30 research outputs found

### Interaction between two Fuzzy Spheres

We have calculated interactions between two fuzzy spheres in 3 dimension. It
depends on the distance r between the spheres and the radii rho_1, rho_2. There
is no force between the spheres when they are far from each other (long
distance case). We have also studied the interaction for r=0 case. We find that
an attractive force exists between two fuzzy sphere surfaces.Comment: Latex file, 13 page

### Dynamical generation of gauge groups in the massive Yang-Mills-Chern-Simons matrix model

It has been known for some time that the dynamics of k coincident D-branes in
string theory is described effectively by U(k) Yang-Mills theory at low energy.
While these configurations appear as classical solutions in matrix models, it
was not clear whether it is possible to realize the k =/= 1 case as the true
vacuum. The massive Yang-Mills-Chern-Simons matrix model has classical
solutions corresponding to all the representations of the SU(2) algebra, and
provides an opportunity to address the above issue on a firm ground. We
investigate the phase structure of the model, and find in particular that there
exists a parameter region where O(N) copies of the spin-1/2 representation
appear as the true vacuum, thus realizing a nontrivial gauge group dynamically.
Such configurations are analogous to the ones that are interpreted in the BMN
matrix model as coinciding transverse 5-branes in M-theory.Comment: 4 pages, 3 figures, (v3) some typos correcte

### Solution of KdV Equation by Haar Integration Method

Haar wavelets is used to get a simplified algebraic form of Korteweg-de Vries (KdV) equation. Here Taylor's series expansion is also used to get some iterative formula which is further used to get numerical solutions. More approximate solutions can be obtained by increasing the order of the matrix of integration

### High Temperature Limit of the $N= 2$ IIA Matrix Model

The high temperature limit of a system of two D-0 branes is investigated. The
partition function can be expressed as a power series in $\beta$ (inverse
temperature). The leading term in the high temperature expression of the
partition function and effective potential is calculated {\em exactly}.
Physical quantities like the mean square separation can also be exactly
determined in the high temperature limit. We comment on SU(3) IIB matrix model
and the difficulties to study it.Comment: Lattice 2000 (Gravity and Matrix Models

### A Unified Treatment of the Characters of SU(2) and SU(1,1)

The character problems of SU(2) and SU(1,1) are reexamined from the
standpoint of a physicist by employing the Hilbert space method which is shown
to yield a completely unified treatment for SU(2) and the discrete series of
representations of SU(1,1). For both the groups the problem is reduced to the
evaluation of an integral which is invariant under rotation for SU(2) and
Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by
applying a rotation to a unit position vector in SU(2) and a Lorentz
transformation to a unit SO(2,1) vector which is time-like for the elliptic
elements and space-like for the hyperbolic elements in SU(1,1). The details of
the procedure for the principal series of representations of SU(1,1) differ
substantially from those of the discrete series.Comment: 31 pages, RevTeX, typos corrected. To be published in Journal of
Mathematical Physic

### The instability of intersecting fuzzy spheres

We discuss the classical and quantum stability of general configurations
representing many fuzzy spheres in dimensionally reduced
Yang-Mills-Chern-Simons models with and without supersymmetry. By performing
one-loop perturbative calculations around such configurations, we find that
intersecting fuzzy spheres are classically unstable in the class of models
studied in this paper. We also discuss the large-N limit of the one-loop
effective action as a function of the distance of fuzzy spheres. This shows, in
particular, that concentric fuzzy spheres with different radii, which are
identified with the 't Hooft-Polyakov monopoles, are perturbatively stable in
the bosonic model and in the D=10 supersymmetric model.Comment: 13 pages, (v3) reference added and some arguments refine

### Dynamical aspects of the fuzzy CP$^{2}$ in the large $N$ reduced model with a cubic term

``Fuzzy CP^2'', which is a four-dimensional fuzzy manifold extension of the
well-known fuzzy analogous to the fuzzy 2-sphere (S^2), appears as a classical
solution in the dimensionally reduced 8d Yang-Mills model with a cubic term
involving the structure constant of the SU(3) Lie algebra. Although the fuzzy
S^2, which is also a classical solution of the same model, has actually smaller
free energy than the fuzzy CP^2, Monte Carlo simulation shows that the fuzzy
CP^2 is stable even nonperturbatively due to the suppression of tunneling
effects at large N as far as the coefficient of the cubic term ($\alpha$) is
sufficiently large. As \alpha is decreased, both the fuzzy CP$^2$ and the fuzzy
S^2 collapse to a solid ball and the system is essentially described by the
pure Yang-Mills model (\alpha = 0). The corresponding transitions are of first
order and the critical points can be understood analytically. The gauge group
generated dynamically above the critical point turns out to be of rank one for
both CP^2 and S^2 cases. Above the critical point, we also perform perturbative
calculations for various quantities to all orders, taking advantage of the
one-loop saturation of the effective action in the large-N limit. By
extrapolating our Monte Carlo results to N=\infty, we find excellent agreement
with the all order results.Comment: 27 pages, 7 figures, (v2) References added (v3) all order analyses
added, some typos correcte

### Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term

Fuzzy spheres appear as classical solutions in a matrix model obtained via
dimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simons
term. Well-defined perturbative expansion around these solutions can be
formulated even for finite matrix size, and in the case of $k$ coincident fuzzy
spheres it gives rise to a regularized U($k$) gauge theory on a noncommutative
geometry. Here we study the matrix model nonperturbatively by Monte Carlo
simulation. The system undergoes a first order phase transition as we change
the coefficient ($\alpha$) of the Chern-Simons term. In the small $\alpha$
phase, the large $N$ properties of the system are qualitatively the same as in
the pure Yang-Mills model ($\alpha =0$), whereas in the large $\alpha$ phase a
single fuzzy sphere emerges dynamically. Various `multi fuzzy spheres' are
observed as meta-stable states, and we argue in particular that the $k$
coincident fuzzy spheres cannot be realized as the true vacuum in this model
even in the large $N$ limit. We also perform one-loop calculations of various
observables for arbitrary $k$ including $k=1$. Comparison with our Monte Carlo
data suggests that higher order corrections are suppressed in the large $N$
limit.Comment: Latex 37 pages, 13 figures, discussion on instabilities refined,
references added, typo corrected, the final version to appear in JHE