Remarks on the realization of the Atiyah-Singer index theorem in lattice gauge theory

We discuss the interplay between topologically non-trivial gauge field configurations and the spectrum of the Wilson-Dirac operator in lattice gauge theory. Our analysis is based on analytic arguments and numerical results from a lattice simulation of QED_2.Comment: Talk given at LATTICE97, 3 pages, 1 figur

Properties of the Fixed Point Lattice Dirac Operator in the Schwinger Model

We present a numerical study of the properties of the Fixed Point lattice Dirac operator in the Schwinger model. We verify the theoretical bounds on the spectrum, the existence of exact zero modes with definite chirality, and the Index Theorem. We show by explicit computation that it is possible to find an accurate approximation to the Fixed Point Dirac operator containing only very local couplings.Comment: 38 pages, LaTeX, 3 figures, uses style [epsfig], a few comments and relevant references adde

One Loop Calculations in Gauge Theories Regulated on an x+x^+-p+p^+ Lattice

In earlier work, the planar diagrams of $SU(N_c)$ gauge theory have been regulated on the light-cone by a scheme involving both discrete $p^+$ and $\tau=ix^+$. The transverse coordinates remain continuous, but even so all diagrams are rendered finite by this procedure. In this scheme quartic interactions are represented as two cubics mediated by short lived fictitious particles whose detailed behavior could be adjusted to retain properties of the continuum theory, at least at one loop. Here we use this setup to calculate the one loop three gauge boson triangle diagram, and so complete the calculation of diagrams renormalizing the coupling to one loop. In particular, we find that the cubic vertex is correctly renormalized once the couplings to the fictitious particles are chosen to keep the gauge bosons massless.Comment: 26 pages, 36 figure

The Hadron Spectrum from Lattice QCD

Determining the hadron spectrum and hadron properties beyond the ground states is a challenge in lattice QCD. Most of these results have been in the quenched approximation but now we are entering the dynamical era. I review some of the ideas and methods of the lattice approach, concentrating on a few examples and on results obtained for Chirally Improved (CI) fermions.Comment: 18 pages, 12 figures, 1 table; Notes based on a lecture at the Int. School of Nuclear Physics, 29th Course, 16-24. Sept. 2007, Erice/Sicily, "Quarks in Hadrons and Nuclei"; minor modification

Light-cone Superstring in AdS Space-time

We consider fixing the bosonic light-cone gauge for string in AdS in the phase space framework, i.e. by choosing $x^+ = \tau$, and by choosing $\sigma$ so that $P^+$ is distributed uniformly (its density is independent of $\sigma$). We discuss classical bosonic string in AdS space and superstring in AdS_5 x S^5. In the latter case the starting point is the action found in hep-th/0007036 where the kappa-symmetry is fixed by a fermionic light cone gauge. We derive the light cone Hamiltonian in the AdS_5 x S^5 case and in the case of superstring in AdS_3 x S^3. We also obtain a realization of the generators of the basic symmetry superalgebra psu(2,2|4) in terms of the AdS_5 x S^5 superstring coordinate fields.Comment: 34 pages, latex. v3: section 5.4 revised. v4: minor corrections, version to appear in NP

Staggered versus overlap fermions: a study in the Schwinger model with Nf=0,1,2N_f=0,1,2

We study the scalar condensate and the topological susceptibility for a continuous range of quark masses in the Schwinger model with $N_f=0,1,2$ dynamical flavors, using both the overlap and the staggered discretization. At finite lattice spacing the differences between the two formulations become rather dramatic near the chiral limit, but they get severely reduced, at the coupling considered, after a few smearing steps.Comment: 15 pages, 7 figures, v2: 1 ref corrected, minor change

Physics, Topology, Logic and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.Comment: 73 pages, 8 encapsulated postscript figure

UHECR as Decay Products of Heavy Relics? The Lifetime Problem

The essential features underlying the top-down scenarii for UHECR are discussed, namely, the stability (or lifetime) imposed to the heavy objects (particles) whatever they be: topological and non-topological solitons, X-particles, cosmic defects, microscopic black-holes, fundamental strings. We provide an unified formula for the quantum decay rate of all these objects as well as the particle decays in the standard model. The key point in the top-down scenarii is the necessity to adjust the lifetime of the heavy object to the age of the universe. This ad-hoc requirement needs a very high dimensional operator to govern its decay and/or an extremely small coupling constant. The natural lifetimes of such heavy objects are, however, microscopic times associated to the GUT energy scale (sim 10^{-28} sec. or shorter). It is at this energy scale (by the end of inflation) where they could have been abundantly formed in the early universe and it seems natural that they decayed shortly after being formed.Comment: 11 pages, LaTex, no figures, updated versio