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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.

2NS-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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2.2T-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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