39,129 research outputs found
Representation Growth of Linear Groups
Let be a group and the number of its -dimensional
irreducible complex representations. We define and study the associated
representation zeta function \calz_\Gamma(s) = \suml^\infty_{n=1}
r_n(\Gamma)n^{-s}. When is an arithmetic group satisfying the
congruence subgroup property then \calz_\Gamma(s) has an ``Euler
factorization". The "factor at infinity" is sometimes called the "Witten zeta
function" counting the rational representations of an algebraic group. For
these we determine precisely the abscissa of convergence. The local factor at a
finite place counts the finite representations of suitable open subgroups
of the associated simple group over the associated local field . Here we
show a surprising dichotomy: if is compact (i.e. anisotropic over
) the abscissa of convergence goes to 0 when goes to infinity, but
for isotropic groups it is bounded away from 0. As a consequence, there is an
unconditional positive lower bound for the abscissa for arbitrary finitely
generated linear groups. We end with some observations and conjectures
regarding the global abscissa
The SO(32) Heterotic and Type IIB Membranes
A two dimensional anomaly cancellation argument is used to construct the
SO(32) heterotic and type IIB membranes. By imposing different boundary
conditions at the two boundaries of a membrane, we shift all of the two
dimensional anomaly to one of the boundaries. The topology of these membranes
is that of a 2-dimensional cone propagating in the 11-dimensional target space.
Dimensional reduction of these membranes yields the SO(32) heterotic and type
IIB strings.Comment: 12 pages, Late
The Effect of Spatial Curvature on the Classical and Quantum Strings
We study the effects of the spatial curvature on the classical and quantum
string dynamics. We find the general solution of the circular string motion in
static Robertson-Walker spacetimes with closed or open sections. This is given
closely and completely in terms of elliptic functions. The physical properties,
string length, energy and pressure are computed and analyzed. We find the {\it
back-reaction} effect of these strings on the spacetime: the self-consistent
solution to the Einstein equations is a spatially closed spacetime with
a selected value of the curvature index (the scale f* is normalized to
unity). No self-consistent solutions with exist. We semi-classically
quantize the circular strings and find the mass in each case. For
the very massive strings, oscillating on the full hypersphere, have {\it independent} of and the level spacing {\it
grows} with while the strings oscillating on one hemisphere (without
crossing the equator) have and a {\it finite} number of
states For there are infinitely many string states
with masses that is, the level spacing grows {\it slower} than
The stationary string solutions as well as the generic string fluctuations
around the center of mass are also found and analyzed in closed form.Comment: 30 pages Latex + three tables and five figures (not included
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