Distributions of dirac operator eigenvalues

The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory

Individual eigenvalue distributions of chiral random two-matrix theory and the determination of F_pi

Dirac operator eigenvalues split into two when subjected to two different external vector sources. In a specific finite-volume scaling regime of gauge theories with fermions, this problem can be mapped to a chiral Random Two-Matrix Theory. We derive analytical expressions to leading order in the associated finite-volume expansion, showing how individual Dirac eigenvalue distributions and their correlations equivalently can be computed directly from the effective chiral Lagrangian in the epsilon-regime. Because of its equivalence to chiral Random Two-Matrix Theory, we use the latter for all explicit computations. On the mathematical side, we define and determine gap probabilities and individual eigenvalue distributions in that theory at finite N, and also derive the relevant scaling limit as N is taken to infinity. In particular, the gap probability for one Dirac eigenvalue is given in terms of a new kernel that depends on the external vector source. This expression may give a new and simple way of determining the pion decay constant F_pi from lattice gauge theory simulations

Wilson loops in N = 4 supersymmetric Yang-Mills theory from random matrix theory

Based on the AdS/CFT correspondence, string theory has given exact predictions for circular Wilson loops in U(N) N = 4 supersymmetric Yang-Mills theory to all orders in a 1/N expansion. These Wilson loops can also be derived from Random Matrix Theory. In this paper we show that the result is generically insensitive to details of the Random Matrix Theory potential. We also compute all higher k-point correlation functions, which are needed for the evaluation of Wilson loops in arbitrary irreducible representations of U(N)

On finite-volume gauge theory partition functions

We prove a Mahoux–Mehta-type theorem for finite-volume partition functions of SU(Nc≥3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V1/mπ4. The theorem allows one to express any k-point correlation function of the microscopic Dirac operator spectrum entirely in terms of the 2-point function. The sum over topological charges of the gauge fields can be explicitly performed for these k-point correlation functions. A connection to an integrable KP hierarchy, for which the finite-volume partition function is a τ-function, is pointed out. Relations between the effective partition functions for these theories in 3 and 4 dimensions are derived. We also compute analytically, and entirely from finite-volume partition functions, the microscopic spectral density of the Dirac operator in SU(Nc) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory

Determination of F_pi from Distributions of Dirac Operator Eigenvalues with Imaginary Density

In the epsilon-regime of lattice QCD one can get an accurate measurement of the pion decay constant F_pi by monitoring how just one single Dirac operator eigenvalue splits into two when subjected to two different external vector sources. Because we choose imaginary chemical potentials our Dirac eigenvalues remain real. Based on the relevant chiral Random Two-Matrix Theory we derive individual eigenvalue distributions in terms of density correlations functions to leading order in the finite-volume epsilon-expansion. As a simple byproduct we also show how the associated individual Dirac eigenvalue distributions and their correlations can be computed directly from the effective chiral Lagrangian

Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition

We introduce a random two-matrix model interpolating between a chiral Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE) and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit) this theory is in one to one correspondence to the partition function of Wilson chiral perturbation theory in the epsilon regime, such as the related two matrix-model previously introduced in refs. [20,21]. For a generic number of flavours and rectangular block matrices in the chGUE part we derive an eigenvalue representation for the partition function displaying a Pfaffian structure. In the quenched case with nu=0,1 we derive all spectral correlations functions in our model for finite-n, given in terms of skew-orthogonal polynomials. The latter are expressed as Gaussian integrals over standard Laguerre polynomials. In the weakly non-chiral microscopic limit this yields all corresponding quenched eigenvalue correlation functions of the Hermitian Wilson operator.Comment: 27 pages, 4 figures; v2 typos corrected, published versio

Topological gravity on the lattice

In this paper we show that a particular twist of $\mathcal{N}=4$ super Yang-Mills in three dimensions with gauge group SU(2) possesses a set of classical vacua corresponding to the space of flat connections of the {\it complexified} gauge group $SL(2,C)$. The theory also contains a set of topological observables corresponding to Wilson loops wrapping non-trivial cycles of the base manifold. This moduli space and set of topological observables is shared with the Chern Simons formulation of three dimensional gravity and we hence conjecture that the Yang-Mills theory gives an equivalent description of the gravitational theory. Unlike the Chern Simons formulation the twisted Yang-Mills theory possesses a supersymmetric and gauge invariant lattice construction which then provides a possible non-perturbative definition of three dimensional gravity.Comment: 10 page

Individual Eigenvalue Distributions for the Wilson Dirac Operator

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac Operator DW. The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours Nf and for non-zero low energy constants W6, W7, W8. It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at fixed chirality nu this expansion truncates after at most nu terms for small lattice spacing "a". Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D5, where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched DW at small "a" we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size nu.Comment: 20 pages, 5 figures; v2: typos corrected, refs added and discussion of W6 and W7 extende

Two dimensional fermions in three dimensional YM

Dirac fermions in the fundamental representation of SU(N) live on the surface of a cylinder embedded in $R^3$ and interact with a three dimensional SU(N) Yang Mills vector potential preserving a global chiral symmetry at finite $N$. As the circumference of the cylinder is varied from small to large, the chiral symmetry gets spontaneously broken in the infinite $N$ limit at a typical bulk scale. Replacing three dimensional YM by four dimensional YM introduces non-trivial renormalization effects.Comment: 21 pages, 7 figures, 5 table

The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential

We discuss chiral perturbation theory for two and three quark flavors in the epsilon expansion at next-to-next-to-leading order (NNLO) including a small imaginary chemical potential. We calculate finite-volume corrections to the low-energy constants $\Sigma$ and $F$ and determine the non-universal modifications of the theory, i.e., modifications that cannot be mapped to random matrix theory (RMT). In the special case of two quark flavors in an asymmetric box we discuss how to minimize the finite-volume corrections and non-universal modifications by an optimal choice of the lattice geometry. Furthermore we provide a detailed calculation of a special version of the massless sunset diagram at finite volume.Comment: 21 pages, 5 figure