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms \alpha. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-K\"ahler structure on the cotangent bundle of a special K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and Introduction, version to appear in J. Geom. Phy

Spin(7)-manifolds and symmetric Yang--Mills instantons

In this Letter we establish a relationship between symmetric SU(2) Yang--Mills instantons and metrics with Spin(7)-holonomy. Our method is based on a slight extension of that of Bryant and Salamon developed to construct explicit manifolds with special holonomies in 1989. More precisely, we prove that making use of symmetric SU(2) Yang--Mills instantons on Riemannian spin-manifolds, we can construct metrics on the chiral spinor bundle whose holonomies are within Spin(7). Moreover if the resulting space is connected, simply connected and complete, the holonomy coincides with Spin(7). The basic example is the metric constructed on the chiral spinor bundle of the round four-sphere by using a generic SU(2)-instanton of unit action; hence it is a five-parameter deformation of the Bryant--Salamon example, also found by Gibbons, Page and Pope.Comment: 10 pages, no figures, LaTeX. More references have been added; but this version differs from the published on

Fivebrane Gravitational Anomalies

Freed, Harvey, Minasian and Moore have proposed a mechanism to cancel the gravitational anomaly of the M-theory fivebrane coming from diffeomorphisms acting on the normal bundle. This procedure is based on a modification of the conventional M-theory Chern-Simons term. We compactify this space-time interaction to the ten-dimensional type IIA theory. We then analyze the relation to the anomaly cancellation mechanism for the type IIA fivebrane proposed by Witten.Comment: 20 pages, Tex, no figures. Added Reference

WDVV Equations as Functional Relations

We discuss the associativity or WDVV equations and demonstrate that they can be rewritten as certain functional relations between the {\it second} derivatives of a single function, similar to the dispersionless Hirota equations. The properties of these functional relations are further discussed.Comment: 9 pages LaTex. Typos in equations (33) and (38) correcte

One size fits all: equilibrating chemically different polymer liquids through universal long-wavelength description

Mesoscale behavior of polymers is frequently described by universal laws. This physical property motivates us to propose a new modeling concept, grouping polymers into classes with a common long-wavelength representation. In the same class samples of different materials can be generated from this representation, encoded in a single library system. We focus on homopolymer melts, grouped according to the invariant degree of polymerization. They are described with a bead-spring model, varying chain stiffness and density to mimic chemical diversity. In a renormalization group-like fashion library samples provide a universal blob-based description, hierarchically backmapped to create configurations of other class-members. Thus large systems with experimentally-relevant invariant degree of polymerizations (so far accessible only on very coarse-grained level) can be microscopically described. Equilibration is verified comparing conformations and melt structure with smaller scale conventional simulations

Apex Exponents for Polymer--Probe Interactions

We consider self-avoiding polymers attached to the tip of an impenetrable probe. The scaling exponents $\gamma_1$ and $\gamma_2$, characterizing the number of configurations for the attachment of the polymer by one end, or at its midpoint, vary continuously with the tip's angle. These apex exponents are calculated analytically by $\epsilon$-expansion, and numerically by simulations in three dimensions. We find that when the polymer can move through the attachment point, it typically slides to one end; the apex exponents quantify the entropic barrier to threading the eye of the probe

Mesoscopic non-equilibrium thermodynamics approach to the dynamics of polymers

We present a general formalism able to derive the kinetic equations of polymer dynamics. It is based on the application of nonequilibrium thermodynamics to analyze the irreversible processes taking place in the conformational space of the macromolecules. The Smoluchowski equation results from the analysis of the underlying diffusion process in that space within the scheme of nonequilibrium thermodynamics. We apply the method to different situations, concerning flexible, semiflexible and rod-like polymers and to the case of more concentrated solutions in which interactions become important.Comment: 13 pages (RevTex). To be published in Physica

Non-Abelian Chern-Simons models with discrete gauge groups on a lattice

We construct the local Hamiltonian description of the Chern-Simons theory with discrete non-Abelian gauge group on a lattice. We show that the theory is fully determined by the phase factors associated with gauge transformations and classify all possible non-equivalent phase factors. We also construct the gauge invariant electric field operators that move fluxons around and create/anihilate them. We compute the resulting braiding properties of the fluxons. We apply our general results to the simplest class of non-Abelian groups, dihedral groups D_n.Comment: 16 pages, 7 figure

Corrections to scaling in multicomponent polymer solutions

We calculate the correction-to-scaling exponent $\omega_T$ that characterizes the approach to the scaling limit in multicomponent polymer solutions. A direct Monte Carlo determination of $\omega_T$ in a system of interacting self-avoiding walks gives $\omega_T = 0.415(20)$. A field-theory analysis based on five- and six-loop perturbative series leads to $\omega_T = 0.41(4)$. We also verify the renormalization-group predictions for the scaling behavior close to the ideal-mixing point.Comment: 21 page