Higher Derivative Brane Couplings from T-Duality

Abstract

The Wess-Zumino coupling on D-branes in string theory is known to receive higher derivative corrections which couple the Ramond-Ramond potential to terms involving the square of the spacetime curvature tensor. Consistency with T-duality implies that the branes should also have four-derivative couplings that involve the NS-NS B-field. We use T-duality to predict some of these couplings. We then confirm these results with string worldsheet computations by evaluating disc amplitudes with insertions of one R-R and two NS-NS vertex operators. Comment: 30 pages, 1 figure, references added

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MIFP-10-29
Higher Derivative Brane Couplings from T-Duality
Katrin Becker, Guangyu Guo and Daniel Robbins
Department of Physics, Texas A&M University,
College Station, TX 77843, USA
Abstract
The Wess-Zumino coupling on D-branes in string theory is known to receive higher
derivative corrections which couple the Ramond-Ramond potential to terms involving
the square of the spacetime curvature tensor. Consistency with T-duality implies that
the branes should also have four-derivative couplings that involve the NS-NS B-field.
We use T-duality to predict some of these couplings. We then confirm these results
with string worldsheet computations by evaluating disc amplitudes with insertions of
one R-R and two NS-NS vertex operators.
July 6, 2010
Email: kbecker, guangyu, robbins@physics.tamu.edu
arXiv:1007.0441v1 [hep-th] 2 Jul 2010

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1 Introduction
In order to make progress towards connecting string theory to real world physics, it is
crucial to understand the vacua of the theory - what ingredients can be used to con-
struct vacua and what consistency conditions constrain the possible ways of assembling
these ingredients. Of particular interest in vacuum construction are D-branes (local-
ized objects on which strings can end) and fluxes (various higher-dimensional analogs
of magnetic fields).
Fluxes have become a very important ingredient in constructing semi-realistic
vacua, as they provide a straightforward mechanism to give masses to the many scalar
fields which describe the geometry of the internal manifold. In type II string theory,
for example, there are fluxes corresponding to the NS-NS B-field, whose field strength
is a three-form H3, and various R-R p-form fields C(p), with field strengths F(p+1). The
term fluxes is most commonly used to describe the discrete topological parameters that
come from integrating these field-strengths over cycles in the internal manifold. In this
paper, however, we are always working locally, and so we won’t be dealing directly
with these topological quantities. We do, however, deal with situations where some of
these potentials (especially B) have non-vanishing derivatives, and so we expect our
results to be important especially in the presence of fluxes.
Another important set of ingredients are D-branes - non-perturbative excitations
in the theory which are localized to a sub-manifold of the ten-dimensional spacetime.
The D-branes that we will be considering carry R-R charge (and hence are stable to
decay), and there are many degrees of freedom localized to their worldvolumes. These
localized degrees of freedom are one of the reasons that D-branes are so attractive in
vacuum building, as they can include chiral matter and non-abelian gauge groups.
D-branes can also be very important in finding consistent compactifications, as they
can sometimes be needed to satisfy an important class of consistency conditions known
as tadpole equations. In particular, tadpole equations can impose discrete topological
constraints on the number and type of D-branes and quantized fluxes.
For instance in type IIB, the equation of motion for C(4)wrapping the directions
of Minkowski space is an internal closed six-form which gets contributions both from
fluxes (terms proportional to F(3)∧ H3) and from delta-function forms corresponding
to localized sources such as D3-branes and O3-planes, and can also receive contribu-
tions from higher-derivative corrections to the action. If the six-form is not exact,
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then there can be a topological obstruction to solving the tadpole equation, and the
compactification would be inconsistent. In fact, it turns out that in some examples of
this sort (as well as in some other contexts), there may be no way to solve the tadpole
constraint at leading order in a momentum expansion. Higher derivative corrections
must then be included that often change the global structure drastically - either by
allowing the existence of solutions, or perhaps by spoiling the consistency of solutions
that otherwise appeared to be fine. For this reason, it is crucial to understand these
corrections and their global properties.
The IIB case mentioned above is an excellent example. The local tadpole equation
gets modified by higher derivative terms which, when integrated over the internal
space, gives a definite topological contribution, proportional to the Euler number of
the auxiliary Calabi-Yau four-fold in F-theory. In a limit in which the compactification
is well described by type IIB with O7-planes and D7-branes, the higher derivative
corrections have precisely the form of a four-derivative correction to the action localized
at these O7 and D7 sources. The leading piece of the action from the D7-branes is a
Wess-Zumino action
SWZ= T7
?
D7
CeB+2πα?F|8−form. (1.1)
At the O7-planes we have something similar, but the pull-back of B vanishes, there’s no
gauge field, and the numerical coefficient is different. These actions do get corrections
depending on derivatives of the bulk metric [1, 2, 3],
CeB+2πα?F|8−form+π2(α?)2
24
CeB+2πα?F|4−form∧ (trRT∧ RT− trRN∧ RN) + O((α?)4),
(1.2)
where RTand RNare curvatures (we will explain our notation more fully in section 2).
As emphasized above, these higher-derivative terms really must be included in order
to accurately judge the consistency of a given solution.
But these terms are not the end of the story. They provide a particular set of
four-derivative couplings on the brane between the bulk spacetime metric and the R-
R potential which contribute crucially to the C(4)tadpole. However, there can be
many other couplings between C(4)and bulk NS-NS fields at this same derivative
order. Indeed, by using T-duality, one can deduce some more couplings which involve
derivatives of B-fields, or will involve R-R fields of different degree, etc. It is not
clear that these couplings will necessarily lead to new topological restrictions, but in
some contexts they might, and they will certainly modify the local tadpole equation.
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Similar couplings have been obtained via U-duality in M-theory and string theory
in [4, 5], where they have been used to avoid no-go theorems in IIA and M-theory flux
compactifications. Clearly, these issues need to be examined more closely than they
have been.
1.1 Summary of Results
In this paper we start with some of the known corrections to the Wess-Zumino term
in the action of a Dp-brane,
SWZ original= Tpπ2(α?)2
24
?
Dp
CeB(trRT∧ RT− trRN∧ RN). (1.3)
By analyzing the conditions which are imposed by consistency with T-duality, we
will show that the action must contain these terms as well as several others at this
order,
SWZ⊃ Tpπ2(α?)2
?1
24
?
Dp
dxa1∧ ··· ∧ dxap+1
?
b
ap−2B
(p − 2)!C(p−1)
+∂
2
1
(p − 3)!C(p−3)
a1···ap−3
−2∂
[b
ap−2h
c]
ap−1∂apbhap+1c+ 2∂
[j
ap−2h
k]
ap−1∂apjhap+1k
?
k]
ap∂ap+1jBi
?
aphi1b∂ap+1jhi2
−∂
j
ap−1∂apbBap+1j+ ∂
?
b
ap−1hij∂apbBap+1j− ∂
(p − 1)!C(p+1)
j
ap−2B
b
ap−1∂apjBap+1b
+
1
a1···ap−2i
2∂
[b
ap−1h
c]
ap∂ap+1bBi
c− 2∂
[j
ap−1h
k
j
ap−1hib∂apjBap+1b
+1
2
1
a1···ap−1i1i2
?
apBi1c∂ap+1[bBi2
−∂
b
aphi1j∂ap+1bhi2
j+ ∂
j
b
−2∂
b
c]+ 2∂
j
apBi1k∂ap+1[jBi2
k]
??
. (1.4)
Here we have expanded around a D-brane with the usual static gauge embedding in
a flat background with no B-field. Indices from the beginning of the alphabet run
over directions along the worldvolume of the D-brane, while indices from the middle of
the alphabet run over the transverse directions. We have included metric fluctuations,
gµν = ηµν+ hµν, and B-field fluctuations, Bµν, as well as fluctuations of the R-R
potentials of degrees (p − 3), (p − 1), and (p + 1), and we have only worked to first
order in R-R fluctuations and quadratic order in NS-NS fluctuations. The first line
inside the curly braces comes from expanding the known couplings (1.3) to this order
in fluctuations. The remaining five lines are new couplings.
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To check these results, we will compute disc amplitudes involving the insertion of
one closed string R-R vertex operator and two NS-NS vertex operators. The results
of this computation will agree with (1.4) up to an overall normalization, which can in
turn be fixed by comparing with (1.3).
In section 2 we will use spacetime T-duality to argue for the presence of these addi-
tional terms, and we will in fact use the Buscher rules to compute several terms which
must be present, eventually arriving at (1.4), which is the key result of this section. In
section 3 we will confirm these predictions by doing three-point disc amplitude com-
putations, involving one R-R field and two NS-NS fields in the presence of a D-brane.
There are subtleties in the computation for general couplings in this type of amplitude
which would require careful addition of boundary terms to settle - without adding the
correct boundary terms, one finds disagreements when performing the computation in
different pictures, for example. However, we are fortunate that the particular terms
which we predicted from T-duality do not require these boundary terms, so we may
proceed with the somewhat naive computation. In order to bolster our assertion that
extra boundary terms are not needed, we perform the computation in several different
pictures and confirm that in each case the results agree with the other cases and with
the spacetime T-duality prediction. We conclude in section 4.
2 Predictions from T-Duality
2.1 Buscher rules
The low-energy bosonic spectrum of type II closed string theory includes the metric
gµν, the two-form potential Bµν, and the dilaton Φ from the NS-NS sector, and p-form
potentials C(p)from the R-R sector, where p is odd for IIA or even for IIB.
In backgrounds which include a U(1) isometry, string theory appears to enjoy a
duality, called T-duality, relating one background which solves the equations of motion
to another. Pick coordinates such that the isometry corresponds to translation in one
coordinate, y, and let the remaining coordinates be labeled by indices µ, ν, etc. Then
the explicit T-duality transformations for the NS-NS fields are given by [6]
g?
yy=
1
gyy,g?
µy=Bµy
gyy,g?
µν= gµν−gµygνy− BµyBνy
gyy
,
B?
µy=gµy
gyy,B?
µν= Bµν−Bµygνy− gµyBνy
gyy
,Φ?= Φ −1
2lngyy, (2.1)
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