Riemannian geometry of quantum groups and finite groups with nonuniversal differentials

We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant differential calculus, using the quantum frame bundle formalism introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras $C[G]$ with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group $C[S_3]$ is worked out in full detail and a unique torsion free and cotorsion free or `Levi-Civita' connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as $S_3$. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form.Comment: 43 pages, 1 figure. Revised August 2001 to cut page length (eg deleted appendix) for publication in CMP. Also fleshed out ex. of q-Killing metric for q-SU_2 previously mentioned (no significant additions

On Superpotentials for D-Branes in Gepner Models

A large class of D-branes in Calabi-Yau spaces can be constructed at the Gepner points using the techniques of boundary conformal field theory. In this note we develop methods that allow to compute open string amplitudes for such D-branes. In particular, we present explicit formulas for the products of open string vertex operators of untwisted A-type branes. As an application we show that the boundary theories of the quintic associated with the special Lagrangian submanifolds Im \omega_i z_i = 0 where \omega_i^5=1 possess no continuous moduli.Comment: 33 pages, 2 figure

Renormalizability of the open string sigma model and emergence of D-branes

Rederiving the one-loop divergences for the most general coupling of the open string sigma model by the heat kernel technique, we distinguish the classical background field from the mean field of the effective action. The latter is arbitrary, i.e. does not fulfil the boundary conditions. As a consequence a new divergent counter term strongly suggests the introduction of another external one-form field (beside the usual gauge field), coupled to the normal derivative at the boundary. Actually such a field has been proposed in the literature for different reasons, but its full impact never seems to have thoroughly investigated before. The beta function for the resulting renormalizable model is calculated and the consequences are discussed, including the ones for the Born-Infeld action. The most exciting property of the new coupling is that it enters the coefficient in front of the normal derivative in Neumann boundary conditions. For certain values of the background fields this coefficient vanishes, leading to Dirichlet boundary conditions. This provides a natural mechanism for the emergence of D-branes.Comment: 24 pages, a reference and discussion (about 1 page, sec. 3.3 and 4.1) added, typos correcte

Robustness of the Quintessence Scenario in Particle Cosmologies

We study the robustness of the quintessence tracking scenario in the context of more general cosmological models that derive from high-energy physics. We consider the effects of inclusion of multiple scalar fields, corrections to the Hubble expansion law (such as those that arise in brane cosmological models), and potentials that decay with expansion of the Universe. We find that in a successful tracking quintessence model the average equation of state must remain nearly constant. Overall, the conditions for successful tracking become more complex in these more general settings. Tracking can become more fragile in presence of multiple scalar fields, and more stable when temperature dependent potentials are present. Interestingly though, most of the cases where tracking is disrupted are those in which the cosmological model is itself non-viable due to other constraints. In this sense tracking remains robust in models that are cosmologically viable

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.Comment: 13 pages, published versio

A maximum principle for the mutation-selection equilibrium of nucleotide sequences.

We study the equilibrium behaviour of a deterministic four-state mutation-selection model as a model for the evolution of a population of nucleotide sequences in sequence space. The mutation model is the Kimura 3ST mutation scheme, and the selection scheme is assumed to be invariant under permutation of sites. Considering the evolution process both forward and backward in time, we use the ancestral distribution as the stationary state of the backward process to derive an expression for the mutational loss (as the difference between ancestral and population mean fitness), and we prove a maximum principle that determines the population mean fitness in mutation-selection balance

Isotropy Representation of Flag Manifolds

Flag manifolds of a classical compact Lie group G considered up to a diffeomorphism are described in terms of painted Dynkin diagrams. The explicit decomposition of the isotropy representation into irreducible components is given