Categorial mirror symmetry for K3 surfaces

We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of \fF(X) is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.Comment: 11 pages, AmsLatex. Exposition (hopefully) improved, one argument simplifie

A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over \C, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.Comment: Revised version, 15 pages AMSTeX with AMSppt.sty v. 2.1

A Fourier transform for sheaves on Lagrangian families of real tori

We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible unitary connections supported on complex submanifolds of the relative Jacobian of $X$ (suitable conditions being verified on both sides).Comment: Latex, 30 pages (in a4wide format), no figures. v2: Minor expository changes, typos corrected. v3: Final version to appear in two parts in J. Geom. Phy

On the irreducibility of some quiver varieties

We prove that certain quiver varieties are irreducible and therefore are isomor-phic to Hilbert schemes of points of the total spaces of the bundles OP1( 12n) for n 65 1

Superlocalization formulas and supersymmetric Yang-Mills theories

By using supermanifold techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions.Comment: 22 pages, Latex2

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure

Frequency-modulated electromagnetic neural stimulation (FREMS) as a treatment for symptomatic diabetic neuropathy: results from a double-blind, randomised, multicentre, long-term, placebo-controlled clinical trial

AIMS/HYPOTHESIS: The aim was to evaluate the efficacy and safety of transcutaneous frequency-modulated electromagnetic neural stimulation (frequency rhythmic electrical modulation system, FREMS) as a treatment for symptomatic peripheral neuropathy in patients with diabetes mellitus. METHODS: This was a double-blind, randomised, multicentre, parallel-group study of three series, each of ten treatment sessions of FREMS or placebo administered within 3 weeks, 3 months apart, with an overall follow-up of about 51 weeks. The primary endpoint was the change in nerve conduction velocity (NCV) of deep peroneal, tibial and sural nerves. Secondary endpoints included the effects of treatment on pain, tactile, thermal and vibration sensations. Patients eligible to participate were aged 18-75 years with diabetes for ≥ 1 year, HbA(1c) <11.0% (97 mmol/mol), with symptomatic diabetic polyneuropathy at the lower extremities (i.e. abnormal amplitude, latency or NCV of either tibial, deep peroneal or sural nerve, but with an evocable potential and measurable NCV of the sural nerve), a Michigan Diabetes Neuropathy Score ≥ 7 and on a stable dose of medications for diabetic neuropathy in the month prior to enrolment. Data were collected in an outpatient setting. Participants were allocated to the FREMS or placebo arm (1:1 ratio) according to a sequence generated by a computer random number generator, without block or stratification factors. Investigators digitised patients' date of birth and site number into an interactive voice recording system to obtain the assigned treatment. Participants, investigators conducting the trial, or people assessing the outcomes were blinded to group assignment. RESULTS: Patients (n = 110) with symptomatic neuropathy were randomised to FREMS (n = 54) or placebo (n = 56). In the intention-to-treat population (50 FREMS, 51 placebo), changes in NCV of the three examined nerves were not different between FREMS and placebo (deep peroneal [means ± SE]: 0.74 ± 0.71 vs 0.06 ± 1.38 m/s; tibial: 2.08 ± 0.84 vs 0.61 ± 0.43 m/s; and sural: 0.80 ± 1.08 vs -0.91 ± 1.13 m/s; FREMS vs placebo, respectively). FREMS induced a significant reduction in day and night pain as measured by a visual analogue scale immediately after each treatment session, although this beneficial effect was no longer measurable 3 months after treatment. Compared with the placebo group, in the FREMS group the cold sensation threshold was significantly improved, while non-significant differences were observed in the vibration and warm sensation thresholds. No relevant side effects were recorded during the study. CONCLUSIONS/INTERPRETATION: FREMS proved to be a safe treatment for symptomatic diabetic neuropathy, with immediate, although transient, reduction in pain, and no effect on NCV. TRIAL REGISTRATION: ClinicalTrials.gov NCT01628627. FUNDING: The clinical trial was sponsored by Lorenz Biotech (Medolla, Italy), lately Lorenz Lifetech (Ozzano dell'Emilia, Italy)

Discrete approaches to quantum gravity in four dimensions

The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the author welcomes any comments and suggestion