Deformations of Toric Singularities and Fractional Branes

Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.Comment: 40 pages, 23 figures; typos correcte

Counting BPS Baryonic Operators in CFTs with Sasaki-Einstein duals

We study supersymmetric D3 brane configurations wrapping internal cycles of type II backgrounds AdS(5) x H for a generic Sasaki-Einstein manifold H. These configurations correspond to BPS baryonic operators in the dual quiver gauge theory. In each sector with given baryonic charge, we write explicit partition functions counting all the BPS operators according to their flavor and R-charge. We also show how to extract geometrical information about H from the partition functions; in particular, we give general formulae for computing volumes of three cycles in H.Comment: 46 pages, 10 figures; comments and clarifications added, published versio

The scaling equation of state of the three-dimensional O(N) universality class: N >= 4

We determine the critical equation of state of the three-dimensional O(N) universality class, for N=4, 5, 6, 32, 64. The N=4 is relevant for the chiral phase transition in QCD with two flavors, the N=5 model is relevant for the SO(5) theory of high-T_c superconductivity, while the N=6 model is relevant for the chiral phase transition in two-color QCD with two flavors. We first consider the small-field expansion of the effective potential (Helmholtz free energy). Then, we apply a systematic approximation scheme based on polynomial parametric representations that are valid in the whole critical regime, satisfy the correct analytic properties (Griffiths' analyticity), take into account the Goldstone singularities at the coexistence curve, and match the small-field expansion of the effective potential. From the approximate representations of the equation of state, we obtain estimates of universal amplitude ratios. We also compare our approximate solutions with those obtained in the large-N expansion, up to order 1/N, finding good agreement for N \gtrsim 32.Comment: 3 pages, 2 figures. Talk presented at Lattice2004(spin), Fermilab, June 21-26, 200

The Baryonic Branch of Klebanov-Strassler Solution: a Supersymmetric Family of SU(3) Structure Backgrounds

We exhibit a one-parameter family of regular supersymmetric solutions of type IIB theory that interpolates between Klebanov-Strassler (KS) and Maldacena-Nunez (MN). The solution is obtained by applying the supersymmetry conditions for SU(3)-structure manifolds to an interpolating ansatz proposed by Papadopoulos and Tseytlin. Other than at the KS point, the family does not have a conformally-Ricci-flat metric, neither it has self-dual three-form flux. Nevertheless, the asymptotic IR and UV are that of KS troughout the family, except for the extremal value of the interpolating parameter where the UV solution drastically changes to MN. This one-parameter family of solutions is interpreted as the dual of the baryonic branch of gauge theory, confirming the expecation that the KS solution corresponds to a particular symmetric point in the branch.Comment: 32 pages, 6 eps figures. v2: Typos fixed. v3: Comments added on the gauge theory interpretation of the solutio

Alignment of the ATLAS inner detector for the LHC Run II

ATLAS a multipurpose experiment at the LHC proton-proton collider. Its physics goals require high resolution, unbiased measurement of all charged particle kinematic parameters. These critically depend on the layout and performance of the tracking system, notably quality of its offline alignment. ATLAS is equipped with a tracking system built using different technologies, silicon planar sensors (pixel and micro-strip) and gaseous drift- tubes, all embedded in a 2T solenoidal magnetic field. For the LHC Run II, the system has been upgraded with the installation of a new pixel layer, the Insertable B-layer (IBL). Offline track alignment of the ATLAS tracking system has to deal with about 700,000 degrees of freedom (DoF) defining its geometrical parameters. The task requires using very large data sets and represents a considerable numerical challenge in terms of both CPU time and precision. The adopted strategy uses a hierarchical approach to alignment, combining local and global least squares techniques. An outline of the track based alignment approach and its implementation within the ATLAS software will be presented. Special attention will be paid to integration to the alignment framework of the IBL, which plays the key role in precise reconstruction of the collider luminous region, interaction vertices and identification of long-lived heavy flavor states. Techniques allowing to pinpoint and eliminate tracking systematics due to alignment as well as strategies to deal with time-dependent variations will be briefly covered. The first results from Cosmic Ray commissioning runs and status from proton-proton collision in LHC Run II will be discussed

Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Given a pair of graphs ? and ?, the problems of deciding whether there exists either a homomorphism or an isomorphism from ? to ? have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where ? and ? are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy