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Comment: 4 pages, 5 figures

Comment: 22pgs; Changed to the ATMP style and minor other changes

Comment: Final version, to appear in JCAP

Comment: 30 pages, Latex, accepted for publication in Eur.Phys.J.C, some typos
corrected

We construct a D2-D8-$\bar{D8}$ configuration in string theory, it can be
described at low energy by two dimensional field theory. In the weak coupling
region, the low energy theory is a nonlocal generalization of Gross-Neveu(GN)
model which dynamically breaks the chiral flavor symmetry $U(N_f)_L \times
U(N_f)_R$ at large $N_c$ and finite $N_f$. However, in the strong coupling
region, we can use the SUGRA/Born-Infeld approximation to describe the low
energy dynamics of the system. Also, we analyze the low energy dynamics about
the configuration of wrapping the one direction of D2 brane on a circle with
anti-periodic boundary condition of fermions. The fermions and scalars on D2
branes get mass and decouple from the low energy theory. The IR dynamics is
described by the $QCD_2$ at weak coupling. In the opposite region, the dynamics
ha...

Comment: 55 pages, 7 figures, LaTeX; v2: typos corrected, references added;
v3: typos corrected

Comment: 25 pages, LaTeX, 10 figures, examples and references added

We generalise the electric-magnetic duality in standard Maxwell theory to its
non-commutative version. Both space-space and space-time non-commutativity are
necessary. The duality symmetry is then extended to a general class of
non-commutative gauge theories that goes beyond non-commutative
electrodynamics. As an application of this symmetry, plane wave solutions are
analysed. Dispersion relations following from these solutions show that general
non-commutative gauge theories other than electrodynamics admits two waves with
distinct polarisations propagating at different velocities in the same
direction.

We rewrite the Born-Infeld Lagrangian, which is originally given by the
determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the
field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4
\cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$
are given by the linear combination of $g$ and $F$, it is found, based on the
representation matrix for the multiplication operator of the Cayley-Dickson
algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter,
where distinguished matrices are not similar matrices. We also give a
reasonable condition to fix the paramete