Download 1 The free boson on the sphere, normal ordering, and all that

Dr. E. Palti
December 14, 2016
String Theory - WiSe 2016/17 Problem Set 8
(Due: Tuesday, December 20, 2016)
1
The free boson on the sphere, normal ordering,
and all that
Consider the action for the free boson
Z
1
S=
dz dz ∂X(z, z) ∂X(z, z)
2πα0
(1)
with energy momentum tensor
T (z) = −
1
: ∂X(z)∂X(z) :
α0
(2)
in terms of the chiral fields X(z, z) = X(z) + X(z).
a) Recall from the lecture the definition of the normal ordering prescription : . . . :
employed above. Give the general relation between normal ordered and radially
ordered operators.
b) Give the correlator hX(z)X(w)i for the field X(z). Compare this with the correlator of two primary fields in a general CFT. Deduce the correlator h∂X(z)∂X(w)i
from hX(z)X(w)i.
c) Review the derivation of the OPE T (z)∂X(w) for the primary field ∂X(w) via
Wick’s theorem as given in the lecture. Following that same logic, compute the OPE
T (z)T (w) =
1
: ∂X(z)∂X(z) : : ∂X(w)∂X(w) :
α0 2
(3)
by writing down all cross-contractions between fields not in the same normal ordered
expression and bring the result into the standard form
T (z)T (w) =
c/2
2T (w)
∂w T (w)
+
+
+ (finite terms) .
(z − w)4 (z − w)2 (z − w)
(4)
Take a minute to enjoy the ease of this derivation of the normal ordering constant
c as compared to a brute-force computation in terms of the modes.
d) Now consider the normal ordered field : eikX(z) : . By rewriting : eikX(z) :=
P∞ (ik)n
n
n=0 n! : X (z) : and using Wick’s theorem, derive the OPE
∂X(z) : eikX(w) :=
−iα0 k : eikX(w) :
+ ....
2
z−w
(5)
Use this to compute also the OPE
T (z) : eikX(w) :=
α0 k 2 : eikX(w) : ∂ : eikX(w) :
.
+
4 (z − w)2
z−w
(6)
What does this tell us about the nature of : eikX(w) :?
2
The OPE of the energy momentum tensor
In the complex plane, the Virasoro generators Ln are given by
I
z. n+1
z T (z) .
Ln =
C0 2πi
(7)
a) Show that
I
[Lm , Ln ] =
C0
w.
2πi
I
Cw
z. m+1 n+1
z
w T (z)T (w) ,
2πi
(8)
where C0 denotes a contour about w = 0 and Cw is a contour about z = w. As
usual the product T (z)T (w) is meant to be the radially ordered product.
(Hint: Write the commutator as a difference of two double contour integrals and
use a contour deformation of the z. integration for fixed w.)
b) Use (8) and the (radially ordered) operator product
T (z)T (w) =
c/2
2T (w)
∂w T (w)
+
+
+ (finite terms) ,
4
2
(z − w)
(z − w)
(z − w)
as well as the Cauchy-Riemann formula,
I
z.
f (z)
1
=
f (n−1) (w) ,
n
2πi
(z
−
w)
(n
−
1)!
Cw
(9)
(10)
to rederive the quantum Virasoro algebra:
[Lm , Ln ] = (m − n)Lm+n +
c
(m3 − m)δm+n,0 .
12
(11)