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arXiv:hep-th/0607055v2 5 Sep 2006
ITP–UU–06/31
SPIN–06/27
hep-th/0607055
Supergravity description of spacetime instantons
Mathijs de Vroome and Stefan Vandoren
Institute for Theoretical Physics and Spinoza Institute
Utrecht University, 3508 TD Utrecht, The Netherlands
M.T.deVroome, vandoren@phys.uu.nl
Abstract
We present and discuss BPS instanton solutions that appear in type II string theory com-
pactifications on Calabi-Yau threefolds. From an effective action point of view these arise
as finite action solutions of the Euclidean equations of motion in four-dimensional N = 2
supergravity coupled to tensor multiplets. As a solution generating technique we make use
of the c-map, which produces instanton solutions from either Euclidean black holes or from
Taub-NUT like geometries.
1Introduction
Black holes in superstring theory have both a macroscopic and microscopic description. On
the macroscopic side, they can be described as solitonic solutions of the effective supergrav-
ity Lagrangian. Microscopically they can typically be constructed by wrapping p-branes
over p-dimensional cycles in the manifold that the string theory is compactified on. The
microscopic interpretation is best understood for BPS black holes.
Apart from this solitonic sector, string theory also contains instantons. Microscopically
they arise as wrapped Euclidean p-branes over p + 1-dimensional cycles of the internal
manifold. The aim of this paper is to present a macroscopic picture of these instantons
as solutions of the Euclidean equations of motion in the effective supergravity Lagrangian.
We focus hereby on spacetime instantons, whose effects are inversely proportional to the
string coupling constant gs. The models that we will study are type II string theories com-
pactified on a Calabi-Yau (CY) threefold. The resulting effective action is N = 2,D = 4
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supergravity coupled to vector multiplets and tensor multiplets. The latter can be dualized
to hypermultiplets, and the geometry of the hypermultiplet moduli space - containing the
dilaton - is known to receive quantum corrections, both from string loops [1] and from
instantons [2]. The instanton corrections are exponentially suppressed and are difficult to
compute directly in string theory. Our results yields some progress in this direction, since
within the supergravity description one finds explicit formulae for the instanton action.1
Related work can also be found in [4, 5], but our results are somewhat different and contain
several new extensions.
Interestingly, there is a relation between black hole solutions in type IIA/B and instanton
solutions in type IIB/A. Microscopically, this can be understood from T-duality between
IIA and IIB. Macroscopically, this follows from the c-map [6, 7], as we will show explicitly.
This defines a map between vector and tensor multiplets and as a consequence, (BPS)
solutions of the vector multiplet Lagrangian are mapped to (BPS) solutions of the tensor-
or hypermultiplet Lagrangian. We will use this mapping in Euclidean spacetimes. Roughly
speaking, there are two classes of solutions on the vector multiplet sector: (Euclidean) black
holes and Taub-NUT like solutions. These map to D-brane instantons and NS-fivebrane
instantons respectively. The distinguishing feature is that the corresponding instanton
actions are inversely proportional to gsor g2
give the explicit solution and the precise value of the instanton action.
The D-brane instantons are found to be the solutions to the equations obtained from c-
mapping the BPS equations of [8]. Their analysis contains also R2interactions, but they
can be easily switched off. The BPS equations then obtained are similar, but not identical
to the equations derived in [9]. In the derivation and description of D-brane instantons
we find it convenient to make the symplectic structure of the theory and its equations
manifest. The NS-fivebrane instantons are derived in a different way, not by using the
c-map. This is because the BPS solutions in Euclidean supergravity coupled to vector
multiplets are not fully classified. We therefore construct the NS-fivebrane instantons by
extending the Bogomol’nyi-bound-formulation of [10].
Ultimately, we hope to get a better understanding of non-perturbative string theory. In
particular, it is expected that instanton effects resolve conifold-like singularities in the hy-
permultiplet moduli space of Calabi-Yau compactifications, see e.g. [11]. These singulari-
ties are closely related - by the c-map - to the conifold singularities in the vector multiplet
moduli space due to the appearance of massless black holes [12]. Moreover, in combination
with the more recent relation between black holes and topological strings [13], it would
srespectively. For both type of instantons, we
1Instanton actions can also be studied from worldvolume theories of D-branes. For a discussion on this
in the context of our paper, we refer to [3]. It would be interesting to find the precise relation to our
analysis.
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be interesting to study if topological string theory captures some of the non-perturbative
structure of the hypermultiplet moduli space. For some hints in this direction, see [14].
Finally, we remark that instantons play an important role in the stabilization of moduli.
For an example related to our discussion, we refer to [15].
This paper is organized as follows: In section 2 we treat NS-fivebrane instantons in the
context of N = 1 supergravity. We use this simple setup to introduce various concepts, e.g.
the c-map, which we use in later sections. Section 3 is devoted to a review of instanton
solutions in the universal hypermultiplet of N = 2 supergravity and their relation to
gravitational solutions of pure N = 2 supergravity. Then in section 4 we consider instanton
solutions to the theory obtained from arbitrary CY compactification of type II superstrings.
Some technical details are provided in appendices at the end of this paper, including a
treatment of electric-magnetic duality in tensor multiplet Lagrangians.
2 NS-fivebrane instantons
In this section, we give the N = 1 supergravity description of the NS-fivebrane instanton.
The main characteristic of this instanton is that the instanton action is inversely propor-
tional to the square of the string coupling constant. In string theory, such instantons
appear when Euclidean NS-fivebranes wrap six-cycles in the internal space, and therefore
are completely localized in both space and (Euclidean) time.
It is well known that Euclidean NS-fivebranes in string theory are T-dual to Taub-NUT or
more generally, ALF geometries [11] (see also [16]). We here re-derive these results from
the perspective of four-dimensional (super-) gravity in a way that allows us to introduce
the c-map conveniently.
2.1A Bogomol’nyi bound
We start with a simple system of gravity coupled to a scalar and tensor in four spacetime
dimensions,
Lm= −1
with
H = dB .
2κ2eR(e) +1
2|dφ|2+1
2e2φ|H|2,(2.1)
(2.2)
We use form notation for the matter fields; see Appendix (A.1) for our conventions.
This model appears as a sub-sector of N = 1 low-energy effective actions in which gravity is
coupled to N = 1 tensor multiplets. In our case we have one tensor multiplet that consists
of the dilaton φ and the NS two-form B. In four dimensions, a tensor can be dualized
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into a scalar, such that only chiral multiplets appear. We will not do this dualization for
reasons that become clear below.
The instanton solution can be found by deriving a Bogomol’nyi bound on the Euclidean
Lagrangian [17],
Le=1
Here, we have left out the Einstein-Hilbert term. It is well known that this term is not
positive definite, preventing us to derive a Bogomol’nyi bound including gravity. In most
cases, our instanton solutions are purely in the matter sector, and spacetime will be taken
flat. The Bogomol’nyi equation then is
2|eφ∗ H ∓ eφde−φ|2∓ d(eφH) .(2.3)
∗H = ±de−φ.(2.4)
This implies that e−φshould be a harmonic function. The ± solutions refer to instantons
or anti-instantons. Notice that the surface term in (2.3) is topological in the sense that it
is independent on the spacetime metric. It is easy to check that the BPS configurations
(2.4) have vanishing energy momentum tensor, so that the Einstein equations are satisfied
for any Ricci-flat metric.
One can now easily evaluate the instanton action on this solution. The only contribution
comes from the surface term in (2.3). Defining the instanton charge as
?
S3H = Q ,(2.5)
with H the three-form field strength, we find2
Sinst=|Q|
g2
s
.(2.6)
Here we have assumed that there is only a contribution from infinity, and not from a
possible other boundary around the location of the instanton. It is easy to see this when
spacetime is taken to be flat. In that case the single-centered solution for the dilaton is
e−φ= e−φ∞+
|Q|
4π2r2,(2.7)
which is the standard harmonic function in flat space with the origin removed. We have
furthermore related the string coupling constant to the asymptotic value of the dilaton by
gs≡ e−φ∞/2.(2.8)
In our notation, this is the standard convention.
2In the tensor multiplet formulation, the instanton action has no imaginary theta-angle-like terms.
They are produced after dualizing the tensor into an axionic scalar, by properly taking into account the
constant mode of the axion. In the context of NS-fivebrane instantons, this was explained e.g. in [18].
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2.2 T-duality and the c-map
We will now re-derive the results of the previous subsection using the c-map. Though
no new results, it will enable us to set the notation and to prepare for more complicated
situations discussed in the next sections.
To perform the c-map, we dimensionally reduce the action (2.1) and assume that all the
fields are independent of one coordinate. This can most conveniently be done by first
choosing an upper triangular form of the vierbein, in coordinates (xm,x3≡ τ),m = 0,1,2,
ea
µ=
?
e−˜φ/2ˆ ei
0
m
e˜φ/2˜Bm
e˜φ/2
?
. (2.9)
The metric then takes the form
ds2= e˜φ(dτ +˜B)2+ e−˜φˆ gmndxmdxn,(2.10)
and we take˜φ,˜Bmand ˆ gmnto be independent of τ. For the moment, we take τ to be one
of the spatial coordinates, but at the end of this section, we will apply our results to the
case when τ is the Euclidean time. In our example, the Wick rotation is straightforward
on the scalar-tensor sector.
We have that e = e−˜φˆ e, and the scalar curvature decomposes as (ignoring terms that lead
to total derivatives in the Lagrangian)
−eR(e) = −ˆ eR(ˆ e) +1
2|d˜φ|2+1
2e2˜φ|˜H|2,(2.11)
Similarly, we require the dilaton and 2-form to be independent of τ. The three-dimensional
Lagrangian then is3
Lm
3= −ˆ eR(ˆ e) +1
2|d˜φ|2+1
2e2˜φ|˜H|2+1
2|dφ|2+1
2e2φ|H|2,(2.12)
where H is the two-form field strength descending from the three-form H in four dimensions
(see also Appendix (A.2)), so we have again that H = dB in three dimensions, where B is
a one-form.
In addition, there is an extra term in the Lagrangian,
Laux
3
= −1
2e2(φ+˜φ)|H −˜B ∧ H|2,(2.13)
which plays no role in the three-dimensional theory. Here H ≡ dB is the three-form
arising from the spatial component of the four-dimensional H. Being a three-form in three
dimensions it is an auxiliary field. This term can therefore trivially be eliminated by its
own field equation.
3For convenience of normalization, we set κ−2= 2.
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Note that the Lagrangian L3has the symmetry
φ ←→˜φ ,B ←→˜B . (2.14)
In fact, careful analysis shows that also Laux
3
is invariant, provided we transform
B →˜B ≡ B −1
2
˜B ∧ B .(2.15)
The transformations in (2.14) and (2.15) define the c-map. The resulting theory can now
be reinterpreted as a dimensional reduction of a four-dimensional theory of gravity coupled
to a scalar˜φ and tensor˜B obtained from the c-map, and vierbein
˜ ea
µ=
?
e−φ/2ˆ ei
0
m
eφ/2Bm
eφ/2
?
, (2.16)
where φ and B are the original fields in (2.1) before the c-map. Our symmetry is related to
the Buscher rules for T-duality [19]. We here derived these rules from an effective action
approach in Einstein frame, similar to [20].
One can apply the c-map to solutions of the equations of motion. Given a (τ-independent)
solution {ea
as described above. This procedure can be done both in Minkowski and in Euclidean
space. In the latter case, we can take the coordinate τ to be the Euclidean time, as time-
independent solutions can easily be Wick rotated. This is precisely the situation we are
interested in. To be more precise, we first formulate the Euclidean four-dimensional theory
based on Euclidean metrics coupled to a scalar and tensor. The dimensional reduction is
still based on the decomposition of the vierbein (2.9) with τ the Euclidean time. After
dimensional reduction over τ, the Einstein-Hilbert term now gives
µ,φ,Bµν}, one can construct another solution after the c-map, given by {˜ ea
µ,˜φ,˜Bµν}
eR(e) = ˆ eR(ˆ e) +1
2|d˜φ|2+1
2e2˜φ|˜H|2,(2.17)
such that the symmetry (2.14) still holds.
2.3 Taub-NUT geometries and NS-fivebrane instantons
To generate instanton solutions, we will start from a time independent solution of pure
Einstein gravity, and perform the c-map. This uplifts to a new solution in four dimen-
sions with generically nontrivial scalar and tensor. In other words, we do a T-duality over
Euclidean time. This of course only makes sense as a solution-generating-technique. How-
ever, such a solution is not an instanton, since it is not localized in τ. We therefore have to
uplift the solution to a τ-dependent solution in four dimensions. This is easy if the original
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A.4Integration of spherically symmetric harmonic functions
?
SD−1
∞
∗d
Q
(D − 2)(V olSD−1)rD−2= (−)DQ . (A.9)
B Electric-magnetic duality
Suppose we have a theory with a set of n + 1 two-forms BIand n + 1 scalars χI(with
I running from 0 to n), possibly accompanied by other fields (collectively denoted by φ).
Furthermore assume we can describe the set of two-forms and scalars by (generalized) field
strengths CIand DI. The CIare three-forms composed of the field strengths of the two-
forms with possible extra terms. The DIare one-forms composed of the ”field strengths”
of the scalars with also extra terms allowed. We then introduce the objects
(∗CI)µ =
1
2
1
?|g|
2
?|g|
δL
δDµI,
(∗DI)αβγ
= −1
6
δL
δCαβγI. (B.1)
In case the theory under consideration has only terms quadratic in CIand/or DIit can
be written as
L = DI∧ CI+ DI∧ CI. (B.2)
We now restrict ourselves to cases where the set of equations formed by the Bianchi iden-
tities of the generalized field strengths and the equations of motion of BIand χIcan be
formulated as
d
?
?
CI
CI
?
?
= α(φ) ∧ ∗
?
?
CI
CI
?
?
+ β(φ) ∧
?
?
DI
DI
?
?
,
d
DI
DI
= γ(φ) ∧ ∗
CI
CI
+ η(φ) ∧
DI
DI
. (B.3)
α(φ) and β(φ) are three-forms, while γ(φ) and η(φ) are one-forms8.
The set of equations (B.3) is invariant under the electric-magnetic duality transformations
?
?
CI
CI
?
?
−→
?
?
˜CI
˜CI
?
?
=
?
?
UI
WIJ
J
ZIJ
V
I
J
??
??
CJ
CJ
?
?
,
DI
DI
−→
˜DI
˜DI
=
UI
WIJ
J
ZIJ
V
I
J
DJ
DJ
, (B.4)
8In section 4 of the main text we have α = γ = η = 0 and β = iH.
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and α, β, γ, and η transforming trivially. We call (B.4) electric-magnetic duality transfor-
mations as Bianchi identities and equations of motion are rotated into each other, which is
similar to the effect of conventional electric-magnetic duality transformations working on
the field strengths and dual field strengths of one-form gauge fields.
A dual Lagrangian is defined by
(∗˜CI)µ =
1
2
1
?|g|
2
?|g|
δ˜L
δ˜DµI,
(∗˜DI)αβγ
= −1
6
δ˜L
δ˜CαβγI. (B.5)
In terms of the old (CI,CI) and (DI,DI) these equations are
1
2
1
?|g|
δ˜L
δDµJ
= (UTW)JK∗ CK
+δDη
µ+ (UTV )
K
J
∗ CµK
δCαβγ
K
δDµJ(ZTV )KL∗ DαβγL
δCαβγ
K
δDµJ(ZTW)K
K
δDµJ(ZTV )KL∗ CηL−1
+δDη
δDµJ(ZTW)K
= −1
δCαβγ
K
δCµρσJ(ZTV )KL∗ DαβγL+
−1
6
6
K
L∗ CL
µρσ−1
η−1
6
L∗ DL
αβγ,
1
2
1
?|g|
δ˜L
δCµρσJ
6(UTW)JK∗ DK
6(UTV )
K
J
∗ DµρσK
δDη
K
δCµρσJ(ZTV )KL∗ CηL
δDη
K
δCµρσJ(ZTW)K
−1
6
δCαβγ
K
δCµρσJ(ZTW)K
L∗ DL
αβγ+
L∗ CL
η. (B.6)
As it turns out this set of equations can only be solved consistently in case the transfor-
mation matrix in (B.4) belongs to Sp(2n + 2,R) when all fields are real, or otherwise a
complexified version thereof. Furthermore it is important to note that generically we get
L(˜CI,˜DI) ?= L(CI,DI), i.e. the Lagrangian does not transform as a function. For purely
quadratic theories the dual Lagrangian becomes
˜L =
˜DI∧˜CI+˜DI∧˜CI.(B.7)
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