Arnold Sommerfeld Center
Ludwig–Maximilians–Universität München
Prof. Dr. Dieter Lüst
Summer term 2015
Exercises for String Theory II (TMP-TD2)
Problem set 10, due July 10, 2015
1 p-form electrodynamics and extended objects
In addition to the fundamental string, superstring theory also contains fundamental extended objects, such
as D-branes, NS5-branes, etc. These objects are electric or magnetic sources for the various p-form fields
present in the bulk of superstring theory. Among these p-form fields are the Kalb-Ramond B-field and the
various R-R-forms. In this problem we want to understand some features of generic higher-electrodynamics
in D-dimensions. This theory can be written as a theory of a p + 2-form Fp+2 that is subject to the Bianchi
identity and the equations of motion,
dFp+2 = 0
e
d ∗ Fp+2 = (−1)p ∗ jp+1
.
(i) Show that the above system of equations can be derived from the action
Z
Z
1
e
Fp+2 ∧ ∗Fp+2 + Ap+1 ∧ ∗jp+1
,
Sp [Ap+1 ] = −
2
where dAp+1 = Fp+2 is a local solution of the Bianchi identity. Determine the ambiguity in the choice
e
of Ap+1 and show that the conservation law d ∗ jp+1
= 0 is necessary for the gauge-invariance of the
e
e
action. Rewrite the conservation law in terms of the components of jp+1
= jp+1,µ
dxµ1 ∧dxµ2 ∧
1 µ2 ...µp+1
. . . ∧ dxµp+1 and the covariant derivative ∇µ .
Solution:
We calculate the variation w.r.t. Ap+1 ,
Z
Z
Z
Z
e
e
δS = − dδAp+1 ∧ ∗Fp+2 + δAp+1 ∧ ∗jp+1
= (−1)p+1 δAp+1 ∧ d ∗ Fp+2 + δAp+1 ∧ ∗jp+1
,
where in the last step we integrated by parts using Stoke’s theorem and assumed that the variation
vanishes at infinity. As the integrals are scalar products for differential forms that are not degenerate,
it follows that
e
d ∗ Fp+2 = (−1)p ∗ jp+1
.
Since Fp+2 = dAp+1 , we can always add a closed p-form to the gauge-field. The equations of motion
do not contain Ap+1 , but only the gauge-invariant field strength. Therefore, the action must be gaugee
invariant, which implies that d ∗ jp+1
= 0 as a necessary condition (otherwise the equations of motion
would depend on the choice of Ap+1 ). Using the definition of the Hodge star for p-forms α and β as
√
α ∧ ∗β = αµ1 µ2 ···µp β µ1 µ2 ···µp −gdD x,
we deduce that the gauge-invariance of the interaction terms can be equivalently expressed as
Z
√
e,µ µ ···µ
0 = (∂µ1 µ2 µ3 ···µp+1 )jp+11 2 p+1 −gdD x,
integrating by parts and using that is arbitrary, we deduce that
√
√
e,µ µ ···µ
e,µ µ ···µ
0 = ∂µ1 ( −gjp+11 2 p+1 ) = −g ∇µ1 jp+11 2 p+1 .
(ii) A closed p-brane Σ can couple electrically to higher electrodynamics via
Z
Sint =
Ap+1 .
Σ
Derive the associated conserved current
e
∗jp+1
and show that it is the Poincaré dual of [Σ] ∈ H• (M, R).
Solution:
The Poincaré dual κ of a homology p-cycle Σ is defined via
Z
Z
φ=
κ ∧ φ,
Σ
M
where φ is an arbitrary closed p-form on M . More generally, one can find a D − p form κ for arbitrary
submanifolds of M of dimension p. Intuitively, this form has δ-support in the normal directions of Σ.
In this case it is possible to find the Poincaré dual of Σ such that the above integral holds for arbitrary
p-forms φ.
We want to find an explicit form for κ. To this end we assume that Σ is a regular, smooth manifold in
M , which can be locally described as the zero set of D − p functions F a . Then, κ will take the form
Y
∗jp+1 = κ =
δ(F a )dF a ,
a
where the ordering depends on the induced orientation of Σ. This is well-defined and coordinate
independent since F a can be chosen as normal coordinates to Σ.
(iii) In vacuum the higher Maxwell equations can be rewritten in terms of a D−p−2-form field strength using
Hodge duality. Conclude that the electrically charged branes for the dual theory are of dimension D −
p − 4. These branes are the magnetic duals of the branes of the original theory. In type IIB superstring
theory, argue that the magnetic dual of the fundamental string is a five-dimensional brane, the so-called
NS5-brane, and that D1-string which couples electrically to the R-R-form C2 is magnetically dual to
a D5-brane, while the D3-brane is its own magnetic dual as it couples to C4 .
Solution:
In vacuum we have a symmetry Fp+2 ↔ ∗Fp+2 . This symmetry exchanges the Bianchi identity
with the Maxwell equations. This means that, locally in vacuum, we can write Fp+2 = dAp+1 and
∗Fp+2 = dAD−p−3 and that adding interaction terms to the free Maxwell action,
Z
Z
e
m
Ap+1 ∧ ∗jp+1
+ AD−p−3 ∧ ∗jD−p−3
,
gives rise to equations of motion that are gauge-invariant under both the electrical and magnetic gauge
symmetry provided that both the electric and magnetic current are conserved so that no new degrees
of freedom are introduced. In vacuum, the magnetic and electric potential are subject to the constraint
∗dAp+1 = dAD−p−3 . Note that there is no action for the new equations of motion,
e
d ∗ dAp+1 = (−1)p ∗ jp+1
m
d ∗ dAD−p−3 = (−1)D−p ∗ jD−p−3
.
The natural objects that give a magnetic current are then D − p − 4-branes. The fundamental string
couples to the B-field for p = 1, the magnetic dual has dimension D − 1 − 4 = 5. This object is called
an N S5-brane. A D3-brane couples electrically to C4 since the field strength is self-dual, it follows
that the magnetic and electric charges are identical so that the duals are also D3-branes. Similarly, a
D1-brane couples to C2 and the dual must be five dimensional, and so it must be a D5-brane.
(iv) The electric charge qe of a region containing some matter is defined via Gauß’s law,
Z
e
qe =
∗jp+1
,
S D−p−1
where S D−p−1 is a sphere surrounding the matter. Using the Maxwell equations, express the electric
charge through the field strength Fp+2 . The magnetic charge qm is defined as the electric charge in
the dual theory. Find an expression for qm in terms of the field strength Fp+2 . By requiring that Sp
gives a single-valued integration measure in the path-integral, derive the Dirac quantization condition
qe qm ∈ 2πZ.
Solution:
A p dimensional object has a world volume with p + 1 parallel and D − p − 1 normal directions. In the
normal directions the object is located at the origin and we can choose a sphere in these directions and
e
extend it to a cylinder along the time direction. The condition d ∗ jp+1
=
R 0 and Stoke’s theorem tell
us now that the integral over the space-like end surfaces of the cyclinder, S D−p−2 ,t ∗Fp+2 only receives
R
e
contributions from the source of the form (−1)p B D−p−1 ∗jp+1
. This is the electric charge. Similarly,
we can find the magnetic charge
Z
qm =
Fp+2 .
S p+2
Here S p+2 is a p + 2 dimensional sphere surrounding the D − p − 4-brane.
In the presence
R of magnetic charges the gauge-potential Ap+1 is not globally defined so that the putative
coupling qe Σ Ap+1 is not invariant. However, a well-defined path-integral measure requires that the
action must only be gauge-invariant upto addition of 2πk. We therefore define the interaction term as
Z
qe
Fp+2 ,
Ω
where Ω is any manifold in vacuum that bounds Σ = ∂Ω. However, such an extension is not unique,
but any two such extensions can be combined to a boundaryless manifold Ω ∪ Ω̄0 . We therefore need
that
Z
qe
Fp+2 ∈ 2πZ.
Ω∪Ω̄0
R
Alternatively, we can say that the qe R Fp+2 ∈ 2πZ for any closed p + 2 dimensional submanifold
R ⊂ M . Since dF = 0 in vacuum, this tells us that the magnetic charges present in the theory have to
satisfy qe qm ∈ 2πZ.
2 S-duality in type IIB supergravity
Type IIB supergravity contains two 2-form field B and C2 and two scalar fields C0 and Φ. These fields can be
grouped such that they transform under SL(2, R). B and C2 are grouped into a linear doublet (B2 , C2 ) ≡ B
with field strengths H ≡ dB. The fields C0 and Φ are organized as as a complex scalar τ = C0 + ie−Φ which
lives in the upper half-plane and transforms as a modular parameter under SL(2, Z). We can define a metric
g = eΦ |τ dx + dy|2
on a complex torus Σ with coordinates x and y. We can think of B as a field taking values in the tangent
space T Σ and C0 and Φ as coordinates on H.
The bosonic part of the type IIB-supergravity action in the Einstein frame reads then as
Z
Z
√
1
1
1
10
µ −1
SIIB = 2
d x −G R + Tr ∂ g ∂µ g − 2
∗Hi ∧ Hj gij
2κ
4
4κ
Z
Z
√
1
2
i
j
10
− 2
d x −G|F5 | + C4 ∧ H ∧ H ij .
8κ
(i) Show that the SL(2, R)-symmetry can be interpreted as a diffeomorphism on the auxiliary torus provided one identifies B with a vector field on Σ.
Solution:
Moreover, we find that
Imτ 0 = Im
ατ + δ
1
=
e−Φ .
γτ + δ
|γτ + δ|2
So that
0
eΦ |τ 0 dx0 + dy 0 |2 = eΦ |(ατ + β)dx + (γτ + δ)dy|2 .
Defining x = α0 x + βy 0 and y = γx0 + δy 0 we see that the metric does not change and that B =
B ∂x + C2 ∂y = B0 is invariant.
(ii) Show that the action is invariant under SL(2, R)-transformations.
Solution:
We can interpret an SL(2, R) as a field independent diffeomorphism on the (x, y)-plane and the action
is manifestly invariant under such diffeomorphisms. Hence, the total action is invariant.
0 −1
(iii) Show that S =
maps strong coupling to weak coupling and vice-versa.
1 0
Solution:
Under S-transformations τ → −1/τ . If we start at a background with C0 = 0, we find that this implies
Φ → −Φ. But since g = eΦ is the string coupling constant, this means that S-transformations map
strongly coupled theories with g 1 to weakly coupled theories with g 0 = 1/g ≈ 0.
(iv) It is expected that the complete superstring theory breaks the SL(2, R) symmetry down to the discrete
subgroup SL(2, Z). Assume that there is a state with electric charges (qeB , qeC ) and magnetic charges
B C
(qm
, qm ). What are the charges after an SL(2, Z) transformation? Is the Dirac quantization condition
still satisfied?
Solution:
The fields C2 and B transform in the fundamental representation of SL(2, R). The magnetic and
electric charges for C2 and B transform then again in that representation. If we assume that electric
and magnetic charges are integral, then we necessarily need to break to SL(2, Z) to preserve the Dirac
quantization condition.
General information
The lecture takes place on
Monday at 14 - 16 c.t. in A449 (Theresienstraße 37) and
Friday at 12 - 14 c.t. in A449 (Theresienstraße 37).
The webpage for the lecture and exercises can be found at
http://www.physik.uni-muenchen.de/lehre/vorlesungen/sose_15/TMP-TD2_StringTheoryII