Generalized Lagrangian Master Equations

We discuss the geometry of the Lagrangian quantization scheme based on (generalized) Schwinger-Dyson BRST symmetries. When a certain set of ghost fields are integrated out of the path integral, we recover the Batalin-Vilkovisky formalism, now extended to arbitrary functional measures for the classical fields. Keeping the ghosts reveals the crucial role played by a natural connection on the space of fields.Comment: LaTeX, 12 pages, CERN--TH-7247/9

Partition Function Zeros of an Ising Spin Glass

We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization flows in such models with frustration, and by the possible connection of the latter with spin glass behaviour. In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class of random polynomials with positive integer coefficients. Some aspects of this problem have been studied in various branches of mathematics, and we show how polynomial mappings which are used in graph theory to classify graphs, may help in characterizing the distribution of zeros. We finally discuss the possible limiting set as the volume is sent to infinity.Comment: LaTeX, 18 pages, hardcopies of 15 figures by request to [email protected], CERN--TH-7383/94 (a note and a reference added

Microscopic Spectrum of the Wilson Dirac Operator

We calculate the leading contribution to the spectral density of the Wilson Dirac operator using chiral perturbation theory where volume and lattice spacing corrections are given by universal scaling functions. We find analytical expressions for the spectral density on the scale of the average level spacing, and introduce a chiral Random Matrix Theory that reproduces these results. Our work opens up a novel approach to the infinite volume limit of lattice gauge theory at finite lattice spacing and new ways to extract coefficients of Wilson chiral perturbation theory.Comment: 4 pages, 3 figures, refs added, version to appear in Phys. Rev. Let

BRST Formulation of Partition Function Constraints

We show that constraints on the generating functional have direct BRST-extensions in terms of nilpotent operators $\Delta$ that annihilate this generating functional, and which may be of arbitrarily high order. The free energy $F$ in the presence of external sources thus satisfies a ``Master Equation'' which is described in terms of a tower of higher antibrackets.Comment: LaTeX, 7 page

Effects of dynamical quarks on the spectrum of the Wilson Dirac operator

Effects of dynamical quarks on the microscopic spectrum of the Wilson Dirac operator are analyzed by means of effective field theory. We consider the distributions of the real modes of the Wilson Dirac operator as well as the spectrum of the Hermitian Wilson Dirac operator, and work out the case of one flavor in all detail. In contrast to the quenched case, the theory has a mild sign problem that manifests itself by giving a spectral density that is not positive definite as the spectral gap closes.Comment: 7 pages 3 figures. Talk given at the XXVIII International Symposium on Lattice Field Theory (Lattice 2010), Villasimius, Italy, June 201

QCD3 and the Replica Method

Using the replica method, we analyze the mass dependence of the QCD3 partition function in a parameter range where the leading contribution is from the zero momentum Goldstone fields. Three complementary approaches are considered in this article. First, we derive exact relations between the QCD3 partition function and the QCD4 partition function continued to half-integer topological charge. The replica limit of these formulas results in exact relations between the corresponding microscopic spectral densities of QCD3 and QCD4. Replica calculations, which are exact for QCD4 at half-integer topological charge, thus result in exact expressions for the microscopic spectral density of the QCD3 Dirac operator. Second, we derive Virasoro constraints for the QCD3 partition function. They uniquely determine the small-mass expansion of the partition function and the corresponding sum rules for inverse Dirac eigenvalues. Due to de Wit-'t Hooft poles, the replica limit only reproduces the small mass expansion of the resolvent up to a finite number of terms. Third, the large mass expansion of the resolvent is obtained from the replica limit of a loop expansion of the QCD3 partition function. Because of Duistermaat-Heckman localization exact results are obtained for the microscopic spectral density in this way.Comment: LaTeX, 41 pages. References and some clarifying remarks adde

New Factorization Relations for Yang Mills Amplitudes

A double-cover extension of the scattering equation formalism of Cachazo, He and Yuan (CHY) leads us to conjecture covariant factorization formulas of n-particle scattering amplitudes in Yang-Mills theories. Evidence is given that these factorization relations are related to Berends-Giele recursions through repeated use of partial fraction identities involving linearized propagators.Comment: 7 pages, 3 figures, version to appear in PR

Exact Vacuum Energy of Orbifold Lattice Theories

We investigate the orbifold lattice theories constructed from supersymmetric Yang-Mills matrix theories (mother theories) with four and eight supercharges. We show that the vacuum energy of these theories does not receive any quantum correction perturbatively.Comment: 14 pages, no figure, LaTeX2e, typos corrected, errors in references corrected, comments adde