Yang-Mills Integrals Research Articles

SL(2,R)matrix model and supersymmetric Yang-Mills integrals

The density of states of Yang-Mills integrals in the supersymmetric case is characterized by power-law tails whose decay is independent of $N$, the rank of the gauge group. It is believed that this has no counterpart in matrix models, but we construct a matrix model that exactly exhibits this property. In addition, we show that the eigenfunctions employed to construct the matrix model are invariant under the collinear subgroup of conformal transformations, $SL(2,\mathbb{R})$. We also show that the matrix model itself is invariant under a fractional linear transformation. The wave functions of the model appear in the trigonometric Rosen-Morse potential and in free relativistic motion on anti-de Sitter space.

Semiclassical Geometry of 4D Reduced Supersymmetric Yang-Mills Integrals

We investigate semiclassical properties of space-time geometry of the low energy limit of reduced four dimensional supersymmetric Yang-Mills integrals using Monte-Carlo simulations. The limit is obtained by an one-loop approximation of the original Yang-Mills integrals leading to an effective model of branched polymers. We numerically determine the behaviour of the gyration radius, the two-point correlation function and the Polyakov-line operator in the effective model and discuss the results in the context of the large-distance behaviour of the original matrix model.

Convergence of the Gaussian Expansion Method in Dimensionally Reduced Yang-Mills Integrals

We advocate a method to improve systematically the self-consistent harmonic approximation (or the Gaussian approximation), which has been employed extensively in condensed matter physics and statistical mechanics. We demonstrate the {\em convergence} of the method in a model obtained from dimensional reduction of SU($N$) Yang-Mills theory in $D$ dimensions. Explicit calculations have been carried out up to the 7th order in the large-N limit, and we do observe a clear convergence to Monte Carlo results. For $D \gtrsim 10$ the convergence is already achieved at the 3rd order, which suggests that the method is particularly useful for studying the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory.

Yang-Mills integrals

Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to analytically evaluate the integrals in cases of low order. This is exhibited by evaluating a Yang-Mills integral over real symmetric matrices of order 3.

Geometry of reduced supersymmetric 4D Yang–Mills integrals

We study numerically the geometric properties of reduced supersymmetric non-compact SU(N) Yang-Mills integrals in D=4 dimensions, for N = 2,3, ..., 8. We show that in the range of large eigenvalues of the matrices A^mu, the original D-dimensional rotational symmetry is spontaneously broken and the dominating field configurations become one-dimensional, as anticipated by studies of the underlying surface theory. We also discuss possible implications of our results for the IKKT model.

Gaussian and mean field approximations for reduced Yang-Mills integrals

In this paper, we consider bosonic reduced Yang-Mills integrals by using some approximation schemes, which are a kind of mean field approximation called gaussian approximation and its improved version. We calculate the free energy and the expectation values of various operators including Polyakov loop and Wilson loop. Our results nicely match to the exact and the numerical results obtained before. Quite good scaling behaviors of the Polyakov loop and of the Wilson loop can be seen under the 't Hooft like large-N limit for the case of the loop length smaller. Then, simple analytic expressions for the loops are obtained. Furthermore, we compute the Polyakov loop and the Wilson loop for the case of the loop length sufficiently large, where with respect to the Polyakov loop there seems to be no known results in appropriate literatures even in numerical calculations. The result of the Wilson loop exhibits a strong resemblance to the result simulated for a few smaller values of N in the supersymmetric case.

Yang–Mills integrals for orthogonal, symplectic and exceptional groups

We apply numerical and analytic techniques to the study of Yang-Mills integrals with orthogonal, symplectic and exceptional gauge symmetries. The main focus is on the supersymmetric integrals, which correspond essentially to the bulk part of the Witten index for susy quantum mechanical gauge theory. We evaluate these integrals for D=4 and group rank up to three, using Monte Carlo methods. Our results are at variance with previous findings. We further compute the integrals with the deformation technique of Moore, Nekrasov and Shatashvili, which we adapt to the groups under study. Excellent agreement with all our numerical calculations is obtained. We also discuss the convergence properties of the purely bosonic integrals.

Yang-Mills integrals

SU(N) Yang-Mills integrals form a new class of matrix models which, in their maximally supersymmetric version, are relevant to recent non-perturbative definitions of 10-dimensional IIB superstring theory and 11-dimensional M-theory. We demonstrate how Monte Carlo methods may be used to establish important properties of these models. In particular, we consider the partition functions as well as the matrix eigenvalue distributions. For the latter we derive a number of new exact results for SU(2). We also report preliminary computations of Wilson loops.

Eigenvalue distributions in Yang-Mills integrals

We investigate one-matrix correlation functions for finite SU( N) Yang-Mills integrals with and without supersymmetry. We propose novel convergence conditions for these correlators which we determine from the one-loop perturbative effective action. These conditions are found to agree with non-perturbative Monte Carlo calculations for various gauge groups and dimensions. Our results yield important insights into the eigenvalue distributions ϱ(λ) of these random matrix models. For the bosonic models, we find that the spectral densities ϱ(λ) posses moments of all orders as N → ∞. In the supersymmetric case, ϱ(λ) is a wide distribution with an N - independent asymptotic behavior ϱ( λ) ∼ λ −3, λ −7, λ −15 for dimensions D = 4,6,10, respectively.