Recent results on charm from E831-FOCUS

E831-FOCUS is a photoproduction experiment which collected data during the 1996/1997 fixed target run at Fermilab. More than 1 million charm particles have been reconstructed. Using this sample we measure the lifetimes of all the weakly decaying singly charmed particles, establishing the charm lifetime hierachy. Then we present recent results on semileptonic decays of charm mesons, including the new s-wave inteference phenomena in D+ to K-pi+mu+nu, and high statistics branching ratio and form factor measurements.Comment: Invited talk at the Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003 (eConf C0304052). 4 pages LaTeX, 2 eps figure

Derived categories of coherent sheaves and motives of K3 surfaces

Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational coefficients. In this paper we prove the conjecture in the case X is a K3 surface admitting an elliptic fibration (a case that always occurs if the Picard rank of X is at least 5) with finite-dimensional Chow motive. We also relate this result with a conjecture by Huybrechts showing that, for a K3 surface with a symplectic involution, the finite-dimensionality of its motive implies that the involution acts as the identity on the Chow group of 0-cycles. We give examples of pairs of K3 surfaces with the same finite-dimensional motive but not derived equivalent.Comment: 18 page

Charm Hadronic Decays From FOCUS: Lessons Learnt

The FOCUS photoproduction experiment took data in the ninenties and produced a wealth of results in charm physics. Some of the studies were seminal for contemporary experiments, and even paved the way for the technology of many charm and beauty analysis tools.Comment: Presented by S.Bianco at CHARM2010, IHEP Beijing. Six pages, 2 figure

Some Further Evidence about Magnification and Shape in Neural Gas

Neural gas (NG) is a robust vector quantization algorithm with a well-known mathematical model. According to this, the neural gas samples the underlying data distribution following a power law with a magnification exponent that depends on data dimensionality only. The effects of shape in the input data distribution, however, are not entirely covered by the NG model above, due to the technical difficulties involved. The experimental work described here shows that shape is indeed relevant in determining the overall NG behavior; in particular, some experiments reveal richer and complex behaviors induced by shape that cannot be explained by the power law alone. Although a more comprehensive analytical model remains to be defined, the evidence collected in these experiments suggests that the NG algorithm has an interesting potential for detecting complex shapes in noisy datasets

AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac-Moody algebra $\widehat{\mathfrak{gl}}_k$ on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for $\widehat{\mathfrak{gl}}_k$, which proves the AGT correspondence for pure $\mathcal{N}=2$ $U(1)$ gauge theory on $X_k$. We consider Carlsson-Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition $\widehat{\mathfrak{gl}}_k\simeq \mathfrak{h}\oplus \widehat{\mathfrak{sl}}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra $\mathfrak{h}$ and primary fields of $\widehat{\mathfrak{sl}}_k$. We use these operators to prove the AGT correspondence for $\mathcal{N}=2$ superconformal abelian quiver gauge theories on $X_k$.Comment: 58 pages; v2: typos corrected, reference added; v3: Introduction expanded, minor corrections and clarifying remarks added throughout, references added and updated; Final version published in Journal of Geometry and Physic