The Microscopic Dirac Operator Spectrum

We review the exact results for microscopic Dirac operator spectra based on either Random Matrix Theory, or, equivalently, chiral Lagrangians. Implications for lattice calculations are discussed.Comment: Lattice2001(Plenary), 9 page

The Dirac Operator Spectrum and Effective Field Theory

When chiral symmetry is spontaneously broken, the low-energy part of the Dirac operator spectrum can be computed analytically in the chiral limit. The tool is effective field theory or, equivalently in this case, Random Matrix Theory.Comment: LaTeX, 12 page

QCD Dirac Spectra With and Without Random Matrix Theory

Recent work on the spectrum of the Euclidean Dirac operator spectrum show that the exact microscopic spectral density can be computed in both random matrix theory, and directly from field theory. Exact relations to effective Lagrangians with additional quark species form the bridge between the two formulations. Taken together with explicit computations in the chGUE random matrix ensemble, a series of universality theorems are used to prove that the finite-volume QCD partition function coincides exactly with the universal double-microscopic limit of chUE random matrix partition functions. In the limit where N_f and N_c both go to infinity with the ratio N_f/N_c fixed, the relevant effective Lagrangian undergoes a third order phase transition of Gross-Witten type.Comment: LaTeX, 6 page

Quenched and Unquenched Chiral Perturbation Theory in the \epsilon-Regime

The chiral limit of finite-volume QCD is the $\epsilon$-regime of the theory. We discuss how this regime can be used for determining low-energy observables of QCD by means of comparisons between lattice simulations and quenched and unquenched chiral perturbation theory. The quenched theory suffers in the $\epsilon$-regime from ``quenched finite volume logs'', the finite-volume analogs of quenched chiral logs.Comment: LaTeX, 7 pages, contribution to LHP200

Dirac Operator Spectra from Finite-Volume Partition Functions

Based on the relation to random matrix theory, exact expressions for all microscopic spectral correlators of the Dirac operator can be computed from finite-volume partition functions. This is illustrated for the case of $SU(N_c)$ gauge theories with $N_c\geq 3$ and $N_f$ fermions in the fundamental representation.Comment: LaTeX, 6 page

Spectral Sum Rules of the Dirac operator and Partially Quenched Chiral Condensates

Exploiting Virasoro constraints on the effective finite-volume partition function, we derive generalized Leutwyler-Smilga spectral sum rules of the Dirac operator to high order. By introducing $N_v$ fermion species of equal masses, we next use the Virasoro constraints to compute two (low-mass and large-mass) expansions of the partially quenched chiral condensate through the replica method of letting $N_v \to 0$. The low-mass expansion can only be pushed to a certain finite order due to de Wit-'t Hooft poles, but the large-mass expansion can be carried through to arbitrarily high order. Results agree exactly with earlier results obtained through both Random Matrix Theory and the supersymmetric method.Comment: LaTeX, 19 pages, misprints correcte

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

Qualitons from QCD

Qualitons, topological excitations with the quantum numbers of quarks, may provide an accurate description of what is meant by constituent quarks in QCD. Their existence hinges crucially on an effective Lagrangian description of QCD in which a pseudoscalar colour-octet of fields enters as a new variable. We show here how such new fields may be extracted from the fundamental QCD Lagrangian using the gauge-symmetric collective field technique.Comment: LaTeX, 12 pages, CERN--TH-7073/9