The D=1 Matrix Model and the Renormalization Group

We compute the critical exponents of $d = 1$ string theory to leading order, using the renormalization group approach recently suggested by Br\'{e}zin and Zinn-Justin.Comment: 8 pages, Latex, CERN-TH-6546/9

Generalized Ginsparg-Wilson algebra and index theorem on the lattice

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form $\gamma_{5}(\gamma_{5}D)+(\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$ where $k$ stands for a non-negative integer. The choice $k=0$ corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been established yet. We suggest that the Wilsonian effective action is essential to avoid infrared singularities encountered in general perturbative analyses.Comment: 11 pages. Talk given at APCTP-Nankai Joint Symposium on Lattice Statistics and Mathematical Physics, Tianjin, China, 8-11 October, 2001. To be published in the Proceedings and in Int. Jour. Mod. Phys.

Bosonization of non-relativstic fermions in 2-dimensions and collective field theory

We revisit bosonization of non-relativistic fermions in one space dimension. Our motivation is the recent work on bubbling half-BPS geometries by Lin, Lunin and Maldacena (hep-th/0409174). After reviewing earlier work on exact bosonization in terms of a noncommutative theory, we derive an action for the collective field which lives on the droplet boundaries in the classical limit. Our action is manifestly invariant under time-dependent reparametrizations of the boundary. We show that, in an appropriate gauge, the classical collective field equations imply that each point on the boundary satisfies Hamilton's equations for a classical particle in the appropriate potential. For the harmonic oscillator potential, a straightforward quantization of this action can be carried out exactly for any boundary profile. For a finite number of fermions, the quantum collective field theory does not reproduce the results of the exact noncommutative bosonization, while the latter are in complete agreement with the results computed directly in the fermi theory.Comment: references added and typos corrected; 21 pages, 3 figures, eps

Wess-Zumino model with exact supersymmetry on the lattice

A lattice formulation of the four dimensional Wess-Zumino model that uses Ginsparg-Wilson fermions and keeps exact supersymmetry is presented. The supersymmetry transformation that leaves invariant the action at finite lattice spacing is determined by performing an iterative procedure in the coupling constant. The closure of the algebra, generated by this transformation is also showed.Comment: 13 pages. Few references added. New appendix on Ward identity added. Version to be published in JHE

Topological susceptibility from the overlap

The chiral symmetry at finite lattice spacing of Ginsparg-Wilson fermionic actions constrains the renormalization of the lattice operators; in particular, the topological susceptibility does not require any renormalization, when using a fermionic estimator to define the topological charge. Therefore, the overlap formalism appears as an appealing candidate to study the continuum limit of the topological susceptibility while keeping the systematic errors under theoretical control. We present results for the SU(3) pure gauge theory using the index of the overlap Dirac operator to study the topology of the gauge configurations. The topological charge is obtained from the zero modes of the overlap and using a new algorithm for the spectral flow analysis. A detailed comparison with cooling techniques is presented. Particular care is taken in assessing the systematic errors. Relatively high statistics (500 to 1000 independent configurations) yield an extrapolated continuum limit with errors that are comparable with other methods. Our current value from the overlap is \chi^{1/4} = 188 \pm 12 \pm 5 \MeV.Comment: 18 pages, 7 figure

Dynamical overlap fermions, results with hybrid Monte-Carlo algorithm

We present first, exploratory results of a hybrid Monte-Carlo algorithm for dynamical, n_f=2, four-dimensional QCD with overlap fermions. As expected, the computational requirements are typically two orders of magnitude larger for the dynamical overlap formalism than for the more conventional (Wilson or staggered) formulations.Comment: 13 pages, 2 figure

Liouville field theory with heavy charges. II. The conformal boundary case

We develop a general technique for computing functional integrals with fixed area and boundary length constraints. The correct quantum dimensions for the vertex functions are recovered by properly regularizing the Green function. Explicit computation is given for the one point function providing the first one loop check of the bootstrap formula.Comment: LaTeX 26 page

Spectral Properties of the Overlap Dirac Operator in QCD

We discuss the eigenvalue distribution of the overlap Dirac operator in quenched QCD on lattices of size 8^{4}, 10^{4} and 12^{4} at \beta = 5.85 and \beta = 6. We distinguish the topological sectors and study the distributions of the leading non-zero eigenvalues, which are stereographically mapped onto the imaginary axis. Thus they can be compared to the predictions of random matrix theory applied to the \epsilon-expansion of chiral perturbation theory. We find a satisfactory agreement, if the physical volume exceeds about (1.2 fm)^{4}. For the unfolded level spacing distribution we find an accurate agreement with the random matrix conjecture on all volumes that we considered.Comment: 16 pages, 8 figures, final version published in JHE

Perturbation Calculation of the Axial Anomaly of a Ginsparg-Wilson lattice Dirac operator

A recent proposal suggests that even if a Ginsparg-Wilson lattice Dirac operator does not possess any topological zero modes in topologically-nontrivial gauge backgrounds, it can reproduce correct axial anomaly for sufficiently smooth gauge configurations, provided that it is exponentially-local, doublers-free, and has correct continuum behavior. In this paper, we calculate the axial anomaly of this lattice Dirac operator in weak coupling perturbation theory, and show that it recovers the topological charge density in the continuum limit.Comment: 25 pages, v2: calculation up to O(g^4) for nonabelian gauge backgroun

Low-energy couplings of QCD from topological zero-mode wave functions

By matching 1/m^2 divergences in finite-volume two-point correlation functions of the scalar or pseudoscalar densities with those obtained in chiral perturbation theory, we derive a relation between the Dirac operator zero-mode eigenfunctions at fixed non-trivial topology and the low-energy constants of QCD. We investigate the feasibility of using this relation to extract the pion decay constant, by computing the zero-mode correlation functions on the lattice in the quenched approximation and comparing them with the corresponding expressions in quenched chiral perturbation theory.Comment: 31 pages. v2: references and a small clarification added; published versio