Iain Banks Culture 06 Inversions
Iain Banks The Br
ustawa_o_transporcie_drogowym_wersja_obowiazujaca_od_01.04.2009_r.
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Czarna Legenda Dziejów Polski â Jerzy Robert Nowak
Frank Kmietowicz Bogomilcy w Polsce
Fred Saberhagen Vlad Tepes 09 A Sharpness on the Neck
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Juliusz Verne W puszczach Afryki (Napowietrzna wioska)
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the M theory spacetime. This description is obviously not invariant under the SL(5, Z)
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duality symmetry of the (2, 0) theory. It is valid only in the region that the fifth toroidal
direction is much smaller than the other four. It then defines the eleven dimensional Planck
scale, [50] . In a generic region of the space of backgrounds, we simply have BPS charges
in the 10 dimensional second rank antisymmetric tensor representation of SL(5, Z). The
breakup into 6 wrapped membrane charges and 4 Kaluza-Klein momenta is only sensible
in regions where an eleven dimensional M theoretic spacetime picture becomes valid. In
any such regime, the (2, 0) theory is well approximated by SYM. At a fundamental level,
the wrapped transverse membrane charges and the transverse momenta are all part of the
BPS central charge which appears in the anticommutator of a dynamical and a kinematical
SUSY generator.
In the limit of noncompact eleven dimensional spacetime, most of the SYM degrees of
freedom decouple, and the system becomes the super quantum mechanics which describes
M theory in eleven noncompact dimensions. However, in the presence of one or more
wrapped membranes we must keep enough of the SYM degrees of freedom to implement
the relation T r[Xm, Xn] = Wmn, with Wmn the membrane wrapping number. This re-
produces the ansatz of [13] . As shown in [32] and elaborated upon in [93] one can also
study low energy fluctuations around these configurations. The enhanced gauge symmetry
which obtains when two membranes approach each other, originally derived in the D-brane
formalism can be rederived directly from the matrix model. This serves as the starting
point for the calculation of [26] .
One can also discuss membranes with one direction wrapped on the transverse torus
and the other around the longitudinal axis. These are configurations which carry a BPS
charge which appears in the anticommutator of two dynamical SUSY generators, and
carries one transverse vector index. As explained in [36] this charge is just the momentum
of the SYM field theory on the dual torus. Indeed, the dynamical SUSY generators in
the IMF are, in those dimensions where the SYM prescription is the whole story, just the
SUSY generator of the SYM theory in temporal gauge. They close on the SYM momentum,
up to a gauge transformation (sometimes there are other central charges for topologically
nontrivial configurations).
If the situation for membranes is eminently satisfactory, the situation for fivebranes is
more obscure. Longitudinal fivebranes were first discussed in [64] . These authors observed
that the D4 brane was the longitudinally wrapped fivebrane of M theory and they could
boost it into the IMF by considering its interactions with an infinite number of D0 branes.
This leads to SUSY quantum mechanics with eight SUSYs containing fields in the vector
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multiplet and hypermultiplets in the adjoint and k fundamental representations (for k
fivebranes). We have discussed this model above in the matrices for matrices ansatz for
the (2, 0) field theory. As we will see this is apt to be the proper definition of longitudinal
fivebranes in eleven dimensional spacetime.
[31] and [32] tried to construct the longitudinal fivebrane as a classical solution of
the matrix model. In particular, [32] observed that an object with nonzero values of
T rXiXjXkXl%Ełijkl (with %Eł the volume form of some four dimensional transverse subspace)
would be a BPS state with the right properties to be the longitudinal fivebrane. Indeed,
the BPS condition is [Xi, Xj] = %Ełijkl[Xk, Xl], which can be realized by the covariant
derivative in a self dual four dimensional gauge connection. Thus [31] and [32] suggested
that the limit of such a configuration in the 4 + 1 dimensional gauge theory, would be the
longitudinal fivebrane. Unfortunately, this definition is somewhat singular for the case of
minimal instanton charge (on a torus, this gives an instanton of zero scale size).
Perhaps it could be improved by going to the (2, 0) theory and defining an instanton
as a minimum energy state with the lowest value of momentum around the fifth toroidal
direction (which defines the Planck scale). At any rate, it is clear that in searching for
infinite BPS branes in eleven dimensions, we are discussing a limiting situation in which
many degrees of freedom are being decoupled. This suggests that the best description
will be the effective quantum mechanics of [64] which keeps just those degrees of freedom
necessary to define the longitudinal fivebrane.
There is no comparable description of the purely transverse fivebrane. Seiberg has
explained why there is no SYM configuration which respresents it even in a singular way.
Matrix Theory on a five torus is the theory of Type II NS fivebranes in the limit of zero
string coupling. This theory does have a limit in which it becomes 5 + 1 dimensional
SYM theory. However, the wrapped M theory fivebrane is a state of this theory which
is translation invariant on the five torus and has an energy density of order the cutoff
scale. Since it is not localized on the torus, it does not give rise to a long range SYM
field. Nonetheless, by carefully taking the infinite radius limit of such a wrapped fivebrane
configuration we should find a description in terms of the matrix quantum mechanics
coupled to some other degrees of freedom, in the spirit of [64] . We do not yet understand
enough about the theory of NS fivebranes proposed in [53] to derive this construction. One
should also understand the connection of these ideas to the proposal of [31] for constructing
the transverse fivebrane wrapped on the three torus.
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10. Conclusions
Matrix Theory is in its infancy. It seems to me that we have taken some correct
first steps towards a nonperturbative formulation of the Hamiltonian which lies behind the
various string perturbation expansions. It is as yet unclear how far we are from the final
formulation of the theory. I would like here to suggest a plan for the route ahead. Like all
such roadmaps of the unknown it is likely to lead to quite a few dead ends and perhaps
even a snakepit or two. But it s the best I can do at the moment to help you on your way
if you want to participate in this journey.
First, we must complete the compactification of the maximally supersymmetric version
of the theory, and this for two reasons. The present situation seems to indicate new
phenomena when there are six or more toroidally compactified dimensions. Surely we will
have to understand these in the controlled setting of maximal SUSY if we are to understand
them in more complicated situations. In addition, the work of Berkooz and Rozali [67]
and Seiberg [53] suggests that at least the passage to half as many SUSYs is relatively [ Pobierz całość w formacie PDF ]
