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We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) . In this approach the DFrFT is defined as the $N \times N$ dimensional unitary representation of the generator of the arithmetic rotational group $SO_{2}[\mathbb{Z}_N]$, which is the finite set of $\bmod N$ integer, $2\times 2$ matrices acting on the points of the discrete toroidal phase space lattice $\mathbb{Z}_N \times \mathbb{Z}_N$, preserving the euclidean distance $\bmod N$. Using known factorization properties of $SO_{2}[\mathbb{Z}_N]$ for any integer $N$ into products of $SO_{2}[\mathbb{Z}_{p^{n}}]$'s where $p^n$ are the prime power factors of $N$ and $n=1,2,\ldots$, it is enough to study the arithmetic rotation group $SO_{2}[\mathbb{Z}_{p^n}]$ which is abelian and cyclic. We show that we can find always an appropriate power of the generator of $SO_{2}[\mathbb{Z}_{p^n}]$, which produces the rotation by $90$ degrees whose the unitary representation is the discrete Fourier transform (DFT). We construct explicitly, using techniques of the Finite Quantum Mechanics (FQM), the $p^n$ dimensional unitary matrix representation of the group $SO_{2}[\mathbb{Z}_{p^n}]$ and especially we work out in detail the one which corresponds to the generator. This is our definition of the arithmetic fractional Fourier transform (AFrFT). Following this definition, we proceed to the construction of efficient quantum circuits for the AFrFT, on sets of $n$ $p$-dimensional qudits with $p$ a prime integer, by introducing novel quantum subcircuits for diagonal operators with quadratic phases as well as new quantum subcircuits for multipliers by a constant. The quantum subcircuits that we introduce provide a set capable to construct quantum circuits for any element of a more general group, the group of Linear Canonical Transformations (LCT), $SL_{2}[\mathbb{Z}_N]$ of the toroidal phase space lattice. As a byproduct, extensions of the diagonal and multiplier quantum circuits for both the qudit and qubit case are given, which are useful alone in various applications. Also, we analyze the depth, width and gate complexity of the efficient AFrFT quantum circuit and we estimate its gate complexity which is of the order $O(n^2)$, its depth which is of the order $O(n)$ with depth $n$, while at the same time it has a structure permitting local interactions between the qudits., Comment: 28 pages, 14 Figures

We show that the recently introduced classical Arnol'd cat map lattice field theories, which are chaotic, are exponentially mixing to all orders. Their mixing times are well-defined and are expressed in terms of the Lyapunov exponents, more precisely by the combination that defines the inverse of the Kolmogorov-Sinai entropy of these systems. We prove by an explicit recursive construction of correlation functions, that these exhibit $l-$fold mixing for any $l= 3,4,5,\ldots$. This computation is relevant for Rokhlin's conjecture, which states that 2-fold mixing induces $l-$fold mixing for any $l>2$. Our results show that 2-fold exponential mixing, while being necessary for any $l-$fold mixing to hold it is nevertheless not sufficient for Arnol'd cat map lattice field theories., Comment: 21 pages LaTeX, uses utphys.bst for bibliography style. v2: Typos and equation layout corrected. v3: References added and clarifying remarks in sections 2 and 6

Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $HW_{2^s}$ over the discrete phase space lattice $Z_{2^s}$ $\otimes$ $Z_{2^s}$ is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation., Comment: 25 pages

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems can be, also, understood from the equations of motion in configuration space, where they describe inverted harmonic oscillators, with the runaway behavior of the potential competing with the toroidal compactification of the phase space. We highlight the spatio-temporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction., Comment: 39 pages, 5 PNG figures, LaTeX2e. Uses utphys.bst for references. v2: Streamlined presentation and added references. v3: Published version

According to the holographic picture of 't Hooft and Susskind, the black hole entropy, $S_{\rm BH}$, is carried by the chaotic microscopic degrees of freedom, that live in the near horizon geometry and have a Hilbert space of states of finite dimension, $d=\exp(S_{\rm BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the discrete, finite, random geometry, AdS$_2[\mathbb{Z}_N]$, where $N$ is proportional to $S_{\rm BH}$. What had remained as an open problem was how the smooth AdS2 geometry can be recovered, in the limit when N goes to infinity. In this contribution we present the salient points of the solution to this problem, which involves embedding AdS$_2[\mathbb{Z}_N]$ in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer, within 2+1 Minkowski spacetime. In this construction $N$ and $M$ can be considered IR and UV cutoffs. The continuum limit, corresponding to the smooth AdS$_2$ geometry, is obtained by taking $N$ and $M$ to infinity in a correlated way, using properties of the Fibonacci and $k$-Fibonacci sequences. This method can be directly applied to higher-dimensional AdS spacetimes, also., Comment: 14 pages LaTeX, 3 figures. Uses PoS style files and JHEP BibTeX. Contribution to the Proceedings of the 2021 Corfu Workshops, "Elementary Particle Physics and Gravity", summarizing arXiv:1908.06641

We study the leading (LO) and the next-to-leading order (NLO) stability of multipole perturbations for a static dielectric M2-brane with spherical topology in the 11-dimensional maximally supersymmetric plane-wave background. We observe a cascade of instabilities that originates from the dipole (j=1) and quadrupole (j=2) sectors (the only unstable sectors of the LO) and propagates towards all the multipoles of the NLO., Comment: 10 pages, 1 figure

We review our recent work on ellipsoidal M2-brane solutions in the large-N limit of the BMN matrix model. These bosonic finite-energy membranes live inside SO(3)xSO(6) symmetric plane-wave spacetimes and correspond to local extrema of the energy functional. They are static in SO(3) and stationary in SO(6). Chaos appears at the level of radial stability analysis through the explicitly derived spectrum of eigenvalues. The angular perturbation analysis is suggestive of the presence of weak turbulence instabilities that propagate from low to high orders in perturbation theory., Comment: 14 pages, 2 figures; contribution to the proceedings of the 13th CHAOS conference, 9 - 12 June 2020, Florence, Italy

According to the 't Hooft-Susskind holography, the black hole entropy,$S_\mathrm{BH}$, is carried by the chaotic microscopic degrees of freedom, which live in the near horizon region and have a Hilbert space of states of finite dimension $d=\exp(S_\mathrm{BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the AdS$_2[\mathbb{Z}_N]$ discrete, finite and random geometry, where $N\propto S_\mathrm{BH}$. It has been constructed by purely arithmetic and group theoretical methods in order to explain, in a direct way, the finiteness of the entropy, $S_\mathrm{BH}$. What has been left as an open problem is how the smooth AdS$_2$ geometry can be recovered, in the limit when $N\to\infty$. In the present article we solve this problem, by showing that the discrete and finite AdS$_2[\mathbb{Z}_N]$ geometry can be embedded in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer. This family can be constructed by an appropriate toroidal compactification and discretization of the ambient $(2+1)$-dimensional Minkowski space-time. In this construction $N$ and $M$ can be understood as "infrared" and "ultraviolet" cutoffs respectively. The above construction enables us to obtain the continuum limit of the AdS$_2^M[\mathbb{Z}_N]$ discrete and finite geometry, by taking both $N$ and $M$ to infinity in a specific correlated way, following a reverse process: Firstly, by recovering the continuous, toroidally compactified, AdS$_2[\mathbb{Z}_N]$ geometry by removing the ultraviolet cutoff; secondly, by removing the infrared cutoff in a specific decompactification limit, while keeping the radius of AdS$_2$ finite. It is in this way that we recover the standard non-compact AdS$_2$ continuum space-time. This method can be applied directly to higher-dimensional AdS spacetimes., Comment: 22 pages, LaTeX2e, many PNG figures. v1: Uses utphys.bst for the references. v2: Clarifications about the precursors, additional figures and references. v3: Further clarifying remarks and references. v4: Streamlined presentation; references added. v5: Further improvements of the presentation, references added. v6: Final version, as published in SIGMA. The displayed abstract is shortened

We explore the stability of a recently found class of spinning dielectric M2-branes in the 11-dimensional maximally supersymmetric plane-wave background. We find two small windows of instabilities in the dipole (j=1) and quadrupole (j = 2) sector of linear multipole perturbations., Comment: 7 pages, 2 figures. Matches published version

We present some basic integer arithmetic quantum circuits, such as adders and multipliers-accumulators of various forms, as well as diagonal operators, which operate on multilevel qudits. The integers to be processed are represented in an alternative basis after they have been Fourier transformed. Several arithmetic circuits operating on Fourier transformed integers have appeared in the literature for two level qubits. Here we extend these techniques on multilevel qudits, as they may offer some advantages relative to qubits implementations. The arithmetic circuits presented can be used as basic building blocks for higher level algorithms such as quantum phase estimation, quantum simulation, quantum optimization etc., but they can also be used in the implementation of a quantum fractional Fourier transform as it is shown in a companion work presented separately., Comment: 34 pages, 18 figures

We investigate the large-N limit of the BMN matrix model by analyzing the dynamics of ellipsoidal M2-branes that spin in the 11-dimensional maximally supersymmetric SO(3)xSO(6) plane-wave background. We identify finite-energy solutions by specifying the local minima of the corresponding energy functional. These configurations are static in SO(3) due to the Myers effect and rotate in SO(6) with an angular momentum that is bounded from above. As a first step towards studying their chaotic properties, we evaluate the Lyapunov exponents of their radial fluctuations., Comment: 7 pages, 8 figures

Scientists are increasingly turning to datacenter-scale computers to produce and analyze massive arrays. Despite decades of database research that extols the virtues of declarative query processing, scientists still write, debug and parallelize imperative HPC kernels even for the most mundane queries. This impedance mismatch has been partly attributed to the cumbersome data loading process; in response, the database community has proposed in situ mechanisms to access data in scientific file formats. Scientists, however, desire more than a passive access method that reads arrays from files. This paper describes ArrayBridge, a bi-directional array view mechanism for scientific file formats, that aims to make declarative array manipulations interoperable with imperative file-centric analyses. Our prototype implementation of ArrayBridge uses HDF5 as the underlying array storage library and seamlessly integrates into the SciDB open-source array database system. In addition to fast querying over external array objects, ArrayBridge produces arrays in the HDF5 file format just as easily as it can read from it. ArrayBridge also supports time travel queries from imperative kernels through the unmodified HDF5 API, and automatically deduplicates between array versions for space efficiency. Our extensive performance evaluation in NERSC, a large-scale scientific computing facility, shows that ArrayBridge exhibits statistically indistinguishable performance and I/O scalability to the native SciDB storage engine., Comment: 12 pages, 13 figures

We study the octonionic selfduality equations for $p=3$-branes in the light cone gauge and we construct explicitly, instanton solutions for spherical and toroidal topologies in various flat spacetime dimensions $(D=5+1,7+1,8+1,9+1)$, extending previous results for $p=2$ membranes. Assuming factorization of time we reduce the self-duality equations to integrable systems and we determine explicitly periodic, in Euclidean time, solutions in terms of the elliptic functions. These solutions describe 4d associative and non-associative calibrations in $D=7,8$ dimensions. It turns out that for spherical topology the calibration is non compact while for the toroidal topology is compact. We discuss possible applications of our results to the problem of 3-brane topology change and its implications for a non-perturbative definition of the 3-brane interactions., Comment: 15 pages, 4 figures

Based on our recent work on the discretization of the radial AdS$_2$ geometry of extremal BH horizons,we present a toy model for the chaotic unitary evolution of infalling single particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single particle dynamics for an observer falling into the BH horizon, with time evolution operator the quantum Arnol'd cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single--particle Hilbert space takes values in the set of Fibonacci integers., Comment: 28 pages LaTeX2e, 8 jpeg figures. Clarified certain issues pertaining to the relation between mixing time and scrambling time; enhanced discussion of the Eigenstate Thermalization Hypothesis; revised figures and updated references. Typos corrected

A new infinite-size limit of strings in RxS2 is presented. The limit is obtained from single spike strings by letting by letting the angular velocity parameter omega become infinite. We derive the energy-momenta relation of omega-infinity single spikes as their linear velocity v-->1 and their angular momentum J-->1. Generally, the v-->1, J-->1 limit of single spikes is singular and has to be excluded from the spectrum and be studied separately. We discover that the dispersion relation of omega-infinity single spikes contains logarithms in the limit J-->1. This result is somewhat surprising, since the logarithmic behavior in the string spectra is typically associated with their motion in non-compact spaces such as AdS. Omega-infinity single spikes seem to completely cover the surface of the 2-sphere they occupy, so that they may essentially be viewed as some sort of "brany strings". A proof of the sphere-filling property of omega-infinity single spikes is given in the appendix., Comment: 35 pages, 14 figures. Matches published version; Contains equation (4.21) that gives the first few finite-size corrections to the energy of omega-infinity single spikes

Non-abelian discrete symmetries are of particular importance in model building. They are mainly invoked to explain the various fermion mass hierarchies and forbid dangerous superpotential terms. In string models they are usually associated to the geometry of the compactification manifold and more particularly to the magnetised branes in toroidal compactifications. Motivated by these facts, in this note we propose a unified framework to construct representations of finite discrete family groups based on the automorphisms of the discrete and finite Heisenberg group. We focus in particular in the $PSL_2(p)$ groups which contain the phenomenologically interesting cases., Comment: 16 pages

We review an explicit regularization of the AdS$_2$/CFT$_1$ correspondence, that preserves all isometries of bulk and boundary degrees of freedom. This scheme is useful to characterize the space of the unitary evolution operators that describe the dynamics of the microstates of extremal black holes in four spacetime dimensions. Using techniques from algebraic number theory to evaluate the transition amplitudes, we remark that the regularization scheme expresses the fast quantum computation capability of black holes as well as its chaotic nature., Comment: 8 pages, 2 JPEG figues. Contribution to the VII Black Holes Workshop, Aveiro PT, Decemeber 2014

We generalize the method of our recent paper on the large-spin expansions of Gubser-Klebanov-Polyakov (GKP) strings to the large-spin and large-winding expansions of finite-size giant magnons and finite-size single spikes. By expressing the energies of long open strings in RxS2 in terms of Lambert's W-function, we compute the leading, subleading and next-to-subleading series of classical exponential corrections to the dispersion relations of Hofman-Maldacena giant magnons and infinite-winding single spikes. We also compute the corresponding expansions in the doubled regions of giant magnons and single spikes that are respectively obtained when their angular and linear velocities become smaller or greater than unity., Comment: 43 pages, 13 figures; Matches published version. Rewritten appendix C

We demonstrate that the large-spin expansion of the energy of Gubser-Klebanov-Polyakov (GKP) strings that rotate in RxS2 and AdS3 can be expressed in terms of Lambert's W-function. We compute the leading, subleading and next-to-subleading series of exponential corrections to the infinite-volume dispersion relation of GKP strings that rotate in RxS2. These strings are dual to certain long operators of N=4 SYM theory and provide their scaling dimensions at strong coupling. We also show that the strings obey a short-long (strings) duality. For the folded GKP strings that spin inside AdS3 and are dual to twist-2 operators, we confirm the known formulas for the leading and next-to-leading coefficients of their anomalous dimensions and derive the corresponding expressions for the next-to-next-to-leading coefficients., Comment: 46 pages, 8 figures; Matches published version; Contains equation (7.3) that gives the finite-size corrections to the dispersion relation of giant magnons at strong coupling

We propose a finite discretization for the black hole geometry and dynamics. We realize our proposal, in the case of extremal black holes, for which the radial and temporal near horizon geometry is known to be AdS$_2=SL(2,\mathbb{R})/SO(1,1,\mathbb{R})$. We implement its discretization by replacing the set of real numbers $\mathbb{R}$ with the set of integers modulo $N$, with AdS$_2$ going over to the finite geometry AdS$_2[N]=SL(2,\mathbb{Z}_N)/SO(1,1,\mathbb{Z}_N)$. We model the dynamics of the microscopic degrees of freedom by generalized Arnol'd cat maps, ${\sf A}\in SL(2,\mathbb{Z}_N)$, which are isometries of the geometry at both the classical and quantum levels. These exhibit well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization $N$, which are crucial for fast quantum information processing. We construct, finally, a new kind of unitary and holographic correspondence, for AdS$_2[N]$/CFT$_1[N]$, via coherent states of both the bulk and boundary geometries., Comment: 33 pages LaTeX2e, 1 JPEG figure. Typos corrected, references added. Clarification of several points in the abstract and the text

We study membrane configurations in AdS_{7/4}xS^{4/7}. The membranes are wrapped around the compact manifold S^{4/7} and are dynamically equivalent to bosonic strings in AdS_5. We thus conveniently identify them as "Stringy Membranes". For the case of AdS_7xS^4, their construction is carried out by embedding the Polyakov action for classical bosonic strings in AdS_5, into the corresponding membrane action. Therefore, every string configuration in AdS_5 can be realized by an appropriately chosen stringy membrane in AdS_7xS^4. We discuss the possibility of this being also the case for stringy membranes in AdS_4xS^7/Z^k (k > 1 or k = 1). By performing a stability analysis to the constructed solutions, we find that the (membrane) fluctuations along their transverse directions are organized in multiple Lam\'{e} stability bands and gaps in the space of parameters of the configurations. In this membrane picture, strings exhibit a single band/gap structure., Comment: 40 pages, 9 figures; published version

In the framework of Nambu Mechanics, we have recently argued that Non-Hamiltonian Chaotic Flows in $ R^{3} $, are dissipation induced deformations, of integrable volume preserving flows, specified by pairs of Intersecting Surfaces in $R^{3}$. In the present work we focus our attention to the Lorenz system with a linear dissipative sector in its phase space dynamics. In this case the Intersecting Surfaces are Quadratic. We parametrize its dissipation strength through a continuous control parameter $\epsilon$, acting homogeneously over the whole 3-dim. phase space. In the extended $\epsilon$-Lorenz system we find a scaling relation between the dissipation strength $ \epsilon $ and Reynolds number parameter r . It results from the scale covariance, we impose on the Lorenz equations under arbitrary rescalings of all its dynamical coordinates. Its integrable limit, ($ \epsilon = 0 $, \ fixed r), which is described in terms of intersecting Quadratic Nambu "Hamiltonians" Surfaces, gets mapped on the infinite value limit of the Reynolds number parameter (r $\rightarrow \infty,\ \epsilon= 1$). In effect weak dissipation, through small $\epsilon$ values, generates and controls the well explored Route to Chaos in the large r-value regime. The non-dissipative $\epsilon=0 $ integrable limit is therefore the gateway to Chaos for the Lorenz system., Comment: 15 pages and 2 figures, corrected typos and added references

Recent cosmological data for very large distances challenge the validity of the standard cosmological model. Motivated by the observed spatial flatness the accelerating expansion and the various anisotropies with preferred axes in the universe we examine the consequences of the simple hypothesis that the three-dimensional space has a global R^2 X S^1 topology. We take the radius of the compactification to be the observed cosmological scale beyond which the accelerated expansion starts. We derive the induced corrections to the Newton's gravitational potential and we find that for distances smaller than the S^1-radius the leading 1/r-term is corrected by convergent power series of multipole form in the polar angle making explicit the induced anisotropy by the compactified third dimension. On the other hand, for distances larger than the compactification scale the asymptotic behavior of the potential exhibits a logarithmic dependence with exponentially small corrections. The change of Newton's force from 1/r^2 to 1/r behavior implies a weakening of the deceleration for the expanding universe. Such topologies can also be created locally by standard Newtonian axially symmetric mass distributions with periodicity along the symmetry axis. In such cases we can use our results to obtain measurable modifications of Newtonian orbits for small distances and flat rotation spectra, for large distances at the galactic level., Comment: 14 pages

Based on our recent work on Quantum Nambu Mechanics $\cite{af2}$, we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of Non-commutative phase space coordinates as Hermitian $ N \times N $ matrices in $ R^{3}$. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving non-dissipative sector survive for long times., Comment: 14 pages, Based on invited talks delivered at: Fifth Aegean Summer School, "From Gravity to Thermal Gauge theories and the AdS/CFT Correspondance", September 2009, Milos, Greece; the Intern. Conference on Dynamics and Complexity, Thessaloniki, Greece, 12 July 2010; Workshop on "AdS4/CFT3 and the Holographic States of Matter", Galileo Galilei Institute, Firenze, Italy, 30 October 2010

We consider quantum mechanical effects of the modified Newtonian potential in the presence of extra compactified dimensions. We develop a method to solve the resulting Schroedinger equation and determine the energy shifts caused by the Yukawa-type corrections of the potential. We comment on the possibility of detecting the modified gravitational bound state Energy spectrum by present day and future experiments., Comment: 12 pages, 2 figures

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $ N \times N $ matrices in $ R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space., Comment: 35 pages, 4 figures, LaTex

We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus T_N^3 represented by elements of the group SL(3,Z_N). These flows can be considered as special motions of the Nambu dynamics (linear Nambu flows) in the three dimensional toroidal phase space and are characterized by invariant vectors, a, of T_N^3. We quantize all such flows which are necessarily restricted on a planar two-dimensional phase space, embedded in the 3-torus, transverse to the vector a . The corresponding maps belong to the little group of the vector a in SL(3,Z_N) which is an SL(2,Z_N) subgroup. The associated linear Nambu maps are generated by a pair of linear and quadratic Hamiltonians (Clebsch-Monge potentials of the flow) and the corresponding quantum maps, realize the metaplectic representation of SL(3,Z_N) on the discrete group of three dimensional magnetic translations i.e. the non-commutative 3-torus with deformation parameter the N-th root of unity. Other potential applications of our construction are related to the quantization of deterministic chaos in turbulent maps as well as to quantum tomography of three dimensional objects., Comment: 13 pages, LaTeX2e

We consider Nambu-Poisson 3-algebras on three dimensional manifolds $ {\cal M}_{3} $, such as the Euclidean 3-space $R^{3}$, the 3-sphere $S^{3}$ as well as the 3-torus $T^{3}$. We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms $SDiff({\cal M}_{3})$ is identical to the Nambu-Poisson algebra on ${\cal M}_{3}$. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation of $ SDiff({\cal M}_{3})$. We propose a quantization prescription for the Nambu-Poisson algebra which provides us with the correct classical limit. As such it possesses all of the expected classical properties constituting, in effect, a concrete representation of Nambu-Lie 3-algebras., Comment: 44 pages, v2:typos corrected, references added

Methods to effectively detect multi-locus genetic association are becoming increasingly relevant in the genetic dissection of complex trait in humans. Current approaches typically consider a limited number of hypotheses, most of which are related to the effect of a single locus or of a relatively small number of neighboring loci on a chromosomal region. We have developed a novel method that is specifically designed to detect genetic association involving multiple disease-susceptibility loci, possibly on different chromosomes. Our approach relies on the efficient discovery of patterns comprising spatially unrestricted polymorphic markers and on the use of appropriate test statistics to evaluate pattern-trait association. Power calculations using multi-locus disease models demonstrate significant gain of power by using this method in detecting multi-locus genetic association when compared to a standard single marker analysis method. When analyzing a Schizophrenia dataset, we confirmed a previously identified gene-gene interaction. In addition, a less conspicuous association involving different markers on the same two genes was also identified, implicating genetic heterogeneity., Comment: 49 pages, 4 tables, 4 figures

We propose a method to obtain new exact solutions of spinning p-branes in flat space-times for any p, which manifest themselves as higher dimensional Euler Tops and minimize their energy functional. We provide concrete examples for the case of spherical topology S^{2}, S^{3} and rotational symmetry \prod_{i}SO(q_{i}). In the case of toroidal topology T^{2}, T^{3} the rotational symmetry is \prod SU(q_{i}) and m target dimensions are compactified on the torus T^{m} . By double dimensional reduction the Light Cone Hamiltonians of T^{2}, T^{3} reduce to those of closed string S^{1} and T^{2} membranes respectively. The solutions are interpreted as non-perturbative spinning soliton states of type IIA-IIB superstrings., Comment: 33 pages, LATEX; more typos corrected; some equation numbering corrections

We show that previous proposals to accommodate the MSSM with string theory N=0 non-supersymmetric compactifications coming from intersecting D6-branes may be made fully consistent with the cancellation of RR tadpoles. In this respect we present the first examples of non-supersymmetric string Pati-Salam model vacua with starting observable gauge group $SU(4)_c \t SU(2)_L \t SU(2)_R$ (SU(2) from Sp(2)'s) that accommodate the spectrum of the 3 generation MSSM with a gauged baryon number with all extra exotics (either chiral or non-chiral) becoming massive and all MSSM Yukawas realized. The N=1 supersymmetry of the visible sector is broken by an extra supersymmetry messenger breaking sector that preserves a different N$^{\prime}$=1 susy, exhibiting the first examples of stringy gauge mediated models. Due to the high scale of the models, these models are also the first realistic examples of carriers of stringy split supersymmetry exhibiting universal slepton/squark masses, massive string scale gauginos, {\em unification of SU(3), SU(2) gauge couplings at} $2.04 \times 10^{16}$ GeV, a stable proton and the appearance of a landscape split SM with chiral fermions and only Higgsinos below the scale of susy breaking; the LSP neutralino candidate could also be only Higgsino or Higgsino-Wino mixture. We also add RR, NS and metric fluxes as every intersecting D-brane model without fluxes can be accommodated in the presence of fluxes. The addition of metric fluxes in the toroidal lattice also stabilizes the expected real parts of all in AdS closed string moduli (modulo D-term affects), leaving unfixed only the imaginary parts of K\"ahler moduli., Comment: 54 pages, v3; minor corrections and references added

We construct unitary evolution operators on a phase space with power of two discretization. These operators realize the metaplectic representation of the modular group SL(2,Z_{2^n}). It acts in a natural way on the coordinates of the non-commutative 2-torus, T_{2^n}^2$ and thus is relevant for non-commutative field theories as well as theories of quantum space-time. The class of operators may also be useful for the efficient realization of new quantum algorithms., Comment: 5 pages, contribution to Lattice 2005 (theoretical developments)

It is shown that a $d$-dimensional classical SU(N) Yang-Mills theory can be formulated in a $d+2$-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. In this paper we present an explicit proof for the case of the torus and the sphere., Comment: 12 pages

We construct quantum evolution operators on the space of states, that realize the metaplectic representation of the modular group SL(2,Z_2^n). This representation acts in a natural way on the coordinates of the non-commutative 2-torus and thus is relevant for noncommutative field theories as well as theories of quantum space-time. The larger class of operators, thus defined, may be useful for the more efficient realization of new quantum algorithms., Comment: 7 pages, LaTeX2e, amssymb

We propose a formulation of d-dimensional SU(N) Yang-Mills theories on a d+2-dimensional space with the extra two dimensions forming a surface with non-commutative geometry. This equivalence is valid in any finite order in the 1/N expansion., Comment: 14 pages LaTeX2e

We discuss the first string theory examples of three generation non-supersymmetric SU(5) and {\em flipped} SU(5) GUTS, which break to the Standard model at low energy, without extra matter and/or gauge group factors. Our GUT examples are based on IIA $Z_3$ orientifolds with D6-branes intersecting at non-trivial angles. These theories necessarily satisfy RR tadpoles and are free of NSNS tadpoles as the complex structure moduli are frozen (even though a dilaton tadpole remains) to discrete values. We identify appropriately the bifundamental Higgses responsible for electroweak symmetry breaking. In this way, the neutrino see-saw mechanism get nicely realized in these constructions. Moreover, as baryon number is not a gauged symmetry gauge mediated dimension six operators do contribute to proton decay; however proton lifetime may be safely enhanced by appropriately choosing a high GUT scale. An accompanying natural doublet-triplet splitting guarantees the suppression of scalar mediated proton decay modes and the stability of triplet scalar masses against higher dimensional non-renormalizable operators., Comment: 26 pages, 5 figures; no changes, one comment added in the introduction

String solitons in AdS_5 contain information of N=4 SUSY Yang-Mills theories on the boundary. Recent proposals for rotating string solitons reproduce the spectrum for anomalous dimensions of Wilson operators for the boundary theory. There are possible extensions of this duality for lower supersymmetric and even for non-supesymmetric Yang-Mills theories. We explicitly demonstrate that the supersymmetric anomalous dimensions of Wilson operators in N=0,1 Yang-Mills theories behave, for large spin J, at the two-loop level in perturbation theory, like log J. We compile the analytic one- and two-loop results for the N=0 case which is known in the literature, as well as for the N=1 case which seems to be missing., Comment: 16 pages, Appendix included in chapter. Version to appear in Nucl.Phys.B

We construct the unitary evolution operators that realize the quantization of linear maps of SL(2,R) over phase spaces of arbitrary integer discretization N and show the non-trivial dependence on the arithmetic nature of N. We discuss the corresponding uncertainty principle and construct the corresponding coherent states, that may be interpreted as non-commutative solitons., Comment: 3 pages LaTeX. Uses espcrc2.sty and amssymb.sty. Contribution to Lattice 2002(theoretical)

We introduce a particular embedding of seven dimensional self-duality membrane equations in C^3\times R which breaks G_2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C^3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a finite one and then they bounce to infinity in finite time. Their motion is periodic. These generalized complex Nahm equations, in the axially symmetric case, lead to extensions of the continuous Toda equation to complex space., Comment: 14 pages, 2 figures

In the lightcone frame, where the supermembrane theory and the Matrix model are strikingly similar, the equations of motion admit an elegant complexification in even dimensional spaces. Although the explicit rotational symmetry of the target space is lost, the remaining unitary symmetries apart from providing a simple and unifying description of all known solutions suggest new ones for rotating spherical and toroidal membranes. In this framework the angular momentum is represented by U(1) charges which balance the nonlinear attractive forces of the membrane. We examine in detail a six dimensional rotating toroidal membrane solution which lives in a 3-torus, $T^3$ and admits stable radial modes. In Matrix Theory it corresponds to a toroidal N-$D_{0}$ brane bound state. We demonstrate its existence and discuss its radial stability., Comment: 7 pages, revtex

We study the Penrose limit of various AdS_p X S^q orbifolds. The limiting spaces are waves with parallel rays and singular wave fronts. In particular, we consider the orbifolds AdS_3 X S^3/\Gamma, AdS_5 X S^5/\Gamma and AdS_{4,7} X S^{7,4}/\Gamma where \Gamma acts on the sphere and/or the AdS factor. In the pp-wave limit, the wave fronts are the orbifolds C^2/\Gamma, C^4/\Gamma and R XC^4/\Gamma, respectively. When desingularization is possible, we get asymptotically locally pp-wave backgrounds (ALpp). The Penrose limit of orientifolds are also discussed. In the AdS_5 X RP^5 case, the limiting singularity can be resolved by an Eguchi-Hanson gravitational instanton. The pp-wave limit of D3-branes near singularities in F-theory is also presented. Finally, we give the embedding of D-dimensional pp-waves in flat M^{2,D} space., Comment: 20 pages, references added

We study Q-balls associated with local U(1) symmetries. Such Q-balls are expected to become unstable for large values of their charge because of the repulsion mediated by the gauge force. We consider the possibility that the repulsion is eliminated through the presence in the interior of the Q-ball of fermions with charge opposite to that of the scalar condensate. Another possibility is that two scalar condensates of opposite charge form in the interior. We demonstrate that both these scenaria can lead to the existence of classically stable, large, gauged Q-balls. We present numerical solutions, as well as an analytical treatment of the ``thin-wall'' limit., Comment: 13 Pages, Latex 2.09, 5 figures, accepted for publication in Phys.Rev.D