hepth/9906018
{centering}
Duality and Instantons in String Theory
ELIAS KIRITSIS^{*}^{*}*email address:
Physics Department, P.O. Box 2208, University of Crete
GR71003 Heraklion, GREECE
Abstract
In these lecture notes duality tests and instanton effects in supersymmetric vacua of string theory are discussed. A broad overview of BPSsaturated terms in the effective actions is first given. Their role in testing the consistency of duality conjectures as well as discovering the rules of instanton calculus in string theory is discussed. The example of heterotic/typeI duality is treated in detail. Thresholds of and terms are used to test the duality as well as to derive rules for calculated with D1brane instantons. We further consider the case of couplings in N=4 groundstates. Heterotic/type II duality is invoked to predict the heterotic NS5brane instanton corrections to the threshold. The couplings of typeII string theory with maximal supersymmetry are analysed and the Dinstanton contributions are described Other applications and open problems are sketched.
May 1999
Based on lectures given at the 1999 Trieste Spring School on String Theory
Contents
 1 Introduction
 2 BPSsaturated terms and Nonrenormalization Theorems
 3 Survey of BPSsaturated terms
 4 Instantons in String Theory
 5 Heterotic/TypeI duality and Dbrane instantons

6 N=4 couplings and fivebrane instantons
 6.1 General remarks
 6.2 Oneloop corrections in sixdimensional type IIA and IIB theories
 6.3 Oneloop gravitational corrections in fourdimensions
 6.4 CPodd couplings and holomorphic anomalies
 6.5 From the typeII to the heterotic string
 6.6 NS5brane instantons
 6.7 Absence of d=4 instanton corrections for in the N=4 heterotic theory
 7 N=8, couplings and Dbrane instantons in type II string theory
 8 Summary and Open Problems
1 Introduction
Nonperturbative dualities have changed our way of thinking about string theory. They have also gave us the possibility to calculate nonperturbative effects that we expected to be there but had no a priori way of setting up, let alone calculate.
There two basic issues that will be addressed in these lecture notes. They are inextricably linked to oneanother. One is “testing” duality conjectures. The second is putting them to work.
We will have to be clear in what we mean by “testing” a duality conjecture. In a theory where we do not have an a priori nonperturbative definition a weakstrong coupling duality is a definition of the strongly coupled theory. In general and for minimal supersymmetry it might not even be a complete nonperturbative definition. The issue becomes nontrivial if there a independent nonperturbative definition of the theory. This however is not the case so far for supersymmetric theories since the only other known nonperturbative definition, namely formulation on a lattice, breaks supersymmetry. Thus, the only issue at stake in the case of duality is consistency of the definition. Even in the case of field theory we know of examples where consistency alone poses constraints in the nonperturbative definition of a theory. It is for example well known that cluster decomposition and defining the nonperturbative theory without including (smooth) monopoles are inconsistent. Thus, what we are testing at best is the consistency of the set of rules that duality uses to define the nonperturbative theory.
The consistency checks are stringent for effective couplings that have special properties. We call them BPSsaturated effective couplings. In a sense that will be made more precise later, supersymmetry constraints the form of their thresholds. They are the most reliable tools in checking consistency of duality conjectures.
There have been many qualitative checks of various nonperturbative dualities, but so far quantitative checks are scarce. In order to do a tractable quantitative test of a nonperturbative duality we need to carefully choose the quantity to be computed. Since usually a weak coupling computation has to be compared with a strong coupling one, one has to choose a quantity whose strong coupling computation can also be done at weak coupling. Such quantities are very special and generally turn out to be terms in the effective action that obtain loop contributions from BPS states only. They are also special from the supersymmetry point of view, since the dependence of their couplings on moduli must satisfy certain holomorphicity or harmonicity conditions. Moreover, when supersymmetry commutes with the loop expansion, they get perturbative corrections from a single order in perturbation theory. Such terms also have special properties concerning instanton corrections to their effective couplings. In particular, they obtain corrections only from instantons that leave some part of the original supersymmetry unbroken. Sometimes, such terms are directly linked to anomalies.
The other role such effective couplings play is in that duality can be used to calculate their nonperturbative corrections. One can then identify the nonperturbative effects responsible for such corrections. For theories with more than N=2 supersymmetry ^{1}^{1}1We count the supersymmetries using fourdimensional language (in units of four supercharges). such nonperturbative effects are due to instantons. Instantons in string theory can be associated to Euclidean branes wrapping around an appropriate compact manifold. By studying such nonperturbative thresholds we can learn the rules of instanton calculus.
While solitons have been studied vigorously in the context of duality, the attention paid to instantons has been lesser and more recent: it includes work on the pointlike Dinstanton of type IIB [133]–[10], on the resolution of the typeIIA conifold singularity by Euclidean 2branes [11]–[13], and on nonperturbative effects associated with Euclidean 5branes [14]–[17]. Here we will first look as an instructive example, at a simpler case, that of Euclidean Dstrings present in typeI string theory [18, 19, 20]: these are physically less interesting, since they are mapped by strong/weakcoupling dualities to standard worldsheet instanton effects on the typeIIB, respectively heterotic side. Our motivation is however to gain a better understanding of the rules of semiclassical Dinstanton calculations, which could prove useful in more interesting contexts. We will further consider the case of couplings in N=4 groundstates [16, 17]. In this case the relevant instanton corrections as we will argue are due to the NS5brane in the heterotic string [16], or the D5brane in the typeI string. the threshold on the other hand is oneloop exact in the typeII dual. Although we will go some way towards interpreting the threshold, a complete instanton calculation is lacking here.
Finally there is another area where duality and Dinstanton corrections are at play. This is the case of the threshold corrections to eightderivative terms in typeII vacua with maximal supersymmetry. A representative eightderivative term is the term. Here one can use Tduality, perturbative Dpbrane dynamics and elevendimensional input to successfully calculate the nonperturbative corrections. This has been done down to sixdimensional compactifications, [3],[5][9],[21][24] with a fairly good understanding of the Dinstanton calculus, but not without some puzzles.
The structure of these lectures is as follows: In section two we provide a general discussion on the nature and properties of BPSsaturated terms. In section three we give a survey of known such terms in theories with varying amounts of global or local symmetry. In section four we discuss the anticipated structure of instantons in string theory as well their similarities and differences with standard field theory instantons. In section five we discuss in detail heterotic/typeI duality in various dimensions. The two theories are compared in perturbation theory in ninedimensions. In eight dimensions there are D1instanton corrections on the typeI side that are mapped by the duality to the perturbative heterotic contribution. We use this to derive the instanton rules. In section six we analyse couplings in N=4 groundstates. The threshold is oneloop exact in the type II string on K3 and related via duality to NS5brane instanton corrections in the heterotic dual. IN section seven we summarize the results on the threshold in typeII groundstates with maximal supersymmetry. Finally section eight contains a short summary as well a survey of open problems.
2 BPSsaturated terms and Nonrenormalization Theorems
Supersymmetry plays an important role in uncovering and testing the consistency of duality symmetries. It provides important constraints into the dynamics. Moreover, it has special nonrenormalization theorems that help in discerning some properties of the strong coupling limit.
A central role is played by the BPS states. These are ‘‘shorter than normal” representations of the appropriate supersymmetry algebra^{2}^{2}2For a more complete discussion, see [25, 26].. Their existence is due to some of the supersymmetry operators becoming “null”, in which case they do not create new states. The vanishing of some supercharges depends on the relation between the mass of a multiplet and some central charges appearing in the supersymmetry algebra. These central charges depend on electric and magnetic charges of the theory as well as on expectation values of scalars (moduli) that determine various coupling constants. In a sector with given charges, the BPS states are the lowest lying states and they saturate the socalled BPS bound which, for pointlike states, is of the form
(2.1) 
where is the central charge matrix.
Massive BPS states appear in theories with extended supersymmetry. BPS states behave in a very special way:
At generic points in moduli space they are absolutely stable. The reason is the dependence of their mass on conserved charges. Charge and energy conservation prohibits their decay. Consider as an example, the BPS mass formula
(2.2) 
where are integervalued conserved charges, and is a complex modulus. This BPS formula is relevant for N=4, SU(2) gauge theory, in a subspace of its moduli space. Consider a BPS state with charges , at rest, decaying into N states with charges and masses , . Charge conservation implies that , . The fourmomenta of the produced particles are with . Conservation of energy implies
(2.3) 
Also in a given charge sector (m,n) the BPS bound implies that any mass , with given in (2.2). Thus, from (2.3) we obtain
(2.4) 
and the equality will hold if all particles are BPS and are produced at rest (). Consider now the twodimensional vectors on the complex plane, with length . They satisfy . Repeated application of the triangle inequality implies
(2.5) 
This is incompatible with energy conservation (2.4) unless all vectors are parallel. This will happen only if is real.^{3}^{3}3It can also happen when the charges are integer multiples of the same charge. In that case the composite states are simple bound states of some “fundamental” states. We will not worry further about this possibility. For energy conservation it should also be a rational number. On the other hand, due to the SL(2,ℤ) invariance of (2.2), the inequivalent choices for are in the SL(2, fundamental domain and is never real there. In fact, real rational values of are mapped by SL(2, to , and since is the inverse of the coupling constant, this corresponds to the degenerate case of zero coupling. Consequently, for finite, in the fundamental domain, the BPS states of this theory are absolutely stable. This is always true in theories with more than eight conserved supercharges (corresponding to N supersymmetry in four dimensions). In cases corresponding to theories with 8 supercharges, there are regions in the moduli space, where BPS states, stable at weak coupling, can decay at strong coupling. However, there is always a large region around weak coupling where they are stable.
Their massformula is supposed to be exact if one uses renormalized values for the charges and moduli. The argument is that quantum corrections would spoil the relation of mass and charges, if we assume unbroken SUSY at the quantum level. This would give incompatibilities with the dimension of their representations. Of course this argument seems to have a loophole: a specific set of BPS multiplets can combine into a long one. In that case, the above argument does not prohibit corrections. Thus, we have to count BPS states modulo long supermultiplets. This is precisely what helicity supertrace formulae do for us [25, 26]. Even in the case of N=1 supersymmetry there is an analog of BPS states, namely the massless states.
Thus, the presence of BPS states can be calculated at weak coupling and can be trusted in several cases at strong coupling. Their massformula is valid beyond perturbation theory, although both sides can obtain nontrivial quantum corrections in N=2 supersymmetric theories. There are no such corrections in N supersymmetric theories.
There is another interesting issue in theories with supersymmetry: special terms in the effective action, protected by supersymmetry. We will generically call such terms “BPSsaturated terms” for reasons that will become obvious in the sequel. We cannot at the moment give a rigorous definition of such terms for a generic supersymmetric theory, due to the lack of offshell formulation. However, for theories with an offshell formulation the situation is better and all BPSsaturated terms are known.
We will try here before we embark in a detailed discussion of various cases to give some generic features of such terms.
(a) Supersymmetry constraints their (modulidependent) coefficients to have a special structure. The simplest situation is complex holomorphicity but there several other cases less common where one would have special conditions associated with quaternions, as well as noncompact groups of the O(p,q) type , etc. We will generically call such constraints “holomorphicity” constraints.
(b) In cases where there is a superfield formulation, such terms are (chiral) integrals over parts of superspace. This is at the root of the holomorphicity conditions of the previous item.
(c) Such terms satisfy special nonrenormalization theorems. It should be stressed here and it would be seen explicitly later that some of the nonrenormalization theorems depend crucially on the perturbation theory setup.
Before we embark further into a discussion of the nonrenormalization theorems we should stress in advance that they are generically only valid for the Wilsonian effective action. The reason is that many nonrenormalization theorems are violated due to IR divergences. There are several examples known, we will mention here the case of N=1 supersymmetry: In the presence of massless of massless contributions a quantum correction to the Kähler potential (not protected by a nonrenormalization theorem) can be indistinguishable from a correction to the superpotential (not renormalized in perturbation theory), [27]. Thus, from now on we will be always discussing the Wilsonian effective action.
To continue further, we would like here to separate two cases.

Absolute nonrenormalization theorems. Such theorems state that a given term in the effective action of a supersymmetric theory does not get renormalized. This should be valid both for perturbative and nonperturbative corrections. A typical example of this is the case of two derivative terms in the effective action of an N=4 supersymmetric theory (with or without gravity).

Partial (or perturbative) nonrenormalization theorems. Such theorems usually claim the absence of perturbative corrections for a given effective coupling, or that the only corrections appear in a few orders only in perturbation theory. Typically this happens at oneloop order but we also know of cases where renormalization can occur at a single, arbitrarilyhigh, loop order. It is also common in the case of oneloop corrections only that the appropriate effective couplings are related to an anomaly via supersymmetry. The appropriate AdlerBardeen type theorem for the anomaly guarantees the absence of higher loop corrections, provided the perturbation theory is set up to respect supersymmetry. An example of such a situation is the case of twoderivative couplings in a theory with N=2 supersymmetry (global or local).
One should note a potentially very interesting generalization of supersymmetric nonrenormalization theorems: their analogues in the case where supersymmetry in softly broken (in field theory) or spontaneously broken (in supergravity or string theory). Although, there are some results in this direction mostly in field theory [28], I believe that much more needs to be done. Moreover this subject is of crucial importance in any unified theory that uses supersymmetry as a solution to the hierarchy problem.
It should be stressed here that the way perturbation theory is setup is crucial for the applicability of such partial nonrenormalization theorems. Many of the nonperturbative string dualities amount simply to different perturbative expansions of the same underlying theory. As we will see in more detail later on the appropriate partial nonrenormalization theorems for the same effective coupling are different in dual versions. In many cases this can be crucial in obtaining the exact result.
There are several examples that illustrate the general discussion above. We will mention some commonly known ones.
1. Heterotic string theory on is dual to type II string theory on K3. The effective coupling, has only a oneloop contribution on the type II side. On the heterotic side apart from the treelevel contribution there are no other perturbative corrections. There are however nonperturbative corrections due to fivebrane instantons [16, 17].
2. Heterotic string theory on K3 is dual to type IIA on a CalabiYau (CY) manifold that is a K3fibration. All two derivative effective couplings are treelevel only on the typeII side. On the other hand they obtain treelevel, oneloop as well as nonperturbative corrections on the heterotic side.
It can sometimes happen that a given perturbative expansion does not commute with supersymmetry. This is the case generically in typeI string theory. One way to see this is to note that the leading correction to the term in ten dimensions comes from the disk diagrams while the term obtains a contribution from oneloop only (GreenSchwarz anomalycanceling term). The two terms are however related by supersymmetry [29].
There is another general property that is shared by BPSsaturated terms: The quantum corrections to their effective couplings can be associated to BPS states. There are two concrete aspects of the statement above:
Their oneloop contributions are due to (perturbative) BPS multiplets only.^{4}^{4}4This was observed in [30] for N=2 twoderivative couplings, and in [31, 26] for N=4 fourderivative couplings and N=8 eightderivative couplings. The way this works out is that the appropriate oneloop diagrams come out proportional to helicity supertraces. The helicity supertrace is a supertrace in a given supersymmetry representation of casimirs of the little group of the Lorentz group [32]. In four dimensions, this is a supertrace of the helicity to an arbitrary even power (by CPT all odd powers vanish):
The helicity supertraces are essentially indices to which only short BPS multiplets contribute [31, 26]. It immediately follows that such oneloop contributions are due to BPS states only. The appropriate helicity supertraces count essentially the numbers of “unpaired” BPS multiplets. It is only these that are protected from renormalization and can provide reliable information in strong coupling regions. In fact, calling the helicity supertraces indices is more than an analogy. They thus provide the minimal information about IRsensitive data. In particular they do not depend on the moduli. Unpaired BPS states in lower dimensions are intimately connected with the chiral asymmetry (conventional index) of the tendimensional (or elevendimensional) theory. It is well known that the tendimensional elliptic genus is the stringy generalization of the Dirac index [33, 34]. Projecting the elliptic genus on physical states in ten dimensions gives precisely the massless states, responsible for anomalies. In lower dimensions, BPS states are determined uniquely by the elliptic genus, as well as the compact manifold data.
We will describe here a bit more the properties of helicity supertraces. Further information and detailed formulae can be found in the appendix of [25].
N=2 supersymmetry. Here we have only one kind of a BPS multiplet the 1/2 multiplet (preserving half of the original N=2 supersymmetry). The trace is nonzero for the 1/2 multiplet but zero from the long multiplets. The long multiplets on the other hand contribute to .
N=4 supersymmetry. Here we have two types of BPS multiplets, the 1/2 multiplets and the 1/4 multiplets. For all multiplets . is nonzero for 1/2 multiplets only. is nonzero for 1/2 and 1/4 multiplets only. Long massive multiplets start contributing only to . A similar stratification appears also in the case of maximal N=8 supersymmetry [25].
Thus, there is a single “index” in the case of an N=2 supersymmetric theory, and it governs the oneloop corrections to the twoderivative action, in standard perturbation theory. In the N=4 case there are two distinct indices. controls oneloop corrections to several fourderivative effective couplings of the , and type. seems to be associated to some six derivative couplings like .
BPSsaturated couplings may receive also instanton corrections. The instantons however that contribute, parallel in a sense the BPS states that contribute in perturbation theory. They must preserve the same amount of supersymmetry. Since in string theory instantons are associated with Euclidean solitonic branes wrapped on compact manifolds, it is straightforward in most situations to classify possible instantons that contribute to BPSsaturated terms. We will see explicit examples in subsequent sections.
Let us summarize here the generic characteristics of BPSsaturated couplings.
(1) They obtain perturbative corrections from BPS states only.
(2) The perturbative corrections appear at a single order of perturbation theory, usually at oneloop.
(3) They satisfy ”holomorphicity constraints”.
(4) They contain simple information about massless singularities.
(5) They obtain instanton corrections from ”BPSinstantons” (instanton configurations that preserve some fraction of the original supersymmetry).
(6) If there exists an offshell formulation they can be easily constructed.
Here we would like to remind the reader of a few facts about string perturbation theory. In particular we focus on heterotic and typeII perturbation theory. Almost nothing is known for typeI perturbation theory beyond one loop.
There are many subtleties in calculating higherloop contributions that arise from the presence of supermoduli. There is no rigorous general setup so far, but several facts are known. As discussed in [35] there are several prescriptions for handling the supermoduli. They differ by total derivatives on moduli space. Such total derivatives can sometimes obtain contributions from the boundaries of moduli space where the Riemann surface degenerates or vertex operator insertions collide. Thus, different prescriptions differ by contact terms. In [36] it was shown that such ambiguities eventually reduce to tadpoles of massless fields at lower orders in perturbation theory. The issue of supersymmetry is also the subject of such ambiguities. It is claimed [35, 36] that in a class of prescriptions supersymmetry is respected genus by genus provided there are no disturbing tadpoles at tree level and one loop. The only exception to this is the case of an anomalous in supersymmetric groundstates. In this case there is a nonzero Dterm at one loop [37].
To conclude, if all (multi) tadpoles vanish at one loop and we use the appropriate prescription for higher loops, we expect supersymmetry to be valid order by order in perturbation theory. It is to be remembered, however, that the above statements apply onshell. Sometimes there can be terms in the effective action that vanish onshell, violate the standard lore above, but are required by nonperturbative dualities. An example was given in [17].
3 Survey of BPSsaturated terms
In this section we will describe the known BPSsaturated terms for a given amount of supersymmetry. We use fourdimensional language for the supersymmetry, which can eventually be translated to various other dimensions bigger than four. For example N=2 fourdimensional theories are related by toroidal compactification to N=1 sixdimensional theories and so on.
3.1 N=1 Supersymmetry
N=1 supergravity in four dimensions contains the supergravity multiplet (it contains the metric and a gravitino) vector multiplets (each contains a vector and a gaugino) and chiral multiplets (each contains a complex scalar and a Weyl fermion).
The ”critical” dimension is four, in the sense that we cannot have an N=1 theory in more than four dimensions.
The full twoderivative effective action is determined by three functions^{5}^{5}5A slightly more extended description can be found in the appendix of [25, 38]. For a detailed exposition as well as a discussion of the nonrenormalization theorems [39] the reader is urged to look in [40].
(a) The Kähler potential . It is an arbitrary real function of the complex scalars of the chiral multiplets. It determines, among other things, the kinetic terms of chiral multiplets (matter) via the Kähler metric .
(b) The Superpotential : It determines the part of the scalar potential associated to the Fterms as . Supersymmetry constraints to be a holomorphic function of the chiral multiplets. Moreover, it should have charge 2 under the U(1) Rsymmetry that transforms the superspace coordinates and the chiral superfields . The reason is that the superpotential is integrated over half of the superspace as . It is thus a ”chiral” density. It turns out that in all cases where and offshell superfield formulation exists all BPSsaturated terms are chiral densities. Both in string theory (supergravity) and in the global supersymmetric limit (decoupling of gravity) it is not renormalized in perturbation theory. In string theory the argument is based on the holomorphicity [41]. The string coupling constant (dilaton), is assembled with the axion (dual of the antisymmetric tensor in four dimensions) into the complex chiral field. The PecceiQuinn symmetry associated to translations of the axion is valid in string perturbation theory, but it is broken by nonperturbative effects since it is anomalous. In perturbation theory, corrections to are multiplied by powers of the coupling constant, . However such corrections must be holomorphic and thus proportional to . This however breaks the PecceiQuinn symmetry in perturbation theory. Thus, no such perturbative corrections can appear. Beyond perturbation theory, instanton effects break the PecceiQuinn symmetry to some discrete subgroup and exponential holomorphic corrections are allowed [14]. In the global limit a similar argument works.
(c) The Wilsonian gauge couplings . They are also holomorphic. The index labels different simple or U(1) components of the gauge group. They are also chiral densities since they appear through in the effective action, where is the spinorial vector superfield. They can obtain corrections only to oneloop in perturbation theory [40]. The argument is similar to that about the superpotential, the anomaly here allowing also a oneloop contribution We should stress here though that the physical effective gauge couplings have corrections to any order in perturbation theory. This is due to the fact that the physical matter fields have a wave function renormalization coming from, the Kähler potential which is not protected from renormalization.
(d) In string theory, there is a possibility of “anomalous” U(1) factors of the gauge group [42]. The term anomalous here is strictly speaking a misnomer. The particular U(1) factor in question will have a nonzero sum of charges for the massless fields. Under normal circumstances it would have been anomalous. In string theory, this anomaly is canceled by an anomalous transformation law of the antisymmetric tensor (or an axion in four dimensions). We will have to distinguish two distinct cases.
In heterotic perturbation theory the anomalycancellation mechanism is a compactification descendant of the GreenSchwarz anomaly cancellation in ten dimensions. The appropriate coupling is which appears at oneloop in the heterotic string. Thus it is the standard antisymmetric tensor that cancels the anomaly. There is a Dterm contribution to the potential. It contains a oneloop contribution that was calculated in [37]. Moreover supersymmetry implies that there should be a twoloop contact term. This was verified by an explicit calculation [43]. Such nontrivial contributions appear since the supersymmetric partner of the antisymmetric tensor (axion) is the dilaton that controls the string perturbative expansion. The states that contribute at one loop are the charged massless particles. In this sense this is an anomaly. As mentioned earlier such states are the closest analogue of BPS states of N=1 supersymmetry (they have half the degrees of freedom compared to massive states). Moreover, upon toroidal compactification to three or two dimensions, they become bonafide BPS states. The twodimensional case was analyzed in [94].
In typeII and typeI perturbation theory the situation is different [45]. The axions that are responsible for canceling the U(1) anomaly come from the RR sector and they are usually more than one. In orbifold constructions they belong to the twisted sector. Since their scalar partners do not coincide with the string coupling constant the Dterm potential appears here at tree level.
(e) There is a series of higher derivative terms which are Fterms and involve chiral projectors of vector superfields [46]. They obtain a perturbative contribution only at gloops.
3.2 N=2 Supersymmetry
N=2 supersymmetry has critical dimension six. The relevant massless multiplets in six dimensions are the supergravity multiplet (graviton, second rank tensor, a scalar a gravitino and a spinhalf fermion), the vector multiplet (a vector and a gaugino), and the hypermultiplet (a fermion and four scalars). The vector multiplets contain no scalars in six dimensions and as such have a unique coupling to gravity. This is not the case with hypermultiplets that have a nontrivial model structure. This structure persists unchanged in four dimensions, and we will discuss it below.
In four dimensions, the supergravity multiplet contains the metric a graviphoton and two gravitini. The vector multiplet contains a vector, two gaugini and a complex scalar while the hypermultiplet is the same as in six dimensions.
We will describe briefly the structure of the effective supergravity theory in four dimensions [47]. The interested reader can consult [48] for further information. Picking the gauge group to be abelian is without loss of generality since any nonabelian gauge group can be broken to the maximal abelian subgroup by giving expectation values to the scalar partners of the abelian gauge bosons. We will denote the graviphoton by , the rest of the gauge bosons by , , and the scalar partners of as . Although the graviphoton does does not have a scalar partner, it is convenient to introduce one. The theory has a scaling symmetry, which allows us to set this scalar equal to one where K is the Kähler potential. We will introduce the complex coordinates , , which will parametrize the vector moduli space (VMS), . The scalars of the generically massless hypermultiplets parametrize the hypermultiplet moduli space and supersymmetry requires this to be a quaternionic manifold^{6}^{6}6In the global supersymmetry limit in which gravity decouples, , the geometry of the hypermultiplet space is that of a hyperkähler manifold.. The geometry of the full scalar manifold is that of a product, .
N=2 supersymmetry implies that the VMS is not just a Kähler manifold, but that it satisfies what is known as special geometry. Special geometry eventually leads to the property that the full action of N=2 supergravity (we exclude hypermultiplets for the moment) can be written in terms of one function, which is holomorphic in the VMS coordinates. This function, which we will denote by , is called the prepotential. It must be a homogeneous function of the coordinates of degree 2: , where . For example, the Kähler potential is
(3.2.1) 
which determines the metric of the kinetic terms of the scalars. We can fix the scaling freedom by setting , and then are the physical moduli. The Kähler potential becomes
(3.2.2) 
where . The Kähler metric has the following property
(3.2.3) 
where . Since there is no potential, the only part of the bosonic action left to be specified is the kinetic terms for the vectors:
(3.2.4) 
where
(3.2.5) 
(3.2.6) 
Here we see that the gauge couplings, unlike the N=1 case, are not harmonic functions of the moduli.
The selfinteractions of massless hypermultiplets are described by a model on a quaternionic manifold (hyperkähler in the global case) A quaternionic manifold must satisfy:
(1) It must have three complex structures , satisfying the quaternion algebra
with respect to which the metric is hermitian. The dimension of the manifold is 4m, . The three complex structures guarantee the the existence of an SU(2)valued hyperkähler twoform K.
(2) There exists a principal SU(2) bundle over the manifold, with connection such that the form K is closed with respect to
(3) The connection has a curvature that is proportional to the hyperkähler form
where is a real number. When the the manifold is hyperkähler. Thus the holonomy of a quaternionic manifold is of the form while for a hyperkähler manifold , with . The existence of the SU(2) structure is natural for the hypermultiplets since SU(2) is the nonabelian part of the N=2 Rsymmetry that acts inside the hypermultiplets. The scalars transform as a pair of spinors under SU(2). When the hypermultiplets transform under the gauge group, then the quaternionic manifold has appropriate isometries (gauging). More information can be found in [48].
Thus, supersymmetry implies (a) that all twoderivative couplings in the vector multiplets are determined by a holomorphic function of the moduli, the prepotential. Here the symmetry is complex holomorphicity (b) All two derivative couplings of the hypermultiplets are determined by quaternionic geometry SU(2) “holomorphicity”.
In N=2 supersymmetry all twoderivative couplings are of the BPSsaturated type. Their precise nonrenormalization properties though depend on the perturbation theory setup.
Global N=2 supersymmetry. It can be obtained by taking the limit of the locally supersymmetric case. Here the holomorphic prepotential that governs the vector multiplet moduli space obtains quantum corrections only at oneloop in perturbation theory. Beyond the perturbative expansion it obtains instanton corrections. On the other hand the massless hypermultiplet geometry does not have any perturbative or nonperturbative corrections. In the quantum theory the only thing that can change are the points where various Higgs branches intersect themselves or the Coulomb branch, [95]. The argument of [95] for this nonrenormalization is supersymmetry and is an adaptation of a similar argument valid in heterotic string theory to be discussed below. In this context the crucial constraint imposed by supersymmetry is that the geometry of the vector moduli space is independent of the geometry of the hypermoduli space. Put otherwise, the only couplings between vector and hypermultiplets are those dictated by the gauge symmetry.
Local N=2 supersymmetry. Here we must distinguish three different types of perturbation theory.
(a) TypeII perturbation theory. This is relevant for type II groundstates with (1,1) fourdimensional supersymmetry. One of the supersymmetries comes from the left moving sector while the other from the right moving one. A typical class of examples much studied is type IIA,B theory compactified on CY threefold. The typeIIA compactification gives an effective theory with vector multiplets and neutral hypermultiplets (see for example [25]). In the typeIIB compactification the roles of and are interchanged. On the other hand such (1,1) groundstates need not be leftright symmetric. What is an important feature of these groundstates is that the dilaton that determines the string coupling constant belongs to a hypermultiplet. This has far reaching consequences for the perturbative expansion. If this fact is combined with the supersymmetric constraint of the absence of neutral couplings between vectors multiplets and hypermultiplets, then we conclude: in typeII (1,1) groundstates the treelevel prepotential is nonperturbatively exact, while the hypermultiplet geometry obtains corrections both in perturbation theory and nonperturbatively. Thus, the exact prepotential in typeIIA groundstates can be obtained by a treelevel calculation, in the appropriate model. It describes the geometry of Kähler structure of the CY manifold. Nontrivial model instantons render this calculation very intricate. On the other hand, in typeIIB groundstates, the exact prepotential is given by the geometry of the moduli space of complex structures, which can be calculated using classical geometry. Mirror symmetry can be further used [50] to solve the analogous typeIIA problem.
An interesting phenomenon, is that generically, CY manifolds develop some conifold (logarithmic) singularities at some submanifolds of their Kähler moduli space. In a type IIA compactification such singularities appear at tree level and cannot be smoothed out by quantum effects since as we have argued there aren’t any. At such conifold points a collection of twocycles shrinks to zero size. It was however been pointed out [51], that at such points, nonperturbative states (D2branes wrapped around the vanishing cycles) become massless. If we included them explicitly in th effective theory, then the singularity disappears. Alternatively, integrating them out reproduces the conifold singularity. The message is the type II perturbation theory gives directly the full quantum effective action after integrating out all massive degrees of freedom.
In the typeIIB groundstate the conifold singularity appears in the hypermoduli space. Here we expect both perturbative and nonperturbative corrections to smoothout the singularity. This has been confirmed in some examples [12].
(b) Heterotic perturbation theory. This type of perturbation theory applies to groundstates of heterotic theory compactified on a sixdimensional manifold of SU(2) holonomy (a prototype is K3T) and typeII asymmetric vacua with (2,0) supersymmetry (ie. both supersymmetries come from the leftmovers or rightmovers). In such vacua, the dilaton belongs to a vector multiplet. Thus supersymmetry implies that the treelevel geometry of the hypermoduli space is exact. On the other hand the geometry of the vector moduli space (prepotential) receives perturbative corrections only at oneloop, as well as nonperturbative corrections due to spacetime instantons. Several such vacua seem to be dual to typeII (1,1) vacua [52]. This duality can be used to determine exactly the geometry both of the vector moduli space as well as the hypermoduli space. Moreover, such a map is consistent with the nonrenormalization theorems mentioned above, and it reproduces the oneloop contribution on the heterotic side [53] as well the SeibergWitten geometry in the global limit [54].
One more property should be stressed here: in heterotictype N=2 vacua, there is no renormalization of the Einstein term. On the other hand there is a gravitational (universal) contribution to the gauge couplings [55, 56] as well as the Kähler potential [57]. This can be thought of as due to worldsheet contact terms [55] or as the gravitational back reaction [56]. Its diagrammatic interpretation (for the gauge coupling) is that of a spacetime contact term where two gauge bosons couple to the dilaton which couples to a generic loop of particles (see [25] for more details).
(c) typeI perturbation theory. This is the perturbation theory of type II orientifolds that contain open sectors. Here we have both open and closed unoriented Riemann surfaces. Here the dilaton belongs partly to a vector multiplet and partly to a hypermultiplet. As a result both the vector moduli space as well as the hypermoduli space receive corrections perturbatively and nonperturbatively. Moreover there is another subtlety: there is no universal renormalization of the gauge couplings and the Kähler potential. On the other hand there is a oneloop (cylinder) renormalization of the the Einstein term [58] consistent with typeI/heterotic duality.
There is a whole series of “chiral” Fterms that generalize the prepotential and the twoderivative effective action, [59, 60, 61, 62]
(3.2.7) 
where
(3.2.8) 
is the supergravity superfield, with the antiselfdual graviphoton field strength and the antiselfdual Riemann tensor. The square is defined as . The superfields X, stand for the vector multiplet superfields, , corresponds to the graviphoton.
In type II perturbation theory, we can go to the model frame by the gauge fixing condition where is the Kähler potential and is the string coupling constant. Then are the true moduli scalars. From supersymmetry we know that must be a homogeneous function of degree . Thus,
(3.2.9) 
This implies that such effective terms obtain a contribution only at the gth order in typeII perturbation theory. This was indeed verified by an explicit computation, [59]. is indeed the prepotential that governs the two derivative interactions. governs among other things, the terms and obtains contributions from one loop only.
The (almost) holomorphic threshold is given by the topological partition function, of a twisted CY model [59]. The mild nonholomorphicity is due to an anomaly and it provides for recursion relations among the various ’s. They have the form
(3.2.10) 
where is the Kähler covariant derivative and are the holomorphic Yukawa couplings. For it is equivalent to
(3.2.11) 
Near a conifold point they all become singular in a different fashion as one approaches the singularity. Their singularity structure was shown to be captured by the c=1 topological matrix model [63].
3.3 N=4 Supersymmetry
The critical dimension of N=4 supersymmetry is ten. In ten dimensions it corresponds to a single MajoranaWeyl supercharge which decomposes to four supercharges upon toroidal compactification to four dimensions. The relevant massless multiplets are the supergravity multiplet (the graviton, second rank tensor a scalar, a MajoranaWeyl gravitino and a Majorana Weyl fermion of opposite chirality in ten dimensions) and the vector multiplet ( a gauge boson and a MajoranaWeyl gaugino). In d dimensions, the supergravity multiplet contains apart from the metric and second rank tensor and original scalar, (10d) vectors (graviphotons). The vector multiplet contains apart from the vector an extra (10d) scalars.
The twoderivative effective action is completely fixed by supersymmetry and the knowledge of the gauge group. Its salient features are that it has no scalar potential in ten dimensions (since there are no scalars in the theory), and it has a ChernSimons coupling of the gauge fields to the secondrank tensor crucial for anomaly cancellation via the GreenSchwarz mechanism. The explicit action and a discussion can be found in [25].
In lower dimensions scalars appear from the components of the metric, second rank tensor and gauge fields. There is a potential for the scalars due in particular to the nonabelian field strengths. The minima of the potential have flat directions parametrized by expectation values of the scalars coming from the supergravity multiplet as well as those coming from the Cartan vectors. These expectation values break generically the nonabelian gauge symmetry to the maximal abelian subgroup. At special values of the moduli massive gauge boson can become massless and gauge symmetry is enhanced.
Supersymmetry constraints the local geometry of the scalars in d dimensions to be that of the symmetric space O(10d,N)/O(10d)O(N) where N is the number of abelian vector multiplets. Moreover, if we neglect the massive gauge bosons, and focus on the generically massless fields then the effective action is invariant under a continuous O(10d,N) symmetry under which the metric, second rank tensor and original scalar are inert, while the vectors transform in the vector while the scalars in the adjoint. The O(10d) part of this symmetry is the Rsymmetry that rotates the supercharges.
The O(10d,N) symmetry is broken by the massive states. In string theory a discrete subgroup remains that is generically a subgroup of O(10d,N,ℤ). The interested reader can find a more detailed discussion of the above in [25]
Since supersymmetry and knowledge of the rank of the gauge group completely fixes the twoderivative effective action of the massless modes, we expect no perturbative or nonperturbative corrections. This has been explicitly verified in various contexts. In the context of field theory (global N=4 supersymmetry) it can be shown that in the Higgs phase perturbative corrections vanish, as well as instanton corrections (due to trivial zero mode counting) [64]. Moreover, in the Higgs phase there are no subtleties with IR divergences. In the local case (string theory) perturbative nonrenormalization theorems have been advanced (see [65]).
The knowledge of BPSsaturated terms for N=4 supersymmetry is scarce. The next type of terms beyond the two derivative ones, are those related by supersymmetry to (CPeven). Among these, there are the CPodd terms (four dimensions) and (six dimensions). In a typeII (1,1) setup, there is no treelevel term [65, 66], but there is a contribution at oneloop. It has been conjectured [16] that there are no further perturbative and nonperturbative corrections. Arguments in favor of this conjecture were advanced in [17]. In particular this conjecture seems to be in agreement with heterotic/type II duality. In the respective heterotic perturbation theory, the term has a tree level contribution [66], but no further perturbative contributions. One could expect nonperturbative contributions in four dimensions due to Euclidean fivebrane instantons wrapped around [16]. This is compatible with the type II oneloop contribution and heterotic/type II duality. The situation in typeI perturbation theory seems to be less clear: it is only known that there is a nontrivial oneloop (cylinder) contribution to the term below ten dimensions [67]. The oneloop correction to the term is proportional to the conformal anomaly. Moreover the conformal anomaly depends on the “duality frame” [68]; although in four dimensions a pseudoscalar is dual to a second rank tensor, they contribute differently to the conformal anomaly; when the scalar contributes 1 the secondrank tensor contributes 91. In the heterotic side we have an antisymmetric tensor and this provides for the vanishing oneloop result while on the typeII side we are in a dual frame and this gives a nonzero oneloop result.
There are several other terms with up to eight derivatives that are of the BPS saturated type. These include and terms [69, 31, 18, 26]. So far, we have been vague concerning the tensor structure of such terms. Here, however, we will be more precise [29, 69, 31]. There are three types of terms in ten dimensions: , and , where is the standard eightindex tensor [65] and is the tendimensional totally antisymmetric symbol. The precise expressions can be found for example in [69]. There are also the , and terms. These different structures can be completed in supersymmetric invariants [29, 69]. The bosonic parts of these invariants are as follows:
(3.3.1a) 
(3.3.1b) 
(3.3.1c) 
As is obvious from the above formulae, apart from the combination, the other fourderivative terms are related to the Green–Schwarz anomaly by supersymmetry. Thus, in ten dimensions, they are expected to receive corrections only at one loop if their perturbative calculation is set up properly (in an Adler–Bardeenlike scheme). The invariant is not protected by supersymmetry. Heterotic/typeII duality in six dimensions implies that it receives perturbative corrections beyond one loop. It is however protected in the presence of supersymmetry [6].
The relevant N=4 string vacua are the following:
TypeI O(32) string theory. It is related by weakstrong coupling duality to the O(32) heterotic string.
The heterotic O(32) and EE strings.
Ftheory on K3. This is an d=8 vacuum and is conjectured to be dual to the heterotic string compactified on .
Type IIA on K3. It is conjectured to be dual to the heterotic string on . There are further typeII N=4 vacua in less than six dimensions [17]. They have either type II or heterotic duals.
We consider first type II N=4 vacua. The case of Ftheory compactifications [70] stands apart in the sense that it has no conventional perturbation theory. The couplings were derived recently [71] using geometric methods that mimic those used in typeII N=2 vacua.
On the other hand there is no computation so far for such terms in the typeII string compactified on K3. This is the obvious quantitative test of the heterotic/typeII duality and it is still lacking.
Most of the information is known for heterotic and typeI vacua. The CPodd terms in (3.3.1c) were explicitly evaluated at arbitrary order of heterotic perturbation theory in [72]. There, by carefully computing the surface terms, it was shown that such contributions vanish for . The CPeven terms are related to the CPodd ones by supersymmetry (except for ). If there are no subtleties with supersymmetry at higher loops, then these terms also satisfy the nonrenormalization theorem. This was in fact conjectured in [72]. In view of our previous discussion on the structure of supersymmetry, we would expect that once supersymmetry is working well at , it continues to work for for a suitable definition of the highergenus amplitudes. It is thus assumed that the CPeven terms do not get contributions beyond one loop. On the other hand, the term (which is nonzero at tree level) is not protected by the anomaly. Thus, it can appear at various orders in the perturbative expansion. It can be verified by direct calculation that it does not appear at one loop on the heterotic side. However, heterotic/typeIIA duality in six dimensions seems to imply that there is a twoloop contribution to this term on the heterotic side. In all of the subsequent discussion for N=4 groundstates , when we refer to terms we will mean the anomalyrelated tensor structures, , , which can always be distinguished from .
In the typeI theory several of these terms have been calculated and match what is expected by heterotic/typeI duality [69]. Most of them appear at tree level (disk) and one loop. There are subtleties though. The heterotic treelevel term implies via duality that there should be a twoloop contribution in the typeI side. That anomaly related terms obtain twoloop contributions is hardly surprising if we recall that supersymmetry (that related CPeven and CPodd terms) is not respected by typeI perturbation theory [18].
In the heterotic theory, the potential instanton contributions must come from configurations that preserve half of the supersymmetry. Thus the only relevant configuration is the heterotic Euclidean fivebrane. It can provide an instanton provided there is a six or higherdimensional compact manifold to wrap it around. Thus, there are no nonperturbative contributions to such terms in on the heterotic side. In four dimensions we generically expect corrections due to NS5brane instantons. There is a prediction of global supersymmetry about thresholds [73]: it states that there are no corrections beyond one loop (in the absence of gravity). We will give an argument in a subsequent section that the same is implied in the local (string) case by heterotic/type II duality.
The situation is different on the typeI side. There the relevant configurations are D1 and D5 branes and provide instanton corrections already in eight dimensions.
No more BPS saturated terms are known in N=4 groundstates. It was conjectured in [18] that in analogy with lower supersymmetry there is an infinite tower of BPSsaturated terms as well in the N=4 case. This rests on the existence of a tower of topological partition functions in the topological model on K3 [107] in analogy with the N=2 case. The leading topological amplitude was shown to correspond to the amplitude of the type II string compactified on K3.
3.4 N=8 supersymmetry
N=8 groundstates are toroidal compactifications of the typeII string. Their critical dimension is eleven, and the master theory is elevendimensional supergravity [75] expected to describe the strong coupling limit of the typeIIA theory. The relevant massless representation in eleven dimensions is the supergravity multiplet and contains the graviton, a threeindex antisymmetric tensor and the gravitino.
Like N=4 supergravity all twoderivative effective couplings receive no renormalization at all. Unlike N=4 supergravity, fourderivative and sixderivative effective couplings do not appear at tree level and do not get renormalized at oneloop. It is expected that they are not renormalized at all (even beyond perturbation theory). There are eightderivative BPSsaturated terms however. as in the N=4 case, but also in this case. There are also related to the coupling of elevendimensional supergravity. These terms obtain tree level plus oneloop corrections in typeII perturbation theory [3, 76, 23, 77]. Beyond perturbation theory they obtain corrections from Dinstantons.
4 Instantons in String Theory
In field theory with supersymmetry instantons are responsible for all nonperturbative corrections (in the Coulomb branch at least). What are instantons in string theory? Despite some prescient papers that touched this issue [78] the subject took a definite shape only after the duality revolution. The central idea is that a string instanton to the zeroth approximation is an instanton of the effective supergravity theory. A very important aspect is that instanton solutions preserve a part of the spacetime supersymmetry.
An important conceptual simplification that occurs in string theory is the direct relation between spacetime instantons and wrapped Euclidean solitonic branes [11]. The concept is rather simple. String theory (or the effective supergravity) contains solitonic branes that usually preserve some amount of supersymmetry (BPS branes). These can be found as classical solutions to the supergravity equations of motion. They include Dbranes, the NS5brane and the M2 and M5 branes of d=11 supergravity. For Dbranes in particular there is an alternative stringy description as Dirichlet branes [79]. An instanton can be produced by wrapping the Euclidean worldvolume of a given brane around an appropriate compact manifold.
What kind of instanton corrections we expect for BPSsaturated terms was discussed case by case in the previous section. Here we will stress that depending on the term we will need instantons with a given number of zero modes. However, this analysis needs care. A typical example is that of multiinstanton configurations. In multiinstanton solutions, there are in general more bosonic moduli describing relative positions and orientation. If the multiinstanton leaves some supersymmetry unbroken, there will be more fermionic zero modes, supersymmetric partners of the bosonic moduli related by the unbroken supersymmetry. If, however their moduli space contains orbifold singularities, then there are contributions localized at the singularities where the number of zero modes is reduced. We will see later an explicit example of this in the case of D1instanton contributions to fourderivative couplings in typeI string theory.
An important question to be answered is: What part of the supersymmetry can an instanton configuration break? The answer to this depends on the particular instanton (Euclidean brane) as well as the number of noncompact dimensions. It depends crucially on the compact manifold, and the way the Euclidean brane is wrapped around it [11, 80].
Can we compare between the instantons we are using in string theory and standard fieldtheory instantons? In field theory, we are usually considering two types of instantons. The first are instantons with finite action, and a typical example is the BPST instanton [81], present in nonAbelian fourdimensional gauge theories. Examples of the other type are provided by the Euclidean Dirac monopole in three dimensions, which is relevant, as shown in [82], to the understanding of the nonperturbative behaviour of threedimensional gauge theories in the Coulomb phase. This type of instanton has an ultraviolet (shortdistance)divergent action, since it is a singular solution to the Euclidean equations of motion. However, by cutting off this divergence and subsequent renormalization, it can contribute to nonperturbative effects. The generalization to the compact gauge theories of higher antisymmetric tensors was also discussed in the context of (lattice) field theory [83]. Another famous case in the same class is the twodimensional vortex of the XY model, responsible for the KT phase transition [84]. In four dimensions we also have the BCD merons [85], with similar characteristics, although their role in the nonperturbative fourdimensional dynamics is not very well understood.
In the context of string theory, we have these two types of instantons. Here, however, the behaviour seems to be somewhat different. Let us consider first the heterotic fivebrane [78]. This solution is intimately connected to BPST instantons in the transverse space and is smooth provided the instanton size is nonzero. At zero size the solution has an exact CFT description but the string coupling is strong. Nonperturbative effects are important and a conjecture has been put forth to explain their nature [86]. Another type of instanton whose effective fieldtheory description is regular is the D3brane of typeIIB theory. On the other hand, the other Dbrane instantons have an effective description that is of the singular type. However, their ultraviolet divergence is cured in their stringy description. This is already clear in the case of the typeI D1brane that will be described in these lectures, where the effective description is singular [87, 88] while the stringy description turns out to be regular and in particular, as we will see later, their classical action is finite.
There seems to be a correspondence of the various fieldtheory instantons to stringy ones. We have already mentioned the example of the heterotic fivebrane, but the list does not stop there. In [89] it was shown that the threedimensional Polyakov QED instanton as well as various nonAbelian merons have an exact CFT description and thus correspond to exact classical solutions of string theory. Moreover, the threedimensional instanton can be interpreted as an avatar of the fivebrane zerosize instanton when the theory is compactified to three dimensions. Similar remarks apply to the stringy merons, which require the presence of fivebranes with fractional charge [89]. In that respect they are solutions of the singular type in the effective field theory. In the context of the string theory, the spectrum of instanton configurations is of course richer, since the theory includes gravity. However, the correspondence of fieldtheory and some stringtheory instantons implies that the fieldtheory nonperturbative phenomena associated with them, are already included in a suitable stringy description.
5 Heterotic/TypeI duality and Dbrane instantons
The conjectured duality [90, 88, 87, 91] between the typeI and heterotic string theories is particularly intriguing. The massless spectrum of both theories, in ten spacetime dimensions, contains the (super)graviton and the (super)Yang Mills multiplets. Supersymmetry and anomaly cancellation fix completely the lowenergy Lagrangian, and more precisely all twoderivative terms and the anomalycanceling, fourderivative GreenSchwarz couplings [65, 69]. One logical possibility, consistent with this unique lowenergy behaviour, could have been that the two theories are selfdual at strong coupling. The conjecture that they are instead dual to each other implies that this unique infrared physics also has a unique consistent ultraviolet extrapolation.
One of the early arguments in favour of this duality [90, 88, 87] was that the heterotic string appears as a singular solution of the typeI theory. Strictlyspeaking this is not an independent test of duality. Since the two effective actions are related by a field redefinition this is not surprising. The real issue is whether consistency of the theory forces us to include such excitations in the spectrum. This can for instance be argued in the case of type II string theory near a conifold singularity of the CalabiYau moduli space [51].
We are not aware of such a direct argument in the case of the heterotic string solution. What is, however, known is that it has an exact conformal description as a D(irichlet) string of typeI theory [91]. In certain ways, Dbranes lie between fundamental quanta and smooth solitons so, even if we admit that they are intrinsic, we must still decide on the rules for including them in a semiclassical calculation. Do they contribute, for instance, to loops like fundamental quanta? And with what measure and degeneracy should we weight their Euclidean trajectories?
Here we will analyse some calculations [31, 18] in which these questions can be answered. The rules consistent with duality turn out to be natural and simple. Dstrings, like smooth solitons, do not enter explicitly in loops ^{7}^{7}7A (light) soliton loop can of course be a useful approximation to the exact instanton sum, as is the case near the strongcoupling singularities of the SeibergWitten solution. For a Dbrany discussion see also [8]., while their (wrapped) Euclidean trajectories contribute to the saddlepoint sum, without topological degeneracy if one takes into account correctly the nonabelian structure of Dbranes.
5.1 Heterotic/TypeI duality in ten dimensions.
We will start our discussion by briefly describing heterotic/typeI duality in ten dimensions. It can be shown [92] that heterotic/typeI duality, along with Tduality can reproduce all known string dualities.
Consider first the O(32) heterotic string theory. At treelevel (sphere) and up to twoderivative terms, the (bosonic) effective action in the model frame is
(5.1.1) 
On the other hand for the O(32) type I string the leading order twoderivative effective action is
(5.1.2) 
The different dilaton dependence here comes as follows: The Einstein and dilaton terms come from the closed sector on the sphere (). The gauge kinetic terms come from the disk (). Since the antisymmetric tensor comes from the RR sector of the closed superstring it does not have any dilaton dependence on the sphere.
We will now bring both actions to the Einstein frame, :
(5.1.3) 
(5.1.4) 
We observe that the two actions are related by while keeping the other fields invariant. This suggests that the weak coupling of one is the strong coupling of the other and vice versa. As mentioned earlier the fact that the two effective actions are related by a field redefinition is not surprising. What is is interesting though is that the field redefinition here is just an inversion of the tendimensional coupling. Moreover, the two theories have perturbative expansions that are very different.
Let us first study the matching of the BPSsaturated high derivative terms in ten dimensions. At tree level, the only fourderivative term is the .It is part of the ChernSimons related combination [66]. Via the duality this term should correspond to a typeI contribution that comes from a genus3 surface. This, of course, has never been computed. At one loop terms would correspond to disk term in the typeI theory. Fortunately, the only nonzero oneloop contribution is of the type and agrees with the disk computation. is zero at oneloop in the heterotic theory, a good thing since it would be impossible to obtain such a term from the disk (that has a single boundary). Similar remarks apply to the and mixed terms.
We should stress again here that the matching of the oneloop heterotic terms with specific disk and oneloop terms in typeI is not a test of duality. It is rather a consequence of N=1 supersymmetry and anomaly cancellation in ten dimensions.
5.2 OneLoop Heterotic Thresholds
As discussed previously, the terms that will be of interest to us are those obtained by dimensional reduction from the tendimensional superinvariants, whose bosonic parts read [29, 69]
(5.2.1) 
These are special because they contain anomalycanceling CPodd pieces. As a result anomaly cancellation fixes entirely their coefficients in both the heterotic and the type I effective actions in ten dimensions. Comparing these coefficients is not therefore a test of duality, but rather of the fact that both these theories are consistent [69]. In lower dimensions things are different: the coefficients of the various terms, obtained from a single tendimensional superinvariant through dimensional reduction, depend on the compactification moduli. Supersymmetry is expected to relate these coefficients to each other, but is not powerful enough so as to fix them completely. This is analogous to the case of N=1 super YangMills in six dimensions: the twoderivative gaugefield action is uniquely fixed, but after toroidal compactification to four dimensions, it depends on a holomorphic prepotential which supersymmetry alone cannot determine.
On the heterotic side there are good reasons to believe that these dimensionallyreduced terms receive only oneloop corrections. To start with, this is true for their CPodd anomalycanceling pieces [72]. Furthermore it has been argued in the past [36] that there exists a prescription for treating supermoduli, which ensures that spacetime supersymmetry commutes with the heterotic genus expansion, at least for vacua with more than four conserved supercharges^{8}^{8}8A notable exception are compactifications with a naivelyanomalous U(1) factor [37, 43].. Thus, we may plausibly assume that there are no higherloop corrections to the terms of interest. Furthermore, the only identifiable supersymmetric instantons are the heterotic fivebranes. These do not contribute in uncompactified dimensions, since they have no finitevolume 6cycle to wrap around. Nonsupersymmetric instantons, if they exist, have on the other hand too many fermionic zero modes to make a nonzero contribution. It should be noted that these arguments do not apply to the sixth superinvariant [29, 69]
(5.2.2) 
which is not related to the anomaly. This receives as we will mention below both perturbative and nonperturbative corrections.
The general form of the heterotic oneloop corrections to these couplings is [93, 94]
(5.2.3) 
where is an (almost) holomorphic modular form of weight zero related to the elliptic genus, and stand for the gaugefield strength and curvature twoforms, is the lattice sum over momentum and winding modes for toroidallycompactified dimensions, is the usual fundamental domain, and
(5.2.4) 
is a normalization that includes the volume of the uncompactified dimensions [31]. To keep things simple we have taken vanishing Wilson lines on the hypertorus, so that the sum over momenta () and windings (),
(5.2.5) 
factorizes inside the integrand. Our conventions are
(5.2.6) 
while winding and momentum are normalized so that and for a circle of radius . The Lagrangian form of the above lattice sum, obtained by a Poisson resummation, reads
(5.2.7) 
with the metric and the (constant) antisymmetrictensor background on the compactification torus. For a circle of radius the metric is .
The modular function inside the integrand depends on the vacuum. It is quartic, quadratic or linear in and , for vacua with maximal, half or a quarter of unbroken supersymmetries. The corresponding amplitudes have the property of saturating exactly the fermionic zero modes in a GreenSchwarz lightcone formalism, so that the contribution from leftmoving oscillators cancels out [94]^{9}^{9}9Modulo the regularization, is in fact the appropriate term in the weakfield expansion of the elliptic genus [93, 94, 33, 34]. In the covariant NSR formulation this same fact follows from function identities. As a result should have been holomorphic in , but the use of a modularinvariant regulator introduces some extra dependence [94]. As a result takes the generic form of a finite polynomial in , with coefficients that have Laurent expansions with at most simple poles in ,
(5.2.8) 
The poles in come from the wouldbe tachyon. Since this is not charged under the gauge group, the poles are only present in the purely gravitational terms of the effective action. This can be verified explicitly in eq. (5.2.9) below. The terms play an important role in what follows. They come from corners of the moduli space where vertex operators, whose fusion can produce a massless state, collide. Each pair of colliding operators contributes one factor of . For maximallysupersymmetric vacua the effective action of interest starts with terms having four external legs, so that . For vacua respecting half the supersymmetries (N=1 in six dimensions or N=2 in four) the oneloop effective action starts with terms having two external legs and thus .
Much of what we will say in the sequel depends only on the above generic properties of . It will apply in particular in the mostoftenstudied case of fourdimensional vacua with N=2. For definiteness we will, however, focus our attention to the toroidallycompactified SO(32) theory, for which [93, 94]
(5.2.9) 
Here is the wellknown tensor appearing in fourpoint amplitudes of the heterotic string [65], and are the Eisenstein series which are (holomorphic for ) modular forms of weight . Their explicit expressions are collected for convenience in the appendices of [19]. The second Eisenstein series is special, in that it requires a nonholomorphic regularization. The entire nonholomorphicity of in eq. (5.2.9), arises through this modified Eisenstein series.
In the toroidallycompactified heterotic string all oneloop amplitudes with fewer than four external legs vanish identically [95]. Consequently eq. (5.2.3) gives directly the effective action, without the need to subtract oneparticlereducible diagrams, as is the case at tree level [66]. Notice also that this fourderivative effective action has infrared divergences when more than one dimensions are compactified. Such IR divergences can be regularized in a modularinvariant way with a curved background [56, 96]. This should be kept in mind, even though for the sake of simplicity we will be working in this paper with unregularized expressions.
5.3 Oneloop TypeI Thresholds
The oneloop typeI effective action has the form
(5.3.1) 
corresponding to the contributions of the torus, Klein bottle, annulus and Möbius strip diagrams. Only the last two surfaces (with boundaries) contribute to the , and terms of the action. The remaining two pure gravitational terms may also receive contributions from the torus and from the Klein bottle. Contrary to what happens on the heterotic side, this oneloop calculation is corrected by both higherorder perturbative and nonperturbative contributions.
For the sake of completeness we review here the calculation of pure gauge terms following refs. [97, 31]. To the order of interest only the short BPS multiplets of the open string spectrum contribute. This follows from the fact that the wave operator in the presence of a background magnetic field reads
(5.3.2) 
where is a nonlinear function of the field, is the spin operator projected onto the plane (12), denotes the momenta in the directions , is a string mass and labels the Landau levels. The oneloop free energy thus formally reads
(5.3.3) 
where the supertrace stands for a sum over all bosonic minus fermionic states of the open string, including a sum over the ChanPaton charges, the center of mass positions and momenta, as well as over the Landau levels.
Let us concentrate on the spindependent term inside the integrand, which can be expanded for weak field
(5.3.4) 
The terms vanish for every supermultiplet because of the properties of the helicity supertrace [31], while to the term only short BPS multiplets can contribute. The only short multiplets in the perturbative spectrum of the toroidallycompactified open string are the SO(32) gauge bosons and their KaluzaKlein dependents. It follows after some straightforward algebra that the special terms of interest are given by the following (formal) oneloop super YangMills expression
(5.3.5) 
where is the lattice of KaluzaKlein momenta on a dimensional torus, and the trace is in the adjoint representation of SO(32).
This expression is quadratically UV divergent, but in the full string theory one must remember to (a) regularize contributions from the annulus and Möbius uniformly in the transverse closedstring channel, and (b) to subtract the oneparticlereducible diagram corresponding to the exchange of a massless (super)graviton between two tadpoles, with the trace being here in the fundamental representation of the group. The net result can be summarized easily, after a Poisson resummation from the openchannel KaluzaKlein momenta to the closedchannel windings, and amounts to simply subtracting the contribution of the zerowinding sector [97, 31]. Using also the fact that we thus derive the final oneloop expression on the typeI side
(5.3.6) 
The conventions for momentum and winding are the same as in the heterotic calculation of the previous section.
The calculation of the gravitational terms is more involved because we have no simple backgroundfield method at our disposal. It can be done in principle following the method described in ref. [58]. There is one particular point we want to stress here: if the oneloop heterotic calculation is exact, and assuming that duality is valid, there should be no worldsheet instanton corrections on the typeI side. Such corrections would indeed translate to nonperturbative contributions in the heterotic string [98], and we have just argued above that there should not be any. The dangerous diagram is the torus which can wrap nontrivially around the compactification manifold. The typeI torus diagram is on the other hand identical to the type IIB one, assuming there are only graviton insertions. This latter diagram was explicitly calculated in eight uncompactified dimensions in ref. [6], confirming our expectations: the CPodd invariants only depend on the complex structure of the compactification torus, but not on its Kähler structure. This is not true for the CPeven invariant .
5.4 Circle Compactification
Let us begin now our comparison of the effective actions with the simplest situation, namely compactification on a circle. There are no worldsheet or Dstring instanton contributions in this case, since Euclidean worldsheets have no finitearea manifold in target space to wrap around. Thus, the oneloop heterotic amplitude should be expected to match with a perturbative calculation on the typeI side. This sounds at first puzzling, since the heterotic theory contains infinitely more charged BPS multiplets than the typeI theory in its perturbative spectrum. Indeed, one can combine any state of the current algebra with appropriate winding and momentum, so as to satisfy the levelmatching condition of physical states. The heterotic theory thus contains short multiplets in arbitrary representations of the gauge group.
The puzzle is resolved by a wellknown trick, used previously in the study of string thermodynamics [99, 100], and which trades the winding sum for an unfolding of the fundamental domain into the halfstrip, . The trick works as follows: starting with the Lagrangian form of the heterotic lattice sum, and
(5.4.1) 
one decomposes any nonzero pair of integers as , where is their greatest common divisor (up to a sign). We will denote the set of all relative primes , modulo an overall sign, by . The lattice sum can thus be written as
(5.4.2) 
Now the set is in onetoone correspondence with all modular transformations,
(5.4.3) 
such that has a solution only if belongs to , and the solution is unique modulo a shift and an irrelevant sign . Indeed the condition
(5.4.4) 
By choosing appropriately we may always bring inside the strip, which establishes the above claim.
Using the modular invariance of , we can thus suppress the sum over and unfold the integration regime for the part of the expression. This gives
(5.4.5) 
There is one subtle point in this derivation [100]: convergence of the original threshold integral, when has a pole^{1}^{1}1(Physical) massless states do not lead to IR divergences in fourderivative operators in nine dimensions, requires that we integrate first in the region. Since constant lines transform however nontrivially under SL(2,Z), the integration over the entire strip would have to be supplemented by a highly singular prescription. The problem could be avoided if integration of the terms in the Lagrangian sum (i.e. those terms that required a change of integration variable) were absolutely convergent. This is the case for , so expression (5.4.5) should only be trusted in this region.
Let us now proceed to evaluate this expression. The fundamental domain integrals can be performed explicitly by using the formula [94]
(5.4.6) 
where
(5.4.7) 
is any modular form of weight which is holomorphic everywhere except possibly for a simple pole at zero. As for the strip integration, it picks up only the term in the expansion of . Modulo the nonholomorphic regularization, only the SO(32) gauge bosons contribute to the elliptic genus at this order, in agreement precisely with the result of the typeI side! For let us define more generally
(5.4.8) 
where is the radius of the compactification circle. The oneloop SO(32) heterotic action takes finally the form
(5.4.9) 
To simplify notation we have written here instead of , instead of etc.
We have expressed the result as an expansion in inverse powers of the compactification volume. Since the heterotic/typeI duality map transforms (model) length scales as
(5.4.10) 
with the openstring loop counting parameter, this expansion can be translated to a genus expansion on the typeI side. The Euler number of an nonorientable surface is given by where is the number of holes, the number of boundaries and the number of crosscaps. The leading term corresponds to the disk and projective plane diagrams and is completely fixed by tendimensional supersymmetry and anomaly cancellation [69]. To check this, one must remember to transform the metric in both and the tensor appropriately. Notice that the typeI sphere diagram, which is the same as in type IIB, only contributes to the invariant which we are not considering here. The subleading o() terms correspond to the annulus, Möbius strip, Klein bottle and torus diagrams, all with . For zero background curvature these agree with the typeI calculation [31] as described in section 5.3.
The last two terms in the expansion (5.4.9) correspond to diagrams with . These contributions must be there if the duality map of ref. [90] does not receive higherorder corrections. Such corrections could anyway always be absorbed by redefining fields on the typeI side, so that if duality holds, there must exist some regularization scheme in which these highergenus contributions do arise. These terms do on the other hand come from the boundary of moduli space. For instance the contribution to the term comes from the boundary of moduli space shown in figure 1. It could thus be conceivably eliminated in favour of some lowerdimension operators in the effective action.
It is in any case striking that a single heterotic diagram contains contributions from different topologies on the typeI side. Notice in particular that the divergent term in the oneloop field theoretic calculation, regularized on the heterotic side by replacing the strip by a fundamental domain, is regularized on the typeI side by replacing the annulus by the disk.
5.5 Twotorus Compactification
The next simplest situation corresponds to compactification on a twodimensional torus. There are in this case worldsheet instanton contributions on the heterotic side, and our aim in this and the following sections will be to understand them as (Euclidean) Dstring trajectory contributions on the typeI side. The discussion can be extended with little effort to toroidal compactifications in lower than eight dimensions. New effects are only expected to arise in four or fewer uncompactified dimensions, where the solitonic heterotic instantons or, equivalently, the typeI D5branes can contribute.
The targetspace torus is characterized by two complex moduli, the Kählerclass
(5.5.11) 
and the complex structure
(5.5.12) 
where and are the model metric and antisymmetric tensor on the heterotic side. The oneloop thresholds now read
(5.5.13) 
where the lattice sum takes the form [101]
(5.5.14) 
The exponent in the above sum is (minus) the Polyakov action,
(5.5.15) 
evaluated for the topologically nontrivial mapping of the string worldsheet onto the targetspace torus,
(5.5.16) 
The entries of the matrix are integers, and both targetspace and worldsheet coordinates take values in the (periodic) interval . To verify the above assertion one needs to use the metrics
(5.5.17) 
The Polyakov action is invariant under global reparametrizations of the worldsheet,
(5.5.18) 
which transform
(5.5.19) 
Following Dixon, Kaplunovsky and Louis [101], we decompose the set of all matrices into orbits of PSL(2,Z), which is the group of the above transformations up to an overall sign. There are three types of orbits,
A canonical choice of representatives for the degenerate orbits is
(5.5.20) 
where the integers should not both vanish, but are otherwise arbitrary. Distinct elements of a degenerate orbit are in onetoone correspondence with the set , i.e. with modular transformations that map the fundamental domain inside the strip, as in section 5.4. In what concerns the nondegenerate orbits, a canonical choice of representatives is
(5.5.21) 
Distinct elements of a nondegenerate orbit are in onetoone correspondence with the fundamental domains of in the upperhalf complex plane.
Trading the sum over orbit elements for an extension of the integration region of , we can thus express eqs. (5.5.13,5.5.14) as follows
(5.5.22) 
The three terms inside the curly brackets are constant, powersuppressed and exponentiallysuppressed in the large compactificationvolume limit. They correspond to treelevel, higher perturbative and nonperturbative, respectively, contributions on the typeI side. The discussion of the perturbative contributions follows exactly the analogous discussion in section 5.4. The only difference is the replacement of eq. (5.4.8) by
(5.5.23) 
where are generalized Eisenstein series. In the openstring channel of the typeI side this takes into account properly the (double) sum over KaluzaKlein momenta [31]. Notice that the holomorphic anomalies in lead again to higher powers of the inverse volume, which translate to highergenus contributions on the typeI side. Notice also that the term has a logarithmic infrared divergence, which must be regularized appropriately, as discussed in the introduction.
We turn now to the novel feature of eight dimensions, namely the contributions of worldsheet instantons. Plugging in the expansion (5.2.8) of the elliptic genus, we are lead to consider the integrals
(5.5.24) 
Doing first the (Gaussian) integral, one finds after some rearrangements