Narrow Results

The gradient flow scheme has emerged as a prominent nonperturbative renormalization scheme on the lattice, where flow time is introduced to define the renormalization scale. In this study, we perturbatively compute the gradient flow coupling for the SU(N) Yang–Mills theory in the large-N limit in terms of the lattice bare coupling up to three-loop order. This is achieved by combining the twisted Eguchi–Kawai model with the numerical stochastic perturbation theory. We analyze the flow time dependence of the perturbative coefficients to determine the perturbative beta function coefficients, successfully computing the one-loop coefficient in the large-N limit using three matrix sizes $N=289,441,529$. However, the higher-order coefficients are affected by large statistical errors. We also explore the potential for reducing these statistical errors through variance reduction combined with the large-N factorization property of the SU(N) Yang–Mills theory, and estimate the required number of samples for the precise determination of the higher-order coefficients.

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

We present a formal derivation of the mean-field expansion for dilute Bose–Einstein condensates with two-particle interaction potentials which are weak and finite-range, but otherwise arbitrary. The expansion allows for a controlled investigation of the impact of microscopic interaction details (e.g. the scaling behavior) on the mean-field approach and the induced higher-order corrections beyond the s-wave scattering approximation.

We study quantum gravitational effect on a two-dimensional open universe with one particle by means of a string bit model. We find that matter is necessarily homogeneously distributed if the influence of the particle on the size of the universe is optimized.

The dilute Bose gas is studied in the large-N limit using functional integration.