Arnold Sommerfeld Center
Dr. Erik Plauschinn
LMU Munich
SoSe 2017
String Theory II (TMP-TD2)
Problem Set 3
due on 17.05.17
1
Boundary conditions for the superstring
Superstring theory is a theory which contains real bosonic fields X µ and real fermionic
fields µ . The action in super-conformal gauge reads as follows

Z
1
2 ↵
µ
S=
d2
⌘ @↵ X µ @ Xµ + 2i ⇢↵ @↵ µ ,
(1)
8⇡
↵0
where the two-dimensional Dirac matrices can be expressed using the Pauli matrices as
⇢0 = i 2 and ⇢1 = 1 .
(i) Derive the closed-string boundary conditions for the fermions by demanding boundary terms to vanish when varying the action with respect to µ .
(ii) Compare with the closed-string boundary conditions for the bosons and explain
the di↵erence.
Let us now turn to the open string.
(iii) Derive the open-string boundary conditions for the fermions by demanding boundary terms to vanish when varying the action.
(iv) List all possible inequivalent combinations of open-string boundary conditions for
the fermions.
In the last problem set, you verified that the action (1) is invariant under the following
infinitesimal supersymmetry transformation
q
q
µ
µ
µ
µ
2
2 1 ↵
X
=
i
✏
,
=
(2)
0
✏
✏
↵
↵0 2 ⇢ @ ↵ X ✏ .
(v) Show that the open-string boundary conditions break half of the world-sheet supersymmetry. Which transformations survive?
1
2
The N = 1 super Virasoro algebra
The generators of conformal and supersymmetry transformations of the superstring in
super-conformal gauge are the energy-momentum tensor T↵ and the fermionic partner
TF ↵ . The mode expansions of these fields (in light-cone coordinates ± and in the NS
sector) are given by
T
=
4⇡ 2 X
Lm e
l2
2⇡i m
/l
,
TF
=
m2Z
and similarly for the
of @ X µ and µ as
Lm =
+
/l
, (3)
part. The modes Lm and Gr can be expressed using the modes
n
n2Z
Gr =
2⇡i r
r2Z+ 2
1X
:↵
2
X
p ⇣ ⇡ ⌘3/2 X
2
Gr e
l
1
:↵
m2Z
m
· ↵n+m : +
1 X ⇣
m⌘
r+
:b
2
2
1
r2Z+ 2
r
· bm+r : ,
(4)
· bm+r : ,
µ
and the only non-trivial (anti-)commutation relations are [↵m
, ↵n⌫ ] = m m+n ⌘ µ⌫ and
µ ⌫
µ⌫
{br , bs } = r+s ⌘ . As usual, the notation · means the contraction of the d-dimensional
indices µ. Furthermore, : . . . : denotes normal ordering which for Lm reads
1 X
1 X
L(↵)
↵ n · ↵n+m +
↵n+m · ↵ n ,
m =
2
2
n<0
n 0
L(b)
m
1 X
=
r+
2
1
m
2
r +2
bm
r
· br
(i) Using the above expressions, show that
⇣m
[Lm , Gr ] =
2
1 X
r+
2
1
r
(5)
m
2
br · bm
r
.
2
⌘
r Gm+r .
(6)
The commutator (6) is part of the N = 1 super Virasoro algebra, which takes the
following form
[Lm , Ln ] = (m
n)Lm+n +
d
m3
8
⌘
r Gm+r ,
2
✓
◆
d 2 1
{Gr , Gs } = 2 Lr+s +
r
2
4
[Lm , Gr ] =
m
⇣m
m+n ,
(7)
r+s .
You are invited to check the remaining relations of this algebra, however, this is not part
of this problem set.
2