finite-N Corrections Research Articles

Holographic index calculation for Argyres–Douglas and Minahan–Nemeschansky theories

Abstract We calculate the superconformal indices of the $\mathcal {N}=2$ superconformal field theories realized on N coincident D3-branes in 7-brane backgrounds with constant axiodilaton via the anti-de Sitter/conformal field theory correspondence. We include the finite-N corrections as the contribution of D3-branes wrapped around 3-cycles in the internal space. We take only single-wrapping contributions into account for simplicity. We also determine the orders of the next-to-leading corrections that we do not calculate. The orders are relatively high, and we obtain many trustworthy terms. We give the results for N = 1, 2, 3 explicitly, and find nice agreement with known results.

Finite-N superconformal index via the AdS/CFT correspondence

Abstract We propose a prescription to calculate the superconformal index of the ${\cal N}=4$U(N) supersymmetric Yang–Mills theory with finite N on the anti-de Sitter side. The finite-N corrections are included as contributions of D3-branes wrapped around three-cycles in ${\boldsymbol S}^5$, which are calculated as the index of the gauge theories realized on the wrapped branes. The single-wrapping contribution has been studied in a previous work, and we further confirm that the inclusion of multiple-wrapping contributions correctly reproduces the higher-order terms as far as we have checked numerically.

A theory of giant vortices

We elaborate a theory of giant vortices [1] based on an asymptotic expansion in inverse powers of their winding number n. The theory is applied to the analysis of vortex solutions in the abelian Higgs (Ginzburg-Landau) model. Specific properties of the giant vortices for charged and neutral scalar fields as well as different integrable limits of the scalar self-coupling are discussed. Asymptotic results and the finite-n corrections to the vortex solutions are derived in analytic form and the convergence region of the expansion is determined.

Finite-N corrections to the M-brane indices

We investigate finite-N corrections to the superconformal indices of the theories realized on M2- and M5-branes. For three-dimensional theories realized on a stack of N M2-branes we calculate the finite-N corrections as the contribution of extended M5-branes in the dual geometry AdS4×S7. We take only M5-brane configurations with a single wrapping into account, and neglect multiple-wrapping configurations. We compare the results with the indices calculated from the ABJM theory, and find agreement up to expected errors due to the multiple wrapping. For six-dimensional theories on N M5-branes we calculate the indices by analyzing extended M2-branes in AdS7×S4. Again, we include only configurations with single wrapping. We first compare the result for N = 1 with the index of the free tensor multiplet to estimate the order of the error due to multiple wrapping. We calculate first few terms of the index of AN−1 theories explicitly, and confirm that they can be expanded by superconformal representations. We also discuss multiple-wrapping contributions to the six-dimensional Schur-like index.

Finite-N corrections to the superconformal index of toric quiver gauge theories

Abstract The superconformal index of quiver gauge theories realized on D3-branes in toric Calabi–Yau cones is investigated. We use the AdS/CFT correspondence and study D3-branes wrapped on supersymmetric cycles. We focus on brane configurations in which a single D3-brane is wrapped on a cycle, and we do not take account of branes with multiple wrapping. We propose a formula that gives finite-$N$ corrections to the index caused by such brane configurations. We compare the predictions of the formula for several examples with the results on the gauge theory side obtained by using localization for small sizes of gauge groups, and confirm that the formula correctly reproduces the finite-$N$ corrections up to the expected order.

Extreme eigenvalue distributions of Jacobi ensembles: New exact representations, asymptotics and finite size corrections

Let W1 and W2 be independent n×n complex central Wishart matrices with m1 and m2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W1+W2)−1W1, which are analogous to those of F matrices W1W2−1 and those of the Jacobi unitary ensemble (JUE). Defining α1=m1−n and α2=m2−n with m1, m2≥n, we derive new exact distribution formulas in terms of (α1+α2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α1 and α2 fixed (i.e., under the so-called “hard-edge” scaling limit). The analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n→∞ in terms of α1- and α2-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations and properties of Legendre polynomials, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.

Finite-N corrections to the superconformal index of S-fold theories

Abstract We study the superconformal index of S-fold theories by using the anti-de Sitter / conformal field theory correspondence. It is known that the index in the large-$N$ limit is reproduced as the contribution of bulk Kaluza–Klein modes. For finite-$N$, D3-branes wrapped around the non-trivial cycle in $\boldsymbol{S}^5/\mathbb{Z}_k$, which corresponds to Pfaffian-like operators, give the corrections of order $q^N$ to the index. We calculate the finite-$N$ corrections by analyzing the fluctuations of wrapped D3-branes. Comparisons to known results show that our formula correctly reproduces the corrections up to errors of order $q^{2N}$.

Brownian regime of finite-N corrections to particle motion in the XY Hamiltonian mean field model

We study the dynamics of the N-particle system evolving in the XY Hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent Brownian noises over a time scale diverging not slower than as , which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings.

Adsorption of a single polymer chain on a surface: Effects of the potential range

We investigate the effects of the range of adsorption potential on the equilibrium behavior of a single polymer chain end-attached to a solid surface. The exact analytical theory for ideal lattice chains interacting with a planar surface via a box potential of depth U and width W is presented and compared to continuum model results and to Monte Carlo (MC) simulations using the pruned-enriched Rosenbluth method for self-avoiding chains on a simple cubic lattice. We show that the critical value U(c) corresponding to the adsorption transition scales as W(-1/ν), where the exponent ν=1/2 for ideal chains and ν≈3/5 for self-avoiding walks. Lattice corrections for finite W are incorporated in the analytical prediction of the ideal chain theory U(c)≈(π(2)/24)(W+1/2)(-2) and in the best-fit equation for the MC simulation data U(c)=0.585(W+1/2)(-5/3). Tail, loop, and train distributions at the critical point are evaluated by MC simulations for 1≤W≤10 and compared to analytical results for ideal chains and with scaling theory predictions. The behavior of a self-avoiding chain is remarkably close to that of an ideal chain in several aspects. We demonstrate that the bound fraction θ and the related properties of finite ideal and self-avoiding chains can be presented in a universal reduced form: θ(N,U,W)=θ(NU(c),U/U(c)). By utilizing precise estimations of the critical points we investigate the chain length dependence of the ratio of the normal and lateral components of the gyration radius. Contrary to common expectations this ratio attains a limiting universal value <R(g[perpendicular])(2)>/<R(g[parallel])(2)>=0.320±0.003 only at N~5000. Finite-N corrections for this ratio turn out to be of the opposite sign for W=1 and for W≥2. We also study the N dependence of the apparent crossover exponent φ(eff)(N). Strong corrections to scaling of order N(-0.5) are observed, and the extrapolated value φ=0.483±0.003 is found for all values of W. The strong correction to scaling effects found here explain why for smaller values of N, as used in most previous work, misleadingly large values of φ(eff)(N) were identified as the asymptotic value for the crossover exponent.

Linear theory and violent relaxation in long-range systems: a test case

In this paper, several aspects of the dynamics of a toy model for long-range Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian mean field (HMF). For special cases, exact finite-N linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, N → ∞, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports the mean field modulus in the vicinity of its equilibrium value within some linear e-folding times.

Large-Nreduction in QCD-like theories with massive adjoint fermions

Large-N QCD with heavy adjoint fermions emulates pure Yang-Mills theory at long distances. We study this theory on a four- and three-torus, and analytically argue the existence of a large-small volume equivalence. For any finite mass, center symmetry unbroken phase exists at sufficiently small volume and this phase can be used to study the large-volume limit through the Eguchi-Kawai equivalence. A finite temperature version of volume independence implies that thermodynamics on R^3 x S^1 can be studied via a unitary matrix quantum mechanics on S^1, by varying the temperature. To confirm this non-perturbatively, we numerically study both zero- and one-dimensional theories by using Monte-Carlo simulation. Order of finite-N corrections turns out to be 1/N. We introduce various twisted versions of the reduced QCD which systematically suppress finite-N corrections. Using a twisted model, we observe the confinement/deconfinement transition on a 1^3 x 2-lattice. The result agrees with large volume simulations of Yang-Mills theory. We also comment that the twisted model can serve as a non-perturbative formulation of the non-commutative Yang-Mills theory.

Finite-n Corrections to Normal Approximation

ABSTRACT We show that the sampling distribution of a statistic which follows the Central Limit Theorem can be approximated by a Normal distribution, modified according to the statistic's first few moments. One can thus achieve an impressive accuracy even for relatively small samples.

Finite-volume corrections to the mean-field solution of the SK model

The authors show in a simple way how noninteger exponents can arise in the study of the finite-N corrections to the free energy density of the mean-field solution of the SK model of spin glasses. The well known mean-field solution obtained with the hierarchical ansatz is valid in the thermodynamical limit N to infinity . They study the fluctuations associated with the longitudinal eigenvalues of the free-energy Hessian in the neighbourhood of the critical point, neglecting the interactions between different modes. They find that the contribution of the positive eigenvalues is of the order of N-3/4 ln N; for the zero modes, they find that they cannot be treated separately from the ones coming from the nonlongitudinal fluctuations because they would give a divergent contribution.