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When writing down the spectrum, it is interesting to note that the normal-ordering constant we ve had to deal with in previous theories is no longer an issue. This is because it cancels out with the bosonic and fermionic modes (this is supersymmetry after all). The mass-shell condition for the type I open string is
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The ground state consists of a bosonic state and its fermionic partner. Both are massless (there is no tachyon state). The bosonic state is a massless vector which is denoted by i where i = 1, , 8 . The massless fermion partner is given by a where a = 1, , 8 as well. We can transform between the two states as follows:
b a = i ab S0 i b i = i ab S0 a
(9.37)
The states are normalized according to i j = ij 1 a b = ( h + )ab 2 (9.38)
Summary
In this chapter, we have extended our discussion of supersymmetric string theory. First, we introduced the notion of super elds and showed how to write an action where supersymmetry was manifest. We then discussed some of the central
String Theory Demysti ed
components of space-time supersymmetry, rst in the context of the point particle which gives the D0-brane action and then in the case of string theory. The discussion presented in this chapter is far from complete. The reader who is interested in studying superstring theory on a serious level is urged to consult the references.
Quiz
1. Verify that A A + A by calculating A = [ Q, A ] . 2. Calculate {DA , QB}. 3. Find 0 0 . 4. Compute { 11 , 0 }.
A Summary of Superstring Theory
Our nal foray into the details of superstring theory will be a cursory look at heterotic superstrings in the next chapter. The heterotic string theories are a kind of hybrid between the 26-dimensional bosonic theory and the 10-dimensional superstring theory with fermions. This probably sounds intractable, but we will see in a minute how this can possibly work out into a consistent theoretical framework. Before we dive into heterotic strings, let s take a step back and qualitatively summarize the overall picture of string theory.
A Summary of Superstring Theory
Before jumping into some mathematical details about heterotic string theory, let s review the basic structure of string theory. This is a good idea because there are several string theories with different states of the string. Five of the theories are
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String Theory Demysti ed
superstring theories, and we also saw that it is possible to construct a theory consisting only of bosons. Actually you can list four different bosonic string theories, which we will do here.
BOSONIC STRING THEORY
We began our look at strings by considering bosonic string theory. This is an unrealistic theory because we know that the real world contains particles that are fermions. Nonetheless bosonic string theory provides an easier framework that can be used to illustrate the key ideas and techniques of string theory. Some key aspects of bosonic string theory you should remember are It introduces the concept of extra spatial dimensions. In order to avoid ghosts (states with negative norm) we were forced to accept that there are 26 space-time dimensions. The ground state (the lowest energy or lowest excitation mode of the string) has a negative mass-squared ( m 2 = 1 / ). This state is called a tachyon. The presence of a tachyon in the theory indicates that the ground state or vacuum is unstable. Note that in relativity, tachyons are particles that travel faster than the speed of light. Therefore the tachyon is a physically unrealistic particle. There is no known way to remove tachyon states from bosonic string theory. Bosonic string theory always includes gravity. This is indicated by the presence of a spin-2 state called the graviton. This is a hint that string theories provide a framework for the uni cation of all known physical interactions. Bosonic string theories also include a state called the dilaton. This is a scalar eld which is denoted by . It is related to the coupling constant g via g = exp , where is the vacuum expectation value of the dilaton eld. If you need to brush up on your quantum eld theory, note that the coupling constant determines the strength of an interaction. The dilaton eld is dynamical (it is space-time dependent), so in string theory we obtain a dramatic result that the string coupling constant can be dynamical. The dilaton is also known as the gravitational scalar eld and may play a role in the recently discovered nonzero cosmological constant. Strings can be either open or closed and can be oriented or unoriented. If a string is oriented, this means that directions along the string are unequivalent. So you can tell which way you re going along the string. By choosing open or closed strings and oriented or unoriented strings, we can actually construct four different bosonic string theories.
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