On small black holes, KK monopoles and solitonic 5-branes
Pablo A. Cano, ?ngel Murcia, Pedro F. Ramírez, Alejandro Ruipérez
OOn small black holes, KK monopolesand solitonic 5-branes
Pablo A. Cano a , ´Angel Murcia b , Pedro F. Ram´ırez c and Alejandro Ruip´erez d,ea Instituut voor Theoretische Fysica, KU LeuvenCelestijnenlaan 200D, B-3001 Leuven, Belgium b Instituto de F´ısica Te´orica UAM/CSIC,C/ Nicol´as Cabrera, 13-15, C.U. Cantoblanco, 28049 Madrid, Spain c Max-Planck-Institut f¨ur Gravitationsphysik (Albert Einstein Institut),Am M¨uhlenberg 1, D-14476 Potsdam, Germany d Dipartimento di Fisica ed Astronomia “Galileo Galilei”, Universit`a di Padova,Via Marzolo 8, 35131 Padova, Italy e INFN, Sezione di Padova,Via Marzolo 8, 35131 Padova, Italy
Abstract
We review and extend results on higher-curvature corrections to different configura-tions describing a superposition of heterotic strings, KK monopoles, solitonic 5-branesand momentum waves. Depending on which sources are present, the low-energy fieldsdescribe a black hole, a soliton or a naked singularity. We show that this property isunaltered when perturbative higher-curvature corrections are included, provided thesources are fixed. On the other hand, this character may be changed by appropriateintroduction (or removal) of sources regardless of the presence of curvature corrections,which constitutes a non-perturbative modification of the departing system. The gen-eral system of multicenter KK monopoles and their 5-brane charge induced by higher-curvature corrections is discussed in some detail, with special attention paid to thepossibility of merging monopoles. Our results are particularly relevant for small blackholes (Dabholkar-Harvey states, DH), which remain singular after quadratic curvaturecorrections are taken into account. When there are four non-compact dimensions,we notice the existence of a black hole with regular horizon whose entropy coincideswith that of the DH states, but the charges and supersymmetry preserved by bothconfigurations are different. A similar construction with five non-compact dimensionsis possible, in this case with the same charges as DH, although it fails to reproducethe DH entropy and supersymmetry. No such configuration exists if d >
5, which weinterpret as reflecting the necessity of having a 5-brane wrapping the compact space. a r X i v : . [ h e p - t h ] F e b ontents A.1 General form of supersymmetric configurations . . . . . . . . . . . . . . . . . 44A.2 Killing spinor equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B First-order α (cid:48) -corrections to the fundamental rotating string solution 48 Many supergravity theories are known to describe certain low energy limit of string theory.It is then natural to expect that, given a solution to the equations of motion of one of thesesupergravity theories, a correspondent description in terms of fundamental objects of stringtheory exists. This identification, however, turns out to pose a challenging problem, un-less the task is somehow facilitated. Simplifications take place when the system preserves1ome of the supersymmetries of the theory. From the field theory perspective, supersymme-try imposes relations between the components of the different fields, such that the allowedconfigurations are described by a reduced set of functions. Restrictions also occur for theequations of motion, with many of them being no longer independent. Typically, it sufficesto solve Maxwell equations and Bianchi identities for some p -form fields, and it follows thatthe Einstein and scalar equations are automatically satisfied [1, 2]. Hence, one can say thata solution is completely determined by the specification of the charge distribution associatedto the corresponding p -forms. On the UV part of the story, one then needs to find super-symmetric states in the spectrum acting as sources of those fields, an information that canbe read from the worldsheet or worldbrane (effective) action. The identification obtained inthis manner can be tested by comparing additional properties, like the number of supersym-metries preserved or the degeneracy. The use of these tools has been very fruitful, playing arole in much progress in string theory. Some noteworthy examples are the discovery of non-perturbative fundamental objects in the spectrum, evidence in favour of a web of dualitiesconnecting seemingly distinct string theories or the identification of the microscopic degreesof freedom responsible for the thermodynamic entropy of certain black holes. A quite limitedlist of references is [3–17].The microscopic derivation of black hole entropy performed by Strominger and Vafafollowed a seminal paper of Sen that studied heterotic small black holes [18] , whose eventhorizon is singular and has zero size. Small black holes provide a toy model that was closeto become the first confirmed description of black hole microstates in quantum gravity and,hence, their study has a special position in the history of the achievements of the theory.Consider states consisting of excitations of a string carrying winding and momentum charges( Q w , Q n ). This system was first studied by Dabholkar and Harvey (DH) in [19] —see also[20]. In the heterotic theory, the degeneracy of these states gives the following value for theentropy in the large charge limit [21, 22] S = 4 π (cid:112) Q n Q w . (1.1)The mass of the DH states grows linearly with the value of the charges. Hence, for largevalues of ( Q w , Q n ) a black hole can be expected to emerge at the effective gravitational fieldtheory [23]. However, when one tries to construct such a black hole, a singular horizon withvanishing area is obtained and the formula (1.1) is not reproduced. Sen argued that, since theeffective theory shall not be valid in regions of large curvature, a “stretched horizon” surfacebeyond which the usual understanding breaks down can be defined. He then postulated thatarea of this stretched horizon would account for the macroscopic entropy of the system, andshowed that the value, remarkably, scales with √Q n Q w .Sen’s insight found two lines of continuation. On the one hand, a string carrying mo-mentum should oscillate, and one can study how many solutions can be constructed suchthat the string’s profile lies within a stretched horizon [24]. Depending on the duality frameused to describe them, the resulting geometries are of singular or solitonic nature. On theother hand, working within the special geometry formulation of effective four-dimensional2upergravity with higher-curvature corrections, it was found in [25, 26] that it is possibleto construct a regular near-horizon geometry reproducing (1.1) such that only two of thelower dimensional vectors carry non-vanishing charge. The two approaches offer a distinctrealization of the macroscopic entropy in the field theory, and a debate was opened regardingthe compatibility of these two ideas [27, 28].In the light of the findings of [25, 26], shortly followed by [29, 30], it emerged the appealingidea that stringy higher-curvature corrections lead to the resolution of the singular smallhorizon. String theory, as candidate to being a consistent theory of quantum gravity,is expected to resolve the singularities that mark the limitations of classical gravitationaltheories. One should be able to describe the collapse and evaporation of a black hole in termsof a unitary evolution free of divergences in a UV-complete theory. But the idea that stringyor quantum corrections may resolve singularities directly in the low-energy (field-theory)approximation goes beyond that expectation. It is, therefore, interesting to explore if this isactually a generic feature of the theory. Arguably, the simplest test that can be performedis to study similar configurations in slightly different situations. However, it turns out thatthe same mechanism that produced the horizon resolution in a few cases, failed in otherswithout a clear explanation. Some examples of the latter case are those of a type II stringwith winding and momentum charges on a toroidal compactification [17], or a heteroticstring with five or more non-compact dimensions [31, 35]. In view of these facts, it is fair toacknowledge that the effect of higher-curvature corrections must be understood better. Inthis article we study the problem by revisiting the original small-black-hole system directly inthe original ten-dimensional heterotic theory, instead of using four-dimensional supergravityformulated in the language of special geometry as was done in [25, 26, 29, 30]. While wewill not have at our disposal the powerful tools based on the attractor mechanism developedin [38–41], in exchange we will have analytic solutions in the complete black hole exteriorregion, with direct control on which are the sources in the equations of motion, which as wewill see facilitates the microscopic interpretation.In the last years an intensive effort to understand the effect of higher-curvature correc-tions to solutions of heterotic string theory has been performed [42–50]. The cases consid-ered include different supersymmetric configurations of strings, momentum, Kaluza-Kleinmonopoles (KK) and solitonic 5-branes (S5), as well as some non-extremal black holes (thatlack a microscopic interpretation to date). The small-black-hole system with four non-compact dimensions was studied in [45], where it was found that the perturbative curvaturecorrections leave the field theory solution singular. Additionally, a curvature-corrected so-lution with a regular horizon and whose Wald entropy coincides with (1.1) was described.It was argued that this field configuration should not be identified microscopically with theDH small black hole, because it contains a KK monopole. Interestingly, this charge doesnot appear explicitly in the entropy formula, although its value needs to differ from zero inorder to have a regular horizon. A crucial ingredient in the construction is the presence oflocalized solitonic 5-brane sources, with a non-trivial charge profile that asymptotes to zero. See also [16, 17, 31–37] and references therein. A five-dimensional heterotic two charge solution with regular horizon exists, but its entropy differs from(1.1). We will discuss this solution in more detail in section 6. ≤ d ≤ In section 2 we describe our default course of action for the construction of solutions ofheterotic string theory at first order in α (cid:48) . Section 3 reviews the supersymmetric black holeswith four (five) non-compact dimensions that result from the superposition of the four (three,without KK monopole) types of sources considered in the article. Besides describing thecomplete solutions in the exterior of the black hole, we use the near-horizon entropy functionformalism as an alternative method to obtain some relevant properties of the solutions. Thepurpose of this is manifolded; on one side, it is useful as a consistency check and facilitatesthe comparison with previous literature, while on the other side it is illustrative to showhow some information beyond the near-horizon background needs to be given in order todistinguish between solutions with 3 charges and 4 charges with unit KK monopole. Insection 4, we study the fields that result from general superposition of KK monopoles andS5 branes. Special attention is paid to the merging of monopoles and possible emergenceof conical defects. It is described that fractional charge contributions induced by curvaturecorrections is a generic property of orbifolded spaces, consequence of the fact that the integralof the Bianchi identity is related to the orbifold Euler character of the space, which typicallyhas non-integer value. The curvature corrections to small black holes and rings (string withwinding and momentum, static or oscillating) in general number of dimensions are computedin section 5. Finally, section 6 describes a very special family of black hole solutions of thekind considered in section 3 with the property that the S5 charge is completely screened,which we call fake small black holes . Some of the properties of the resulting field configuration(but, crucially, not all of them) coincide with those of the DH states. Most importantly, whileDH (and, hence, small black holes) are 1 / / It is with relative frequency that problems need to be approached perturbatively. In someoccasions the equations that need to be solved are known, but they are too complicated tobe directly treated. In that case, it may happen that those can be expressed as a smallmodification, in some appropriate sense, of a set of simpler equations, for which analyticsolutions can be found. The system is then expressed in terms of a series expansion, possiblywith infinite terms, where the zeroth order term corresponds to the simpler set of equations.Another common situation, which we will encounter in this work, is that only a pertur-bative description of the system is known, with the complete non-perturbative formulationinexistent or unknown. A schematic representation of such a problem is ∞ (cid:88) n =0 α n f n,i [ φ a , O ( φ a )] = 0 . (2.1)Here i labels a number of independent equations for the variables φ a , with O ( φ a ) collectivelyrepresenting any possible operator acting on the variables. The expansion is controlled bythe presence of the parameter α , whose power serves to label the order of the correction.The functional form of terms of higher order n > k could be unknown, or simply it may becomputationally convenient to truncate the series at a certain order. Perturbative solutionsto the system at kth -order are expressions of the form φ a = φ a + k (cid:88) n =1 α n φ an , (2.2)such that, when substituted in (2.1), the equations are not necessarily identically satisfied,but the non-vanishing terms are of order k + 1 or higher in the expansion parameter. Suchexpression is usually interpreted as a good approximation of the real solution of the fullsystem if some requirements are fulfilled, which include an estimation of how small thenon-vanishing part of the equations is.The zeroth-order term in (2.2), φ a , plays a special role. It is an exact solution of thezeroth-order system of equations that serves as the starting point in the construction of thesolution. In order to obtain it, boundary conditions need to be given for the variables. Theseboundary conditions are considered to be part of the specification of the zeroth-order system.The subsequent terms in the expansion φ ak are progressively computed using the previouslyobtained values for φ am , with m < k , as input in equation (2.1), which is then solved up to5erms of order α k +1 . The perturbative solution is therefore built order by order from φ a ,which can be used as a sort of label to identify the configuration.The variables we shall be interested in are fields defined on a manifold. In the problems wefind in this article, boundary conditions can be chosen following different approaches. In firstplace, we need to specify the asymptotic structure of the manifold (i.e. its topology) and theassumed isometries. The remaining information can be specified through the introduction oflocalized sources in the equations of motion or, alternatively, indicating the asymptotic fall-off behaviour of (independent) fields. In this work, this corresponds either to the electionof sources signalling the presence of fundamental objects of the heterotic theory, or theindependent charges carried by the field configuration. Both possibilities are technicallyvalid and, most frequently, they define inequivalent perturbative expansions. The reason isthat, in certain configurations, some of the higher-order terms behave as delocalized sourcesof charge in the equations of motion, affecting the original relation between localized sourcesand asymptotic charges of the fields. Hence, if one of these properties is kept constant in theconstruction of the perturbative solution, the other one will change, and viceversa. Whenconstructing these solutions, it is fundamental to identify these relations appropriately andunderstand their implications, as we emphasize at several stages in this work.In the perturbative constructions presented below, the boundary conditions are fixed byspecifying the localized sources in the system. The advantage of this approach is that, for thesystems in which there is a ST interpretation, the fundamental constituents of the solutionremain fixed, so it gives us information of how higher-curvature corrections modify a givenstringy configuration. The low-energy limit of heterotic string theory is described by an effective field theory forits massless modes —the metric g µν , the dilaton φ , the Kalb-Ramond (KR) 2-form B µν , anda set of non-Abelian Yang-Mills fields A Aµ with gauge group fixed to be either SO(32) orE × E — which involves a double perturbative expansion in α (cid:48) , the string length square,and g s , the string coupling. In this work, we will only deal with the α (cid:48) expansion, assumingwe are in a regime where g s - or loop corrections can be neglected. To first order in α (cid:48) , the bosonic part of the effective action of the heterotic string is given by[52–54] We will however work with a consistent truncation in which all the Yang-Mills vector fields are trivial. Of course, this is something that must be checked a posteriori . The solutions described in Section 4.2have a divergent dilaton, we refer to [51] for more information about this issue. = g s πG (10)N (cid:90) d x (cid:112) | g | e − φ (cid:20) R − ∂φ ) + 12 · H − α (cid:48) R ( − ) µν ab R ( − ) µν ba + . . . (cid:21) , (2.3)where R ( − ) ab = dω ( − ) ab − ω ( − ) ac ∧ ω ( − ) cb is the curvature of the torsionful spin connection,defined as ω ( − ) ab ≡ ω ab − H cab e c , where ω ab is the spin connection. The 3-form field strength H associated to the Kalb-Ramond 2-form B is given by H = dB + α (cid:48) L( − ) , (2.4)where Ω L( − ) = dω ( − ) ab ∧ ω ( − ) ba − ω ( − ) ab ∧ ω ( − ) bc ∧ ω ( − ) ca , (2.5)is the Chern-Simons 3-form of ω ( − ) ab . The Bianchi identity is obtained by taking the exteriorderivative of eq. (2.4), getting dH = α (cid:48) R ( − ) ab ∧ R ( − ) ba . (2.6)The equations of motion at first order in α (cid:48) can be obtained by varying the action (2.3)with respect to the metric, dilaton and Kalb-Ramond 2-form. In doing so, we can ignore implicit occurrences of these fields through the torsionful spin connection, which accordingto the Bergshoeff-de Roo lemma yield terms of second order in α (cid:48) [53]. The set of equationsthat one obtains is R µν − ∇ µ ∂ ν φ + 14 H µρσ H ν ρσ − α (cid:48) R ( − ) µρab R ( − ) ν ρ ba = O ( α (cid:48) ) , (2.7)( ∂φ ) − ∇ φ − · H + α (cid:48) R ( − ) µν ab R ( − ) µν ba = O ( α (cid:48) ) , (2.8) d (cid:0) e − φ (cid:63)H (cid:1) = O ( α (cid:48) ) . (2.9)Although it is not explicitly written in (2.6)-(2.9), the equations are allowed to have localizedsources in the form of Dirac delta functions. These appear at zeroth-order in the perturbative It is worth to emphasize that this only holds if one works perturbatively in α (cid:48) . Later on, we shall be interested in studying the conditions that must be satisfied by ourconfigurations in order to preserve a certain amount of supersymmetry. Therefore, we needto know the supersymmetry transformations of the fermionic fields, the gravitino ψ µ andthe dilatino λ . Their explicit form also receive α (cid:48) corrections, but fortunately to us, theyappear at cubic order in α (cid:48) , see for instance [53]. Hence, for the purposes of this work, thesupersymmetry transformations of the fermionic fields reduce to δ (cid:15) ψ µ = (cid:18) ∂ µ − ω (+) µab Γ ab (cid:19) (cid:15) , (2.10) δ (cid:15) λ = (cid:18) ∂ a φ Γ a − H abc Γ abc (cid:19) (cid:15) , (2.11)where ω (+) ab = ω ab + H cab e c . Regular supersymmetric black-hole solutions to supergravity theories have five or four non-compact dimensions . The simplest black holes of this kind that one can obtain as solutionsof the effective equations of motion of the heterotic string have the following form [55, 56] ds = 2 Z − du (cid:18) dt − Z + du (cid:19) − Z dσ − dz α dz α , Higher-dimensional supersymmetric solutions may describe black strings with a null isometry in a non-compact direction of spacetime. = (cid:63) σ d Z + d Z − − ∧ du ∧ dt ,e φ = e φ ∞ Z Z − , (3.1)where dσ = H − ( dη + χ ) + H d(cid:126)x , dχ = (cid:63) (3) d H , (3.2)is the metric of a four-dimensional Gibbons-Hawking (GH) space [57, 58], where the functions Z , + , − are defined. It turns out that this field configuration, as it stands, preserves at leastfour of the sixteen supersymmetries of the theory.The coordinates z α ∼ z α +2 π(cid:96) s parametrize a four-dimensional torus, T , with no internaldynamics, whereas z ≡ t − u ∼ z + 2 πR z parametrizes an internal direction, S z , whosedynamics is non-trivial.The equations of motion are satisfied if Z , + , − are harmonic functions in the GH space. The choices that yield the black-hole solutions we are interested in are Z , + , − = 1 + q , + , − r , H = (cid:15) + q H r , (3.3)where (cid:15) is either 0 or 1 and where r denotes the radial coordinate of E : r ≡ (cid:126)x (3) · (cid:126)x (3) . The (cid:15) = 0 and (cid:15) = 1 cases will give raise to five- and four-dimensional black holes respectively.Let us consider each case separately. Static, spherically-symmetric, three-charge black holes in five dimensions
In the (cid:15) = 0 case, which implies there are five non-compact dimensions, the change ofvariables ρ = 4 q H r and ψ = ηq H allows us to rewrite the metric as dσ = dρ + ρ (cid:0) dψ + dφ + dθ + 2 cos θ dψdφ (cid:1) , (3.4)where one can recognize the factor multiplied by ρ as the metric of the round 3-sphere S .Hence, this (trivial) choice of the GH function gives the metric of E . One may notice thatthe parameter q H has no physical significance. Note that the Gibbons-Hawking function H is also harmonic in E and in GH. ds = ( Z Z + Z − ) − / dt − ( Z Z + Z − ) / (cid:0) dρ + ρ d Ω (cid:1) . (3.5)It represents an extremal black hole with three electric charges. The horizon is placed at ρ = 0 and the Bekenstein-Hawking entropy is given by S BH5d = π G (5)N (cid:112) ˜ q ˜ q + ˜ q − , (3.6)where ˜ q , + , − ≡ q H q , + , − . In addition to the metric, the compactification yields three vectorfields, with A i = − Z − i dt , and two scalars. The gauge-invariant conserved electric chargecarried by a vector field inside a co-dimension 2 compact spacelike surface (usually taken tobe a 3-sphere) is defined, up to a normalization constant, as the integral over the surface ofthe variation of the Lagrangian with respect to the rt component of the field strength . In thezeroth-order theory, this is the integral of the dual field strength. The evaluation gives Q i ∼ ˜ q i , which is the reason why the poles of the harmonic functions are often referred as “charges”.However, when corrections are incorporated, the previous definition of charge may includeadditional terms such that the result is not just the pole of the harmonic function. Beingof higher-order in derivatives, these terms become subleading in the asymptotic expansion,and we get lim r →∞ Z i = 1 + c i Q i r + O ( r − ) , (3.7)for some convenient normalization constants c i , whose value can be inferred from the discus-sion below. This issue is treated with greater care in the following subsection.Two of the three types of charges —namely, Q − and Q — correspond to the electric andmagnetic (or S5-brane) charge associated to the Kalb-Ramond 2-form B µν . They can beunderstood as being produced by fundamental strings, which are electrically-charged withrespect to B µν , and by a stack of N solitonic five-branes, which carry instead magneticcharge. In the lower-dimensional description, these objects act as point-like sources, as theyare wrapped along the internal directions. Concretely, fundamental strings are wrappedalong S z with total winding number w and solitonic five-branes wrap the five-dimensionaltorus T × S z . Finally, the charge Q + is associated to the momentum n of a gravitational wavewhich travels along the z direction. Introducing these sources in the equations of motionvia Dirac delta functions with appropriate coefficients [42, 44], it is possible to obtain a General five-dimensional supergravities contain gauge Chern-Simons terms that give additional contri-butions, such that magnetic fields become electric sources. The requirement of spherical symmetry impliesthat 2-forms cannot have magnetic sources, so these terms do not contribute here. q i and the number of fundamental objects in the microscopicinterpretation, ˜ q − = g s α (cid:48) w , ˜ q = α (cid:48) N , ˜ q + = g s α (cid:48) R z n . (3.8)In this case, the charges of the system are equal to the localized sources, Q − = w , Q = N , Q + = n . However, it is important to bear in mind that, in general, these quantities aredifferent, as will become evident when we include α (cid:48) corrections. Now, using G (5)N = G (10)N (2 π(cid:96) s ) πR z = πg s α (cid:48) R z , ( G (10)N = 8 π g s α (cid:48) ) , (3.9)the Bekenstein-Hawking entropy gives S BH5d = 2 π √ nwN = 2 π (cid:112) Q + Q − Q , (3.10) Static, spherically-symmetric, four-charge black holes in four dimensions
Let us now consider the (cid:15) = 1 case, which means there are four non-compact dimensions. TheGibbons-Hawking 1-form χ is determined by solving eq. (3.2). A possible local expression is χ = q H cos θ dφ , (3.11)where we have introduced the spherical coordinates θ and φ , defined in terms of the Cartesiancoordinates (cid:126)x (3) = ( x , x , x ) as x = r sin θ cos φ , x = r sin θ sin φ , x = r cos θ . (3.12)The resulting metric has a Dirac-Misner string singularity, as χ is ill-defined at θ = 0 , π . Itis well-known that this string can be removed if η is a compact coordinate ( η ∼ η + 2 πR η )and q H obeys the quantization condition q H = R η W , W ∈ N . (3.13)The resulting GH metric describes an orbifold with a conical singularity for integer valuesof W other than 1. This can be seen by studying the r → σ (cid:12)(cid:12)(cid:12) r → = dρ + ρ (cid:2) ( d (Ψ /W ) + cos θdφ ) + dθ + sin θdφ (cid:3) , (3.14)where we have introduced the radial coordinate ρ = 2 R η W r and the angular coordinateΨ = 2 η/R η , whose periodicity is 4 π . Then, we see that near r = 0 the GH metric is that ofthe orbifold E / Z W . This conical singularity, however, is not present in the ten-dimensionalmetric (3.1) as long as ˜ q (cid:54) = 0, in which case the conformal factor behaves as Z | r → ∼ ˜ q /ρ and we are left with the metric of the lens space S / Z W , which is perfectly regular.The four-dimensional geometry (in modified Einstein frame) that one obtains when com-pactifying the solution over the internal manifold T × S z × S η is the following ds = ( Z Z + Z − H ) − / dt − ( Z Z + Z − H ) / (cid:0) dr + r d Ω (cid:1) , (3.15)and describes an extremal black hole with four charges: Q , Q + , Q − and Q H = W . Theadditional charge with respect to the five-dimensional case, Q H , is the magnetic charge ofthe Kaluza-Klein vector associated to the compactification over the isometric direction ofthe GH space. Therefore, the system described by this black-hole solution contains, apartfrom extended objects present in the five-dimensional case, a KK monopole. The relationbetween the parameters q i and the number of fundamental objects is now given by q − = g s α (cid:48) w R η , q = α (cid:48) N R η , q + = g s α (cid:48) n R z R η , q H = R η W . (3.16)The horizon of these black holes is again at r = 0 and the Bekenstein-Hawking entropy interms of the source parameters and the charges reads S BH4d = 2 π √ nwN W = 2 π (cid:112) Q + Q − Q Q H , (3.17)where we have made use of eq. (3.16). The first-order α (cid:48) corrections to the black holes presented in the previous section have beencomputed in an analytic fashion in [42–44]. Let us summarize here the main results of thesepapers. The reason for the modification with respect to the expressions in the five-dimensional solution is thatthe stringy objects are now smeared over the transverse direction η which forms part of the GH space.
12 first result is that the corrected solutions have the same form as the zeroth-orderexpressions given in eqs. (3.1)-(3.2). This fact can be interpreted as a consequence of super-symmetry, which strongly constrains the form of the field configuration. The functions Z + and Z receive α (cid:48) corrections, while Z − and H remain unmodified. In the five-dimensionalcase, the corrections to these functions take the following form Z + = 1 + ˜ q + ρ + α (cid:48) q + ( ρ + ˜ q + ˜ q − )˜ q ( ρ + ˜ q ) ( ρ + ˜ q − ) + O ( α (cid:48) ) , Z = 1 + ˜ q ρ − α (cid:48) ρ + 2˜ q ( ρ + ˜ q ) + O ( α (cid:48) ) , (3.18)whereas in the four-dimensional case it was found that they are given by Z + = 1 + q + r + α (cid:48) q + q H q r + r ( q + q − + q H ) + q H q + q H q − + q q − ( r + q H ) ( r + q ) ( r + q − ) + O ( α (cid:48) ) , Z = 1 + q r − α (cid:48) [ F ( r ; q ) + F ( r ; q H )] + O ( α (cid:48) ) , (3.19)where F ( r ; q ) ≡ ( r + q H ) ( r + 2 q ) + q q H ( r + q H ) ( r + q ) . (3.20)On the one hand, the new terms in the functions Z and Z + are everywhere finite, whichimplies that the α (cid:48) corrections do not change neither the location of the horizon nor thenear-horizon geometry. This is consequence of the fact that R ( − ) ab vanishes in these limits,so at this order the equations of motion remain uncorrected in that region. On the otherhand, as we describe below, the value of the charges is modified.Before doing so, it is worth noticing a subtle point about the perturbative constructionof the solution. As we constructed it, the near-horizon geometry is entirely determined bythe choice of boundary conditions, in terms of delta functions, when solving the equations ofmotion. This is convenient in the case at hands, as these functions have a direct interpretationin the microscopic theory in terms of localized sources of fundamental objects. Still, it wouldbe possible to follow an alternative approach and solve the equations of motion fixing theboundary conditions in terms of asymptotic properties of the solution. With that choice, onewould keep the charges fixed, but the near-horizon geometry and the localized sources wouldbe modified. Since these sources are in correspondence with the microscopic interpretationof the solution, this alternative approach is inconvenient if one wants to study how a given13tring theory system behaves when α (cid:48) corrections are incorporated . This is the reason whywe fix the boundary conditions at r = 0.We begin now the discussion about the charges that receive corrections. First, we startnoting that in presence of higher-curvature terms the definition is not unique. Consider theBianchi identity (2.6). In presence of external sources, it is modified as dH − α (cid:48) R ( − ) ab ∧ R ( − ) ba = (cid:63)J S , (3.21)where J S is a six-form current satisfying the conservation law d (cid:63) J S = 0, which followsfrom the well-known fact that R ( − ) ab ∧ R ( − ) ba = d Ω L( − ) . In the case at hands, this current isproduced by a stack of S5-branes and its integral over the transverse space to the S5-branes, M , is proportional to the number of S5-branes, N : (cid:90) M (cid:18) dH − α (cid:48) R ( − ) ab ∧ R ( − ) ba (cid:19) = (cid:90) M (cid:63)J S = 4 π α (cid:48) N . (3.22)Therefore, the relation between the harmonic poles (˜ q and q ) and N can be obtained afterevaluation of the left-hand side of this equation. As the sources are fixed, one obtains thesame result than in previous section [42, 44],˜ q = α (cid:48) N , q = α (cid:48) N R η . (3.23)The number of S5-branes N coincides with the notion of brane-source charge of ref. [60].It is worth stressing that the notion of brane-source charge that we have just defined isconserved. This is different from what happens in other scenarios, like in type II, where theChern-Simons terms appearing in the Bianchi identities of the RR field strengths are notnecessarily closed forms if one allows for external sources.In addition, one can define a second notion of charge, which is usually called the Maxwellcharge. Unless otherwise stated, this is the notion that we use in the article when we talkabout charges — as opposed to sources . In the S5-brane case, it is denoted by Q and it isgiven by Q = 14 π α (cid:48) (cid:90) ∂ M H . (3.24) A clear example of the importance of this observation can be found in the study of isolated KK monopolesolution, which dates back to the 90 (cid:48) s [59], as emphasized in section 4.1. This remark plays a central role inour discussion of small black holes.
14s the brane-source charge, the Maxwell charge is also conserved and gauge-invariant. Themain difference between these two charges lies in the fact that the brane-source chargeis localized while the Maxwell charge is not, as it gets contributions from the quadratic-curvature corrections, which behave as effective delocalized sources of S5-brane charge in theBianchi identity. Evaluating (3.24) and using (3.23), in the five-dimensional case we obtain Q = ˜ q − α (cid:48) α (cid:48) = N − , (3.25)while in the four-dimensional case, Q = 2 R η α (cid:48) (cid:18) q − α (cid:48) q H (cid:19) = N − W . (3.26)We observe that, up to a normalization constant, the Maxwell charge can be read from theasymptotic expansion of the function Z , with Z = 1 + c Q /r + O ( r − ) for large r . Hence,we see that this also gives the charge carried by the lower-dimensional vectors, as defined inprevious subsection.Let us now discuss the momentum charge. Analogously to the S5-brane case, the coef-ficients ˜ q + and q + —which control the leading (divergent) term of the function Z + in thenear-horizon limit— are related to the momentum n exactly as in the zeroth-order solution,see eqs. (3.8) and (3.16). The α (cid:48) corrections to this function give a contribution to theasymptotic charge carried by the lower-dimensional Kaluza-Klein vector, which can be readoff from the asympotic expansion of Z + , which is Z + = 1 + c + Q + /r + O ( r − ), with: Q + = n (cid:18) N (cid:19) , (3.27)in the five-dimensional case and Q + = n (cid:18) N W (cid:19) , (3.28)in the four-dimensional case. An interesting task for the future would be to investigate howthese two different notions of Kaluza-Klein momentum charge appear when compactifyingthe α (cid:48) -corrected action (2.3) on a circle [61, 62]. The remaining charges do not receivecorrections, hence Q − = w , Q H = W , (3.29)as in the zeroth-order solution. 15t is possible to compute the entropy of these black holes using directly Wald’s formula.The presence of the Riemann curvature tensor in the field strength H makes this a subtleproblem, which has been addressed in previous literature [32, 63]. For the family of blackholes we are interested in, the result was obtained in [47]. For the five- and four-dimensionalsolutions it was found, respectively S W5d = 2 π √ nwN (cid:18) N (cid:19) = 2 π (cid:112) Q + Q − ( Q + 3) , (3.30) S W4d = 2 π √ nwN W (cid:18) N W (cid:19) = 2 π (cid:112) Q + Q − ( Q Q H + 4) . (3.31)The expressions in terms of the sources coincide for both kind of solutions when W = 1,which is consequence of the fact that their near-horizon limit is identical, i.e. AdS × S × T .In other words, it is not possible to distinguish a 4 d black hole with Q H = W = 1 froma 5 d black hole if only the near-horizon fields are obtained. Indeed, the distinction is onlypossible if information beyond the near-horizon region is somehow taken into account. Inthe presence of a general KK monopole of charge W , one can write in convenient coordinates ds = ρ g s α (cid:48) w du (cid:20) dt − g s α (cid:48) nR z ρ du (cid:21) − α (cid:48) N W (cid:20) dρ ρ + d Ω / Z W (cid:21) − d(cid:126)z ,e − φ = wN ,H = 1 g s α (cid:48) w ρdρ ∧ du ∧ dt + α (cid:48) N θdθ ∧ dψ ∧ dϕ . (3.32)Heterotic string theory on this background was studied in [64], where the left and rightcentral charges were determined to be c l = 6 Q − ( k + 2) , c r = 6 Q − k , with k the total (cid:92) SL (2)level for the right-movers ( k + 2 for the left-movers). Upon use of Cardy’s formula, themicroscopic entropy (to all orders in the α (cid:48) expansion in the large charge approximation)that one obtains is S C = 2 π (cid:112) Q + Q − ( k + 2) . (3.33) In general, the application of Wald’s formula to the heterotic theory gives a gauge dependent expression.This problem can be solved for the family of solutions considered, where is possible to write the actionin a covariant form after imposing symmetries and adding boundary terms. In order to deal with Chern-Simons terms, the approach takes the dual Kalb-Ramond field strength (which transforms as a tensor) asthe fundamental field, see [47]. The configuration given in (3.32) with W = 1 can be interpreted in two different manners: a near-horizonbackground of a black hole with three or four independent charges. However, each interpretation is onlyconsistent when, outside the horizon, there is respectively a five- or four-dimensional non-compact space.
16e notice that this expression matches both (3.30) and (3.31) if the level is identified withthe
AdS curvature radius in string units as k = N W (with the understanding that W = 1in the five-dimensional case). As observed in [64], consistency of the bosonic (cid:92) SU (2) CFTtheory on this background requires that its level, which was found to be k −
2, must be theproduct of W and another integer, which implies that Q is quantized. Our expressions (3.31)and (3.30) in terms of the charges also match those of [17] (identically) and the perturbativeexpansion obtained in [65], respectively. Since R ( − ) a b is zero in the near-horizon background,no corrections are expected in this region in the higher-curvature expansion, such that theexpressions (3.30) and (3.31) in terms of the sources would also be exact in the α (cid:48) expansion.Before continuing, we recall that the derivation of Wald’s formula from the first law ofblack hole mechanics assumes that all fields in the theory behave as tensors under generalcoordinate transformations, although this is only true for the metric and the dilaton (not forthe Kalb-Ramond B µν , which includes gauge and Nicolai-Townsend [66] transformations).A proof of the first law taking into account this property of the heterotic theory at first orderin α (cid:48) has been only recently found [67, 68], obtaining a manifestly gauge invariant generalentropy formula. When applied to the solutions at hand, the result reproduces (3.30), (3.31).In the following section we derive these expressions for the entropies and the chargesusing a near-horizon approach. Some aspects of the α (cid:48) -corrected black holes we have just presented have been previouslystudied in [32, 35, 36, 69] making use of the entropy function formalism developed by Sen etal. [17, 69], which provides a useful method to find the near-horizon geometry and the entropyof extremal black holes. The aim of this subsection is to review this formalism and checkits compatibility with the results presented in previous subsections. From a ten-dimensionalperspective, the near-horizon geometry of both types of black holes is essentially the same:AdS × S × T in the five-dimensional case and AdS × S / Z W × T in the four-dimensionalcase. Therefore, for most of the discussion it is enough to study the near-horizon geometryof the four-dimensional black holes. The five-dimensional one will be carefully recoveredsetting W = 1 and taking into account the implications of having one additional non-compact coordinate in the asymptotic space, such that one obtains (3.25) for the S5-branecharge instead of (3.26). As a warm-up exercise, let us first consider the leading-order computation, ignoring for thetime being the α (cid:48) corrections. In this approximation, the effective action is that of ten-dimensional N = 1 supergravity compactified on T × S z × S η . The compactification on the17rivial T is straightforward and yields S = g s πG (6)N (cid:90) d ˆ x (cid:112) | ˆ g | e − φ (cid:18) ˆ R − ∂φ ) + 112 H (cid:19) , (3.34)where ˆ R is the Ricci scalar of ˆ g ˆ µ ˆ ν , the six-dimensional metric, and ˆ x ˆ µ = { x µ , z, η } , with µ = { , , , } , denote the coordinates of the six-dimensional spacetime. Further compacti-fication on z and η yields the STU model of N = 2 , d = 4 supergravity S = g s πG (4)N (cid:90) d x (cid:112) | g | s (cid:18) R − a ij ∂ µ φ i ∂ µ φ j − t F (1)2 − u F (2)2 − u s F (3)2 − t s F (4)2 (cid:19) , (3.35)where φ i = { s, t, u } are the three scalar fields present in this model. The relation betweenthe lower- and the higher-dimensional fields is g µν = ˆ g µν − ˆ g µz ˆ g νz ˆ g zz − ˆ g µη ˆ g νη ˆ g ηη ,A (1) = − ˆ g µz g zz , A (2) = − ˆ g µη g ηη , A (3) = ˜ B zµ , A (4) = ˜ B ηµ ,s = e − φ (cid:112) ˆ g zz ˆ g ηη , t = (cid:112) | ˆ g zz | , u = (cid:113) | ˆ g ηη | , (3.36)where ˜ B ˆ µ ˆ ν is the dual of the Kalb-Ramond 2-form B ˆ µ ˆ ν , defined as d ˜ B = ˜ H ≡ e − φ (cid:63) H . (3.37) Ansatz for the near-horizon geometry . We shall restrict ourselves to study the near-horizon geometry of static, extremal, spherically-symmetric black holes assuming that notonly the metric but all fields are invariant under the SO(2 , × SO(3) isometry group. Themost general ansatz consistent with this symmetry and the four type of charges we want todescribe is 18 s = v (cid:18) r dt − dr r (cid:19) − v (cid:0) dθ + sin θdφ (cid:1) ,F (1) = e dr ∧ dt , F (2) = p sin θdθ ∧ dφ , F (3) = e dr ∧ dt , F (4) = p sin θdθ ∧ dφ ,s = u s , t = u t , u = u u , (3.38)where v , v , e , e , p , p and (cid:126)u ≡ ( u s , u t , u u ) are constants. The election of electric or mag-netic character of F ( i ) is motivated by the stringy interpretation of the solution.The above configuration can be straightforwardly uplifted to six dimensions by makinguse of eqs. (3.36). We obtain: d ˆ s = v (cid:18) r dt − dr r (cid:19) − v (cid:0) dθ + sin θdφ (cid:1) − u t ( dz − e rdt ) − u u ( dη + 2 p cos θdφ ) , ˜ H = 2 e dt ∧ dr ∧ dz + 2 p sin θdθ ∧ dη ∧ dφ ,e φ = u t u u u s . (3.39) Extremization of the entropy function . Following [63], we define the function f ( v , v , (cid:126)u, e i , p i )as the integral over the angular coordinates of the (four-dimensional) Lagrangian evaluatedon the ansatz (3.38): f ( v , v , (cid:126)u, e i , p i ) ≡ (cid:90) dθdφ ( (cid:112) | g |L ) | (3.38) . (3.40)It can be shown that the metric and scalar equations of motion reduce to the extrem-ization of the function f with respect to v , v and (cid:126)u , while the equations of motion of thevector fields and the Bianchi identities are trivially satisfied for this ansatz. Hence, theextremization of the function f gives five equations which fix v , v and (cid:126)u in terms of theelectric and magnetic charges of the black hole. The latter are defined as Q I = G (4)N g s (cid:90) dθdφ δδF ( I ) rt ( (cid:112) | g |L ) , P I = 14 π (cid:90) dθdφ F ( I ) θφ . (3.41)Then, Q I = G (4)N g s ∂f∂e I , and P I = p I . (3.42)19et us now define the entropy function E as the Legendre transform of f , E ( v , v , (cid:126)u, Q I , P I ) = 2 π (cid:32) g s G (4)N Q I e I − f ( v , v , (cid:126)u, e I , P I ) (cid:33) , (3.43)where the parameters e I should be regarded as functions of the electric charges, e = e ( Q ).The entropy function evaluated at the extremum of f gives Bekenstein-Hawking entropy asa function of the electric and magnetic charges [63], S BH ( Q, P ) = E ( v , ext ( Q, P ) , v , ext ( Q, P ) , (cid:126)u ext ( Q, P ) , Q, P ) . (3.44)In the case at hands, the function f is found to be equal to f ( v , v , (cid:126)u, e i , P i ) = g s G (4)N (cid:20) u s (cid:18) e u t v v − p u u v v + v − v (cid:19) + e u u v − p u t v u s v v (cid:21) , (3.45)and it has an extremum at v , ext = v , ext = 4 Q P , e , ext = (cid:115) Q P P Q , e , ext = (cid:115) Q P P Q ,(cid:126)u ext = (cid:32)(cid:115) Q P Q P , (cid:114) Q P , (cid:114) Q P (cid:33) . (3.46)Substituting the values of v , v and (cid:126)u at the extremum of the entropy function in the six-dimensional ansatz (3.39) yields d ˆ s = 4 Q P (cid:18) r dt − dr r − dθ − sin θdφ (cid:19) − Q P (cid:32) dz − (cid:115) Q P P Q rdt (cid:33) − Q P ( dη + 2 P cos θdφ ) , ˜ H = 2 (cid:115) Q P P Q dt ∧ dr ∧ dz + 2 P sin θdθ ∧ dη ∧ dφ ,e φ = Q P . (3.47)20e can now make a comparison with the near-horizon limit of the solutions studied in theprevious subsections to extract the relation between the electric and magnetic charges ( Q, P )and the source parameters: Q = α (cid:48) n R z R η , Q = α (cid:48) N R η , P = W R η , P = α (cid:48) w R η . (3.48)Plugging these values back into (3.47), d ˆ s = α (cid:48) N W (cid:18) r dt − dr r − dθ − sin θdφ (cid:19) − α (cid:48) nR z w (cid:32) dz − R z (cid:114) wN Wn rdt (cid:33) − α (cid:48) NR η W (cid:18) dη + W R η θdφ (cid:19) , ˜ H = α (cid:48) R z (cid:114) nwWN dt ∧ dr ∧ dz + α (cid:48) w R η sin θdθ ∧ dη ∧ dφ ,e φ = Nw , (3.49)that matches (3.32) after a coordinate redefinition. Let us note an interesting property ofthe near-horizon limit, which is that the 3-form field strength is selfdual (with respect tothe orientation (cid:15) trθφηz = +1) in six dimensions, namely ˜ H = (cid:63) ˜ H = e − φ H . This, as we willdiscuss in section 6, is directly related to supersymmetry.Finally, the entropy is obtained by evaluating E at the extremum. The function f vanishesthere, and we simply have S BH ( Q , Q , P , P ) = 2 πg s G (4)N ( Q e + Q e ) | ext = 4 πg s G (4)N (cid:112) Q Q P P = 2 π √ nwN W , (3.50)in agreement with eq. (3.17). α (cid:48) corrections Rewriting of the α (cid:48) -corrected action . Let us now take into account the α (cid:48) corrections to thesupergravity action (2.3). Since the trivial T plays absolutely no role in the discussion, wecan directly work in six dimensions after integrating over the internal directions associatedto the torus, 21 = g s πG (6)N (cid:90) d x (cid:112) | g | e − φ (cid:18) R − ∂φ ) + 112 H + α (cid:48) R ( − ) µνρσ R ( − ) µνρσ (cid:19) , (3.51)where G (6)N = G (10)N (2 π(cid:96) s ) − . It is well-known that the Chern-Simons term in the localdefinition of H , eq. (2.4), hampers the application of the entropy function formalism, asthis field depends on the curvature. Fortunately, at least in the cases of interest to us, it ispossible to deal with this problem, see for instance [17, 32, 35, 47]. Let us note, nevertheless,that the application of Wald’s formula to theories that contain fields that do not transformas tensors is not justified in terms of the first law of thermodynamics, and it would beinteresting to develop a more rigorous treatment of the entropy function formalism in thelight of Refs. [67, 68]. The first step is to rewrite the action in terms of the dual 2-form ˜ B defined in (3.37). To achieve this purpose, we add the following total derivative to the action(3.51) ˜ S = S − g s πG (6)N (cid:90) (cid:18) H − α (cid:48) L( − ) (cid:19) ∧ ˜ H . (3.52)The variation of the action with respect to ˜ B gives the Bianchi identity of H , and thevariation with respect to H gives (3.37), which can be used to eliminate H in terms of ˜ H everywhere. As a result, the dependence on the Riemann tensor has been made explicit,although now we have to deal with the non-covariant form of the Lagrangian. The resultingaction can be split in three contributions:˜ S = (cid:90) d x (cid:112) | g | ( ˜ L + ˜ L + ˜ L ) , (3.53)where ˜ L = g s πG (6)N (cid:20) e − φ (cid:0) R − ∂φ ) (cid:1) + e φ
12 ˜ H (cid:21) , ˜ L = g s πG (6)N e − φ α (cid:48) R ( − ) µνρσ R ( − ) µνρσ , ˜ L = g s πG (6)N α (cid:48) (cid:15) µ µ µ ν ν ν (3!) (cid:112) | g | Ω L( − ) µ µ µ ˜ H ν ν ν . (3.54)The only contribution that it is not manifestly covariant is the last one, ˜ L , but under someassumptions a total derivative can be added to recast it in a manifestly covariant form. Wedenote the resulting Lagrangian as (cid:112) | g | ˘ L = (cid:112) | g | ˜ L + total derivative. It will be the sum22f two contributions ˘ L = ˘ L (cid:48) + ˘ L (cid:48)(cid:48) , corresponding to the following split of the Chern-Simons3-form Ω L( − ) = A + Ω L , (3.55)where A = 12 d (cid:0) ω ab ∧ H ba (cid:1) + 14 H ab ∧ DH ba − R ab ∧ H ba + 12 H ab ∧ H bc ∧ H ca . (3.56)and Ω L is the Chern-Simons 3-form associated the Levi-Civita spin connection ω ab . The firstcontribution ˘ L (cid:48) is obtained from the first term in (3.55), after adding a total derivative thatcancells the one in (3.56), namely (cid:112) | g | ˘ L (cid:48) = g s πG (6)N α (cid:48) (cid:15) µ µ µ ν ν ν (3!) ˜ A µ µ µ ˜ H ν ν ν , (3.57)with ˜ A = A − d (cid:0) ω ab ∧ H ba (cid:1) .We are left with the second contribution due to Ω L . For this we can use that, in thefamily of solutions considered, the six-dimensional metric (3.39) is the sum of two three-dimensional metrics —parametrized by the coordinates { t, r, z } and { θ, φ, η } respectively—and that also ˜ H is the sum of two contributions according to this splitting of the metric.Then, we have (cid:112) | g | ˜ L (cid:48)(cid:48) = g s πG (6)N α (cid:48) (cid:15) µ µ µ ν ν ν (3!) Ω L µ µ µ ˜ H ν ν ν = g s πG (6)N α (cid:48) (cid:16) Ω L θφη ˜ H trz − Ω L trz ˜ H θφz (cid:17) , (3.58)where we have chosen the orientation (cid:15) trθφηz = +1. The last information we need in orderto write this in a manifestly covariant form is that for three-dimensional metrics admittinga spacelike isometry, ds = λ [ h αβ dx α dx β − ( dx (cid:93) + V α dx α ) ] , α, β = 1 , . (3.59)the Chern-Simons 3-form Ω L is given by [70]Ω L12 (cid:93) = R ( dV ) − ( dV ) ( dV ) ( dV ) + ∂ [1 V , (3.60)23here R is the Ricci scalar of the two-dimensional metric h αβ and V is a certain 1-formwhich involves the spin-connection associated to h αβ . Again, (cid:112) | g | ˘ L (cid:48)(cid:48) is obtained by addinga total derivative to the action that cancels the last term in (3.60). Corrections to the entropy function . The most convenient way to find the corrections tothe entropy function is to evaluate the six-dimensional Lagrangian on the ansatz (3.39), f ( v , v , (cid:126)u, e i , p i ) = (cid:90) dzdηdθdφ (cid:104)(cid:112) | g | (cid:16) ˜ L + ˜ L + ˘ L (cid:48) + ˘ L (cid:48)(cid:48) (cid:17)(cid:105) (3.39) . (3.61)Therefore, the function f will be now the sum of four contributions, f = f + f + f (cid:48) + f (cid:48)(cid:48) :1. The first contribution is exactly the same as the one we obtained in (3.45).2. The second contribution will not be displayed since we do not need it to computethe first-order corrections. This is due to the fact that the curvature tensor R ( − ) µνρσ vanishes when evaluated at the extremum (3.46). Then, this contribution must be atleast of second order in α (cid:48) , so we can simply ignore it.3. The third contribution (3.57) is f (cid:48) = g s α (cid:48) G (4)N (cid:20) u t p u s v (cid:0) u t p + u s (cid:0) u t e − v (cid:1)(cid:1) + u u e u s v (cid:0) u u e + u s (cid:0) u u p − v (cid:1)(cid:1)(cid:21) . (3.62)4. Finally, the last contribution (3.58) is f (cid:48)(cid:48) = g s α (cid:48) G (4)N (cid:20) e p u u (2 p u u − v ) v − e p u t ( v − e u t ) v (cid:21) . (3.63)It is straightforward to check that (3.49) is also an extremum of the corrected entropyfunction, as expected. However, the relation between the electric charges carried by thelower-dimensional vector fields and the source parameters is no longer the one we found atzeroth order in α (cid:48) , eq. (3.48). Now, taking into account the corrections to f , we find Q = G (4)N g s ∂f∂e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ext = α (cid:48) R z R η n (cid:18) N W (cid:19) ,Q = G (4)N g s ∂f∂e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ext = α (cid:48) R η (cid:18) N − W (cid:19) , (3.64)which agree with the value of the Maxwell charges obtained in the previous subsection.Finally, evaluating the corrected entropy function E at the extremum, we get Wald’s entropy24 W4d = 2 π √ nwN W (cid:18) N W (cid:19) = 2 π (cid:112) Q + Q − ( Q Q H + 4) , (3.65)as previously reported in [32, 35, 47]. Five-dimensional three-charge black holes . The above steps can be repeated to obtainthis near-horizon solution after the following modifications are taken into account. In firstplace, since there are now less independent parameters, the appropriate ansatz is d ˆ s = v (cid:18) r dt − dr r (cid:19) − v (cid:0) dθ + sin θdφ (cid:1) − u t ( dz − e rdt ) − v ( dψ + cos θdφ ) , ˜ H = 2 e dt ∧ dr ∧ dz + 2 p sin θdθ ∧ dη ∧ dφ ,e φ = u t √ v u s . (3.66)Additionally, the first term in the right-hand-side of (3.58) needs to be set to zero. Thereason is the following. First, we notice that this term only depends on spatial componentsof the Riemann tensor, hence it cannot play a role in the computation of the entropy fromWald’s formula. Second, the Chern-Simons 3-form of a 3-sphere is zero when evaluated usingthe Christoffel symbols, while reduces to a total derivative when evaluated using the spinconnection. Hence, the inclusion of this term depends on the boundary conditions of theconfiguration. This leads us to the third and last consideration; from the structure of (3.58),it is clear that this term has the interpretation of magnetic source of the Kalb-Ramond field(or electric source of ˜ H ) produced by the geometry of the Gibbons-Hawking space. In thefour-dimensional solution, this term is responsible for a factor of − /W in the screening ofthe S5-brane charge (the other − /W comes from (3.57)) produced by the KK gravitationalinstanton —more details about this are given in the following section. Since in the five-dimensional solution the KK instanton number is zero, there can be no contribution fromthis term.Once these observations are considered, it is straightforward to check that (3.49) with W = 1 (which in this case does not have the physical interpretation of a charge, just like itdoes not indicate the presence of a KK monopole) gives again an extremum of the entropyfunction. Its evaluation gives S W5d = 2 π √ nwN (cid:18) N (cid:19) = 2 π (cid:112) Q + Q − ( Q + 3) . (3.67)For the charges, one gets 25 = α (cid:48) R z n (cid:18) N (cid:19) , Q = α (cid:48) N − , P = α (cid:48) w . (3.68) In the previous section we have described two families of regular black-hole solutions, withfour and five non-compact dimensions respectively. Before describing the singular case ofsmall black holes made by strings and momentum, it is convenient to study first the systemformed by Kaluza-Klein monopoles and solitonic 5-branes.At zeroth order, the unit charge Kaluza-Klein monopole is a well-known solution of stringtheory in which all fields are trivial except for the metric, that reads ds = dt − dz α dz α − H − ( dη + χ ) − H (cid:0) dr + r d Ω (cid:1) , H = 1 + R η r , χ = R η θdϕ , η ∼ η + 2 πR η . (4.1)It is straightforward to check that this solution can be obtained from the family consideredin section 3.1 setting n = w = N = 0, W = 1. On the other hand, at first sight it mightnot be obvious that the α (cid:48) -corrected Kaluza-Klein monopole is not automatically obtainedperforming the same operation on the solution described in section 3.2. There are severalreasons why such procedure fails. In first place, it is unclear how to treat the n/N → / α (cid:48) -correction to Z + , see (3.19). More importantly,the term F ( r ; q ) in Z collapses into a harmonic pole that causes, among other effects, adivergence in the dilaton. On the other hand, the direct computation of the corrections tothe original background (4.1) gives a regular configuration ds = dt − dz α dz α − Z (cid:2) H − ( dη + χ ) + H (cid:0) dr + r d Ω (cid:1)(cid:3) ,e φ = e φ ∞ Z , H = (cid:63) σ d Z , with Z = 1 − α (cid:48) F ( r ; R η ) , (4.2)with H and χ unchanged. Recall that F ( r ; q ) = ( r + q H ) ( r + 2 q ) + q q H ( r + q H ) ( r + q ) , q H = R η W . (4.3)26 relevant property of the α (cid:48) -corrected heterotic KK monopole is that it carries -1 unitsof solitonic 5-brane charge, as defined in (3.24). This observation dates back to [59], thatarrived to this conclusion without explicitly finding (4.2), but using the fact that the KKmonopole is a gravitational instanton with unit instanton number. The argument goes asfollows. The Kalb-Ramond Bianchi identity has the form dH = α (cid:48) R ( − ) a b ∧ R ( − ) b a . (4.4)In absence of matter fields at zeroth order, the right hand side is proportional to the Pon-tryagin density. Hence, upon integrating this equation over a four-dimensional Riemannianspace, we obtain that the magnetic charge carried by H is proportional to the gravitationalinstanton number. Working out the details, the aforementioned factor of − r →∞ Z = 1 − α (cid:48) r + . . . .As emphasized by Sen in [59], the fact that the KK monopole carries this S5 charge isa necessary condition for the consistency of S-duality of heterotic string theory. Thus, inorder to properly understand the corrections to this string theory system, the perturbativesolution must be constructed fixing the sources at r = 0, while the asymptotic charges areallowed to vary. Observe that it is the asymptotic Maxwell charge of the Kalb-Ramondfield strength the one that contains the information about the microscopic S5 charge (the S5brane source charge vanishes for this configuration, N = 0). Additionally, one notices thatthe truncation of sources directly in α (cid:48) -corrected solutions can produce a wrong answer; hadwe simply set n = w = N = 0, W = 1 directly in the general corrected solution of section3.1, we would not have obtained the appropriate value of S5 charge.The previous discussion extends straighforwardly to a multicenter configuration of KKmonopoles. In (4.1) we can use a multicenter harmonic function, H = 1 + (cid:80) mi =1 R η r i , where r i represents now the three-dimensional Euclidean distance measured from some point (cid:126)x i ,interpreted as the location of a monopole. Likewise, χ must be appropriately modified . Itis well-known that the resulting space is a regular gravitational instanton, with instantonnumber given by the number of poles of the harmonic function, m . From the previousargument, one concludes that the multicenter configuration carries − m units of S5 charge.The backreacted solution is still of the form of (4.2), with the already mentioned multicenterexpressions for H , χ and with Z = 1 − α (cid:48) (cid:80) mi =1 F ( r i ; R η ).The most general configuration is that of multicenter KK monopoles, each with genericcharge. So far in this section, we have restricted to unit charge monopoles by setting thecoefficient of all harmonic poles to R η /
2. Together with the fact that the coordinate η hasperiod 2 πR η , this ensures that the metric is locally flat at the centers (cid:126)x i . On the contrary,a monopole with general charge is obtained if the coefficient is W i R η /
2, with W i a positiveinteger. The resulting space presents conical singularities at the centers whenever the chargeis larger than one, as described after (3.14). In what follows, we offer a detailed computation Its expression is not important for our discussion and can be readily found in the literature.
27f the instanton number when these defects are present. The gravitational instanton numberis defined as n = − π (cid:90) M R a b ∧ R b a , (4.5)where M denotes the four extended dimensions where the metric is non-trivial. In fact,since the solution is purely four-dimensional and the instanton number is independent ofconformal rescalings of the metric, we can just evaluate the integral above in the metric ds M = H − ( dη + χ ) + H (cid:0) dr + r d Ω (cid:1) . (4.6)For simplicity, let us perform the computation in the case of one center, so that H = 1 + Wr , χ = W cos θdϕ , W = 1 , , . . . , (4.7)where we are setting units such that R η = 2. Now, the curvature tensor is self-dual, theinstanton number can be expressed as n = 132 π (cid:90) M d x √ g X , (4.8)where X = R µνρσ R µνρσ − R µν R µν + R is the Gauss-Bonnet density. If M were a manifold,this would be nothing but its Euler characteristic, but in our case one has to be careful withthis interpretation due to the presence of a conic defect at r = 0. In order to perform thecomputation we may first split M in two regions r > r and r < r , that we may call M ∞ and M , respectively. The instanton number is then the sum of “Euler characteristics” n = X ( M ) + X ( M ∞ ) , (4.9)where now, since each part M p has a boundary, we have to take into account the boundaryterms: X ( M p ) = 132 π (cid:90) M p d x √ g X + 316 π (cid:90) ∂ M p d x √ h (cid:18) K [ i [ i R jk ] jk ] − K [ i [ i K jj K k ] k ] (cid:19) . (4.10)Here h ij is the induced metric on the boundary r = r , R is the intrinsic curvature and K is the extrinsic curvature, defined as The self-duality of the Riemann tensor also implies that the vanishing of the Ricci tensor, R µν = 0. ij = 12 L n h ij , (4.11)where L n is the Lie derivative with respect to the normal vector n . Since the normal vectorsto M and M ∞ are opposite, it is obvious that in eq. (4.9) the boundary terms cancel outand one is left with the integration in the whole volume, hence recovering eq. (4.8). Theevaluation of X ( M ∞ ) is straightforward and it yields X ( M ∞ ) = − W (4 r + W )( r + W ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ r − W (4 r + W )( r + W ) = 0 . (4.12)This actually follows from the fact that M ∞ is topologically S × S × [0 ,
1) and from thefactorization property of the Euler characteristic.Let us now consider M . We already mentioned that near r = 0 the KK monopolebecomes the orbifold E / Z W . Therefore, M is is topologically B / Z W , this is, a 1 /W sliceof the unit ball in E , with the sides identified as illustrated in Fig. 1. One can then applyeq. (4.10) to this space in order to compute X ( M ). Notice that, if the sides were notidentified, one would need to take them into account in the boundary integral and therewould be additional contributions coming from the vertices, so that the result would be 1, i.e. , the Euler characteristic of any simply-connected open set in E . However, once theyare identified they do not form part of the boundary and, for the same reason, there are nocontributions from any of the vertices. Let us also stress that the curvature of this space isidentically zero at all points, so that no bulk contribution can come from the cone at r = 0as well. Thus, the only contribution to eq. (4.10) comes from the boundary at r . It is clearthat adding up W times that result one would get the corresponding value for the Eulercharacteristic of the disc, which is 1. Therefore, we conclude that n = X ( M ) = 1 W . (4.13)Note that this is, in fact, the orbifold
Euler characteristic of E / Z W . Orbifold Euler numbersare naturally rational, and it has been known for long that the Gauss-Bonnet formula appliedto orbifolds gives this result rather than the standard Euler characteristic [71]. Therefore,our fractional result for the instanton number of the higher-charge KK monopole should notcome as a surprise.This result can be straightforwardly generalized to an arbitrary number of centers, inwhich case each center contributes as before and we get n = m (cid:88) i =1 W i . (4.14)Note that, once again, this is the orbifold Euler characteristic of the multicenter KK monopole.29 π W Figure 1: Orbifold B / Z W . This is a slice of the unit ball in E where the sides (red dashedlines) are identified. The boundary is only composed of the arc of the circumference r = 1(red solid line).While not a surprise, the fact that the result is a fractional number might feel uncom-fortable when thinking about charge quantization. Additionally, there seems to exist a quiteextended expectation that the charges carried by m unit charge KK monopoles should bethe same than those carried by one monopole of charge m . While obviously the KK chargescoincide, the former has ( − m ) S5 charge while the latter has ( − /m ) if we use (4.14). Thiscould lead to the proposal that, in the presence of conical singularities, the S5 charge shouldnot simply be the instanton number, but an additional contribution should be added. Suchputative term should amount to ( W i − /W i ) for every defect, such that the S5 charge isalways minus the KK charge. We do not know of any argument supporting the introductionof such term, and hence we will not do it here.On the contrary, there are hints that the S5 charge might not necessarily be given in allcases by − m . On one side, let us note that the fractional KK contribution to the S5 chargeis crucial in order to obtain the correct value for the black hole entropy in (3.30), and in thatcase there is no conical singularity whatsoever. Besides this, we can find another possibleargument in the study of the moduli space of the effective worldvolume theory of heteroticKaluza-Klein monopoles, which was argued in [72] to be that of BPS monopoles in a SU (2)gauge theory . The claim strongly relies on the fact that, after taking into account higher-curvature corrections, a collection of m separated, unit charge monopoles has ( m, − m ) KKand S5 charges, respectively. This is a unique feature of the heterotic theory. The conicalsingularity that appears in (4.2) when several KK monopoles coincide is well understood in This is our personal perception of the issue, after having discussed about it with a respectable numberof researchers. We do not know about any bibliographical support of this fact. As described in [72], in M and type II theories the field content determines the effective theory of theKK worldvolume theories, that correspond respectively to N = 1 ( U (1) m ) gauge theories (M and type IIA)and N = (2 ,
0) tensor multiplet (type IIB).
30 and type IIA theories, where it produces an enhancement of the gauge symmetry groupof the worldvolume theory, as well as in type IIB, where tensionless strings appear [72]. Onthe other hand, a relevant property of the moduli space of BPS monopoles is that it hasno singularities [73]. Hence, there would seem to be a contradiction between this fact andthe possibility of having higher-charge heterotic KK monopoles with conical singularities. Away out of this problem would be that the S5 charge they carry is not the same as whenthey are separated, so that higher-charge KK monopoles have different quantum numbersthan BPS monopoles.
In previous section we have shown that higher-charge KK monopoles, if alone, have a discretecharge spectrum which does not obey standard quantization rules. A plausible interpretationcould be that the corresponding solution to the equations of motion does not correspond toany actual state of string theory. Just like in classical electromagnetism, there are solutions tothe field equations which are discarded once Dirac quantization is imposed, i.e. we shall onlyconsider solutions in which the charge is an integer. In the case at hands, this implies thatwe need to add S5 branes. Remarkably, this addition also resolves the orbifold singularityand produces a regular, geodesically complete manifold. The solution has still the form givenin (4.2), now with Z = 1 + q r − α (cid:48) [ F ( r ; q ) + F ( r ; q H )] + O ( α (cid:48) ) , (4.15)where, we recall, q = α (cid:48) N R η . Contrary to the situation in the previous section, the rightsolution is obtained if we truncate n = w = 0 in the α (cid:48) -corrected black hole of section 3.2.Now, when we approach the r → ds = dt − dz α dz α − (cid:0) dβ + α (cid:48) N W d Ω / Z W (cid:1) , φ = − β √ α (cid:48) N W . (4.16)with d Ω / Z the metric on the Lens space S / Z W , which has the form of the metric on the3-sphere but its volume is only a 1 /W fraction of it. The radial coordinate has been redefinedas β = √ α (cid:48) N W log r , such that r → β → −∞ . For fixed values of β , t , z α , the geometry is that of a 3-sphere with pointsidentified under the action of a discrete Z W group that has no fixed points.The computation of the S5 charge, defined in (3.24), yields31 = N − W , (4.17)which, in light of the discussion below eq. (3.33), we assume to be an integer. In particular,this means that the localized brane source charge N can be fractional. This somewhatunusual value is a consequence of the Z W quotient performed at the sphere that surroundsthe brane. The topology of the space coincides with that of the previous subsection, wherethe conical singularity has been mapped to the asymptotic near-brane region r →
0, andhence the KK gravitational instanton screens the S5 charge with a factor of − /W . Anadditional factor of − /W comes from the introduction of the S5-brane localized sources.This latter effect, which has been mostly ignored in the literature, is a consequence of usingthe supersymmetric formulation of the heterotic theory given in [53], as described in [43].Recall that in this formulation the torsion component of the spin connection is − H µ a b dx µ .Due to the presence of this term, a stack of S5 branes produces a new gravitational instantonthat backreacts as a source in the α (cid:48) -corrected Bianchi identity. An elementary observationthat, nevertheless, must be stressed is that, once the S5 branes are included, there is nouncertainty in the computation of the S5-brane charge, as the manifold has no singularityanymore.In summary, we have seen that a non-perturbative modification of the higher-charge KKmonopole, that involves the introduction of S5 branes, allows to solve the problem of chargequantization and simultaneously resolves the conifold singularity. The most relevant effectof higher-curvature corrections is to modify the charges of the zeroth-order solution, whichmust be allowed to vary in the perturbative expansion. We have also seen that the truncationof charges in a general corrected solution may produce an incorrect result. After having discussed those solutions made up of Kaluza-Klein monopoles and solitonic5-branes, we now turn our attention into the study of small black holes and rings, consistingsolely of fundamental strings wrapping an compact direction (denoted by z ) with momentumflowing along them.Regarding small black holes, special attention has been paid to the four-dimensionalones. At leading order in the α (cid:48) expansion, they were shown to be solutions of the heteroticeffective action characterised by a singular horizon with vanishing area [22]. The inclusionof quadratic-curvature corrections was studied in detail in [45], where it was found that theydo not regularize the singular supergravity solution. A similar analysis was carried out forfive-dimensional small black rings, obtaining analogous conclusions [50]: small black rings infive dimensions are singular in the supergravity approximation and the α (cid:48) corrections do not32ure this behaviour. The aim of this section is to present a general treatment and extendthese results to any number of dimensions.Let us begin with a discussion of the heterotic backgrounds constructed in the mid-1990sin [74, 75] which describe heterotic strings carrying arbitrary right-moving momentum waves,generalizing those of [20, 76]. These solutions preserve half of the spacetime supersymmetries(see Appendix A for further details) and their form is the following ds = 2 Z − du (cid:18) dt + ω − Z + du (cid:19) − ds ( E d − ) − d(cid:126)z − d ) ,B = Z − − du ∧ ( dt + ω ) ,e − φ = e − φ ∞ Z − , (5.1)where ds ( E d − ) represents the metric of E d − and Z − =1 + q − || (cid:126)x − (cid:126)F || d − , Z + =1 + q + + q − ˙ F m ˙ F m || (cid:126)x − (cid:126)F || d − ,ω m = q − ˙ F m || (cid:126)x − (cid:126)F || d − , (5.2)where (cid:126)x ∈ E d − , q ± are constants and F m = F m ( u ) are arbitrary functions of u = t − z .Derivatives with respect to this coordinate are denoted with a dot. Finally, (cid:126)z (9 − d ) representthe coordinates over which the solution has been smeared and parametrize a torus T − d without internal dynamics. The position of the string in the non-compact directions isdetermined parametrically by (cid:126)x = (cid:126)F ( u ) . (5.3)For this solution to represent a closed string, we must demand that (cid:126)F ( u ) is periodic. Wedenote the periodicity of this function by (cid:96) , which does not necessarily coincide with theperiodicity of the compact coordinate z . Instead, we allow the function F to be multi-valuedon S z . All we demand is that the string closes after a finite number of revolutions along z .Therefore, (cid:96) = 2 πR z w , where w = 1 , , . . . represents the winding number along z .Following [77], we can further smear the solution over the compact coordinate z , whichyields the following solution 33 − =1 + (cid:90) (cid:96) q − || (cid:126)x − (cid:126)F || d − du , Z + =1 + (cid:90) (cid:96) q + + q − ˙ (cid:126)F · ˙ (cid:126)F || (cid:126)x − (cid:126)F || d − du ,ω m = (cid:90) (cid:96) q − ˙ F m || (cid:126)x − (cid:126)F || d − du , (5.4)which has no dependence on u anymore. Hence, it can be dimensionally reduced (on T − d × S z ) to d dimensions through a standard Kaluza-Klein reduction. The lower-dimensionalmetric that one obtains, in the Einstein frame, is ds d ) = ( Z + Z − ) − dd − ( dt + ω ) − ( Z + Z − ) d − ds ( E d − ) . (5.5)As we will see next, the lower-dimensional solutions can describe small black holes andrings for particular choices of (cid:126)F ( u ). Before discussing these choices, we shall make use of theresults of [50], where the first-order α (cid:48) corrections to the above class of backgrounds werecomputed. The form of the corrected solution turns out to be the same, so no other fieldcomponents are activated by the corrections. This is in fact a consequence of supersymmetry.As we show in Appendix A (see also [78]), (5.1) is the most general field configuration thatone can write down for the DH states. The curvature corrections only modify the form ofthe function Z + . Then, the corrected solution is (5.1) with Z − , Z + , ω given in terms of thezeroth-order solution (which we now denote as {Z (0) − , Z (0)+ , ω } ) by the following expressions. Z − = Z (0) − + O ( α (cid:48) ) , Z + = Z (0)+ + α (cid:48) Ω (0) mn Ω (0) mn − ∂ m Z (0)+ ∂ m Z (0) − Z (0) − + O ( α (cid:48) ) ,ω = ω (0) + O ( α (cid:48) ) , (5.6)where Ω = dω .We shall now examine the subsequent black hole and black ring solutions arising fromconsideration of two particular profile functions (cid:126)F . We start by considering a constant (cid:126)F , which corresponds to a static fundamental string.Without loss of generality, we can always set (cid:126)F = 0 after an appropriate change of coordi-34ates. Plugging this static ansatz into (5.6), we find that Z + = 1 + ˜ q + ρ d − − (3 − d ) α (cid:48) q + ˜ q − ρ d − ( ρ d − + ˜ q − ) + O (cid:0) α (cid:48) (cid:1) , Z − = 1 + ˜ q − ρ d − + O (cid:0) α (cid:48) (cid:1) , (5.7)where we have made the definitions ˜ q + = q + (cid:96) , ˜ q − = q − (cid:96) and ρ = || (cid:126)x || . The Killing vector ∂ t is timelike for positive values of ρ , becoming null in the ρ → tt componentof the metric (5.5) vanishes, thus signaling the presence of an event horizon at ρ = 0 whosearea is given by A H = ( d − π d − Γ (cid:0) d − (cid:1) (cid:112) − α (cid:48) ˜ q + ˜ q − , (5.8)We see, as anticipated, that the area of the horizon vanishes if curvature corrections areignored (setting α (cid:48) → k one gets uponcompactification of the z -coordinate takes the form k = k ∞ Z / Z d − d − − , ⇒ k ( ρ → ∼ ρ − d − d − . (5.9)Hence, it diverges at the horizon. This tells us that we cannot trust the d -dimensionalmetric (5.5) as it has been obtained through a singular dimensional reduction. This singularbehavior can also be detected directly in ten dimensions, where the divergence of the KKscalar is reflected in a divergence of the ten-dimensional Ricci scalar, whose explicit form is R = − d − (˜ q − ) ρ (˜ q − + ρ d − ) . (5.10) Now let us consider that the string has a non-trivial profile function (cid:126)F . We shall restrict, asin the previous literature (see e.g. [33, 74, 77, 79]), to circular profiles of the form Note that in order to have a regular ( d -dimensional) metric we must ensure that Z + > ρ ∈ R + ,which in turn requires q + < − b ( q − ; d ), where b ( q − ; d ) is a certain positive-definite function of q − and d whichwas determined numerically in [42] for the particular case of d = 5. Remember that the ( d + 1)-dimensional metric is expressed in the Einstein frame. = R cos (cid:18) π W u(cid:96) (cid:19) , F = R sin (cid:18) π W u(cid:96) (cid:19) , F = · · · = F d − = 0 . (5.11)Such a configuration corresponds to a string winding a 2-torus spanned by z and the polarangle ψ in the x − x plane. More concretely, this yields a helix profile for the string,which swirls around the z -direction while turning round along the circle ( x ) + ( x ) = R .This radius R can be related to the momentum carried by the string [77] and W (not tobe confused with the charge of the KK monopole, denoted by W in the previous section)represents the number of times the string is wrapped along the ψ direction.It was shown in [33] (see also [79]) that this configuration, when reduced to d > α (cid:48) expansion. We shall now investigate the effect ofthe first-order corrections on these solutions. To this aim, we first re-derive the explicitform of zeroth-order solution for the above circular profile which was presented in [33]. It isconvenient to introduce the following set { ξ, ψ, η, φ , . . . , φ d − } of new coordinates x = ξ cos ψ , x = ξ sin ψ ,x = η cos( φ ) , . . . x d − = η sin( φ ) . . . sin( φ d − ) . (5.12)After some routine computations, one finds Z (0) ± = 1 + ˜ q ± ( ξ + η + R ) d − F (cid:18) d − , d −
14 ; 1; 4 R ξ ( ξ + η + R ) (cid:19) ,ω (0) = ( d − π ˜ q − W R ξ (cid:96) ( ξ + η + R ) d − F (cid:18) d − , d + 14 ; 2; 4 R ξ ( ξ + η + R ) (cid:19) dψ , (5.13)where F ( a, b ; c ; z ) denotes the hypergeometric function and where we have defined ˜ q − ≡ q − (cid:96) and ˜ q + ≡ q + (cid:96) + 4 π W R q − /(cid:96) . Note that these results are strictly equivalent to those pre-sented at [33] after identifying their notation { f f , f p , A m } with our notation {Z (0) − , Z (0)+ , − ω m } .Regarding future manipulations, it is convenient to rewrite it in terms of the so-called ringcoordinates [80], which are defined as ξ = (cid:112) y − x − y R , η = √ − x x − y R , (5.14)and whose range is −∞ ≤ y ≤ − − ≤ x ≤
1. The metric (5.5) (of the zeroth-ordersolution) in this coordinates reads 36 s d ) = ( Z (0)+ Z (0) − ) − dd − ( dt + ω (0) ) − R ( Z (0)+ Z (0) − ) d − ( x − y ) (cid:20) dy y − y − dψ + dx − x + (1 − x ) d Ω d − (cid:21) , (5.15)where d Ω d − denotes the metric of S ( d − and Z (0) ± = 1 + ˜ q ± (cid:18) y − x R y (cid:19) d − F (cid:18) d − , d −
14 ; 1; 1 − y (cid:19) ,ω (0) = ( d − π ˜ q − W R ( y − (cid:96) ( x − y ) (cid:18) y − x R y (cid:19) d − F (cid:18) d − , d + 14 ; 2; 1 − y (cid:19) dψ . (5.16)It is not difficult to see that the norm of the Killing vector ∂ t vanishes at y → −∞ . However,this does not correspond to a regular horizon since this null hypersurface has vanishing areaand, furthermore, the curvature blows up there, exactly what one finds for the static smallblack holes discussed in the previous subsection.Let us then take into account the corrections. Given this zeroth-order solution, it isstraightforward to use (5.6) to find the corrections to Z + . Since its explicit form involveslong expressions which are not particularly illuminating, we relegate it to Appendix B, seeeq. (B.7). It suffices to know that the near-horizon behavior of the function Z + is modifiedby the α (cid:48) corrections as Z + ∼ y →−∞ | y | d − + α (cid:48) | y | d − , (5.17)while Z − ∼ y →−∞ | y | d − , ω ∼ y →−∞ | y | d − dψ . (5.18)Then, we find that the area of the would-be horizon scales as A H ∼ lim y →−∞ (cid:32) ( Z + Z − ) d − R y (cid:33) d − y ∼ (cid:112) α (cid:48) ( (cid:96) ˜ q + − π ˜ q − R W ) ∼ √ nw − J W , (5.19)where n represents the units of momentum carried by the fundamental string and J itsangular momentum. The last expression is obtained upon use of eqs. (A.4) and (A.13) of3733], which relate the parameters ˜ q ± and J with n, w and W as follows:˜ q − πG ( d ) N = Γ (cid:0) d − (cid:1) d − π d − R z wα (cid:48) , ˜ q + πG ( d ) N = Γ (cid:0) d − (cid:1) d − π d − nR z , J = R W α (cid:48) . (5.20)Note that the result (5.19) is in agreement with the scaling argument of Op. Cit. and [18].In another vein, we check that (5.19) vanishes at leading order (as we anticipated) whilereceiving a finite correction once the first-order α (cid:48) corrections are included. However, oneshould be aware of the fact that this finiteness is only a mirage, and it actually comes fromthe combination of two divergences. In order to see this explicitly, let us first carry out thechange of coordinates [81] r = − Ry , x = cos θ , (5.21)which maps the y → −∞ hypersurface to the r → + hypersurface. Using this coordinates,our metric (5.5) reads ds d ) = ( Z + Z − ) − dd − ( dt + ω ) − ( Z + Z − ) d − (1 + r cos θR ) (cid:20) (cid:18) − r R (cid:19) R dψ + dr − r R + r d Ω d − (cid:21) , (5.22)where d Ω d − = dθ + sin θd Ω d − and where it is assumed that Z + , Z − and ω are expressedin terms of the new coordinates (5.21). Surfaces of constant r have topology S × S d − , wherethe S is charted by the ψ -coordinate. By using the near-horizon behavior of Z + , Z − and ω ,we find that the radii R ψ and R d − associated to S and S d − scale near the horizon as R ψ ∼ r → + r − dd − ∼ | y | d − d − , R d − ∼ r → + r d − ∼ | y | − d − . (5.23)Consequently, the radius R d − vanishes in the horizon while R ψ diverges. However, R ψ ( R d − ) d − ,which is proportional to the area, is indeed finite in this limit, what justifies why the expres-sion for the area does not diverge.This unusual behaviour of R ψ and R d − clearly indicates that the metric (5.5) is singularat y → −∞ ( r → + ). We can additionally check the existence of such singularity bycomputing its Ricci scalar R d . Indeed, at zeroth order the Ricci scalar already divergesas R d ∼ y →−∞ | y | d − , and after including the first order-corrections such behaviour is notregularized, since we find that R d ∼ y →−∞ ( α (cid:48) ) − d − | y | d − . This signals the persistence of thesingularity, as well as the breakdown of the perturbative expansion.38 Fake small black holes
The conclusion that one extracts from the previous section is that small black holes arenot regularized by higher-curvature corrections. On the other hand, previous results inthe literature have described the existence of regular black holes with a reduced number ofcharges when the curvature corrections are included, while at zeroth order the solutions withthese reduced number of charges are singular. The possible compatibility of these seeminglycontradictory facts is studied below.
Let us recall some of the results described in section 3.2. The Wald entropy of the four-dimensional black hole in terms of the asymptotic charges has the following expression, S W4d = 2 π (cid:112) Q + Q − ( Q Q H + 4) . (6.1)Looking at this formula only, one sees that if any of the solitonic 5-brane or the Kaluza-Kleincharges is set to zero, we obtain S W4d | Q Q H =0 ? = 4 π (cid:112) Q + Q − , (6.2)whose expression coincides with the microscopic degeneracy of the DH system (1.1). If itwere possible to truncate both of the two charges in a consistent manner and, particularly,such that these expressions hold, this could be interpreted as a resolution of the horizon offour-dimensional small black holes via higher-curvature terms.However, in previous sections of the paper, we have illustrated how the truncation (oraddition) of sources (and, consequently, of charges) in a particular solution is a procedurethat needs to be handled with care. Indeed, if we remove the KK monopole from the generalfour-dimensional solution described in section 3, the result will be a singular space. Thecorrections to the functions Z and Z + —see eqs. (3.19)— diverge when the KK-monopolecharge vanishes, which tells us that this limit must be taken before computing the α (cid:48) correc-tions. Doing so, one finds that the functions that determine the solution are given by Z + = 1 + q + r − α (cid:48) q + q − r ( r + q ) ( r + q − ) + O (cid:0) α (cid:48) (cid:1) , Z = 1 + q r − α (cid:48) q r ( r + q ) + O (cid:0) α (cid:48) (cid:1) , − = 1 + q − r + O (cid:0) α (cid:48) (cid:1) , H = 1 + O (cid:0) α (cid:48) (cid:1) , (6.3)which yield singularities in the spacetime and matter fields. We notice that if one furthertruncates the S5-brane charge, which here is achieved by imposing q = 0, one recovers the(singular) solution derived in section 5.1 for the particular case of d = 4. Clearly, the formula(6.2) is not correct for the resulting configuration.On the other hand, we recall that, as outlined in section 3.2, the vanishing of the MaxwellS5-brane charge does not necessarily imply the absence of S5-branes when KK monopolesare present, as we can have contributions from the higher-curvature terms in the Bianchiidentity. Concretely, for the four-dimensional system we have that Q = 0 if the followingrelation between the sources holds, N W = 2 . (6.4)As a result, we get a black hole with a regular horizon and whose S5-brane charge is com-pletely screened . In this case, the expression for the entropy in (6.2) is correct, and itsvalue coincides with that of the DH states, 4 π √Q + Q − . However, since the KK monopolecharge is necessarily non-vanishing (otherwise, the functions would be given by (6.3)), thesolution cannot be interpreted as a small black hole with regular horizon. Indeed, it is an ordinary black hole with four type of sources which is already regular at zeroth-order in α (cid:48) , with the special property that its S5 brane charge is screened by the higher-curvaturecorrections. Additionally, we point out that the presence of S5-branes, even if its charge isscreened, influences the amount of supersymmetry preserved by these solutions, which is 1 / / S W4d = A/ G , the same relation that was found forthe solutions described in [26, 29, 30].A similar construction is also possible if there are five non-compact dimensions, in whichcase it is possible to have a regular horizon without KK monopole. If we set Q = 0 inthe general solution of section 3.2, which according to (3.25) implies N = 1, we obtain aconfiguration with the same charges than the DH states. However, in this case the numericalfactor of 4 π in the DH entropy is not reproduced. Instead, upon use of (3.30), one has S W5d | Q =0 = 2 √ π (cid:112) Q + Q − . (6.5)Therefore, although there exists a regular five-dimensional fake small black hole —that is, ablack hole with only two asymptotic charges, Q + and Q − —, its entropy does not reproducethe degeneracy of the DH states. This mismatch is a natural consequence of the fact that When a black hole has this property, we call it a fake small black hole . We have just seen that it is possible to have regular supersymmetric black holes with lessthan four (three) Maxwell charges in four (five) dimensions, provided these contain four(three) non-vanishing brane-source charges, which signals the presence of S5-branes and (in4 d ) KK-monopoles.The goal of this section is to show that it is not possible to have regular, supersymmetricnear-horizon geometries in the heterotic theory compactified on T − d × S z if d ≥
6. Whenthere are d ≥ T − d has trivial dynamicsand that the near-horizon limit of the solutions enjoys a SO(2 , × SO( d −
1) symmetry.With these assumptions in mind, we proceed to write down the most general ansatz for thenear-horizon geometry. For the sake of convenience, we use the ( d + 1)-dimensional fields, ds d +1) = v (cid:18) r dt − dr r (cid:19) − v d Ω d − − u k ( dz − erdt ) ,e − φ = u k u φ , ˜ H = p ω S d − , (6.6)where ˜ H is the ( d − H , ˜ H ≡ e − φ (cid:63) ( d +1) H , and ω S d − is the volume form of the round S d − sphere. By applying the entropy function formalism, one can check that there are no regularextrema of the entropy function at zeroth order in α (cid:48) . However, when higher-curvaturecorrections are taken into account, the system of algebraic equations that one has to solvebecomes much harder to study. This is probably the reason why small black holes in d ≥ i.e., we get rid of the torus T − d , which does not play any rˆole here. (cid:63) ( d +1) denotes the ( d + 1)-dimensional Hodge star operator.
41n spite of this, we can follow an alternative route to show that there is no regular,supersymmetric (small) black holes with the assumed SO(2 , × SO( d −
1) isometry in thenear-horizon limit. The argument goes at follows. The dilatino Killing spinor equation isgiven by (cid:18) ∂ a φ Γ a − H abc Γ abc (cid:19) (cid:15) = 0 . (6.7)Since the dilaton is constant, the above equation reduces to H abc Γ abc (cid:15) = 0 and, for our ansatz,this equation can only be satisfied by a non-vanishing Killing spinor (cid:15) if p = 0. Since p isproportional to the winding charge of the fundamental string, this already proves that thereare no regular, supersymmetric solutions describing the near-horizon of higher-dimensionalsmall black holes. However, it is possible to go an step further and show that there are noregular supersymmetric solutions of this form at all. To do this, we can use the integrabilitycondition of the gravitino Killing spinor equation, which reduces to R abcd Γ cd (cid:15) = 0 , (6.8)since the 3-form field strength H vanishes. Contracting this equation with Γ b and usingeq. (2.6) of [82], we arrive to R ab Γ b (cid:15) = 0 . (6.9)It is not difficult to see by explicitly computing the Ricci tensor of the metric (6.6) that theabove integrability condition cannot be satisfied in our configuration for any choice of theparameters.Let us explain why this argument only holds if d ≥
6. Notice that if d = 5, ˜ H is a 3-formand therefore the ansatz (6.6) is not the most general one, as one can write down an electricterm for ˜ H of the form ˜ H = 2˜ e dt ∧ dr ∧ dz + p ω S , (6.10)which precisely indicates the presence of S5-branes, as they are electric sources of the dualKalb-Ramond field strength. The argument works exactly in the same way for d = 4 non-compact dimensions, see eq. (3.39). At the extremum of the entropy function, ˜ H turns outto be self-dual and the dilatino KSE is solved by a non-vanishing Killing spinor (cid:15) satisfying (cid:0) − Γ (cid:1) (cid:15) = 0 . (6.11)42 Conclusions
In this article, we have studied supergravity field configurations that can be interpreted assourced by the presence of different combinations of fundamental heterotic strings (carryingwinding and momentum), solitonic 5-branes and Kaluza-Klein monopoles at first order inthe higher-curvature expansion and in several dimensions.The most relevant effect producedby the higher-curvature corrections is to introduce non-linear couplings between fields, suchthat there are delocalized sources in some of the equations of motion. This produces a shiftin the mass and in some of the charges of the solution, which can have a negative character.An interesting phenomenon is that, in few specific cases ( Q = 0 in d = 5 and Q = − d = 4), the value of the charges does not uniquely determine the solution, and moreinformation is needed for that purpose.Depending on which sources are present, the solutions describe a black hole, a soliton or anaked singularity. The inclusion of first order corrections in the higher-curvature expansionof the effective theory does not change the character of the solution, but the addition ortruncation of sources may do it. This latter operation is intrinsically non-perturbative andmodifies substantially the properties of the fields at zeroth-order in the expansion. We showthat, for this reason, the computation of curvature corrections and the truncation of sourcesare two processes that do not always commute; starting from a zeroth-order solution, thesame result is not necessarily obtained if the two operations are performed in different order.In addition, we have shown that one gets a consistent string theoretic interpretation ofa perturbative solution when the sources are kept fixed in the higher-curvature expansion,allowing variations in the value of the charges. This plays a fundamental role in the studyof Kaluza Klein monopoles (as first noticed more than 20 years ago in [59]) and small blackholes. As a consequence of these observations, small (2-charge) black holes corresponding tothe DH system remain singular when quadratic curvature corrections are included.On the other hand, we note that the corrections imply the existence of regular 3-chargeblack holes in four dimensions, and 2-charge ones in five, where the vanishing charge is thatof the S5 brane. Since all 3- and 2-charge solutions are singular at zeroth order in α (cid:48) , itmight seem that the corrections resolve the singularities. However, following our previousdiscussion, what really happens is that the system described by these solutions is alreadyregular at zeroth-order, and the effect of the corrections is to screen the charge of the S5brane, which has a localized source. In d = 4, the corresponding 3-charge solution has thesame entropy as the DH system, but neither the charges, the sources nor the supersymmetryare equal to those of a string carrying momentum, and hence we refer to it as a fake smallblack hole. The matching of the entropies is most likely a coincidence, since in the five-dimensional case the regular 2-charge solution does not reproduce the entropy of the DHsystem. In turn, there is another 2-charge solution with no localized sources of S5 braneswhich is singular — this is the one describing the DH system. Likewise, in higher dimensionsall 2-charge solutions are singular due to the absence of S5 branes.43 cknowledgments We are thankful to Tom´as Ort´ın for years of collaboration and guidance in this field. Wethank Atish Dabholkar and Ashoke Sen for useful discussions. The work of PAC is sup-ported by a postdoctoral fellowship from the Research Foundation - Flanders (FWO grant12ZH121N). The work of ´AM is funded by the Spanish FPU Grant No. FPU17/04964. ´AMwas further supported by the MCIU/AEI/FEDER UE grant PGC2018-095205-B-I00 and bythe “Centro de Excelencia Severo Ochoa” Program grant SEV-2016-0597. The work of PFRis supported by the Alexander von Humboldt Stiftung. The work of AR is supported by theDepartment of Physics and Astronomy Galileo Galilei with funds of the project PRIN 2017”Supersymmetry Breaking with Fields, Strings and Branes”.
A Supersymmetry analysis
The main purpose of this appendix is to show that the ansatz used in Section 5 to describesmall black holes is the most general one with no dependence on the coordinate u preservinghalf of the spacetime supersymmetries at first order in α (cid:48) . We shall make a wide use of theresults of [83] but we also refer to [82, 84] where supersymmetric heterotic backgrounds havebeen studied and classified using different techniques as those employed in [83]. The resultsof this appendix are contained in the general classification of half-supersymmetric heteroticbackgrounds of [78]. Here we offer a different re-derivation of some of the results containedin this reference. A.1 General form of supersymmetric configurations
According to this reference, the metric of a supersymmetric configuration can always bewritten as ds = 2 f ( du + β ) [ dt + K ( du + β ) + ω ] − h mn dx m dx n , (A.1)where ω = ω m dx m and β = β m dx m are 1-forms on the eight-dimensional space charted bythe coordinates x m and f and K are functions defined on this manifold . It is convenientto introduce the following zehnbein basis e + = f ( du + β ) , e − = dt + K ( du + β ) + ω , h mn dx m dx n = e m e n δ mn . (A.2) In general, objects occurring in the metric may also depend on u . We assume no dependence on thiscoordinate in order to perform a standard KK reduction over this internal direction. ω ab in this basis are ω + − = − ∂ m log f e m , (A.3) ω + m = − f − ∂ m K e + − ∂ m log f e − + 12 ( Kdβ + dω ) nm e n , (A.4) ω − m = − ∂ m log f e + + f dβ ) nm e n , (A.5) ω mn = 12 ( Kdβ + dω ) mn e + + f dβ ) mn e − + (cid:36) pmn e p , (A.6)where we have defined (cid:36) mnp to be the spin connection associated to h mn , which satisfies that de m = − (cid:36) mn ∧ e n .In order for a configuration to be supersymmetric, several conditions need to be accom-plished. First, the torsionful spin connection Ω (+) ab ≡ ω ab + H cab e c must fulfil that Ω (+)[ ab ] − = 0 , (A.7)Ω (+) am − = 0 , (A.8) ∇ (+) a Ω mnpq = 0 , (A.9)where Ω mnpq is a 4-form which can be interpreted as a Spin(7) structure. As such, it possessesthe following properties Ω m m m p Ω n n n p = − m m n n δ m n + 6 δ m m m , n n n , (A.10)Ω m m p p Ω n n p p = − m m n n + 12 δ m m , n n , (A.11)Ω m n p p Ω n m p p = +4Ω m m n n + 6 δ m m , n n . (A.12)Ω m m n n Ω m m n n = − m ··· m , (A.13)Ω mp p p Ω np p p = 42 δ mn , (A.14)Ω m ··· m Ω m ··· m ≡ Ω = 14 · . (A.15)Apart from eqs. (A.7), (A.8) and (A.9), there exists another set of conditions to be imposed.They constrain the components of H abc as follows: H ( − ) − mn = 0 , (A.16) H ( − )+ mn = 148 Ω ms s s ∇ + Ω ns s s , (A.17) H ( − ) mnp = 17 (2 ∂ q φ − H + − q ) Ω qmnp , (A.18)where we have made use of the projectors acting on 2-forms Θ mn and 3-forms Ψ mnp definedin [83]: Θ mn = Θ (+) mn + Θ ( − ) mn , Θ ( ± ) mn = Π ( ± ) mnpq Θ pq , (A.19) In our conventions, we have that de a = + ω ab ∧ e b , with a, b = + , − , m . We note there is an error in eq. (3.17) of [83] since Ω (+) − mn = H − mn , which is non-vanishing in general.Therefore, the only constraint on Ω (+) − mn comes from eq. (A.16). We follow the convention of [83] and indices with same latin letter m i , n i , . . . are totally antisymmetrized. mnp = Ψ (+) mnp + Ψ ( − ) mnp , Ψ ( ± ) mnp = Π ( ± ) mnpqrs Ψ qrs , (A.20)where Π (+) mnpq = 34 (cid:18) δ mn,pq + 16 Ω mnpq (cid:19) , (A.21)Π ( − ) mnpq = 14 (cid:18) δ mn,pq −
12 Ω mnpq (cid:19) , (A.22)Π (+) m m m n n n = 67 (cid:18) δ m m m ,n n n + 14 Ω m m n n δ m n (cid:19) , (A.23)Π ( − ) m m m n n n = 17 (cid:18) δ m m m ,n n n −
32 Ω m m n n δ m n (cid:19) . (A.24)Let us analyse all eqs. (A.7), (A.8), (A.9), (A.16), (A.17) and (A.18) carefully. First, werealize that conditions (A.7) and (A.8) tell us that all the components of Ω (+) ab − vanish ,implying that the components H ab − get fixed in terms of the objects that occur in the metric.We find H m + − = ∂ m log f , H mn − = f ( dβ ) mn . (A.25)Let us postpone the study of eq. (A.9) for the moment. Regarding eq. (A.16), we see itimposes that ( dβ ) ( − ) mn = 0 , (A.26)so that the connection β is that of an Abelian octonionic instanton [85, 86]. On the otherhand, we check that eq. (A.17) can be rewritten by use of eqs. (A.11) and (A.14) as H ( − )+ mn = − ω ( − )+ mn = − ( Kdβ + dω ) ( − ) mn = − ( dω ) ( − ) mn . (A.27)Therefore H + mn can always be expressed as H + mn = H (+)+ mn − ( dω ) ( − ) mn = − ( dω ) mn + K ( dβ ) mn + f − ξ mn , (A.28)for some two-form ξ = ξ mn e m ∧ e n satisfying that ξ ( − ) mn = 0. Consequently, the general formof H for supersymmetric configurations with no dependence on u is H = d log f ∧ e + ∧ e − + f e − ∧ dβ + e + ∧ (cid:0) − dω + Kdβ + f − ξ (cid:1) + 13! H mnp e m ∧ e n ∧ e p , (A.29)with H mnp satisfying (A.18), which can be rewritten by virtue of (A.25) as H ( − ) mnp = 17 ∂ q (2 φ − log f ) Ω qmnp . (A.30)Now it is the moment to study condition (A.9). From the a = ± components of eq. (A.9)and taking into account that Ω mnpq is independent of u and t , we find The component Ω (+)++ − = ω ++ − is not fixed by these equations but it vanishes for the coordinates wehave chosen, see (A.3). (+) ± [ m | s Ω s | npq ] = 0 . (A.31)Contracting this equation with Ω rnpq , we arrive to the equivalent conditionΠ ( − ) mnpq Ω (+) ± pq = 0 , (A.32)which reduces to the self-duality conditions already derived for dβ and ξ . If instead we take a = m at eq. (A.9), we deduce that Ω (+) mnp has special holonomy G ⊆ Spin(7). This lastcondition can be expressed in a fairly compact way if one chooses a basis { e m } for whichthe components of Ω mnpq are constant, which is known to always exist locally since Ω mnpq defines a Spin(7) structure. In particular, in such a basis, we obtain the conditionΠ ( − ) mnrs Ω (+) prs = 0 . (A.33) A.2 Killing spinor equations
Let us now study the Killing spinor equations (KSEs). It was proven in [83] that they arefulfilled by a constant spinor (cid:15) satisfyingΓ + (cid:15) = 0 , (A.34)Π ( − ) (cid:15) = 0 . (A.35)where Π ( − ) = 78 (cid:18) − m ...m Γ m ...m (cid:19) . (A.36)The first condition (A.34) annihilates half of the spacetime supersymmetries and, albeit ingeneral half-supersymmetric configurations do not necessarily satisfy it, the class of half-supersymmetric solutions we are interested in (those describing superpositions of fundamen-tal strings with momentum) does [78]. Hence, if we want to preserve exactly this amountof supersymmetry, we must find a way to avoid using (A.35) but still solving the dilatinoand the gravitino KSEs. For that, we are going to impose extra conditions on the fields toensure that such KSEs hold even if (A.35) does not.To this aim, we first concentrate on the dilatino KSE. It is convenient to use the rewritingof such KSE provided in eq. (3.47) of [83]: (cid:20)
12 (2 ∂ m φ − H + − m ) Γ m − (cid:0) H mnp Γ mnp + 3 H − mn Γ − Γ mn (cid:1)(cid:21) Π ( − ) (cid:15) = 0 , (A.37)where we have already required (A.34). Since we do not want to impose (A.35), the termbetween brackets must vanish necessarily. Therefore, φ = φ + 12 log f , H mnp = 0 , ( dβ ) mn = 0 , (A.38) We correct a innocent typo in eq. (3.47) of [83]. φ is an integration constant.We now move to the gravitino KSE. For a constant spinor satisfying (A.34), it is onlynecessary to check that Ω (+) amn Γ mn (cid:15) = 0 . (A.39)As we derived before at eq. (A.33), in a basis { e m } where the components of Ω m ...m areconstant, we have that Ω ( − )(+) amn = 0. Therefore, we can use eq. (A.46a) of [83] to show thatΩ (+) amn Γ mn (cid:15) = Ω (+) amn Γ mn Π ( − ) (cid:15) = 0 , (A.40)which implies that either Ω (+) amn or Π ( − ) (cid:15) must vanish. Since we do not want on any accountto impose (A.35), we require Ω (+) amn = 0 and this, in turn, demands (cid:36) mnp = 0 , ξ mn = 0 , (A.41)so that e m = dx m and h mn = δ mn . Consequently, the most general configuration preservinghalf of the spacetime supersymmetries with no dependence on u is given by ds = 2 e φ − φ ) du ( dt + Kdu + ω ) − dx m dx m , (A.42) H = 2 e φ − φ ) dφ ∧ du ∧ ( dt + ω ) − e φ − φ ) du ∧ dω . (A.43)Note that these results are identical to those presented at eq. (8.10) of Ref. [78] if weeliminate all dependence on his coordinate v (which we have called u instead). This concludesthe proof of the fact that the ansatz used in Section 5 is the most general ansatz to constructheterotic string backgrounds consisting of supersymmetric superpositions of fundamentalstrings with momentum along them. B First-order α (cid:48) -corrections to the fundamental rotat-ing string solution We present in this Appendix the precise form of the first-order α (cid:48) corrections to the small-black-ring solution presented in subsection 5.2. For that, we just use eqs. (5.16) and plugthem into eq. (5.6) to obtain the α (cid:48) -corrected solution. Following the notation of the maintext, we encounter that Z + = Z (0)+ + α (cid:48) Z (1)+ + O ( α (cid:48) ) , (B.1) Z − = Z (0) − + O ( α (cid:48) ) , (B.2) ω = ω (0) + O ( α (cid:48) ) , (B.3) The 1-form β can always be removed via the coordinate transformation u → u − χ , where dχ = β , sinceby (A.38) β is closed. Z (0)+ = 1 + ˜ q + (cid:18) y − x R y (cid:19) d − F (cid:18) d − , d −
14 ; 1; 1 − y (cid:19) , (B.4) Z (0) − = 1 + ˜ q − (cid:18) y − x R y (cid:19) d − F (cid:18) d − , d −
14 ; 1; 1 − y (cid:19) , (B.5) ω (0) = ( d − π ˜ q − W R ( y − (cid:96) ( x − y ) (cid:18) y − x R y (cid:19) d − F (cid:18) d − , d + 14 ; 2; 1 − y (cid:19) dψ , (B.6) Z (1)+ = 1 (cid:96) C ( x, y ; d ) (cid:34) A ( x, y ; d ) F (cid:18) d − , d −
14 ; 1; 1 − y (cid:19) F (cid:18) d + 14 , d + 34 ; 2; 1 − y (cid:19) + A ( x, y ; d ) F (cid:18) d − , d + 14 ; 2; 1 − y (cid:19) F (cid:18) d − , d + 54 ; 2; 1 − y (cid:19) + A ( x, y ; d ) F (cid:18) d − , d −
14 ; 1; 1 − y (cid:19) + A ( x, y ; d ) F (cid:18) d − , d + 14 ; 2; 1 − y (cid:19) + A ( x, y ; d ) F (cid:18) d − , d + 54 ; 2; 1 − y (cid:19) (cid:35) , (B.7)with the definitions, C ( x, y ; d ) = − − − d y − − d ( d − ˜ q − R − d ( y − x ) d − − d ˜ q − ( R y ) − d ( y − x ) d − F (cid:16) d − , d − ; 1; 1 − y (cid:17) , (B.8) A ( x, y ; d ) = 8( d − (cid:96) ˜ q + xy ( − y ) , (B.9) A ( x, y ; d ) = − d + 1) (cid:0) π ˜ q − R W y (cid:0) ( d − xy − ( d − y + 2 x (cid:1) − (cid:96) ˜ q + (cid:0) y − (cid:1) ( x − y ) (cid:1) , (B.10) A ( x, y ; d ) = 16 (cid:96) ˜ q + y ( x + y ) , (B.11) A ( x, y ; d ) = 4 (cid:0) π ˜ q − R W y (cid:0) ( d − d − xy − ( d − y − d − y + 4 x (cid:1) − (cid:96) ˜ q + (cid:0) y − (cid:1) ( x − y ) (cid:1) , (B.12) A ( x, y ; d ) = − ( d + 1) ( x − y ) (cid:0) (cid:96) ˜ q + (cid:0) y − (cid:1) − π ˜ q − R W y (cid:1) . (B.13)The different hypergeometric functions appearing at the corrected solution have the fol-lowing behaviour (as y → −∞ ): F (cid:18) d − , d −
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