String Theory 230A Homework # 2 Solutions
Richard Eager
Department of Physics
University of California; Santa Barbara, CA 93106
Problem 1 - Lorentz Symmetry
(a) Noether Currents
Consider a global symmetry
+
with an innitesimal parameter. Imagine variatio
String Theory 230A Homework # 8 Solutions
Richard Eager
Department of Physics
University of California; Santa Barbara, CA 93106
Problem 1
Given the BRST transformation
B cc =
1 ab
f ca cb
2 c
show that
2 cc = 0.
In the operator language
2
B cc = [Q, cfw_
String Theory 230A Homework # 5 Solutions
Richard Eager
Department of Physics
University of California; Santa Barbara, CA 93106
Problem 1
(a)
We need to compute the OPE of the vertex operator
O = f : X X exp(ik X )
with the stress energy tensor
T =
1
: X
String Theory 230A Homework # 6 Solutions
Richard Eager
Department of Physics
University of California; Santa Barbara, CA 93106
Problem 1
(a)
1
2i
n
i
=
z
2i
n =
1
2i
iX(z)z n dz
iX(z)z n dz
(b)
We use Ginspargs convention
X(z)X(w) =
1
(z w)2
which diers
String Theory 230A Homework # 4 Solutions
Richard Eager Department of Physics University of California; Santa Barbara, CA 93106
Problem 1 (Polchinski 2.1)
ln |z|2 = (ln z + ln z ) = 1 z
Away from (z, z ) = (0, ln |z|2 is C and we can freely apply equali
1
Homework set 4, due Feb 8th
1. Verify that
log |z|2 =
1
1
= = 2 2 (z, z )
z
z
(1)
2. Consider a theory of a free scalar eld, with the standard correlator
between the X elds. Dene a (modied) stress energy tensor by the
formula
1
T (z) = : XX : (z) + 2
1
Homework set 3, due Feb 1st
1. Consider an open string with Neumann boundary conditions on both
ends. In lightcone quantization, at level one, one has a massless vector
particle with D 2 polarizations (in the end D = 26 for the critical
string). At leve
1
Homework set 6, due Feb 22nd
1. Consider a free (left-moving) boson eld, which is characterized by the
holomorphic operator of weight (1, 0), X and by the correlator
X(z)X(w) =
1
(z w)2
(1)
Dene the mode expansion of X by the following Laurent series
n
1
Homework set 2, due Jan 25
1. Lorentz symmetry
(a) Find the Noether currents on the worldsheet associated to Lorentz
transformations.
(b) Write an expression for the angular momentum of the string in
static gauge.
2. Folded string solution
(a) Show that
1
Homework set 5, due Feb 15th
1. Consider the vertex operator
f : X X exp(ikX) :
(1)
(a) Using the OPE with the stress energy tensor of many free scalar
elds, what are the constraints on f that need to be satised in
order for this vertex operator to be a
Homework 8
1. General BRST work: Show that if
1
BRST cc = f abc ca cb c c
2
ab
where f c are the structure constants of a Lie algebra. then
2
BRST cc = 0
Remember that the ca are anticommutting elds (so called a-numbers).
Since BRST is an anticommuting ge