SUPERSTRINGS! Supersymmetric Strings 
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There are two types of particles in nature - fermions and bosons. A 
fundamental theory of nature must contain both of these types. When we 
include fermions in the worldsheet theory of the string, we 
automatically get a new type of symmetry called supersymmetry which 
relates bosons and fermions. Fermions and bosons are grouped together 
into supermultiplets which are related under the symmetry. This is the 
reason for the "super" in "superstrings". 
A consistent quantum field theory of superstrings exists only in 10 
spacetime dimensions! Otherwise there are quantum effects which render 
the theory inconsistent or 'anomalous'. In 10 spacetime dimensions the 
effects can precisely cancel leaving the theory anomaly free. It may 
seem to be a problem to have 10 spacetime dimensions instead of the 4 
spacetime dimensions that we observe, but we will see that in getting 
from 10 to 4 we actually find some interesting physics. 

In terms of weak coupling perturbation theory there appear to be only 
five different consistent superstring theories known as Type I SO(32), 
Type IIA, Type IIB, SO(32) Heterotic and E8 x E8 Heterotic. 
 　

     　
String Type Closed Closed Closed Closed Open 　
(& closed) 
10d Supersymmetry N=2 　
(chiral) N=2 　
(non-chiral) N=1 N=1 N=1 
10d Gauge groups none none E8 x E8 SO(32) SO(32) 
D-branes -1,1,3,5,7 0,2,4,6,8 none none 1,5,9 

 　
Type I SO(32): 
This is a theory which contains open superstrings. It has one (N=1) 
supersymmetry in 10 dimensions. Open strings can carry gauge degrees 
of freedom at their endpoints, and cancellation of anomalies uniquely 
constrains the gauge group to be SO(32). It contains D-branes with 1, 
5, and 9 spatial dimensions. 
 　
Type IIA: 
This is a theory of closed superstrings which has two (N=2) 
supersymmetries in ten dimensions. The two gravitini (superpartners to 
the graviton) move in opposite directions on the closed string world 
sheet and have opposite chiralities under the 10 dimensional Lorentz 
group, it is a non-chiral theory. There is no gauge group. It contains 
D-branes with 0, 2, 4, 6, and 8 spatial dimensions. 
 　
Type IIB: 
This is also a closed superstring theory with N=2 supersymmetry. 
However in this case the two gravitini have the same chiralities under 
the 10 dimensional Lorentz group, so this is a chiral theory. Again 
there is no gauge group, but it contains D-branes with -1, 1, 3, 5, 
and 7 spatial dimensions. 
 　
SO(32) Heterotic: 
This is a closed string theory with worldsheet fields moving in one 
direction on the world sheet which have a supersymmetry and fields 
moving in the opposite direction which have no supersymmetry. The 
result is N=1 supersymmetry in 10 dimensions. The non-supersymmetric 
fields contribute massless vector bosons to the spectrum which by 
anomaly cancellation are required to have an SO(32) gauge symmetry. 
 　
E8 x E8 Heterotic: 
This theory is identical to the SO(32) Heterotic string, except that 
the gauge group is E8 X E8 which is the only other gauge group allowed 
by anomaly cancellation.
We see that the Heterotic theories don't contain D-branes. They do 
however contain a fivebrane soliton which is not a D-brane. The IIA 
and IIB theories also contain this fivebrane soliton in addition to 
the D-branes. This fivebrane is usually called the "Neveu-Schwarz 
fivebrane" or "NS fivebrane". 
It is worthwhile to note that the E8 x E8 Heterotic string has 
historically been considered to be the most promising string theory 
for describing the physics beyond the Standard Model.  It was 
discovered in 1987 by Gross, Harvey, Martinec, and Rohm and for a long 
time it was thought to be the only string theory relevant for 
describing our universe.  This is because the SU(3) x SU(2) x U(1) 
gauge group of the standard model can fit quite nicely within one of 
the E8 gauge groups.  The matter under the other E8 would not interact 
except through gravity, and might provide a answer to the Dark Matter
problem in astrophysics.  Due to our lack of a full understanding of 
string theory, answers to questions such as how is supersymmetry 
broken and why are there only 3 generations of particles in the 
Standard Model have remained unanswered.  Most of these questions are 
related to the issue of compactification (discussed on the next page). 
What we have learned is that string theory contains all the essential 
elements to be a successful unified theory of particle interactions, 
and it is virtually the only candidate which does so.  However, we 
don't yet know how these elements specifically come together to 
describe the physics that we currently observe. 




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