String Theory Demystified

Summary
114 String Theory Demystifi ed
Conformal fi eld theory is a tool that allows us to transform the physics of the string
worldsheet to the complex plane. Calculations are much easier to do using the
techniques of complex variables. In this chapter we introduced some of the basic
terminology and techniques. In the following chapters, we will continue our
exploration of conformal fi eld theory in the context of string theory by exploring
Kac-Moody algebras, minimal models, vertex operators, and BRST quantization.
Quiz
1. Calculate [ , ] m n .
2. Does Ta z
az
()= 1
satisfy the group composition property?
1+
z
3. Consider Ta( z)=
with a pure imaginary. Find the generator of the
transformation.
1+
az
µ ν
4. Following the text, calculate 0 XLXL 0 .
µ ν
5. For a closed string, calculate ∂zX (, z z), ∂ z′
X ( z′ , z′
).
µ ν
6. Calculate 0 : X ( z, z) X ( z′ , z′
): 0 .
µ ν
7. Find 0 X (, z z) X ( z′ , z′
) 0 .
8. By exploiting the properties of the natural logarithm function, and using
µ ν ν µ
the fact that 0 X (, z z) X ( z′ , z′ ) 0 = 0 X ( z′ , z′ ) X (, z z)
0,
fi nd a
µ ν
compact expression for 0 X (, z z) X ( z′ , z′
) 0 .
ik⋅X ( w)
9. Find the operator product expansion of R( T ( z): e :) .
zz