Lattice supersymmetry in 1D with two supercharges

A consistent formulation of a fully supersymmetric theory on the lattice has been a long standing challenge. In recent years there has been a renewed interest on this problem with different approaches. At the basis of the formulation we present in the following there is the Dirac-Kahler twisting procedure, which was proposed in the continuum for a number of theories, including N=4 SUSY in four dimensions. Following the formalism developed in recent papers, an exact supersymmetric theory with two supercharges on a one dimensional lattice is realized using a matrix-based model. The matrix structure is obtained from the shift and clock matrices used in two dimensional non-commutative field theories. The matrix structure reproduces on a one dimensional lattice the expected modified Leibniz rule. Recent claims of inconsistency of the formalism are discussed and shown not to be relevant.Comment: 14 pages, Presented by SA and AD at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, German

Power grids vulnerability: a complex network approach

Power grids exhibit patterns of reaction to outages similar to complex networks. Blackout sequences follow power laws, as complex systems operating near a critical point. Here, the tolerance of electric power grids to both accidental and malicious outages is analyzed in the framework of complex network theory. In particular, the quantity known as efficiency is modified by introducing a new concept of distance between nodes. As a result, a new parameter called net-ability is proposed to evaluate the performance of power grids. A comparison between efficiency and net-ability is provided by estimating the vulnerability of sample networks, in terms of both the metrics.Comment: 16 pages, 3 figures. Figure 2 and table II modified. Typos corrected. Version accepted for publication in Chao

Self-similarity of higher order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponentH of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis)

Twisted SUSY Invariant Formulation of Chern-Simons Gauge Theory on a Lattice

We propose a twisted SUSY invariant formulation of Chern-Simons theory on a Euclidean three dimensional lattice. The SUSY algebra to be realized on the lattice is the N=4 D=3 twisted algebra that was recently proposed by D'Adda et al.. In order to keep the manifest anti-hermiticity of the action, we introduce oppositely oriented supercharges. Accordingly, the naive continuum limit of the action formally corresponds to the Landau gauge fixed version of Chern-Simons theory with complex gauge group which was originally proposed by Witten. We also show that the resulting action consists of parity even and odd parts with different coefficients.Comment: 22 pages, 5 figures; v2,v3 added references, v4 added two paragraphs and one figure in the summar

Deformed matrix models, supersymmetric lattice twists and N=1/4 supersymmetry

A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar supersymmetries under a twisted version of the supersymmetry algebra. We then discuss a succinct Q=1 deformed matrix model regularization of N=4 SYM in d=4, which is equivalent to a non-commutative $A_4^*$ orbifold lattice formulation. Motivated by recent progress in supersymmetric lattices, we also propose a N=1/4 supersymmetry preserving deformation of N=4 SYM theory on $\R^4$. In this class of N=1/4 theories, both the regularized and continuum theory respect the same set of (scalar) supersymmetry. By using the equivalence of the deformed matrix models with the lattice formulations, we give a very simple physical argument on why the exact lattice supersymmetry must be a subset of scalar subalgebra. This argument disagrees with the recent claims of the link approach, for which we give a new interpretation.Comment: 47 pages, 3 figure

Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra

The lattice superalgebra of the link approach is shown to satisfy a Hopf algebraic supersymmetry where the difference operator is introduced as a momentum operator. The breakdown of the Leibniz rule for the lattice difference operator is accommodated as a coproduct operation of (quasi)triangular Hopf algebra and the associated field theory is consistently defined as a braided quantum field theory. Algebraic formulation of path integral is perturbatively defined and Ward-Takahashi identity can be derived on the lattice. The claimed inconsistency of the link approach leading to the ordering ambiguity for a product of fields is solved by introducing an almost trivial braiding structure corresponding to the triangular structure of the Hopf algebraic superalgebra. This could be seen as a generalization of spin and statistics relation on the lattice. From the consistency of this braiding structure of fields a grading nature for the momentum operator is required.Comment: 45 page

A new large N phase transition in YM2

Inspired by the interpretation of two dimensional Yang-Mills theory on a cylinder as a random walk on the gauge group, we point out the existence of a large N transition which is the gauge theory analogue of the cutoff transition in random walks. The transition occurs in the strong coupling region, with the 't Hooft coupling scaling as alpha*log(N), at a critical value of alpha (alpha = 4 on the sphere). The two phases below and above the transition are studied in detail. The effective number of degrees of freedom and the free energy are found to be proportional to N^(2-alpha/2) below the transition and to vanish altogether above it. The expectation value of a Wilson loop is calculated to the leading order and found to coincide in both phases with the strong coupling value.Comment: 23 pages, 3 figure

A Method for Measuring the Witten Index Using Lattice Simulation

We propose a method to measure the Witten index using lattice simulation. A requirement for the lattice model is that it has at least one exact supersymmetry at finite lattice spacing. We prove the validity of the method in case of the supersymmetric quantum mechanics, where the index is well known.Comment: 23 pages, 9 figures; v2: references revised, section 2 improved and fig. 3 updated; v3: a version to appear in NP

Matrix formulation of superspace on 1D lattice with two supercharges

Following the approach developed by some of the authors in recent papers and using a matrix representation for the superfields, we formulate an exact supersymmetric theory with two supercharges on a one dimensional lattice. In the superfield formalism supersymmetry transformations are uniquely defined and do not suffer of the ambiguities recently pointed out by some authors. The action can be written in a unique way and it is invariant under all supercharges. A modified Leibniz rule applies when supercharges act on a superfield product and the corresponding Ward identities take a modified form but hold exactly at least at the tree level, while their validity in presence of radiative corrections is still an open problem and is not considered here.Comment: 25 page