Содержание к диссертации
Введение
1. Introduction 2
2. Preliminary material 8
2.1 Quaternion Kahler and hyperkahler spaces 8
2.1.1 Quaternionic Kahler spaces in dimension higher than four 9
2.1.2 Quaternion Kahler manifolds in dimension four 12
2.2 Hypercomplex structures 14
2.2.1 Basic concepts 14
2.2.2 Some explicit examples 16
2.3 Hyperkahler spaces in four dimensions 18
2.4 Quaternion Kahler and hyperkahler metrics in d = 4 with at least one isometry 21
2.4.1 Hyperkahler metrics with Killing vectors that are not self-dual 21
2.4.2 Integrability of the axial continuum Toda equation 23
2.4.3 Einstein-Weyl structures and hyperkahler metrics 25
2.4.4 Self-dual structures with one Killing vector 27
2.4.5 The Joyce spaces 29
2.4.6 Identification of the quaternionic-Kahler metrics with at least one isometry 31
2.4.7 Examples of toric quaternion Kahler spaces 33
2.5 Higher dimensional hypergeometry 39
2.5.1 Construction of in hyperkahler manifolds with T" tri-holomorphic isometry 39
2.5.2 Quaternion Kahler spaces in quaternion notation 41
2.5.3 The Swann extension 43
2.6 Spaces with C?2 holonomy 45
2.6.1 The group C?2 and the octonions 45
2.6.2 G% holonomy and self-duality 51
2.6.3 The Bryant-Salamon construction 53
2.6.4 Weak G
2.7 f?2 holonomy spaces and M-theory compactifications 59
3. Heterotic geometry without isometries 62
3.1 Introduction 62
3.2 Hyperkahler torsion manifolds 65
3.2.1 Main properties 65
3.2.2 Relation with the Plebanski-Finley conformal structures 67
3.3 The general solution 70
3.4 Discussion 75
4. D-instanton sums for matter hypermultiplets 77
4.1 Ooguri-Vafa solution 77
4.2 Pedersen-Poon Ansatz 79
4.3 D-instanton sums 80
4.4 Discussion 82
5. Hyperkahler spaces with local triholomorphic U(l) x [/(1) isometry and su pergravity solutions 83
5.1 Toric hyperkahler metrics of the Swaim type 83
5.2 Supergravity solutions related to hyperkahler manifolds 86
6. G2 toric metrics and supergravity backgrounds 89
7. Conclusion 93
7 LINK1 Hypercomplex structures LINK1
As we have seen, the problem of classifying the possible quaternion Kahler spaces appearing in d = 4 is equivalent to find all the Einstein spaces with self-dual Weyl tensor. In this section we focus in four dimensional spaces for which the second property hold namely, self-dual spaces. Let us consider a metric g and denote with [g] the family of metrics obtained from g by arbitrary conformal transformation g -» Q2g. Because the Weyl tensor is conformally invariant it follows that the self-duality of g implies the self-duality of [g], the converse is also true. An important example of self-dual families are the hypercomplex structures [60], [61]. To consider them has several advantages for our purposes. It is easier to understand the Ashtekar-Jacobson-Smolin [149] and the Plebanski [76] description of hyperkahler spaces once the properties of hypercomplex structures are understood. Moreover we will show that hypercomplex structures and weak torsion hyperkahler structures are the same concept in four dimensions. As it follows from equation (2.7), for hyperkahler spaces there exist a frame for which that is, for which the Kahler triplet is closed. Conversely, it was shown in [76] that any space in d = 4 satisfying (2.34) is automatically hyperkahler. As we will see below this implies the curvature tensor Щы of an hyperkahler space is automatically self-dual. A simple calculation show that if we make the transformation g = Sl2g then the Kahler triplet is transformed as J = fi27 and (2.34) became Relation (2.35) is not the same as (2.34) and therefore (2.34) is not invariant under conformal transformations. A conformally invariant generalization of (2.34) is being a an arbitrary 1-form. Under conformal transformation we have a-f 2dlog(Q) and (2.36) is unaltered [61]. This property define a conformal family [g] with all the element sharing the property (2.36). If a is a gradient (a — "Чф, being ф certain function depending on all the coordinates), then it will exist a representative g of [g] with self-dual curvature and therefore hyperkahler. From the expression (2.32) of the Weyl tensor it is possible to check that if R is self-dual, then W is also self-dual. Thus the family [g] corresponding to an hyperkahler metric will have self-dual Weyl tensor (2.32). But the explicit calculation of the Wabcd for a metric satisfying (2.36) shows that a disappear in its calculation.
Therefore the condition to be gradient is not necessary to ensure W- = 0 and condition (2.36) defines a self-dual structure in general [61]. One family of metrics satisfying (2.36) are the so called hypercomplex structures [60]. To define them, let us consider as before Riemanian space M of real dimension 4n endowed with a metric g and a set of three almost complex structures J (i = 1,2,3) satisfying the quaternionic algebra and for which the metric g satisfies for any X,Y in TXM. The complex structure is said to be integrable if the Niejenhuis tensor vanish for every pair of vector fields A , Y. An hypercomplex structure is a conformal structure [g] for which the triplet J1, J2 and J3 are integrable. The integrability condition is conformally invariant. In four dimensions we can select the self-dual complex structures up to an SU(2) transformation. The action of (2.40) over the tangent space TMX is defined by The annulation of the tensor N (X, Y) will be equivalent to the conditions for some set of functions Лі,Л2,Л3, Л4 [59]. This is a direct consequence of (2.41) and (2.39). We immediately recognize that in the case Aa — 0 we recover the Ashtekar-Jacobson-Smolin quadratic equations for self-dual spaces [149]. If we consider the conformal transformation g — a2g, then it is seen that (2.42) is also satisfied with Aa — Aa + 2ealog(fi). This mean that condition (2.42) define a conformal family of metrics [g] and therefore the definition of hypercomplex structures is consistent. Hypercomplex condition implies, but is not implied by, that [g] is self-dual [59]-[60]. This is because (2.42) implies the self-duality condition (2.36). To see the last statement clearly consider the connection us given by It is well known that the antisymmetric part wp is related to the structure functions defined by the Lie Consider now the form This is equivalent to (2.36) with a — A — x- The same formula holds for J and J . Therefore we reach to the conclusion that hypercomplex structures satisfies (2.36) and therefore are self-dual [59]. It is straightforward to prove that under the conformal change g —У Q2g the forms a, A and \ transform Hypercomplex structures arc not the only self-dual structures in four dimensions. Counterexamples are for instance the Joyce spaces that will be introduced in another section. It will be instructive to present some examples of hypercomplex spaces. Let us look now for self-dual structures [g] two commuting [/(1) Killing vectors satisfying (2.42). The representatives [g] of such structures take the Gowdy form The latin and greek indices takes values 1 and 2. Both gab and ga$ are supposed to be independent of the coordinates xa = (0, ip). Then the Killing vectors are д/дв and д/дц and are commuting, so there By Gauss theorem there exists a local scale transformation g - Q2g which reduce (2.46) to being gQp function of the new coordinates (p,rj). We can express where Ax and B, are unknown functions of (p, rj) and the factor p was introduced by convenience. But instead to work with (2.48) we will work with the following equivalent expression Anzatz (2.49) looks more complicated than (2.48), but has the advantage that the inverse einbeins ЄІ takes the simple form Inserting (2.50) and (2.51) into the quadratic system (2.42) gives Q = 1 and the Cauchy Riernman equations and the same equations for J% and By.
This means that the metric (2.49) is described in terms of two holomorphic functions We can find the same result in another way. Let us consider four functions fi,..., f and .9ь ---,94. depending on the coordinates x1 = p and x2 — 77 and let us define the vector fields Clearly ea are the most general vector fields for a metric with two commuting isometries up to a conformal scaling. Introducing this expressions into (2.57) gives the system of equations /i)„ - ШР = 0, Ы„ - {92), = 0 and from the first we see that /3 = {H)p and /4 = (H)4 for some function H(p 77) and therefore we can make the coordinate change (p, 77, в, tp) —у (p — H, 77, в, p) and eliminate /3 and /4. The same holds for g$ and 54 and therefore we are dealing with the case described in (2.51). The corresponding hypcrkahler metric is then conformal to which is the same metric as above with fl = 1. We have redefined e\ = A0, e2 = BQ, f\ = Ai, /2 = B2 in (2.55). There exists another family of hypercomplex structures that can be constructed in terms of holomorphic functions. It is direct to check that Ashtekar equations (2.57) can be cast in the following complex form Let М be a complex surface with holomorphic coordinates (z , z2) and let us define four vector fields ЄІ as being fj a complex function on M. Then (2.56) implies that dfj/dzk = 0 and therefore we can construct an hypercomplex structure using four arbitrary holomorphic functions or two holomorphic vector In a well known work [149] Ashtekar, Jacobson and Smolin introduced a formulation for self-dual manifolds in which they reduced the problem to solve certain quadratic equations for the dual vector fields and a volume form preserving condition. Here we review their construction in the context of hypercomplex structures. Proposition 3 Consider an oriented manifold M and four vector fields в\, e2, єз ап& e4 forming an oriented basis for TM at each point. Let us suppose that the fields satisfies the quadratic equations and the volume preserving condition ЄцО = 0 for some -form 0. Then the vectors ea are conformal to an orthonormal frame of a self-dual Ricci-flat metric. Before to explain Proposition 3 let us note that is (2.57) with the J40 S equal to zero. Then (3.328) implies that x = a- Then if x were a gradient, then the form a also would be a gradient and there will exist a conformal change taking the condition to d T = 0. Therefore the corresponding metric will be conformal to an hyperkahler one, which are automatically self-dual Ricci-flat. As we will see, this is essentially the content of the volume preserving condition of proposition 3, In general the covariant divergence of an arbitrary vector field V is obtained by the formula The divergence of a tetrad e a is given by being c6 defined by Let us define a factor Q through and take the derivative of (2.60) along eQ = 2-1ea. Using the condition CaQ = 0 together with (2.58) The last implication is just the definition (3.328). The transformation rule (2.45) implies that under д and therefore a is a gradient. Then g is conformal to an hyperkahler metric.
- Hypercomplex structures
- Quaternion Kahler spaces in quaternion notation
- Hyperkahler torsion manifolds
- Discussion
Введение к работе
Актуальность темы. Модель Салама-Вайнберга - это квантовая теория поля, которая успешно объясняет два из четырех известных взаимодействий природы, а именно электромагнитные и слабые взаимодействия. Сильное взаимодействие в свою очередь описывается квантовой хромодинамикой, которая успешно согласуется с данными по неупругому рассеянию лептонов на нуклонах. Последнее, четвертое взаимодействие - гравитация - описывается на классическом уровне с помощью теории относительности Эйнштейна. Стандартная модель (модель Салама-Вайнберга плюс квантовая хромодинамика) с одной стороны, и гравитация Эйнштейна с другой, становятся несовместимыми на малых расстояниях или, что одно и то же, для высоких энергий, порядка массы Планка Mpi « 1019ГэВ. Такие масштабы энергии пока являются неисследованной областью физики.
Теории суперструн пытаются объяснить физические явления в этом диапазоне энергий, и поскольку данные теории формулируются в пространствах с числом измерений > 4, существует большой интерес к вопросу редукции размерности за счет компактификаций лишних измерений. Многообразия с уменьшенной голономией играют существенную роль как компактифицированные многообразия суперструн и М-теории, сохраняющие определенное число супер-симметрий после компактификаций, а также в качестве геометрического пространства модулей суперсимметричной сигма-модели, возникающего после компактификаций. Настоящая диссертация связана с приложениями гиперкэлеро-вых и кватернионно-кэлеровых многообразий (которые являются 4п-мерными пространствами с голономиями реализованными группами Ли Sp(n) х Sp(l) и Sp(n) соответственно) в струнной теории. Кроме того в диссертации рассматриваются многообразия с G2 и Spin(7) голономиями. Данная диссертация также имеет дело с гиперкэлеровой геометрией с кручением и SU(3) структурами в шести измерениях.
В течение последних двадцати лет стало ясно, что кватернионно-кэлерова
и гиперкэлерова геометрия являются эффективным инструментом в квантовой теории поля независимо от наличия суперсимметрии. Например, пространство модулей магнитных монополей или пространство модулей инстантонов в теории Янга-Миллса в плоском пространстве являются гиперкэлеровыми [3]. Гиперкэлерова геометрия также возникает в сигма-моделях с N=4 суперсим-метриями [10]. Если в действие сигма-модели включено слагаемое типа Весса-Зумино, то target- пространство (пространство модулей) будет гиперкэлеровым с кручениями (ГКК). Когда суперсимметрия является локальной, гипермуль-типлеты взаимодействуют с гравитонами и итоговое пространство модулей -это кватернионно-кэлерово пространство [4]. Эта геометрия также связана с построением фона (классических решений) одиннадцатимерной супергравитации в виде D-браны. Метрика бозонного target- пространства для D = 4 суперсимметричной сигма модели, полученной компактификацией суперструнной теории типа ИА на многообразие Калаби-Яу, также является гиперкэлеровой [13] (даже с учетом поправок за счет D-инстантонов). Таким образом, возникает интересная задача построить кватернионно-кэлерово и гиперкэлерово пространства, в том числе и с сингулярностями.
Одним из последних достижений в гиперкэлеровой геометрии является построение гиперкэлерова фактора (quotient) [5], простейший пример которого найден в рамках ADHM конструкции (это фактор гиперкэлерова многообразия более высокой размерности по определенной группе три-голоморфных изомет-рий). Важно напомнить, что идея этого построения имеет физическое и интуитивное основание, связанное с дуальностями суперсимметричной сигма-модели. Обратное построение было найдено Шваном [6]. Он показал как кватернионно-кэлерова метрика при D=4n может быть расширена до метрики кватернионных и гиперкэлеровых пространств в D=4(n +1). Метод Швана использован для построения гиперкэлеровых пространств, применяемых в теориях с глобальными N=2 суперсимметриями. Это построение также важно для целей настоящей диссертации
Задача классификации возможных групп голономий для римановых и псев-
доримановых пространств достаточно стара. Она была поставлена Картаном и частично решена Бергером, который представил список возможных групп голономий для этих геометрий [1]. Математическая особенность уменьшения групп голономий n-мерного пространства от SO(n) к подгруппе - это наличие глобально определенных тензоров а, которые являются инвариантными при параллельном переносе. Это свойство эквивалентно исчезновению их ковари-антной производной, т.е. Da = 0. Такой глобально определенный тензор является в некотором смысле аналогом инвариантного подпространства для группы Ли. Наличие инвариантного подпространства подразумевает редукцию SO(n) к ее подгруппе Ли с более низкой размерностью. Общая особенность уменьшенной голономий - наличие ковариантной постоянной р-формы. Например, в D = 4 имеет место изоморфизм SO{A) ^ SU(2)l х SU(2)r и пространство с го-лономией SU(2) характеризуется ковариатно постоянным правосторонним (или левосторонним) спинором «я, определенным на всем многообразии. Присутствие бд подразумевает, что кривизна является самодуальной и что существует вращение системы отсчета, приводящее анти-самодуальную часть связности w к нулю. Примеры - многообразия Егучи-Хансона и Тауба-Нута. Также для многообразия с голономией Gi можно выбрать ортогональную систему отсчета е*, в которой октонионная 3-форма
Ф = е1 Л е2 Л е7 + е1 Л е3 Л е6 + е1 Л е4 Л е5 + е2 Л е3 Л е5 + е4 Л е2 Л е6
+ е3 Л е4 Л е7 + е5 Л е6 Л е7
и его дуальная 4-форма *Ф замкнуты. Эти формы являются действительно инвариантными относительно действия группы (?2, и их замкнутость эквивалентна присутствию ковариянтно постоянного спинора г\, такого что D,r) = 0. Напомним, что d - это на самом деле подгруппа 50(7), которая выделяется с помощью одномерного инвариантного подпространства. Существование этого подпространства - самая важная особенность с физической точки зрения, потому что число суперсимметрий, сохраняемое обычной процедурой Калуца-Клейна связано с числом таких спиноров на внутреннем многообразии.
Вообще говоря, если данная риманова метрика размерности п и допускает по крайней мере один ковариатно постоянный спинор т), удовлетворяющий D,t/ = О, то группа голономий будет S(7(|), Sp(j), G2 либо Sptn(7). Два последних случая соответствуют семи и восьми измерениям. Для метрики с голо-номией G2 (и Spin{7)) существует только один г). После публикации работы Бергера вопрос о существовании метрики с голономией G2 и Spin(7) оставался открытым. Тридцать лет спустя после появления работы Бергера эта задача была решена Бриантом и Саламоном [7], которые доказали ее существование, а также построили некоторые явные примеры [8]. Они нашли интересную связь между кватернионно-кэлеровой метрикой и пространством с уменьшенной голономией и показали, что большой класс метрик с уменьшенной голономией может быть построен как расширение четырехмерных кватернионно-кэлеровых пространств. Это называется расширением Бриант-Саламона, и имеет некоторую аналогию с построением Швана.
Пространства с G2 голономией являются Ричи-плоскими. Их тензор кривизны Rabat удовлетворяет обобщению условия самодуальности в пространстве D = 7, а именно,
!_саЬЫ р
Структурные константы октонионов саьы в D = 7 играют роль, аналогичную символам Кронекера е,^ в D = 4. Самодуальность подразумевает G2 голоно-мию и то, что тензор кривизны является Ричи-плоским (то же самое утверждение имеет место в восьмимерном случае для голономий Spin(7)). Хотя условие самодуальности в итоге приводит к нелинейной системе уравнений, она, тем не менее, может быть решена в случаях с подходящей симметрией.
После работы Брианта были построены несколько явных примеров компактных и некомпактных G2 метрик. В литературе есть примеры так называемых "слабых G2 голономий" [12], которые снова являются фонами 11-мерной супергравитации, и которые также сохраняют N=1 суперсимметрию в D — 4. В этом случае существует нековариантно постоянный спинор т\ удовлетворяющий условию D,rj ~ А-у,??. Условие Я,3 = 0 (Ричи-плоские многообразия) заменяется
в этом случае условием Ду ~ \g,j. В пределе А —» 0 получаем G^ как ограниченную группу голономий. Хитчин показал, что при определенных условиях этот вид пространства можно описать с помощью конической метрики с голо-номией Spin(7) [10]. Физическая интерпретация слабого 6 пространства - это суперсимметричные фоны в теориях суперструн при наличии специальнных тензорных полей (fluxes).
Несмотря на большой прогресс в исследовании многообразий с уменьшенной голономией, все еще остаются открытые вопросы, в особенности связанные с теориями Калуца-Клейна. Реалистичная компактификация должна объяснить наличие киральной материи (киральных фермионов), а такая материя не может быть получена применением процедуры Калуца-Клейна на гладких Gi многообразиях. В гладком случае гармоническое разложение Калуца-Клейна одиннадцатимерной супергравитации будет приводить к N=1 четырехмерной супергравитации, включающей абелевы векторные мультиплеты (без киральных фермионов). Отметим, что киральные фермионы могут появиться, как указано в [11], только если компактифицированное многообразие является сингулярным. Т.о , оказывается, что для построения реалистичной модели нужно исследовать динамику суперструн на орбифолдах.
Еще одно требование состоит в том, что внутреннее пространство должно быть компактным. Существование компактных пространств со специальной голономией и сингулярнями (орбифолдов) было строго доказано Джойсом в [9]. Но явная метрика для таких пространств до сих пор не известна. Тем не менее предполагается, что пространства (типа орбифолда) с D = 8 по одной из его изометрий [11]. Это дает еще одну связь между "гипергеометрией" и пространствами с G2 голономией.
Несмотря на то, что явная Gi метрика для компактных многообразий не известна, в [12] были построены некоторые примеры со слабой G% голономией, которые являются компактными и допускают некоторые виды сингулярности. Кроме того Виттен показал, что физика вблизи сингулярности является по
существу локальной, то есть не зависит от глобальных свойств (типа компактности) внутреннего пространства [11]. По этой причине все еще существует большой интерес к построению многообразий со специальной голономией и коническими сингулярностями без требования компактности.
Целью работы:
Настоящая диссертация представляет собой исследование многообразий с редуцированной голономией с упором на построение явных примеров, которые применимы для компактификаций в суперструнных и М- теориях, а также в суперсимметричных сигма моделях.
Научная новизна:
В работе впервые представлена полная классификация слабых гиперкэлеро-вых пространств с кручением посредством их идентификации с гиперкомплексными пространствами, а также с использованием результатов исследований самодуальных пространств в контексте комплексной геометрии.
Получены новые суперсимметричные фоны суперструнной теории с использованием некоторых новых результатов в кватернионно-кэлеровой геометрии. Также построенно обобщение метрики target- пространства D = 4 сигма модели (метрики Оогури-Вафа), полученной при компактификаций ПА-струн, при наличии нескольких идентичных гипермультиплетов (принимая во внимание также и D-инстантонные поправки).
Апробация работы. Результаты, представленные в диссертации, докладывались и обсуждались на научных семинарах Лаборатории теоретической физики им.Н.Н.Боголюбова Объединенного института ядерных исследований, а также представлялись и докладывались на: Международном коллоквиуме "X International Colloquium Quantum Groups and Integrable Systems" (2001, Прага,
Чешская республика); Международном коллоквиуме "XIII International Colloquium Quantum Groups and Integrable Systems" (2004, Прага, Чешская республика) и Международной конференции " Xl-th International Conference Symmetry Methods in Physics" (2004, Прага, Чешская республика).
Публикации. По материалам диссертации опубликовано 4 работы.
Обьем и структура диссертации. Диссертация состоит из введения, 5 глав и заключения. Общий объем 104 станиц, включая список литературы.
На защиту выдвигаются следующие результаты:
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Показано, что четырехмерные слабые гиперкэлеровы пространства с кручением (СГКК) - это то же самое, что гиперкомплексные пространства и то же самое, что некоторые пространства, рассмотренные Плебански и Финли. С помощью такого отождествления найдена наиболее общая локальная форма для метрики гиперкэлерова пространства с кручением в четырех измерениях. Для СГКК пространств предложена формулировка типа Аштекара-Якобсона-Смолина, в которой проблема построения СГКК метрики сводится к решению квадратичной системы дифференциальных уравнений. Показано, что существует более общее решение (нежели решение Калана-Харви-Строминжера), для которого не существует сохраняющаяся форма объема. Построены несколько явных примеров таких метрик с изометриями [18].
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Найдена наиболее общая форма метрики target- пространства для D — 4 сигма модели (с несколькими идентичными гипермультиплетами), которая возникает в теории суперструн типа НА, компактифицированной на некоторое многообразие Калаби-Яу (вместе с непертурбативными вкладами в метрику от D-инстантонов). Рассматриваемый метрический тензор является "торически гиперкэлеровым", если не принимать во внимание гравитационные поправки.
Этот результат обобщает решение Оогури-Вафа. Мы нашли, что квантовые поправки к классическим (логарифмическим) членам экспоненциально подавлены факторами, полученными от вкладов D-инстантона в квази-классическом описании [17].
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Построено семейство метрик для пространств с голономией Spin(7), G2 а также для пространств со слабой G2 голономией. Результат был также распространен на случай фонового решения для супергравитации, сохраняющего одну суперсимметрию. Наличие торических симметрии дает возможность редукций к фоновым решениям типа ПА путем обычной редукции по одному из векторов Киллинга [15].
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С помощью математического приема, называемого расширением Швана, построено семейство торических гиперкэлеровых метрик для восьмимерного случая. Найдена система координат, для которой метрика имеет вид типа метрики Гиббонса-Хокинга. Решение было расширено до одиннадцатимерной супергравитации, сохраняющей определенное число суперсимметрий, независимо от присутствия потоков. Некоторые типы НА и НВ решений были найдены с помощию редукций (вдоль изометрии), и при помощи правил Т-дуальности [16].
Quaternion Kahler spaces in quaternion notation
The previous sections were concerned with quaternion Kahler and hypcrkahler manifolds in d = A. In this section we deal with higher dimensional ones. We present the most general 4n-diniensional hypcrkahler metrics with n commuting tri-holomorphic Killing vectors [142], namely the Pedersen-Poon metrics. This result generalize the Gibbons-Hawking one (2.68) to d — An. We also present a construction allowing to extend an arbitrary quaternion Kahler metric in d = An to other quaternion Kahler and an hypcrkahler ones in d = A(n + 1). This is the Swann construction [22]. As we will sec, the Svvann extension of the Calderbank-Pedcrsen metrics (2.128) preserves the Killing commuting Killing vectors and they are tri-holomorphic with respect to the new metric. Therefore the extension is of a Pedersen-Poon type. Once this is understood it is straightforward to construct certain classical supergravity solutions. This is the subject of the next section and is part of the thesis work. Let us remind that in section 3 we have described all the four dimensional hyperkahler spaces with at least one self-dual Killing vector namely, the Gibbons-Hawking metrics. We have seen that for such spaces there always exist a coordinate system for which the metric take the form and is therefore a Killing vector. The tensor К , — V K satisfies and for this reason К is called self-dual in the terminology of the references [75]-[74]. The vector К = dt also preserve the hyperkahler triplet J of (2.147). Explicitly J is given and from (2.150) its is direct to check that In general a vector preserving T as in (2.152) is called tri-holomorphic. If instead we have that then К preserve the complex structures and is called tri-hamiltonian. It can be shown that any two of the conditions given above for К implies the third. This mean in particular that for the metric (2.147) the vector К — dt is Killing, tri-holomorphic and tri-hamiltonian. For the spaces (2.81) this condition docs not hold, the Gibbons-Hawking metrics (2.147) are the only one with this properties in d = A. The higher dimensional analogs of (2.147) to d — An dimensions are the hyperkahler spaces with n commuting tri-holomorphic Killing vectors dti. A plausible anzatz is obtained by introducing a set on 3n coordinates x1 = {rrj., г = 1,..,n r = 1,2,3} and the metric where the symmetric matrix 1 and then its inverse C/1J are independent of the coordinates V. The n one forms Aj have the form where the matrix E,j also do not depend on the t, coordinates. The hyperkahler triplet corresponding to (2.153) is given by Here x denote the exterior product of forms (for instance the 3-component of dx1 x dx? is dx\ x dx3 ).
Then it is directly seen that ai J — 0, therefore the Killing vectors dti are tri-holomorphic and tri-hamiltonian. For n 1 (2.153) reduce to the Gibbons-Hawking metrics (2.147). Let us now find the conditions should be satisfied for (2.153) to be an hyperkahler metric, ф In order to use the 4n-bein formalism we can express the matrix U as for a non singular matrix K, which is defined by (2.156) up to an SO(n) rotation. Then we can select the 4n-bein as being Ka the inverse of Kaj. Then if (2.153) is hyperkahler, then N%(X,Y) — 0 for every of the complex structures J1 corresponding to (3.302). In An dimensions the conditions (2.101) are generalized to a system of equations with a An vector potential ЛІ;..., A n that we will not write for simplicity. But introducing (2.157) into such system gives finally the following system of equations 17( = (..-, Щ, being F u the strength tensor corresponding to the "vector potential" Aa. If the last conditions holds then (2.153) is hyperkahler. It was also shown that (2.153) are unique. This mean that for any Tn hyperkahler metrics there exists a coordinate system in which they can be cast in the form (2.153) [142]-[167j. Such system is the momentum map system. In general momentum maps are related to a compact Lie group G acting over an hyperkahler manifold M by tri-holomorphic Killing vectors X, i.e, Killing vectors satisfying The last condition implies that X preserves the hyperkahler forms J , that is щ Here %xJ denotes the contraction of X with the hyperkahler forms. By supposition M is hyperkahler, then dj = 0 and d(ixJk) = 0. This mean that ixJ can be expressed locally as the differential of certain function x%, namely The functions x are called momentum maps and are defined up to a constant. In the case (2.153) the isomctries are д/дЦ and the hypcrkahler form corresponding to (2.162) is [134] Prom the last expression it follows that and therefore (x],...,x") are really the momentum maps of the isometrics. AH the results of this section are stated in the following proposition [142]. Proposition 9 For any hyperkahler metric in D = An with n commuting tri-holomorphic [/(1) isometries there exists a coordinate system in wJiich takes the form where (Uij,Ai) are solutions of the generalized monopole equation and the coordinates (a; ,...,x") with і = 1,2,3 are the momentum maps of the tri-holomorphic vector fields d/dti. The spaces in the form(2.162) are known as the Pcdersen-Poon metrics. Equations (2.163) are known as monopole equations by interpreting Fxixj as a vector potential and t/j as certain Higgs fields. Some Pedersen-Poon examples will be constructed in the following section. It is convenient to introduce a notation for quaternion Kahler manifolds in terms of quaternions, that has the advantage to be more compact [22]. It is fundamental to observe that the holonomy of a quaternion Kahler space is in Sp(ri) x Sp(l) and therefore from the very beginning the connection ш will take values in the algebra sp(n) ф sp(l) with splitting и OJ+ + w_. The Lie algebra sp(n) of Sp(n) can be expressed in terms of quaternion valued matrices A with the property A + A = 0. Let us consider now two arbitrary H-valued 1-forms being I, J and К unit quaternions.
We define the quaternionic wedge product as +/i0 Л щ -f / і Лі/і+/і2Лг/2 + № Л і/3 and in particular for two pure quaternionic forms we have with components only in Im H. This product can be extended to H" valued 1-forms, with H-components given by The wedge product Л of the forms (2.166) is defined as and is clearly seen that fiKv is H-valued. Consider now a four dimensional metric g = e1 e1 + e2 e2 + є3 e3 + e4 e4. with an orthonormal tetrad e . We can extend such tetrad to a quaternionic valued one e e. The hyperkahler triplet associated to g tensor From the definition (2.165) it follows that In An dimensions we select the H"-valucd einbein e defined by The metric is then g = ea e". Taking into account the representation of Sp(n) in terms of quaternions we can write ш = ш+ + Ш- also as a H71 one-form. Then it can be shown that the first Cartan equation translates into and that the hyperkahler triplet is The quaternionic expression of the fundamental relations (2.9) and (2.7) in D=4n are Introducing the second (2.173) into the first to give expressing J and dJ entirely in terms of the sp{l) connection w_ and its differentials. The Swann construction allow to extend any quaternionic Kahler in dimension D = An to another 4(n + l)-dimensional quaternion Kahler one [22]. Before to present it let us consider an An dimensional hyperkahler manifold M with metric g = 5abea cb. From the definition g has holonomy in Sp(n). We make the trivial extension to a 4(n + l)-dimensionaI metric given by being {UQ,UI,U2,UZ) four new coordinates. The metric (2.175) is the direct sum of g plus a flat metric. Then the holonomy Г of (2.175) lies in Sp(n) С Sp(n+ 1), and thus (2.175) is a trivial extension of g to a higher dimensional hyperkahler one. Now let us generalize the trivial anzatz (2.175) by extending a quaternionic Kahler metric g = Sabe" eb to another one in D = 4(n + 1) of the form We have introduced the quaternions and the radius being I, J, К the unit quaternions. It is clear that if we consider an hyperkahler base g then uf_ — 0 and we recover (2.175) from (2.176) if the functions /(u2) and 7(M2) are simply constants. In general the explicit form of the metric is 9s = зИ2д + f[(du0 - «iwL)2 + {dm + uaiJ_ + е щшк_)2] (2.177) although the expression (2.176) will be better for the following. If gs is quaternion Kahler space then в is closed (see the discussion below (2.19)) and this gives a system of equation defining / and g. The fundamental four form (2.19) for gs is explicitly being a = du + іш_. If we want to impose d& = 0 to (2.176) we need the identities The Lie group G2 was considered by Berger [1] as one of the possible holonomy groups of a Riemmanian Ricci-flat space. He also proved that G2 holonomy is possible only in seven dimensions. The existence of G2 holonomy metrics was proved rigorously in [27] 30 years after the Berger work. In particular explicit non compact examples in [29], This spaces have direct application to Kaluza-Klein compactifications of Mheory. This is due to the fact that G2 holonomy manifolds have globally defined one Killing spinor 77 satisfying DiTj — 0 and therefore as internal spaces they preserve certain amount of supersymmetries after compactification. In fact the presence of such spinor is the reason for the reduction of the holonomy from 50(7) to G2. One of the main requirements in order to obtain a realistic four dimensional theory after compactification is that the internal space should be compact [26].
Hyperkahler torsion manifolds
We present some properties of hyperkahler torsion (or heterotic) geometry in four dimensions that makes it even more tractable than its hyperkahler counterpart. We show that in d = 4 hypercomplex structures and weak torsion hyperkahler geometries are the same. We present two equivalent formalisms describing such spaces, they are stated in the propositions of section 1. The first is reduced to solve a non linear system for a doblet of potential functions, which was found first Plebanski and Finley. The second is equivalent to find the solutions of a quadratic Ashtekar-Jacobson-Smolin like system, but without a volume preserving condition. Is in this second sense that heterotic spaces are simpler than usual hyperkahler ones. We also analyze the version of this geometry. Certain examples are presented, some of them are of the Callan-Harvey-Strominger type and others are not. In the conclusion we discuss the benefits and disadvantages of both formulations in detail. Supersymmetric a models in two dimensions with N = 2 supersymmetry occur on Kahler manifolds while N = 4 occurs on hyperkahler spaees [44]. More general supersymmetric о models can be constructed by including Wess-Zuminc-Witten type couplings in the action [94]-[48]. These couplings can be interpreted as torsion potentials and the relevant geometry is a generalization of the Kahler and hyperkahler ones with closed torsion. Such spaces are known as Kahler and hyperkahler torsion (HKT) geometries. The first examples of N—(4,4) SUSY sigma model with torsion (and the corresponding HKT geometry) were found in [176] and developed further in [177]-[179] by using of the harmonic superspace formalism. Hyperkahler torsion geometry is also an useful mathematical tool in order to construct heterotic string models [49]-[50]. In particular heterotic (4,0) supersymmetric models are those that lead to hyperkahler torsion geometry [51]-[53]. If the torsion is not closed we have a weak hyperkahler geometry and this case has also physical significance [55]-[57]. A direct way of classifying the possible HKT spaces is to find the most general weak hy- perkahler spaces and after that to impose the strong condition, i.e, the closure of the torsion. Callan, Harvey and Strominger noticed that under a conformal transformation any usual hyperkahler geometry is mapped into one with torsion [55]; such examples are sometimes called minimal in the literature [68] and they are not the most general [132]. As there is a particular interest in constructing spaces with at least one tri-holomorphic Killing vector due to applications related to dualities [58], there were found heterotic extensions of the known hyperkahler spaces and, in particular, the Eguchi-Hanson and Taub-Nut ones. The heterotic Eguchi Hanson geometry was shown to be conformal to the usual one, while heterotic Taub-Nut is a new geometry [132].
One of the main properties of hyperkahler torsion spaces is the integrability of the complex structures, that is, the annulation of their Nijenhuis tensor. In four dimensions this implies that the Weyl tensor of such manifolds is self-dual [59]-[61]. The converse of this statement is not true in general. On the other hand there exist a one to one correspondence between four dimensional self-dual structures with at least one isometry and 3-dimensional Einstein-Weyl structures [153]. The Einstein-Weyl condition is a generalization of the Einstein one to include щ conformal transformations, and weak hyperkahler spaces should correspond with certain special Einstein-Weyl metrics. In [G7]-[68] it was shown that the self-dual spaces corresponding to the round three sphere and the Berger sphere (which are Einstein-Weyl) are of heterotic type. Arguments related to the harmonic superspace formalism suggest that indeed there are more examples [69]-[70]. The present work is related to the construction of weak heterotic geometries in d = 4 without Killing vectors. This problem is of interest also because any weak space with isometrics should arise as subcases of those presented here. For the sake of clarity we resume the result presented in this letter in the following two equivalent propositions. Proposition 10 Consider a metric g defined on a manifold M together with three complex structures J1 satisfying the algebra J1 P = —Sij+eijiiJk and for which the metric is quaternion hermitian, i.e, g(X, Y) = g(J X, J Y), Define the conformal family ofmetrics [g] consisting of all the metrics g related to g by an arbitrary conformal transformation. a) Then we have the equivalence whereT and Nl(X,Y) are the Kahler form and the Niejenhuis tensor associated to J1, d is the usual exterior derivative and a is a 1-form. If any of (3.295) hold for g, then (3.295) is also satisfied for any g of the conformal structure [g], i.e, (3.295) is conformally invariant. b) Any four dimensional weak hyperkahler torsion metric is equivalent to one satisfying (3.295) and there exists a local coordinate system (x,y,p, q) for which tlie metric take the form up to a conformal transformation g —) u2g. The potentials Ф and Ф satisfy the non-linear system c) Conversely any metric (3.296) defines a confonnal family [g] in which all the elements g are weak hyperkahler torsion metrics. The torsion T corresponding to (3.296) is given by where S Фх — Фу. Under the conformal transformation g —» w2g the torsion is transformed as T - Proposition 10 should not be considered as a generalization of the Kahler formalism for weak HKT spaces. Although (Ф, Ф) is a doublet potential, the metric (3.296) is not written in complex coordinates.
The use of holomorphic coordinates for such spaces is described in detail in [132]. The following is an Ashtekar-Jacobson-Smolin like formulation for the same geometry. The action of the Hodge star g is defined by Proposition 11 Consider a representative g = 8аьеа еь of a conformal family [g] defined on a manifold M as at the beginning of Proposition 1, ea being tetrad 1-forms for which the metric is diagonal. a) Then all the elements g of [g] will be weak hyperkahler torsion iff [еье4] + [е2,ез] = -Л4Є1 - Аъе2 +A2e3 + Л1Є4 where ea is the dual tetrad of ea and Ai are arbitrary functions on M. b) Conversely any solution of (3.299) defines a conformal family [g\ in which all the elements g are weak hyperkahler torsion metrics. The torsion T corresponding to (3.296) is given by where clb are the structure functions defined by the Lie bracket [e0,et] — ccabec. The transformation of the torsion under g —t u2g follows directly from Proposition 1. To conclude, we should mention that the properties of hyperkahler torsion geometry in higher dimension were considered for instance, in [83]-[89], In particular, the quotient construction for HKT was achieved in [85]. Also it was found that when a sigma model is coupled to gravity the resulting target metric is a generalization of quaternionic Kahler geometry including torsion [87]. This result generalizes the classical one given by Witten and Bagger [?], who originally did not include a Wess-Zumino term to the action. A generalization of the Swann extension for quaternion Kahlcr torsion spaces was achieved [84]. To the knowledge of the authors the harmonic superspace description of quaternion Kahler geometry was already obtained in [180], [181] but the extension to the torsion case is still an open problem. The present part is organized as follows. In section 3.2 we define what is weak and heterotic geometry. We show that the problem to finding weak examples is equivalent to solving a conformal extension of the hyperkahler condition (namely, the first (3.295) given above) together with the integrability condition for the complex structure. In section 3.3 we present the consequences of (3.295) following a work of Plebanski and Finley [61]. We found out that integrability and the conformal extension of the hyperkahler condition arc equivalent in d = 4, which is the point a) of Proposition 10. Therefore weak heterotic geometry and hypercomplex structures are exactly the same concept. We also show that representatives of a given heterotic structure are determined by a conformal factor satisfying an inhomogeneous Laplace equation. Just in the case when the structure contains an hyperkahler metric it is possible to eliminate the inhomogeneous part by a conformal transformation. We discuss our results in the conclusions, together with possible applications. For completeness, we show in the appendix how this geometry arise in the context of supersymmetric sigma models.
Discussion
In our final results (4.383) and (4.384) the quantum corrections to the classical (logarithmic) terms (4.385) are exponentially suppressed by the D-instanton factors (in the semiclassical description). Unlike the case of a single matter hypermultiplet, the multi-hypermultiplet moduli space metric is also sensitive to the phases of іщ (i.e. not only to their absolute values). There are no perturbative (type IIA superstring) corrections, while all D-instanton numbers contribute to the solution. Our results can also be applied to an explicit construction of quaternionic metrics in real A(n — 1) dimensions out of known hyperkahler metrics in real An dimensions, which is of particular importance to a description of D-instantons in N=2 supergravity [22]. For instance, the four-dimensional quaternionic manifolds with toric isomctry can be fully classified [128] in terms of the PP-potential F obeying equation (4.377). It would be also interesting to connect our results to perturbative superstring calculations in the D-instanton background, which is still a largely unsolved problem. In this section some examples of toric hyperkahler metrics in eight dimensions are constructed. The Swann construction of hyperkahler metrics is applied to the Calderbank Pedersen spaces in order to find hyperkahler examples with /(1) x U(l) isometry. The connection with the Pedcrscn-Poon toric hyperkahler metrics is explained by finding the momentum map coordinate system. This hyperkahler examples are lifted to solutions of the D=ll supcrgravity with and without fluxes. Type IIA and IIB backgrounds are found by use of dualities. The Swann metric (2.186) is in general quaternion Kahlcr. In the special case with b = 0 # corresponds to an hyperkahler metric with explicit form By use of (5.387) it is direct to extend the Calderbank-Pedersen metrics (2.128) to an hyperkahler one. The quaternionic base is (2.128) and the corresponding w%_ are [128] The resulting expression for the hyperkahler metric is The Calderbank-Pedersen metrics (2.128) have two commuting Killing vectors д/д$ and д/д р. Clearly this are also Killing vectors of the spaces (5.389). The Kahler triplet corresponding to (2.128) This mean that j - and -щ are triholomorphic and therefore (5.389) is a toric hyperkahler metric of the type of the Proposition 9, where we should identity t\=0 and t? = ip. For applications to supergravity we will write the metric (5.389) in the form (2.162). Therefore we should find the momentum maps Similarly for д/д р it is found in accordance with [128]. The next step is to determine the matrix U for (2.177).
This is easily found by noticing that from (2.162) it follows that Then introducing the expression for (5.389) into the first (5.395) gives To find Ai one should obtain from (5.389) and (2.128) that 5( ,0 = ( + )(29, - )+2( + where A1 is given in (5.388) in terms of F. Then from the second (5.395) it is obtained Therefore we have reduced (5.389) with this data to the form (2.162). Formulas (5.397), (5.398) and (5.399) define a class of solutions of the Pedersen Poon equations (2.163) and an hyperkahler metric (2.162) simply described in terms of an unknown function F satisfying (2.129). We must recall however that this simplicity is just apparent because Щ depends explicitly on the coordinates (p, TJ, \q\2), which depends implicitly on the momentum maps {хг х\) by (5.393) and (5.394). Therefore Щ is given only as an implicit function of the momentum maps. For physical applications it is important to find solutions which in this limit tends to for a constant invertiblc matrix Uf? [134]. Formulas (5.393) and (5.394) shows that the asymptotic limit xg —У oo or Xp —У oo corresponds to q — со or fpF — 0. In consequence from (5.397) and (5.396) it follows that /IJ -л со and Щ 0 asymptotically, which is not the desired result. This problem can be evaded defining a new metric (2.162) with and with the same one-forms (5.398) and (5.399) and coordinate system (5.393) and (5.394). Clearly to add this constant do not affect the solution and this data is again a solution of the Pedersen-Poon equation (2.163) for which Ujj — ІІц and XT3 — U. Explicitly we have with inverse The hyperkahler spaces defined by (5.393), (5.394), (5.397), (5.398) and (5.399) can be extended to 11-dimensional supergravity solutions and to IIA and IIB backgrounds by use of dualities. This section present them following mainly [134], more details can be found there and in references therein. We consider D = 11 supergravity solutions with vanishing fermion fields and F ap. Such solutions are of the form that is, a direct sum of the flat metric on R3 and an hyperkahler space in eight dimensions. If all the fields in 11 dimensional supergravity are invariant under the action of an /(1) Killing vector then the 11-dimensional supergravity can be reduced to the type IIA one along the isomctry. In order to make such reduction we should write the 11-dimensional metric (5.404) as [175] The field Au is the 3-form potential and x are the coordinates of the D=10 spacctime. The NS & NS sector of the IIA supergravity is (ф, g , B ) and the R R sector is (C , A p).
After reduction of (5.404) along the isometry tx = p we find that the the non-vanishing fields arc The Killing spinor of the 11 dimensional supergravity was also independent of ti and the reduced spinor is again a IIA Killing spinor. The field ф is independent of 9 and С satisfies G — 0. Then one can use T-duality rules to construct a IIB supergravity solution [51] where I is the IIB pscudosealar, B is the Ramond-Ramond 2-form potential and D is the IIB 4-form potential. The non vanishing IIB fields resulting from the application of the T-duality are and #io is the Einstein frame metric satisfying The examples that we have presented till know are obtained by use of the isometries of the internal spaces. But more backgrounds can be obtained by reducing (5.404) along one of the space directions E2 1 and it is obtained the IIA solution with the other fields equal to zero. After T-dualizing in both angular directions it is obtained where A 1 = Xі, t . This solution can be lifted to a D—ll supergravity solution It is possible to generalize (5.404) to include a non vanishing 4-form F. The result is the membrane solution where H is an harmonic function on the hyperkahler manifold, i.e, satisfies We have seen in the last section that every entry of U,j is an harmonic function and so such H can be generated with an hyperbolic eigenfunction F. After reduction along ip it is found the following IIB solution If instead (5.421) is dimensionally reduced along a fiat direction it is obtained the IIA solution A double dualization gives a new IIA solution The lifting to eleven dimensions gives All the backgrounds presented in this section can be constructed with a single F satisfying (2.129), but the dependence on (xg,x,p) remains implicit. If the Bryant-Salamon extension is applied to the Calderbank-Pcdcrscn spaces the /(1) x U{\) isometry is preserved and the result is a toric ( holonomy metric. In this subsection will be constructed the G-i holonomy metrics corresponding to the examples A and В of subsection 5.7. We select such cases by simplicity, but the procedure could be applied to any quaternion Kahler space as well. By use of the results of the preceding section we extend this toric G2 holonomy metrics to vacuum configurations of the eleven dimensional supergravity preserving at least one supersymmetry. Also type IIA backgrounds will be found by reduction along one of the isometrics. It is straighforward to show that the explicit form of (2.247) when the base space (and, in consequence, the total one) has torus symmetry is