Prof. Dr. Lust
Summer2006
Assignment # 5
(Due June 12, 2006)
1) Normal ordering and the quantum Virasoro algebra
Inthis exerise, we willshow that, inthe quantized bosoni stringtheory, the normal
ordered Virasoro generators
L
m
= 1
2 1
X
n= 1 :
m n
n :
satisfythe Virasoro algebrawith a entral harge 1
:
[L
m
;L
n
℄=(m n)L
m+n +
D
12 m(m
2
1)Æ
m+n :
In order to beome more familiar with the normal ordering presription, we will do
this by brute foremethods, i.e., by simplyusing the denition of the normal ordered
generators L
m
and then alulating their ommutators. We will proeed in several
smallersteps.
a)Explain why the normalorderinginL
m
onlyaets L
0
and whythe Virasoro gener-
ators L
m
an bewritten inthe following form:
L
m
= 1
2 0
X
n= 1
n
m n +
1
2 1
X
n=1
m n
n
(1)
b) Using [X;YZ℄ = [X;Y℄Z +Y[X;Z℄ and [
m
;
n
℄ = mÆ
m+n
prove that, for all
m;n2Z,
[
m
;L
n
℄=m
m+n :
)Deompose the sum
1
X
n= 1
= 0
X
n= 1 +
1
X
n=1
1
Aentralharge,T0,ofaLiealgebraisageneratorthatommuteswithallgeneratorsoftheLiealgebra,
[T
a
;T
0
℄=0,butappearsonthe right handsideofsome ommutators,[T
a
;T
b
℄=T
0
+::: ,for someT
a and
Tb,withbeingaonstant. IntheaboveVirasoroalgebra, ther^oleofT0 isplayedbythetermproportional
toÆm+n,whihshouldbeviewedasanadditionalgeneratorinadditiontotheLm.
show that
[L
m
;L
n
℄ = 1
2 0
X
l = 1
f(m l)
l
m+n l +l
n+l
m l g
+ 1
2 1
X
l =1
f(m l)
m+n l
l +l
m l
n+l
g: (2)
d)Make the substitutionp=n+l in the seond and fourth term in(2) and verify
[L
m
;L
n
℄ = 1
2 f
0
X
l = 1
(m l)
l
m+n l +
n
X
p= 1
(p n)
p
m+n p
+ 1
X
l =1
(m l)
m+n l
l +
1
X
p=n+1
(p n)
m+n p
p
g: (3)
e)From now on, we willrestrit ourselves to the ase n > 0, as the other ases n < 0
and n =0 are ompletelyanalogous. Show, therefore, that, for n > 0, the expression
(3)ind) isequal to
[L
m
;L
n
℄ = 1
2 f
0
X
p= 1
(m n)
p
m+n p +
n
X
p=1
(p n)
p
m+n p
+ 1
X
p=n+1
(m n)
m+n p
p +
n
X
p=1
(m p)
m+n p
p
g (4)
Whihof these four terms are already normal-ordered?
f) Prove
n
X
p=1
(p n)
p
m+n p
= n
X
p=1
(p n)
m+n p
p +
n
X
p=1
(p n)pDÆ
m+n
and insert this for the seond term inthe expression (4)of part e).
g)Showthat your result from part e)is nowequivalent to
[L
m
;L
n
℄= 1
2 1
X
l = 1
(m n) :
l
m+n l :+
1
2 D
n
X
l =1 (l
2
nl)Æ
m+n :
h)Prove,e.g. by indution, the followingidentities:
n
X
q=1 q
2
= 1
6
n(n+1)(2n+1)
n
X
q=1 q =
1
2
n(n+1) (5)
[L
m
;L
n
℄=(m n)L
m+n +
D
12 m(m
2
1)Æ
m+n
fromthe expression inpart g).
2) The quantum Virasoro algebra as a Lie algebra
Inhomework assignment#4, youshowed that the lassialVirasoro generators forma
Liealgebra with respet tothe Poisson braket. In the quantized version, the Poisson
brakets have now beome true ommutators between operators, and we also have a
entral harge:
[L
m
;L
n
℄=(m n)L
m+n +
D
12 m(m
2
1)Æ
m+n
(6)
a)Showthat these ommutationrelationsstill denea Lie algebra.
b)Do L
0 , L
1
and L
1
stillforma Liesubalgebra?
)Do the L
m
with m>0 forma Liesubalgebra? And the L
m
with m<0?
d)WehavealulatedthequantumVirasoroalgebrabyusingnormalorderedgenerators
L
m
. This hoie need not be the physially orret one. One might therefore now
wonder,whether aredenition
L
m
!
~
L
m
=L
m Æ
m;0
with a onstant that parameterizes a dierent ordering in L
0
ould perhaps remove
the troublesome entral harge term in the Virasoro algebra. Show that this annot
happen, i.e., showthat suh a redenition in(6) ould at most hange the linear term
inm in the entralharge, but not the ubi one.
