Domain Wall Fermions and Chiral Symmetry Restoration Rate

Domain Wall Fermions utilize an extra space time dimension to provide a method for restoring the regularization induced chiral symmetry breaking in lattice vector gauge theories even at finite lattice spacing. The breaking is restored at an exponential rate as the size of the extra dimension increases. As a precursor to lattice QCD studies the dependence of the restoration rate to the other parameters of the theory and, in particular, the lattice spacing is investigated in the context of the two flavor lattice Schwinger model.Comment: 3 pages, LaTex, 5 ps figures, contribution to LATTICE97 proceeding

Domain Wall Fermions and MC Simulations of Vector Theories

It is known that domain wall fermions may be used in MC simulations of vector theories. The practicality and usefulness of such an implementation is investigated in the context of the vector Schwinger model, on a 2+1 dimensional lattice. Preliminary results of a Hybrid Monte Carlo simulation are presented.Comment: Talk presented at LATTICE96(chirality in qcd), 3 pages in LaTex, 4 Postscript figure

Chiral determinant on the lattice -- Anomalies and Instantons

An expression for the lattice effective action induced by chiral fermions in any even dimensions in terms of an overlap of two states is shown to have promising properties in two and four dimensions: The correct abelian anomaly is reproduced and gauge field configurations with non-zero topological charge are completely suppressed.Comment: 3 pages, ps-fil

Weak coupling expansion of a chiral gauge theory on a lattice in the overlap formulation

Weak coupling expansion of a chiral gauge theory on a lattice is discussed in the overlap formulation. We analyze the fermion propagator and the fermion-fermion-gauge boson vertex in the one loop level. The chiral properties of the propagator and vertex are correctly preserved without tuning the parameters involved even after the one-loop renormalization, and the ultraviolet divergent parts agree with the continuum theory. Our analysis, together with the existing studies on the vacuum polarization and the gauge boson n-point functions, completes the proof of the renormalizability of this formulation to the one loop level.Comment: LaTeX, 23 page

Gauge Freedom in Chiral Gauge Theory with Vacuum Overlap

Dynamical nature of the gauge degrees of freedom and its effect to fermion spectrum are studied at $\beta=\infty$ for two- and four-dimensional nonabelian chiral gauge theories in the vacuum overlap formalism. It is argued that the disordered gauge degrees of freedom does not contradict to the chiral spectrum of lattice fermion.Comment: 3 pages. LaTeX with espcrc2. Talk given at Lattice '97, Edinburg

Random discrete concave functions on an equilateral lattice with periodic Hessians

Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average value $- s$ concentrate around a quadratic function. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian, with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is then periodic with period $n \mathbb L$, and view this as a mean zero function $g$ on the set of vertices $V(\mathbb{T}_n)$ of a torus $\mathbb{T}_n := \frac{\mathbb{Z}}{n\mathbb{Z}}\times \frac{\mathbb{Z}}{n\mathbb{Z}}$ whose Hessian is dominated by $s$. The resulting set of semiconcave functions forms a convex polytope $P_n(s)$. The $\ell_\infty$ diameter of $P_n(s)$ is bounded below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$. Our main result is that under certain conditions, that are met for example when $s_0 = s_1 \leq s_2$, for any $\epsilon > 0,$ we have $$\lim_{n \rightarrow 0} \mathbb{P}\left[\|g\|_\infty > n^{\frac{7}{4} + \epsilon}\right] = 0$$ if $g$ is sampled from the uniform measure on $P_n(s)$. Each $g \in P_n(s)$ corresponds to a kind of honeycomb. We obtain concentration results for these as well.Comment: 56 pages. arXiv admin note: substantial text overlap with arXiv:1909.0858