Hitchin Functionals, Generalized Geometry and an Exploration of Their Significance for Modern Theoretical Physics

SEPTEMBER 21, 2012

Hitchin Functionals, Generalized Geometry and an exploration of their significance for Modern Theoretical Physics

Supervised by Prof. Daniel Waldram

Cyrus Mostajeran

Submitted in Partial fulfilment of the requirements for the degree of Master of Science of Imperial College London.

Acknowledgments

I should like to thank Professor Waldram for his supervision and encouragement throughout the writing of this dissertation. It was a privilege to have access to the guidance of someone who has been actively involved in researching many of the topics touched upon in this dissertation. I would also like to thank the Theoretical Physics Group at Imperial for creating a warm atmosphere in which to work and in particular for their efforts in organising a range of consistently interesting seminars to which we were all made to feel welcome.

Special thanks to my parents for their constant love and support.

CONTENTS

I. Introduction 1.1 Mathematical foundations 1.2 Supersymmetry 1.3 Supergravity 1.4 Superstrings 1.5 M-theory

II. Hitchin Functionals 2.1 Special holonomy 2.2 Stable forms 2.3 Geometric structures as critical points of functionals 2.4 The geometry of stable three-forms in six dimensions 2.5 The geometry of stable three-forms in seven dimensions 2.6 Constrained critical points and weak holonomy 2.7 A Hamiltonian flow

III. Topological String Theory 3.1 = (2,2) supersymmetry 3.2 The “twisting” procedure and the topological string 풩 3.3 The topological A and B models 3.4 Hitchin’s and functionals in six dimensions 3.5 The functional as the A model 푉퐻 푉푆 3.6 The functional as the B model 푉푆

푉퐻 IV. BPS Black Holes 4.1 Black hole entropy 4.2 BPS black holes 4.3 = 2 supergravity 4.4 The microscopic description of black holes 푁 4.5 The attractor mechanism 4.6 BPS black holes in four dimensions 4.7 BPS black holes in five dimensions

V. The Geometry of Calabi-Yau Moduli Spaces 5.1 Calabi-Yau moduli spaces 5.2 Deformations of Calabi-Yau three-folds 5.3 The moduli space metric 5.4 The complex-structure moduli space 5.5 The Kahler-structure moduli space

̈ VI. Applications of Hitchin’s Hamiltonian Flow 6.1 Hitchin’s V functionals (3) 6.2 structures7 and metrics of holonomy 6.3 Topological M-theory on × 푆푈 퐺2 6.4 Canonical quantization of topological6 1 M-theory 푀 푆 6.5 Half-flat manifolds in = 1 string compactifications

풩 VII. Generalized Geometry 7.1 The linear algebra of 7.2 The Courant bracket ∗ 7.3 The structure of generalized푉 ⊕ 푉 geometries 7.4 The generalized tangent bundle 7.5 Generalized metrics

VIII. Generalized Calabi-Yau Manifolds 8.1 Linear generalized complex structures 8.2 Generalized complex structures on manifolds 8.3 Spinors 8.4 Generalized Calabi-Yau Manifolds 8.5 B-field transforms 8.6 Structure group reductions

IX. The Generalized Hitchin Functional 9.1 An invariant quartic function 9.2 Generalized geometry from a variational perspective 9.3 The topological B-model at one loop

X. Type II Supergravity in the language of Generalized Geometry 10.1 Type II supergravity 10.2 Conformal split frames 10.3 Generalized connections 10.4 The (9,1) × (1,9) structure 10.5 Torsion-free, compatible generalized connections 10.6 Fermionic푂 degrees푂 of freedom 10.7 RR fields 10.8 The equations of motion

XI. Final Remarks 11.1 Related topics in the existing physics literature 11.2 Directions for future research

I. Introduction

1.1 Mathematical foundations

The two key geometrical concepts that provide the mathematical foundations for this dissertation are Hitchin functionals and Generalized geometry, both of which arose from the work of the mathematician Nigel Hitchin. [1] The common historical origin of both of these ideas lies in the observation that in dimensions 6 and 7 there exists a well-defined notion of non-degeneracy of a differential three-form by virtue of the fact that if is a real vector space of dimension 6 or 7, then the group ( ) has an open orbit. In 푉 particular, in 6 dimensions there exists an open orbit with stabilizer conjugate to (3, ) 퐺퐿 푉 and in 7 dimensions there exists an open orbit with stabilizer conjugate to the exceptional 푆퐿 ℂ Lie group . A 3-form that lies in either of these open orbits thus determines a reduction of the structure group of the manifold on which it is defined to the 퐺2 Ω corresponding stabilizer subgroup. Such a 3-form is said to be stable and it is this notion of stability that we take as the generalization of the푀 non-degeneracy of a differential form.

The key point is that if the 3-form lies in an open orbit everywhere and is also a critical point of certain diffeomorphism-invariant functionals defined on the space of 3-forms on Ω the manifold , then the reduction of the structure group is integrable in the sense that we obtain a complex threefold with trivial canonical bundle in 6 dimensions, and a 푀 Riemannian manifold with holonomy in 7 dimensions. These diffeomorphism- invariant functionals whose critical points2 yield such geometrical structures in this way are known as Hitchin functionals. Hitchin퐺 functionals have appeared in various contexts in the recent physics literature and part of this dissertation is devoted to an exploration of their significance as far as theoretical physicists are concerned. In particular, we shall consider their relation to topological string theories, black holes and Calabi-Yau moduli spaces.

Generalized geometry incorporates complex and symplectic geometry as special cases of a unifying geometrical structure. The starting point of generalized geometry is the replacement of the tangent bundle with the direct sum of the tangent and cotangent bundles , and use of the Courant bracket on sections of the bundle as the 푇 natural generalization∗ of the Lie bracket defined on sections of . This gives rise to∗ a 푇 ⊕ 푇 푇 ⊕ 푇 geometry with the novel feature that it transforms naturally under both the action of a 푇 closed 2-form and the usual diffeomorphism group. It turns out that generalized geometry leads to a language ideally suited for the systematic description of supergravity backgrounds. In퐵 particular, we will show how the supersymmetry transformations and equations of motion of Type II supergravity can be written in a simple, elegant and manifestly (9,1) × (1,9) covariant form.

푆푝푖푛 푆푝푖푛

In light of the effectiveness of generalized geometry in describing supergravity backgrounds, it is interesting to note that its historical origin can be found in the very different context of Hitchin’s work on characterizing special geometry in low dimensions using invariant functionals of differential forms. In particular, in six dimensions Hitchin constructs an invariant functional whose critical points yield generalized Calabi-Yau manifolds which belong to a particular class of geometrical structure that is compatible with the general framework of generalized geometry. Indeed, it was within this context that the basic structure of generalized geometry was discovered. This generalized Hitchin functional thus provides the link between Hitchin functionals and generalized geometry in the special 6-dimensional case. Unlike Hitchin functionals, however, generalized geometry is not restricted to specific low dimensions and can indeed be explained without any reference to invariant functionals of differential forms. Nonetheless, it is an interesting observation that generalized geometry essentially has its historical origin in work that relied on the simple mathematical fact known for more than a century [2] that if is a real vector space of dimension 6 or 7, then the group ( ) has an open orbit.

푉1.2 Supersymmetry 퐺퐿 푉

Supersymmetry is a deep symmetry between boson and fermion fields. Consistency of string theory requires supersymmetry, without which one is only left with the nonsupersymmetric bosonic string theories which are clearly unrealistic since they do not incorporate fermions. Any string theory that includes fermions relies on local supersymmetry for its mathematical consistency. Supersymmetry is also perhaps the key concept that is a common feature of all of the theories from across the theoretical physics landscape that we shall consider in this dissertation.

1.3 Supergravity

Supergravity is an extension of Einstein’s general relativity which inlcludes supersymmetry. Supergravity can be seen as naturally arising in the process of making supersymmetry transformations local. It turns out that this process requires the introduction of a graviton field and its supersymmetric partner the gravitino. This theory of local supersymmetry is known as supergravity.

1.4 Superstrings

Superstring theory is a version of string theory that incorporates supersymmetry. Consideration of both left-moving and right-moving modes using superstring formalism leads one to two different types of superstring theory known as type IIA and type IIB. A third type of superstring theory can be derived from type IIB using a procedure known as orientifold projection, which leaves only strings that are unoriented.

The first superstring revolution was sparked in the 1980s with the discovery that consistency of a ten-dimensional = 1 supersymmetric theory with quantum mechanics requires a local Yang-Mills gauge symmetry based on one of two possible Lie algebras: 푁 (32) or × . Using the formalism of the 26-dimensional bosonic string for the left-moving modes8 8 and the formalism of the 10-dimensional superstring theory for the right푆푂 -moving퐸 modes퐸 leads to two other types of supersymmetric string theory known as heterotic theories.

Thus, in total we have five distinct types of ten-dimensional supersymmetric string theory. Type I and the two heterotic theories have = 1 supersymmetry. The type I theory has the gauge group (32), while the heterotic theories realize both (32) and 푁 × . There are 16 conserved supercharges in each of these three theories. Type IIA 푆푂 푆푂 and type IIB theories on the other hand have = 2 supersymmetry with 32 supercharges. 퐸8 퐸8 Later it was discovered that some of these theories푁 can be related via certain dualities. In particular, T-duality relates the two type II theories to each other as well as the two heterotic theories. As part of the second superstring revolution of the 1990s, S-duality was discovered as another kind of duality linking some of the different superstring theories at strong string coupling. For example, it was found that strongly coupled type I theory is equivalent under S-duality to weakly coupled (32) heterotic theory. S-duality also provides a way of relating the type IIB superstring theory to itself, thereby essentially establishing a symmetry in this case. 푆푂

1.5 M-theory

S-duality explains the behaviour of type IIB, type I and the (32) heterotic superstring theory at strong string coupling . For large , the other two types of theory 푆푂 are understood to grow an eleventh dimension of size , where is the fundamental 푔푠 푔푠 string length scale. In type IIA this new dimension is a circle, whereas in the heterotic 푔푠푙푠 푙푠 theory it is a line interval. When is large, non-perturbative techniques are required which lead to a new type of quantum푠 푠 theory in 11 dimensions, known as M-theory. At low energies, M-theory is approximated푔 푙 by a classical field theory known as 11- dimensional supergravity. M-theory and the dualities described above provide a web of dualities that in effect unify the five superstring theories into a single theory.

II. Hitchin Functionals

In this chapter we define the notion of stability of a differential form as an extension of the non-degeneracy of bilinear forms and see how stable -forms only exist for specific values of in certain low dimensions. We then introduce Hitchin functionals as 푝 diffeomorphism-invariant functionals whose critical points on the space of stable forms yield interesting푝 geometrical structures. In particular, Hitchin functionals are constructed in six and seven dimensions which generate Calabi-Yau algebraic three-folds and holonomy manifolds, respectively. We move on to introduce the idea of weak holonomy2 and consider its relation to certain constrained variational problems. Finally, we discuss퐺 a Hamiltonian flow construction by Hitchin that provides a bridge between (3) structures and holonomy manifolds. 푆푈 2.1 Special holonomy퐺2

In this section we introduce the notion of special holonomy and show how it naturally leads us to the idea of stable differential forms which provide the starting point for the construction of Hitchin functionals. Riemannian manifolds with special holonomy also play an important role in string theory compactifications as we shall explain shortly. We begin by defining the holonomy group of a Riemannian manifold at a point.

Definition 2.1: Let ( , ) be an n-dimensional Riemannian manifold with an affine connection . Let p be a point in ( , ) and consider the set of closed loops at , 푀 푔 { ( ): 0 1, (0) = (1) = }. By parallel transporting a vector X along 훻 푀 푔 푝 a loop we obtain a new vector . In this way the loop ( ) and the connection 훾 푡 ≤ 푡 ≤ 훾 훾 푝 ∈ 푇푃푀 define a linear transformation on the tangent space at . The set of these transformations 훾 푋훾 ∈ 푇푃푀 훾 푡 is called the holonomy group at . 푝 Berger’s work on holonomy groups푝 of Riemannian manifolds in the 1950’s led to a classification of the holonomy groups of manifolds which are not locally a product space. It was essentially discovered that under this irreducibility assumption, the manifold is either locally a Riemannian symmetric space [5] or the holonomy group is given by one of the following:

(1) ( )

(2) 푆푂( 푛/2): Kahler manifolds

(3) 푈 푛( /2): Calabï -Yau manifolds

(4) 푆푈(1푛) ( /4): quaternion Kahler manifolds

(5) 푆푝( /푆푝4): 푛hyperkahler manifolds̈

푆푝 푛 ̈

(6) : special manifolds in dimension 7

(7) 퐺2 (7): special manifolds in dimension 8. [6]

Among푆푝푖푛 these the manifolds with holonomy groups , (7), ( /2), ( /4) are Ricci-flat in the sense that they have vanishing Ricci curvature. 퐺2 푆푝푖푛 푆푈 푛 푆푝 푛 In general, the holonomy group is a subgroup of ( , ). If is a metric connection so that ( , ) = ( , ) for all , then the holonomy group is a subgroup of 퐺퐿 푛 ℝ ∇ ( ) provided ( , ) is orientable and Riemannian. That is, for an -dimensional 푔 푋훾 푋훾 푔 푋 푋 푋 ∈ 푇푃푀 Riemannian manifold M, we have ( ) ( ). 푆푂 푛 푀 푔 푛 Definition 2.2: The manifolds with퐻표푙 special푀 hol⊆ 푆푂onomy푛 are those whose holonomy groups satisfy the relation ( ) ( ).

These structures are퐻표푙 characterized푀 ⊂ 푆푂 by푛 the existence of covariantly constant spinors; that is, spinor fields satisfying

휓 = 0. (2.1)

It is the existence of such spinors that lies∇휓 behind the significance of manifolds with special holonomy in string theory. The covariantly constant character of the spinor field ensures that the superstring compactification on preserves a definite fraction of the original 32 supercharges. 푀 It is also a property of special holonomy manifolds that they possess certain invariant p- forms know as calibrations. These are closed forms on M that define special kinds of minimal submanifold known as calibrated submanifolds and are non-trivial only for specific values of . We may use the covariantly constant spinor to construct a calibration on as follows: 푝 휓 ( ) 푀 = … . (2.2) 푝 † 푖1 푖푝 Our interest in these objects lies in휔 the fact휓 that훾 they휓 provide an example of how geometric structure can be characterized by differential forms and action functionals. To elucidate this further, we recall that for a minimal submanifold of real dimension , we may calculate the volume as Vol(S) = from knowledge of the metric . 푆 ⊂ 푀 On the other hand, via the calibration ( ) we may푝 compute the volume as 푝 ∫푆 푑 푥�푔 푔 푝 Vol휔(S) = ( ) (2.3) 푝 ∫푆 휔

9 using a differential form and without explicit use of the metric. As we shall see, the idea of characterizing geometric structure by means of invariant functionals of differential forms lies at the heart of this chapter.

2.2 Stable forms

We seek to arrive at a general notion of non-degeneracy of a p-form. To do this we turn to symplectic geometry for inspiration. A symplectic manifold of dimension is a differentiable manifold endowed with a closed, non-degenerate 2-form where non- 푛 degeneracy means / 0. An equivalent way of capturing this notion of non- 휔 degeneracy of 푛 2 at each point is in terms of the action of ( ) on , where 휔 ≠ is the tangent space 2at the∗ point. This alternative viewpoint is expressed by the2 ∗ 휔 ∈ Λ 푉 퐺퐿 푉 Λ 푉 requirement that the orbit of be open under the above action. This brings us in a 푉 position to define a suitable notion of non-degeneracy for a general p-form, which we shall refer to as stability, as described휔 in [8].

Definition 2.3: Let be the tangent space at a point , and be the space of p-forms at x. Then a p-form is said to be stable at if ( ) lies푝 ∗ in an open orbit of the 푉 푥 훬 푉 ( ) action on . We푝 say∗ is a stable form if it is stable at every point. 푝휌 ∗∈ 훬 푉 푥 휌 푥 The퐺퐿 푉word “stability”훬 푉 in the above휌 definition refers to stability of in the sense of deformation invariance: all forms in a neighbourhood of are equivalent to by a local 휌 ( ) action. 휌 휌 퐺퐿 How푉 often do stable forms occur? To answer this question we look at the stabilizer subgroups of in ( ). For a stable 2-form in even dimension n the stabilizer is the symplectic group푝 ∗ ( ). This simply corresponds to the fact that the 2-form defines 휌 ∈ Λ 푉 퐺퐿 푉 a presymplectic structure as one would expect. Denoting the stabilizer subgroup of by 푆푝 푛 , so that = { ( ): ( ) = }, we have: 휌 휌 휌 퐺 퐺 푔 ∈ 퐺퐿 푉 푔 휌= 휌 ( ) . (2.4) 푝 ∗ 휌 Since the growth of 푑푖푚 퐺= !/푑푖푚! ( 퐺퐿 푉)!− is typically푑푖푚 훬 푉 much faster than that of ( ) = , it is clear푝 that∗ in the vast majority of cases GL(V) cannot act locally 푑푖푚 훬 푉 푛 푝 푛 − 푝 transitively on the2 space of p-forms. Hence we see that stability of differential forms is 푑푖푚indeed퐺퐿 a 푉rather 푛unique phenomenon which is only seen non-trivially for certain combinations of values of n and p. In fact, other than the obvious cases of p = 1 and 2, we find that up to certain formal dualities there are only three other cases. These are when p = 3 and n = 6, 7, 8. As we will only be interested in the geometric structures that arise in the cases n = 6 and 7, we will not discuss the case n = 8 any further. With the combination p = 3, n = 6 we find for any 3-form :

휌

= ( ) 푝 ∗ 휌 푑푖푚 퐺 = 푑푖푚 퐺퐿!/ 푉! (− 푑푖푚)! 훬 푉 2 = 푛36 − 푛20 푝= 16푛 .− 푝

We note that the Lie group SL(3, ) has real− dimension 16 and it can be shown that indeed SL(3, ) is the stabilizer group as the dimension counting argument above would suggest. ℂ A similar calculation for the case p = 3, n = 7 for a 3-form gives dim = 49 – 35 = 14. ℂ We note that the exceptional Lie group G2 has dimension 14 and it turns out that in this 휌 퐺휌 case G2 is the stabilizer of the three-form. [8]

Before continuing to the next section we briefly describe how we can deduce the existence of stable ( )-forms from knowledge of the existence of stable p-forms in dimensions. If ( ) has an open orbit on then it also does on the dual space 푛 − 푝 푛 . As scalars act non-trivially푝 ∗ for 0, it follows that there is an 퐺퐿 푉 Λ 푉 open푝 orbit푛 −on푝 ∗ 푛. In particular, we can also consider stable 4-forms in 7 Λ 푉 ≅ Λ 푉 ⊗ Λ 푉 푝 ≠ dimensions. 푛 − 푝 ∗ Λ 푉 2.3 Geometric structures as critical points of functionals

The stabilizers of discussed in the previous section, as well as the symplectic group ( ), each preserve a volume element ( ) . Hence if we have a stable p-form 휌 defined on a compact oriented manifold , we can푛 ∗integrate the volume form ( ) 푆푝 푛 휙 휌 ∈ Λ 푉 over to obtain a volume ( ). This provides a natural way of defining the action of a 휌 푀 휙 휌 p-form on a manifold. The stability of the differential form , in the sense that it lies in an 푀 푉 휌 open orbit of ( ), ensures that nearby forms are also stable so that the volume 휌 functional is defined and smooth on an open set of forms. 푈 퐺퐿 푉 The volume form ( ) associated to determines a ( )-invariant map . 푛 ∗ The derivative of 휙 at휌 is an invariantly휌 defined element퐺퐿 푉 of ( ) 휙 ∶.푈 As→ Λ 푉 ( ) , the derivative lies in . Therefore,푝 ∗ ∗ there푛 ∗exists a 휙 휌 Λ 푉 ⊗ Λ 푉 unique푝 ∗ ∗ 푛 − 푝 ∗ such 푛that 푛 − 푝 ∗ Λ 푉 ≅ Λ푛 − 푝푉∗ ⊗ Λ 푉 Λ 푉 휌 � ∈ Λ 푉 ( ) = . (2.5)

Now invariance applied to the action퐷휙 of휌 thė scalar휌� ∧ 휌 matriceṡ implies

( ) = ( ). (2.6) 푝 푛 That is, is homogeneous of degree휙 휆 /휌 . Thus,휆 휙 writing휌 = in (2.5) and using Euler’s formula for a homogeneous function we find that 휙 푛 푝 휌 휌̇

= ( ). (2.7) 푛 푝 Now let M be a closed, oriented n-dimensional휌� ∧ 휌 휙 manifold.휌 If ( ) is a globally defined stable p-form, we can define a volume functional as 푝 휌 ∈ Ω 푀 ( ) = ( ). (2.8)

By virtue of the stability of it follows푉 휌 that∫ forms푀 휙 휌 in an open neighbourhood of will also be stable and so we can differentiate the functional. 휌 휌 Theorem 2.1: A closed stable form ( ) is a critical point of ( ) in its cohomology class if and only if = 0. 푝 휌 ∈ 훺 푀 푉 휌 Proof: The first variation yields:푑 휌�

( ) = ( ) = , where we have used (2.5). We훿푉 consider휌̇ ∫푀 variations퐷휙 휌̇ restricted∫푀 휌� ∧ 휌̇ to closed stable p-forms within a fixed cohomology class so that = . Hence,

( ) = 휌̇ 푑훼= ± by Stokes’ theorem. 훿푉 휌̇ ∫푀 휌� ∧ 푑훼 ∫푀 푑휌� ∧ 훼

We see that the variation vanishes for arbitrary if and only if = 0.

푑훼 푑 휌�

Recall that a symplectic manifold was defined as a differentiable manifold equipped ∎ with a closed, non-degenerate 2-form . The skew-symmetric character of differential forms combined with the fact that skew-symmetric matrices are singular in odd dimensions implies that symplectic manifolds휔 only exist in even dimensions. Let M be a symplectic manifold of dimension 2 . We take the Liouville volume form ( ) = and find that the volume ( ) = ( ) is then constant on a fixed cohomology ! 1 푚 휙 휔 class푚 [ ]. Thus we see that we can encode the geometry of a symplectic manifold by 푚 휔 푉 휔 ∫푀 휙 휌 means of a volume functional. Although this example may seem somewhat trivial since the functional휔 we have constructed has turned out to be constant, it does nonetheless illustrate the manner in which geometric structures can arise as critical points. We now turn to the much more interesting cases of geometries defined by the action of three- forms in 6 and 7 dimensions. In the next two sections we will be exploring the work done in [8] in some detail.

2.4 The geometry of stable three-forms in 6 dimensions

Let be a 6-dimensional real vector space and . Let and consider the interior product ( ) . We have ( ) 3 ∗ . The exterior product 푊 Ω ∈ Λ 푊 푤 ∈ 푊 pairing 2 ∗implies the isomorphism 5 ∗ . We may ı 푤 Ω ∈ Λ 푊 ı 푤 Ω ∧ Ω ∈ Λ 푊 now define∗ a linear5 transformation∗ 6 ∗ : 5 by ∗ 6 ∗ 푊 ⊗ Λ 푊 → Λ 푊 6 Λ∗ 푊 ≅ 푊 ⊗ Λ 푊 (퐾Ω) 푊= →( (푊)⊗ Λ 푊), (2.9) where : 퐾 denotesΩ 푤 the퐴 ıisomorphism푤 Ω ∧ Ω provided by the natural exterior product5 ∗ pairing. 6 ∗ 퐴 Λ 푊 → 푊 ⊗ Λ 푊 Using we define ( ) ( ) as 6 ∗ 2 Ω 퐾 휆 Ω ∈ Λ 푊 ( ) = . (2.10) 1 2 Ω Now for a real one-dimensional vector휆 Ω space 6L,푡푟 we퐾 define a vector = as positive ( > 0) if = for some and negative if is positive. In what2 푢 ∈ 퐿 ⊗ 퐿 퐿 follows we shall make use of the following lemma, the details of the proof of which can be found 푢in [8]. 푢 푠 ⊗ 푠 푠 ∈ 퐿 −푢

Lemma 2.1: Assume ( ) 0 for . Then 3 ∗ i) ( ) > 0 if and휆 훺 only≠ if =훺 ∈ +훬 푊 where , are real decomposable 3-forms and 0; 휆 훺 훺 훼 훽 훼 훽 ii) ( ) < 0 if and only if = + where ( ) is a complex 훼 ∧ 훽 ≠ decomposable 3-form and 0. 3 ∗ 휆 훺 훺 훼 훼� 훼 ∈ 훬 푊 ⊗ ℂ Furthermore, these decomposable 3훼-forms∧ 훼� ≠ are unique up to ordering.

The lemma ensures that the following is well-defined.

Definition 2.4: Let be oriented and be such that ( ) 0. Writing in terms of decomposable 3-forms ordered by the3 orientation,∗ we define the complementary 푊 훺 ∈ 훬 푊 훺 ≠ 훺 3-form by 3 ∗ i) 훺if� ∈(훬 )푊> 0, and = + then = ; ii) if ( ) < 0, and = + then = ( ). 휆 훺 훺 훼 훽 훺� 훼 − 훽 The complementary휆 훺 3-form훺 has훼 the훼� defining훺� property푖 훼 − 훼 �that if ( ) > 0 then + is decomposable and if ( ) < 0 then + is decomposable. We also note that = . Ω� 휆 Ω Ω Ω� From here on we shall휆 mainlyΩ be interestedΩ 푖Ω� in the open set = { (Ω� ) <−0Ω}. 3 ∗ 푈 Ω ∈ Λ 푊 ∶ 휆 Ω

If ( ) < 0, then is the real part of the complex form of type (3,0):

휆 Ω Ω = + . (2.11) 푐 We can also show that Ω Ω 푖Ω�

= 2 ( ). (2.12)

This leads us to a natural symplecticΩ ∧ interpretationΩ� �−휆 Ω of ( ) which we shall explore a bit further before proceeding to state and prove the main theorem of this section. 휆 Ω Pick a non-zero vector and note that is preserved by the group of linear transformations ( ) 6( ∗). We endow with the symplectic form defined 휖 ∈ Λ 푊 휖 by 3 ∗ 푆퐿 푊 ⊂ 퐺퐿 푊 Λ 푊 휔 ( , ) = . (2.13) 6 ∗ By the invariance of under휔 Ω the1 Ωaction2 휖 ofΩ 1 ∧(Ω2)∈ weΛ have푊 the existence of the moment map : ( ) where ( ) is the Lie algebra of ( ). We may identify this 휔 푆퐿 푊 Lie algebra3 with∗ its dual∗ via the bi-invariant form ( ), which then allows us to write 휇 Λ 푊 → 픰픩 푊 햘햑 푊 푆퐿 푊 the linear map defined by (2.9) in terms of the moment map simply as ( ) = . We 푡푟 푋푌 now see that ( ) is an ( )- invariant function which by (2.10) can be expressed as 퐾Ω 휇 Ω 퐾Ω 휆 Ω 푆퐿 푊 ( ) = ( ( ) ). (2.14) 1 2 Restriction of the symplectic form휆 Ωto the 6open푡푟 휇 setΩ = { ( ) < 0} defines the structure of a symplectic manifold. On we define a smooth3 function∗ as 푈 Ω ∈ Λ 푊 ∶ 휆 Ω ( ) = 푈 ( ). 휙 (2.15)

We rewrite (2.12) using this smooth휙 Ω function�− 휆as Ω = 2 so that 2 = . = 2 = = = ( ). Recalling that we find that Ω ∧ Ω� 휙휖 . Hence휙� Ω��휖 Ω� ∧ Ω� � The function Ω� defines−Ω a Hamiltonian휙 vector�Ω��휖 field− Ω� ∧ onΩ .Ω ∧ Ω� 휙�Ω�� 휙 Ω

휙 We now turn 휙to our main result in this section. Let 푋 be a푈 closed, oriented 6-dimensional manifold. Let be a 3-form on which defines at each point a vector in and thus a 푀 global section ( ) of ( ) . As | ( )|defines a non-negative continuous3 section∗ of Ω 푀 Λ 푇 ( ) , we may obtain a6 section∗ 2 of by taking the square root and indeed find that 휆 Ω Λ 푇 휆 Ω |6 (∗ )2| is a smooth continuous 6-form6 for∗ ( ) 0. We define a functional on Λ 푇 Λ 푇 ( ) by integrating this 6-form over : �∞휆 Ω3 ∗ 휆 Ω ≠ Φ 퐶 Λ 푇 ( ) = 푀 | ( )|. (2.16)

Φ Ω ∫푀 � 휆 Ω

The (6, )-invariance of implies that the functional is invariant under the action of orientation-preserving differomorphisms. 퐺퐿 ℝ 휆 Φ Theorem 2.2: Let be a compact complex 3-dimensional manifold with trivial canonical bundle and be the real part of a non-vanishing holomorphic 3-form. Then is a 푀 critical point of the functional restricted to the cohomology class [ ] ( , ). 훺 3훺 Conversely, if is a critical point훷 on a cohomology class of an oriented훺 ∈closed퐻 푀 6-ℝ dimensional manifold and ( ) < 0 everywhere, then is the real part of a non- 훺 vanishing holomorphic 3-form. 푀 휆 Ω 훺 Proof: Let be a non-vanishing 6-form on and the real part of a non-vanishing holomorphic 3-form. Thus is a closed 3-form with ( ) < 0 and writing 휖 푀 훺 ( ) = ( ) we have 2 2 훺 휆 Ω 휆 Ω − 휙 훺 휖 ( ) = ( ) . (2.17)

As ( ) 0, is smooth enablingΦ usΩ to take∫푀 the휙 훺first휖 variation of this functional:

Ω ≠ 휙 = . (2.18)

Using the symplectic interpretation훿Φ �weΩ̇ � discuss∫푀 푑휙ed �earlierΩ̇ �휖 we have = ( , ) and = is the Hamiltonian vector field, so that = . Therefore 푑휙�Ω̇ � 휔 푋휙 Ω̇ 휙 푋 − Ω � = 푑휙�Ω. ̇ � 휖 − Ω� ∧ Ω ̇ (2.20)

Since the variation is within a fixed훿Φ� cohomologyΩ̇ � − ∫푀 Ω� class∧ Ω̇ we may write = for a 2-form . Hence Ω̇ 푑휑 휑 = = (2.21) using Stokes’ theorem. If훿Φ follows�Ω̇ � that− ∫푀 Ω� ∧= 푑휑0 at ∫ for푀 푑 allΩ� ∧ 휑 if and only if = 0.

If is a complex manifold with a non훿-vanishingΦ Ω holomorphic휑 3-form +푑Ω� , then (since a holomorphic 3-form is closed) = = 0 and so is a critical point of . 푀 Ω 푖Ω� For the converse, we assume that푑 Ω +푑Ω� = 0 and Ω( ) < 0. For ( ) <Φ0 we may define an almost complex structure by = where is the linear map 푑�Ω 푖Ω�� ( )휆 Ω 휆 Ω 1 defined in (2.9). Now a complex 1-form퐼Ω 퐼 Ωis of �type−휆 Ω(1퐾,0Ω) with respect퐾Ω to if and only if +휃 = 0. 퐼 Ω (2.22)

Taking the exterior derivative of 휃this∧ �equationΩ 푖Ω�� yields

+ = 0, (2.23) implying that has no (0,2) component.푑휃 ∧ �Ω By푖Ω� the� Newlander-Nirenberg theorem we deduce that is integrable. As + is of type (3,0), we have 푑휃 Ω 퐼 0 = �Ω +푖Ω�� = + . (2.24)

That is, + is a holomorphic푑� 3Ω-form.푖Ω�� This휕 completes̅�Ω 푖Ω�� the proof.

Ω 푖 Ω �

We recall that a Calabi-Yau 3-fold can be defined as a Kahler manifold with a covariant∎ constant holomorphic 3-form. Thus we see that the structure of a Calabi-Yau complex algebraic 3-fold can be encoded as a critical point of a diffeomorphism̈ -invariant functional in dimension 6.

2.5 The geometry of stable three-forms in 7 dimensions

We now present a parallel treatment of the 7 dimensional case. Let be a 7-dimensional real vector space and . Define : by 3 ∗ 7 ∗ 푊 Ω ( Ω, ∈)Λ=푊 ( ) 퐵( 푊) ⊗ 푊, → Λ 푊, (2.25) 1 Ω where the interior퐵 products푣 푤 (−)6 ı, 푣( Ω)∧ ı 푤 Ω ∧ .Ω By the∀ properties푣 푤 ∈ 푊 of the exterior product we see that the 2-forms ( ) and ( 2) ∗commute in the expression in (2.25) ı 푣 Ω ı 푤 Ω ∈ Λ 푊 and so the equation is symmetric under the interchange of and . Thus is a ı 푣 Ω ı 푤 Ω symmetric bilinear form on with values in the one-dimensional space and so we 푣 푤 퐵Ω may define a linear map : which satisfies 7 ∗ 푊 7 ∗ Λ 푊 퐾 Ω 푊 → 푊 ⊗ Λ(푊 ) . (2.26) 7 ∗ 9 If 0, we find that an orientation푑푒푡 퐾Ω ∈ is Λinduced푊 on (as the exponent 9 is an odd number) with respect to which we take the positive root ( ) / . This 푑푒푡 퐾Ω ≠ 푊 allows us to define a natural inner product 1 9 7 ∗ 푑푒푡 퐾Ω ∈ Λ 푊 ( , ) = ( , )( ) / (2.27) −1 9 with volume form ( 푔)Ω /푣. 푤 퐵Ω 푣 푤 푑푒푡 퐾Ω 1 9 Recall that in our earlier푑푒푡 퐾 discussionΩ of stability of differential forms we noted the existence of open orbits of forms under the action of ( ) as a rather special feature of a vector space of dimension 6 or 7. Here we will need to define a particular open orbit 퐺퐿 푉 which will play a significant role in the remainder of our analysis. We begin by choosing 푉 a basis , , of and its dual basis , , in . Define the 3-form ∗ 푤1 ⋯ 푤7 푊 휃1 ⋯ 휃7 푊

= ( ) + ( ) + ( ) + . 휑 휃1 ∧ 휃2 − 휃3 ∧ 휃4 ∧ 휃5 휃1 ∧ 휃3 − 휃4 ∧ 휃2 ∧ 휃6 휃1 ∧ 휃4 − 휃2 ∧ 휃3 ∧ 휃7 The special significance휃4휃6 휃of7 is that it has the property that when substituted in (2.27) it yields the Euclidean metric = . We denote the open orbit of by and 휑 define a form to be positive if it lies in2 this orbit. 3 ∗ 푔휑 ∑ 휃푖 휑 푈 ⊂ Λ 푊 Definition 2.5: Pick a basis vector and let be a positive 3-form. Define ( ) by ( ) = ( ) / . 7 ∗ 1 9휖 ∈ Λ 푊 Ω Let휙 Ω ∈ ℝ 휙 Ω ϵ be푑푒푡 the퐾 HodgeΩ star operator associated with the metric . Using the 3-form3 ∗ it can4 be∗ shown that the invariant satisfies = 6 . ∗∶ Λ 푊 → Λ 푊 푔Ω The non-degenerate휑 pairing provided by the exterior휙 productΩ ∧ sets∗ Ω up an휙 isomorphism ( ) . Hence is a -valued function. 3 ∗ ∗ 4 ∗ 4 ∗ TheΛ 푊 Hodge≅ starΛ 푊 identifies 푑휙 andΛ 푊 in an invariant way. It has been shown in the mathematics literature that 3 , ∗which is4 the∗ stabilizer group of , has only the orthogonal Λ 푊 Λ 푊 projection onto as an invariant of its action on . [11] Therefore we must have 퐺2 Ω = ( ) for some real function of3 ∗. Ω Λ 푊 We푑휙� noteΩ̇ � by퐶 DefinitionΩ ∗ Ω ∧ Ω2.5̇ that since is homogeneous퐶 Ω of degree 3, is also homogeneous and of degree 3 × = .Ω Taking = in = ( ) and 7 퐾7 휙 exploiting the homogeneity of , we obtain 9 3 Ω Ω̇ 푑휙�Ω̇ � 퐶 Ω ∗ Ω ∧ Ω̇ ( )( 휙 ) = ( ) = ( ). (2.28) 7 퐶 Ω ∗ Ω ∧ Ω 푑휙 Ω 3 휙 Ω Comparison with = 6 shows that = . That is, is a real constant. 7 We now define theΩ relevant∧ ∗ Ω functional휙 in dimension퐶 18 7 and study퐶 its critical points. We have established that if is a 3-form on a closed 7-dimensional manifold , then ( ) / defines a section of which is a volume form of the metric whenever Ω 푀 109 7 ∗ ( ) Ω . We define a functional on as Ω 푑푒푡 퐾 Λ 푇 ∞ 3 ∗ 푔 푑푒푡 퐾 Ω ≠ ( )Φ= 퐶( Λ 푇 ) / . (2.29) 1 9 Ω Theorem 2.3: Let be a closedΦ 7-Ωdimensional∫푀 푑푒푡 manifold퐾 with a metric with holonomy , with defining 3-form . Then is a critical point of the functional restricted to the 푀 퐺2 cohomology class [ ] ( , ). Ω 3 Ω Φ Ω ∈ 퐻 푀 ℝ

Conversely, if is a critical point on a cohomology class of a closed oriented 7- dimensional manifold such that is always positive, then defines a metric with Ω holonomy G2 on . Ω Ω Proof: Let be a푀 positive closed 3-form and pick a non-vanishing 7-form on . Using Definition 2.5 in the expression (2.29) for the functional , we find Ω 휖 푀 ( ) = ( ) . Φ (2.30)

The first variation gives: Φ Ω ∫푀 휙 Ω ϵ

= = . (2.31) 7 ̇ 푀 ̇ 18 푀 ̇ Since the variation is within훿Φ a� Ωfixed� ∫cohomology푑휙�Ω�휖 class∫ we∗ Ωmay∧ Ω rewrite = for some 2-form . We see that the variation vanishes precisely when ( ) = 0 for all . Ω̇ 푑휑 That is, = 0 if and only if ( ) = 0. 휑 ∫푀 푑 ∗ Ω ∧ 휑 휑 If is covariant훿Φ constant with푑 respect∗ Ω to the metric , so too is . This implies that

is closed and thus a G2-manifold gives a critical point of . Ω 푔Ω ∗ Ω It∗ Ωis shown in [11] that if ( ) = ( ) = 0 then is covariantΦ constant for which proves the converse part of the theorem and so completes the proof. 푑 Ω 푑 ∗ Ω Ω 푔Ω

2.6 Constrained critical points and weak holonomy ∎

Recall that in our introduction to Riemannian manifolds with special holonomy, we noted that these manifolds are characterized by the existence of covariantly constant spinor fields , satisfying = 0. We now define a second category of special metrics known as metrics with weak holonomy. 휓 ∇휓 Definition 2.6: A Riemannian manifold is said to be of weak holonomy if there exist spinor fields satisfying = .

These manifolds휓 can be listed∇푋휓 as: 휆푋휓

(1) The spheres

(2) Einstein-Sasakian manifolds in odd dimensions

(3) 3-Sasakian manifolds in dimension 4 + 3

(4) Weak holonomy (3) manifolds in 푘dimension 6

푆푈

(5) Weak holonomy manifolds in dimension 7. [12]

We shall be interested퐺 2in the 6-dimensional weak holonomy (3) and the 7-dimensional weak holonomy manifolds. We discuss a natural approach to these structures 푆푈 involving invariant2 functionals of differential forms subject to certain constraints. This elegant characterization퐺 is independent of spinor fields and Riemannian metrics.

We begin by considering a compact oriented manifold of dimension . There exists a non-degenerate pairing between the spaces of forms ( ) and ( ) provided by 푝푀 푛−푝 푛 , Ω 푀 Ω 푀 (2.32) for ( ) and ( ). ∫푀 훼 ∧ 훽 푝 푛−푝 If 훼 is∈ exactΩ 푀 so that 훽 =∈ Ω for 푀some ( ), then Stokes’ theorem gives 푝−1 훼 훼 푑훾 훾 ∈=Ω( 1)푀 (2.33) 푝 Thus, ∫푀 푑훾 ∧ 훽 − ∫푀 훾 ∧ 푑훽

= 0, ( ) = 0, (2.34) 푝−1 which establishes the ∫non푀 푑훾-degenerate∧ 훽 pairing∀훾 ∈ Ω 푀 ⟺ 푑훽

( ) ( )/ ( ). (2.35) 푝 푛−푝 푛−푝 The exterior derivative definesΩ푒푥푎푐푡 an푀 isomorphism≅ Ω 푀 betweenΩ푐푙표푠푒푑 푀 ( )/ ( ) and 푛−푝 ( ) which finally yields the result 푛−푝 푑 Ω 푀 Ω푐푙표푠푒푑 푀 푛−푝+1 푒푥푎푐푡 Ω 푀 ( ) ( ). (2.36) 푝 ∗ 푛−푝+1 Recall that as part of our discussionΩ푒푥푎푐푡 of the푀 stability≅ Ω푒푥푎푐푡 of differ푀 ential forms we showed how one can deduce the existence of stable ( )-forms from knowledge of the existence of stable p-forms in dimensions. In particular, the existence of stable 3-forms in 7 푛 − 푝 dimensions implies the existence of stable 4-forms in 7 dimensions and indeed we may use this duality to푛 translate our results regarding the geometry of 3-forms from the previous section to those of the geometry of 4-forms. With = 7 and = 4, (2.36) gives

( ) ( ). 푛 푝 (2.37) 4 ∗ 4 This ensures that the non-degenerateΩ푒푥푎푐푡 quadratic푀 ≅ Ωform푒푥푎푐푡 푀on ( ) given by 4 ( ) = 푄, Ω 푒푥푎푐푡 푀 (2.38)

푄 푑훾 ∫푀 훾 ∧ 푑훾

19 is well-defined. We can now state and prove our key theorem characterizing metrics with weak holonomy .

Theorem 2.4: Let퐺 2 be an exact stable 4-form on a compact 7-dimensional manifold M and ( ) a volume form preserved by the stabilizer of . Following the analysis 휎 in section 2.3, we7 ∗can define a volume functional as 휙 휎 ∈ Λ 푉 휎 ( ) = ( ).

Now is a critical point of ( ) subject푉 휎 to the∫푀 constra휙 휎 int ( ) = , if and only if defines a metric with weak holonomy . 휎 푉 휎 푄 휎 푐표푛푠푡푎푛푡 Proof:휎 The derivative of at is a linear 퐺map2 from to . Thus, 푝 ∗ 푛 ∗ 휙 휎 ( )퐷휙 Λ. 푉 Λ 푉 (2.39) 푝 ∗ ∗ 푛 ∗ The identification ( ) 퐷휙 ∈ Λ 푉 implies⊗ Λ 푉the existence of a unique 푝 ∗ ∗ 푛−푝 ∗ 푛 Λ 푉 ≅ Λ 푉 ⊗ Λ 푉 , (2.40) 푛−푝 ∗ such that 휎� ∈ Λ 푉

( ) = . (2.41)

The analysis so far is valid for all admissible퐷휙 휎̇ values휎� ∧ 휎 ̇of and . We note that in general the map 푛 푝 ( ) (2.42) need not be linear. For = 7, = 3 휎or� ∶ 4휎, however,⟼ 휎� 휎 this map coincides with the Hodge star operator for the inner product on V. That is, here we have 푛 푝 ∗ = . (2.43)

The first variation of ( ) at = is휎 � ∗ 휎

푉 휎 ( 휎 ) =푑훾 ( ) = (2.44)

훿푉 푑 훾̇ ( ∫푀)퐷휙= 푑(훾̇ ) ∫푀 휎�. ∧ 푑 훾 ̇ (2.45)

The first variation of the quadratic⟹ 훿푉 푑form훾̇ ∫(푀 ∗) 휎= ∧ 푑훾̇ is

( )푄=푑훾2 ∫푀 훾 ∧. 푑훾 (2.46)

Using a Lagrange multiplier, the훿푄 constrained푑훾̇ critical∫푀 훾̇ ∧ 푑훾point is given by

( ) = . (2.47)

This is precisely equivalent to the structure푑 ∗ 휎 of a휆휎 manifold with weak holonomy . [13]

퐺 2

We now consider a constrained variational problem in the case = 6, = 3. For this ∎ combination of and , (2.36) gives the non-degenerate pairing: 푛 푝 푛 푝 ( ) ( ). (2.48) 3 ∗ 4 This pairing can be made explicitΩ 푒푥푎푐푡for an푀 exact≅ 3Ω-form푒푥푎푐푡 푀= and an exact 4-form = via 휌 푑훼 휎 푑 훽 , = = . (2.49)

The stability of the 4-form 〈implies휌 휎〉 the∫푀 훼existence∧ 휎 − of∫푀 a 휌symplectic∧ 훽 2-form satisfying = . As a consequence of its stability, defines a reduction of the structure 1 휎 휔 group of the tangent bundle of the 6-dimensional manifold to (6, ), and similarly 휎 2 휔 ∧ 휔 휎 defines a reduction to (3, ). Taken together and provide a reduction to (3), 푀 푆푝 ℝ 휌 which may be viewed as an intersection of these two groups provided the following two 푆퐿 ℂ 휌 휎 푆푈 compatibility conditions for and are satisfied:

휌 휎 = 0, (2.50) and for some constant , 휔 ∧ 휌

푐 ( ) = ( ). (2.51)

Definition 2.7: Let be a stable 3-form휙 and휌 푐휙= 휎 a stable 4-form. We say that the 1 pair ( , ) is of positive type if the almost complex structure defined by turns 휌 휎 2 휔 ∧ 휔 ( , ) into a positive definite form. 휌 휎 퐼휌 휌 We휔 푋 now퐼휌푋 have the following Theorem:

Theorem 2.5: Let ( , ) be a pair of exact stable forms of positive type defined on a compact 6-dimensional manifold M. The pair ( , ) is a critical point of 3 ( ) + 8 ( ) 휌 휎 subject to the constraint ( , ) = , = , if and only if and define a 휌 휎 푉 휌 푉 휎 metric with weak holonomy (3). 푃 휌 휎 〈휌 휎〉 푐표푛푠푡푎푛푡 휌 휎 Aside: We note that the coefficients푆푈 3 and 8 in the linear combination of ( ) and ( ) have no special significance as far as the validity of the above statement is concerned, other than the fact that they are both positive real numbers. These specific푉 values휌 were푉 휎 chosen to simplify some of the expressions in the proof below.

Proof: Reasoning as in the first few lines of the proof of Theorem 2.4, the first variation of 3 ( ) + 8 ( ) gives

푉 휌 푉 휎 3 + 4 . (2.52)

The first variation of ( , ) gives ∫푀 휌� ∧ 휌̇ ∫푀 휔 ∧ 휎̇

푃 휌 휎 ( , ) = + , (2.53) for = and = . Introducing훿푃 휌̇ 휎̇ a Lagrange∫푀 휌̇ ∧ 훽 multiplier∫푀 휎̇ ∧ 훼12 , we find the constrained critical point is determined by 휌 푑훼 휎 푑훽 휆 = 2 (2.54) 2 and 푑 휌� =−3 휆휔. (2.55)

The equations (2.54) and (2.55) together푑휔 imply휆휌 a metric of weak holonomy (3), which is also known as a nearly K hler metric. For the technical details of the last step of the 푆푈 proof deducing a metric of weak holonomy (3) from equations (2.54) and (2.55), the 푎̈ interested reader is referred to [10]. 푆푈

Finally, we note that (2.54) and (2.55) taken together also imply the important ∎ compatibility conditions necessary for the combination of and to give an (3) structure on . 휌 휎 푆푈 2.7 A Hamiltonian푀 Flow

Let be a fixed cohomology class in ( , ) and a fixed cohomology class in ( , ). and are affine spaces whose3 tangent spaces are naturally isomorphic to 풜 퐻 푀 ℝ ℬ 4 ( ) and ( ), respectively. We consider the Cartesian product × 퐻 푀 ℝ 풜 ℬ whose3 tangent space4 at each point is isomorphic to ( ) × ( ). We Ω푒푥푎푐푡 푀 Ω푒푥푎푐푡 푀 풜 ℬ construct a symplectic form on × using the pairing3 , defined4 in (2.49) as Ω푒푥푎푐푡 푀 Ω푒푥푎푐푡 푀 follows 휔 풜 ℬ 〈∙ ∙〉 ( , ), ( , ) = , , . (2.56)

We now present our final휔� t휌heorem1 휎1 of휌2 this휎2 chapter� 〈휌1 which휎2〉 − provides〈휌2 휎1〉 a bridge between (3) structures and holonomy metrics. 푆푈 Theorem 2.6: Let퐺2 ( , ) × be a pair of exact stable forms of positive type that satisfy the compatibility conditions = 0 and ( ) = 2 ( ) at time = . Define 휌 휎 ∈ 풜 ℬ the functional by 휔 ∧ 휌 휙 휌 휙 휎 푡 푡0 퐻

= ( ) 2 ( ), (2.57) and allow the pair ( , ) to evolve퐻 with푉 respect휌 − to푉 the휎 Hamiltonian flow determined by the vector field satisfying = , where denotes an interior product. The 3- 휌 휎 form 푋 푖푋휔 퐷퐻 푖푋휔 = + , (2.58) now defines a metric with holonomy휑 on푑푡 the∧ 휔 7-dimensional휌 manifold × ( , ).

Proof: We rewrite the equation =퐺2 as 푀 푎 푏

푖푋휔 퐷퐻 = . (2.59)

Differentiating , we find ∫푀 휌̇ ∧ 훽 − ∫푀 휎̇ ∧ 훼 퐷퐻

퐻 ( , ) = , (2.60) which combines with (2.59)퐷퐻 to푑훼 give:푑훽 ∫푀 휌� ∧ 푑훼 − ∫푀 휔 ∧ 푑훽

( ) ( + ) = 0, (2.61) for all and , following∫푀 an휌̇ application− 푑휔 ∧ 훽 −of ∫Stokes’푀 휎̇ 푑theorem.휌� ∧ 훼 That is,

훼 훽 = (2.62) and, using = ( ) = ,휕휌 ⁄휕푡 푑휔 휕 1 휎 ̇ 휕푡 2 휔 ∧ 휔 휔 ∧ =휔̇ = . (2.63)

A key step in the proof is to 휕휎show⁄휕푡 that휔 the∧ compatibility휕휔⁄휕푡 −푑 휌conditi� ons are preserved under time evolution. This ensures that the (3) geometry on is not altered as the time is allowed to evolve. To do this for the first condition, we define 푆푈 푀 푡 ( , ) = (2.67)

푋 푋 as a function of the vector field 휇. We휌 휎have, ∫푀 푖 휎 ∧ 휌

( ,푋) = +

푋 푋 푋 푑 휇 휌 ̇ 휎 ̇ = ∫푀 푖 휎̇ ∧ 휌 +∫푀 푖 휎 ∧ 휌̇

푋 푋 = − ∫푀 휎̇ ∧ 푖 +휌 ∫푀 푖 휎 ∧, 휌 ̇ (2.68)

푋 푋 where we have used = 0 in the ∫step푀 푖 from휌 ∧ 휎 ̇the ∫first푀 푖 휎to ∧the휌̇ second lines.

휎̇ ∧ 휌

Recalling that the Lie derivative can be written as = + and using the fact that and are closed, we have by the nilpotency of the exterior derivative that ℒ푋 ℒ푋 푑푖푋 푖푋푑 휌 휎 ( , ) = ( , ) (2.69) and thus we can rewrite (2.68) inℒ푋 the휌 symplectic휎 푑푖푋휌 form푑푖푋 휎

( , ) = ( , ), ( , ) . (2.70)

The map : is thus 푑the휇푋 moment휌̇ 휎̇ map〈ℒ푋 of휌 휎 휌(̇ 휎̇).〉 Rewriting (2.67) in terms of the non-degenerate 2-form gives 휇 푋 ⟼ 휇푋 퐷푖푓푓 푀 휔 ( , ) = , (2.71)

푋 푋 which implies = 0 for all 휇 if and휌 휎 only ∫if푀 푖 휔 ∧=휔 0∧. 휌

The diffeomorphism휇푋 invariance푋 of the functionals휔 ∧ 휌 ( ) and ( ) ensures that commutes with with respect to the Poisson bracket. It follows that the Hamiltonian 푉 휌 푉 휎 퐻 flow of and the subset of × consisting of the pairs ( , ) for which uniformly 휇푋 vanishes for all , are tangential. This subset of × is precisely the space 퐻 풜 ℬ 휌 휎 휇푋 characterized by = 0 and so we have established that if = 0 is satisfied at 푋 풜 ℬ = , it remains satisfied for all values of . It also follows that = 0 for all . 휔 ∧ 휌 휔 ∧ 휌 Recall푡 푡0 that the equation ( ) = uniquely푡 determines for휔 a∧ given휌� . As 푡( ) is homogeneous of degree = = 2, its derivative is homogeneous of degree one. 퐷휙 휌̇ 6 휌� ∧ 휌̇ 휌� 휌 휙 휌 Thus, 푛⁄푝 3 퐷휙 ( , ) = ( ) = . (2.72) 2 A key observation in proving퐷 the휙 preservation휌 휌̇ 퐷휙 휌oḟ the휌 �second∧ 휌̇ compatibility condition is that the derivative of can be expressed in terms , since itself is defined by . Hence, 2 휌� 퐷 휙 휌� 퐷휙 = ( , ), (2.73) 2 which combined with (2.72) gives휕휌�⁄ 휕푡 ∧ 휌 퐷 =휙 휌 휌̇ and thus

( ) 휕휌�⁄=휕푡2∧ 휌 휌� ∧ 휕휌= ⁄2휕푡 , (2.74) where we have used the 휕evolution휌� ∧ 휌 ⁄ equation휕푡 휌� ∧ 휕휌⁄휕푡= .휌� Taking∧ 푑휔 the exterior derivative of = 0 yields = and so 휕휌⁄휕푡 푑휔 휔 ∧ 휌 � 휌 � ∧ 푑휔 −( 푑휌� ∧)휔 = 2

휕 휌 � ∧ 휌 ⁄ 휕푡 = −2 푑휌� ∧ 휔

휔 ∧ 휕휔⁄휕푡 ∧ 휔

= . (2.75) 2 3 3 휕휔 ⁄휕푡 That is, = 0 from which it follows that if ( ) = 2 ( ) holds at 2 = , then it does so for3 all . Therefore, the flow respects the (3) structure on . 휕 �휌� ∧ 휌 − 3 휔 �⁄휕푡 휙 휌 휙 휎 푡Now푡 0consider = + 푡 . We have 푆푈 푀

휑 푑푡 ∧ 휔 휌 = = = 0, (2.76) where we have used (2.62) and푑휑 푑푡= 0∧. 푑휔Using 푑푡the∧ Hodge휕휌⁄휕푡 star operator defined by the metric determined by , we have 푑휌 퐺2 휑 = , (2.77) and thus ∗ 휑 푑푡 ∧ 휌� − 휎

= = = 0, (2.78) using (2.63) and = 0푑. We∗ 휑 recall푑푡 ∧that푑휌� −=푑푡 ∧ 휕휎=⁄휕푡0 is precisely the condition under which defines a metric of holonomy on a 7-dimensional manifold which in this 푑휎 푑휑 푑 ∗ 휑 case is of the form = × ( , ) for some interval ( , ). 휑 퐺2 푁 푁 푀 푎 푏 푎 푏

Finally, we note that the converse to the above theorem also holds true as the steps in∎ the proof are essentially reversible. Specifically, if is a 7-dimensional manifold of holonomy that is foliated by equidistant compact hypersurfaces diffeomorphic to a 6- 푁 퐺2 dimensional manifold , the defining closed forms and restricted to each hypersurface evolve as the Hamiltonian flow of the functional = ( ) 2 ( ). 푀 휌 휎 퐻 푉 휌 − 푉 휎

III. Topological String Theory

We provide an introduction to topological strings and show how the and models naturally arise as the two distinct types of topological string theory. After briefly 퐴 퐵 reviewing the relevant material from Chapter II, we construct the and Hitchin functionals which generate symplectic and holomorphic structures, respectively. We then 푉푆 푉퐻 consider certain conjectured relations between these functionals and the topological and models, which we shall support with classical arguments. 퐴 3.1 퐵 = ( , ) supersymmetry

The퓝 stringퟐ ퟐ theories with which we shall be concerned involve maps from a surface to a target space . In string theory we weigh each map by the Polyakov action and 휙 integrate over all such and over all metrics on . If we only integrate over we obtain Σ 푋 휙 a 2-dimensional quantum field theory whose saddle points are locally area-minimizing 휙 Σ 휙 surfaces in . This 2-dimensional theory is known as a “sigma model into ” and has what is known as = (2,2) supersymmetry. As we shall see, the definition of the 푋 푋 topological string rests upon this (2,2) supersymmetry. 풩 = 2 supersymmetry means there are 4 worldsheet currents and prescribed operator product relations. These worldsheet currents are 풩 , , , (3.1) + − and have spins 1, , , 2 respectively. 퐽In퐺 the퐺 case푇 of the sigma model into , these are 3 3 analogous to the operators , , , acting on the space of differential forms on the 2 2 푋 loop space of . This analogy suggests† the following important operator product relations 푑푒푔 휕̅ 휕̅ ∆ which do in fact hold true. 푋 ( ±) ~ 0, (3.2) 2 퐺 ~ + . (3.3) + − We will now explain the structure of퐺 퐺= (2푇,2) supersymmetry퐽 which will be vital to the rest of our discussion. Let be Calabi-Yau so that the sigma model is conformal. Here 풩 we can write each of the currents , , , as a sum of two commuting holomorphic 푋 and antiholomorphic copies, effectively+ producing− two copies of the = 2 algebra. We 퐽 퐺 퐺 푇 write these as ( , , , ) and ( , , , ) and refer to this split structure as 풩 = (2,2) supersymmetry.+ − The =+(2,−2) structure of superconformal field theory 퐽 퐺 퐺 푇 퐽̅ 퐺�� 퐺�� 푇� should be viewed as an invariant of the Calabi-Yau manifold . [15] 풩 풩 푋

3.2 The “twisting” procedure and the topological string

We consider a procedure referred to as “twisting” whereby we can produce a topological theory from the = (2,2) superconformal field theory discussed in the previous section. We can think of the transition from our ordinary field theory to its 풩 topological version as analogous to the move from the de Rham complex ( ) to its cohomology ( ). Inspired by this analogy and in view of ( ) ~ 0, we ∗encounter an Ω 푋 immediate obstacle∗ when we attempt to form the cohomology of+ the2 zero mode of : to 퐻 푋 퐺 obtain a scalar zero mode we must begin with an operator of spin 1, whereas has +spin 퐺 3/2. + 퐺 The resolution to this problem lies in “twisting” the sigma model as described in detail by Edward Witten in [16] and [17]. The twist can be described as a shift in the operator which has the effect of changing the spins of all the operators by an amount′ proportional to their (1) charge : ′ 푇 → 푇 푆 푆 → 푆 푈 푞

= (3.4) ′ 1 and 푇 푇 − 2 휕퐽

= . (3.5) ′ 1 Following this twisting procedure푆 the 푆operators− 2 푞 ( , ) have spin 1 and ( , ) have spin 2. We note that although ± now possess integer +spin they of course continue− to 퐺 퐽 푇 퐺 retain their fermionic nature. We also point out that in (3.4) we have made a choice between one of two possibilities퐺 in the direction of shift. We could equally well-have chosen to shift in the opposite direction: = + . For now we will suffice it 1 to say that this degree′ of freedom will have a profound′ consequence for the number of 푇 → 푇 푇 푇 2 휕퐽 ways we can construct a topological string from a given Calabi-Yau manifold. We continue with the choice made in (3.4) and take = . We note that is well-defined on arbitrary , fermionic and nilpotent: = 0. We restrict+ our attention to observables 푄 퐺0 푄 which are annihilated by . 2 Σ 푄 Here we note that the twisted푄 = 2 algebra is isomorphic to a subalgebra of the symmetry algebra of the bosonic string. This isomorphism can be written as 풩 ( , , , ) ( , , , ) (3.6) + − 퐺 퐽 푇 퐺 ↔ 푄 퐽푔ℎ표푠푡 푇 푏

where ( , ) and ( , ) are currents of spin 1 and 2 respectively, and ( , ) are the BRST current and antighost corresponding to the diffeomorphism symmetry on the 푄 퐽푔ℎ표푠푡 푇 푏 푄 푏 bosonic string worldsheet.

By performing the Faddeev-Popov procedure we can reduce the integral over metrics on to an integral over the moduli space of genus Riemann surfaces, where the measure is provided by the ghosts. The genus free energy of the bosonic string in the Σ ℳ푔 푔 > 1 case is found in this way to be 푏 푔 푔 ( ) . (3.7) 3푔−3 2 ℳ푔 〈 ∏푖=1 푖 〉 This expression requires only slight∫ modifications� 푏 휇 �for the = 0, 1 cases related to the fact that the sphere and torus admit nonzero holomorphic vector fields. In (3.7), the 푔 denotes a conformal field theory correlation function and the are “Beltrami 〈⋯ 〉 differentials”. These are antiholomorphic one-forms on with values in the holomorphic 휇푖 tangent bundle. Σ We can now define a “topological string” from the correlation functions of the = (2,2) superconformal field theory on fixed by replacing by in (3.7) due to the 풩 isomorphism (3.6). That is, − Σ 푏 퐺 = ( ) . (3.8) 2 3푔−3 − 푔 ℳ푔 〈 ∏푖=1 푖 〉 The full topological string free퐹 energy∫ is� then defined퐺 휇 to� be

= . (3.9) ∞ 2−2푔 푔=0 푔 This expression is to be understoodℱ as an∑ asymptotic휆 퐹 series in , where is the “string coupling constant” weighing contributions at each genus. Thus we arrive at a definition of 휆 휆 the topological string partition function = exp .

3.3 The topological and models 푍 ℱ

Recall that when defining퐀 the퐁 shift by (3.4) we noted that we could just as well have performed the shift in the opposite direction′ so that 푇 → 푇 = + (3.10) ′ 1 and 푇 푇 2 휕퐽

= + . (3.11) ′ 1 푆 푆 2 푞

Following this twist, will have spin 1 and will have spin 2 so that we may now take as our BRST operator− . Thus we see that +we find two distinct paths to the 퐺 퐺 construction− of a topological string theory from a given Calabi-Yau . We refer to these 퐺 푄 topological string theories as the A and B models corresponding to the choices ( , ) 푋 and ( , ) for the BRST operators, respectively. We also note the existence of +A and+ B 퐺 퐺�� models+ arising− from a similar freedom in the choice of shift direction in the 퐺 퐺�� antiholomorphic sector. However, since these are simply related to the A and B models via overall complex conjugation we are essentially only left with two distinct types of topological string theory.

In the A model, the combination + is the d operator on and its cohomology is the de Rham cohomology. It is natural to impose a further “physical state” constraint which 푄 푄� 푋 leads to considering only the degree (1,1) part of this cohomology which correspond to a deformation of the Kahler form on . The observables of the A model which we are considering are deformations of the Kahler moduli of . ̈ 푋 In the B model, the complex is the cohomologÿ with푋 values in . Imposing the “physical state” constraint once again leads us to considering only ∗the degree (1,1) part 휕̅ Λ 푇푋 of the cohomology. We find that the observables are (0,1)-forms taking values in . That is, the observables in this case are the Beltrami differentials on . These differentials 푇푋 span the space of infinitesimal deformations of the operator on and so the observables 푋 of the B model are deformations of the complex structure of . 휕̅ Σ 3.4 Hitchin’s and functionals in six dimensions 푋

Recall that푽 in푺 Chapt푽er푯 II we defined the notion of stability of a differential -form in dimensions as an extension of the non-degeneracy of 2-forms. We also explained how the 푝 푛 existence of stable -forms in dimension implies the existence of stable ( )-forms. We then noted in particular the obvious existence of stable 2-forms in even dimension 푝 푛 푛 − 푝 = 2 as well as the more interesting fact that stable 3-forms exist in dimensions 6 and 7. Thus in six dimensions we find that we can have stable 4-forms by virtue of the existence푛 푚 of stable 2-forms via the duality mentioned above. We now consider in six dimensions the action functionals ( ) and ( ), depending on a stable 4-form and a stable 3-form , which generate symplectic and holomorphic structures respectively. 푉푆 휎 푉퐻 휌 휎 We begin with휌 ( ). For a 4-form the stability condition can be expressed as the requirement that 푆there exists a 2-form such that = ± . Take the + case and 푉 휎 휎 1 define by 푘 휎 2 푘 ∧ 푘 푆 푉 = , (3.12) 1 푉푆 6 ∫푀 푘 ∧ 푘 ∧ 푘

where is our 6-dimensional manifold. We may express this functional in terms of as

푀 = / , (3.13)휎 1 3 2 푆 or in components as: 푉 6 ∫푀 휎

= , (3.14) 1 1 푎1푎2푎3푎4푎5푎6 푏1푏2푏3푏4푏5푏6 1 2 1 2 3 4 3 4 5 6 5 6 푉푆 6 ∫푀 �384 휎푎 푎 푏 푏 휎푎 푎 푏 푏 휎푎 푎 푏 푏 휖 휖 where is the Levi-Civita tensor in six dimensions. 푎1⋯푎6 Keepin휖g the cohomology class [ ] ( , ) fixed so that = + , where = 0 we have 4 휎 ∈ 퐻 푀 ℝ 휎 휎0 푑훼 0 푑 휎 = = ( + ) (3.15) 1 1 푆 0 Invoking Theorem 2.1 we find푉 that3 ∫푀 the휎 ∧condition푘 3 ∫푀 that휎 푑훼is extremal∧ 푘 under variations of is 푉푆 훼 = 0. (3.16)

Hence we see that when extremized, 푑푘 generates a symplectic structure on .

We now consider the functional ( 푉)푆 which was in fact studied in detail in푀 Chapter II. Here we will use Theorem 2.1 to arrive at the result that a holomorphic structure on 푉퐻 휌 can be obtained as a critical point of this functional. Suppose is a stable 3-form and that 푀 it satisfies the condition ( ) < 0 where was defined in section 2.4 of Chapter II. Here 휌 we will refer to such as a positive stable 3-form. The important point is that such is 휆 ρ 휆 fixed by a subgroup of (6, ) isomorphic to (two copies of) (3, ) and so it 휌 휌 determines a reduction of (6, ) to (3, ), which is equivalent to an almost complex 퐺퐿 ℝ 푆퐿 ℂ structure on . This means that we can find complex 1-forms , =1, 2, 3 such that 퐺퐿 ℝ 푆퐿 ℂ 푖 푀 = ( + ), 휁 푖 (3.17) 1 2 1 2 3 �1 �2 �3 where the almost complex휌 structure휁 ∧ 휁is determined∧ 휁 휁 ∧ by휁 the∧ 휁 . Construct a (3,0)-form on using the complex 1-forms as 휁푖 Ω 푀 휁 푖 = . (3.18)

We may rewrite in terms of asΩ 휁1 ∧ 휁2 ∧ 휁3

Ω 휌 = + ( ), (3.19)

where is given by Ω 휌 푖휌� 휌

휌�

= ( ). (3.20) 푖 2 1 2 3 �1 �2 �3 If there exist local complex휌� coordinates− 휁 ∧ 휁 ∧ such휁 − that휁 ∧ 휁 =∧ 휁 , then the almost complex structure is integrable and so defines a complex structure. We note that this integrability 푧푖 휁푖 푑푧푖 condition is clearly equivalent to = 0.

Define ( ) to be the holomorphic푑Ω volume

퐻 푉 휌 ( ) = = ( ) . (3.21) 푖 1 퐻 This can be written in components푉 휌 − 4as∫ 푀 Ω ∧ Ω� 2 ∫푀 휌� 휌 ∧ 휌

( ) = , (3.22) 푖 1 푏 푎 푉퐻 휌 − 4 ∫푀 �− 6 퐾푎 퐾푏 where = . The “positivity” condition on the 3-form 푏 1 푎1푎2푎3푎4푎5푏 that was defined earlier1 2 3 ensures4 5 that the square root in (3.22) gives a real number. 퐾푎 12 휌푎 푎 푎 휌푎 푎 푎휖 휌 Once again we keep the cohomology class fixed so that = + with = 0 and

perform the first variation of ( ) = ( ) . By Theorem0 2.1 we find 0that the 1 휌 휌 푑훽 푑휌 variation vanishes precisely when 푉퐻 휌 2 ∫푀 휌� 휌 ∧ 휌 = 0. (3.23)

From = + we also have =푑0휌�. Combining these two results we arrive at = 0, which was found to be the condition for the integrability of the almost complex 휌 휌0 푑훽 푑휌 structure. Thus we see how ( ) generates a complex structure with holomorphic 푑Ω 3-form. 푉퐻 휌 In summary, we have found that extremizing the two functionals ( ) and ( ) respectively yield a symplectic form and a closed holomorphic (3,0)-form on a 6- 푉푆 휎 푉퐻 휌 dimensional manifold . This seems to suggest a possible relation to the topological A 푘 Ω and B models. For the remainder of this chapter, we will aim to establish a link between 푀 the Hitchin functionals , and these topological models.

3.5 The functional as푉푆 the푉퐻 model

It is argued푽푺 in [18] that the A model퐀 can be reformulated in terms of a topologically twisted (1) gauge theory on , whose bosonic action contains the observables

푈 = 푀 + . (3.24) 푔푠 푆 3 ∫푀 퐹 ∧ 퐹 ∧ 퐹 ∫푀 푘0 ∧ 퐹 ∧ 퐹

In this theory the partition function is a function of the fixed class and the path integral can be expressed as a sum over Kahler geometries with quantized Kahler class 푘0 ̈ = + . ̈ (3.25)

Now for some coefficients and 푘 consider푘0 the푔푠퐹 action

S훼= 훽 , (3.26)

where is a 2-form and is a α4-∫form푀 퐹� ∧ restricted퐹� ∧ 퐹� − to훽 ∫a푀 fixed휎 ∧ 퐹 �cohomology class.

The equation퐹� of motion for휎 is

퐹 � 3 = 0, (3.27)

which immediately implies that is훼 a퐹� stable∧ 퐹� − 4훽휎-form. Substituting this back into the action (3.26) we find that it is equal to the Hitchin functional ( ) = / . 휎 1 3 2 푆 We now return to the action (3.26) and instead consider푉 the휎 equations6 ∫푀 휎 of motion for .

We note that [ ] ( , ) is fixed so that we can write = + to find 휎 4 휎 ∈ S퐻= 푀 ℝ 휎 휎0 . 푑훾 (3.28)

0 It is clear that for 0α, ∫푀 퐹�=∧0퐹 �if∧ and퐹� − only훽 ∫푀 if휎 ∧ 퐹(� −)훽 ∫푀 푑훾= ∧ 퐹� = 0; i.e.

훽 ≠ 훿푆 = 0∫. 푀 푑 훿훾 ∧ 퐹 � − ∫ 푀 훿훾 ∧ 푑 퐹 � (3.29)

Hence the action for can be written as푑 퐹�

퐹 � S = . (3.30)

0 By introducing a change of variablesα ∫푀 퐹� ∧ 퐹�=∧ 퐹� − 훽 ∫,푀 where휎 ∧ 퐹 � is a parameter and is the background Kahler form satisfying = , we can rewrite (3.30) as 퐹 퐹� − 휉푘0 휉 푘0 S = ̈ + 3 휎0 푘0 ∧+푘0 (3 ). (3.31) 2 0 0 0 0 Comparingα ∫푀 with퐹 ∧ (3.24)퐹 ∧ 퐹 we see휉훼 ∫that푀 퐹 for∧ 퐹 ∧=푘 /3∫푀 and휉 훼=푘 ∧=푘1/∧ 퐹 this− 훽 action휎 ∧ 퐹 is exactly equal to that of the (1) gauge theory action (3.24). Furthermore, the change of variables 훼 푔푠 휉 훽 푔푠 = with = 1/ implies 푈 퐹 퐹 � − 휉 푘 0 휉 푔 푠 = , (3.32)

which is equivalent to the quantization 훿푘condition푔푠퐹 (3.25).

Now consider the rescaling operation . Under this rescaling (3.26) becomes

퐹� → 훾퐹�

S = . (3.33) 3 푀 � � � 푀 � For = 3/ and , as aboveα훾 ∫ , we퐹 ∧find퐹 ∧ that퐹 − both훽훾 ∫terms휎 ∧ of퐹 the action have the same coefficient 1/ = 3/ and thus the semiclassical limit 0 corresponds precisely to 훾 √ 푔푠 훼 훽 푆 the weak coupling limit 2 0. Therefore, the gauge theory action (3.24) is equivalent to ℏ 푔푠 ℏ → ( ) in the weak √coupling limit. 푔푠 → 푉Finally푆 휎 we turn to the connection between ( ) and the topologicallt twisted version of the supersymmetric (1) gauge theory with action (3.26) which was just shown to be 푉푆 휎 equivalent to the (1) gauge theory with action (3.24). In the topological (1) gauge 푈 theory on , the partition sum over Kahler geometries with quantized Kahler class (3.25) 푈 푈 can be written as a vacuum expectation value 푀 ̈ ̈ exp + , (3.34) 푔푠 1 2 where are the topological observables〈 � 3 ∫ 풪 ∫ 풪 �〉

풪 푖 = ,

풪1 = 퐹 ∧ 퐹 ∧ 퐹 , (3.35)

풪 2 푘 0 ∧ 퐹 ∧ 퐹

It was shown by Baulieu and Singer in [19]⋮ how the action of this 6-dimensional topological theory can be reconstructed by considering the BRST symmetries that preserve the expectation value (3.34) and the observables (3.35). This construction leads to two gauge fields , and three ghost fields , and . The action of the BRS operator on these fundamental fields is then given by 퐴 퐹 푐 휓 휙 푠 = ,

푠퐴+ [휓,−]퐷푐= , 1 푠푐+ [2 , 푐 ]푐= 휙 ,

푠휓 +푐[휓, ] =−0퐷휙,

푠휙+ [ ,푐 ]휙= , where = + [ , . ] is the gauge푠퐹 covariant푐 퐹 derivative.−퐷휓 [20]

For the퐷 purposes푑 퐴of our analysis we begin by writing locally as the field strength of a gauge potential , 퐹 퐴

= . (3.36)

Now (3.34) and (3.35) are not only invariant퐹 푑퐴under the gauge transformation = but also under the more general transformation 훿퐴 푑휖0 = (3.37)

where the parameter is an infinitesimal훿퐴 1-form휖1 on . Gauge fixing with respect to (3.37) leads to a 1-form ghost field and consideration of the additional symmetry under 휖1 푀 + gives rise to a bosonic 0-form . The only such 6-dimensional topological 휓 theory containing a gauge field and a scalar is a maximally supersymmetric theory with 휖1 → 휖1 푑휆 휙 = 2 topological supersymmetry.

푇 풩We now return to the functional ( ) = / . It was shown how in the weak 1 coupling limit the field is identified with the K3a2hler form via = . We thus find 푉푆 휎 6 ∫푀 휎 that 퐹 ̈ 훿푘 푔푠퐹 = , (3.38) where = . Once more varying 훿푘 within푑휖 0a fixed cohomology class so that = + for = 0 we see that is indeed invariant under the symmetry (3.38): 휎 푘 ∧ 푘 휎 0 0 푆 휎 휎 푑훾 푑 휎 =푉 3 푆 훿 푉 = 32 ∫푀 푘 ∧ 훿휎

= 3 ∫푀 푘 ∧ 푘 ∧ 훿푘

0 = ∫3푀 휎 ∧ 푑휖 3 = 0. (3.39) 2 0 0 0 3.6 The functional as the model − ∫푀 푑휎 ∧ 휖 − ∫푀 푑 훾 ∧ 휖

The B푽 푯model on a Calabi-Yau퐁 3-fold can be constructed by topologically twisting the physical theory with a fixed “background” complex structure given by a holomorphic 푀 3-form . As defined in [21] the partition function of the theory is the generating functional of the correlation functions of the marginal operators , = 1, , , . The Ω 푍퐵 marginal operators are the topological observables of the theory and represent 휙푖 푖 ⋯ ℎ2 1 infinitesimal deformations of the complex structure. Introducing the variables 휙푖 , = 1, , , , where the label infinitesimal deformations, we find that for fixed 푖 ( , , ) is a holomorphic 푖function of from the holomorphic tangent space 푥 푖 ⋯ ℎ2 1 푥 Ω0 to the moduli space of complex structures. Closer examination of the dependence 푍퐵 푥 푔푠 Ω0 푥 푇Ω0ℳ of leads us to a wavefunction interpretation of as is explained in [21]. Beginning by ℳ Ω0 combining and into a “phase space” of dimension , + 1 we arrive at the result 푍퐵 푍퐵 푔푠 푥 ℎ2 1

that from ( , ) we can obtain any other ( , ) by taking an appropriate Fourier

transform. In퐵 this 0interpretation is a wavefunction퐵 0 arising from the quantization of the 푍 푥 Ω ( , ) 푍 푥 Ω ′ symplectic phase space 퐵. 3 푍 Now consider the partition퐻 function푋 ℝ ([ ]) of the 6-dimensional theory with action given by the Hitchin functional . We may write this partition function in terms of the 푍퐻 휌 formal expression 푉퐻 ([ ]) = exp ( + ) , (3.40)

which makes clear in particular푍퐻 the휌 dependence∫ 퐷훽 of�푉 퐻 휌 on [푑훽] � ( , ). Here we note that is independent of the particular choice of symplectic coordinates3 for ( , ), 푍퐻 휌 ∈ 퐻 푋 ℝ whereas has different representations for different choices of symplectic coordinates.3 푍퐻 퐻 푋 ℝ Thus we see that and cannot be equal. 푍퐵 To overcome 푍this퐻 obstacle푍퐵 we combine the B and B models, where we recall that these two are related via overall complex conjugation. Consider = , where is of � course the B model partition function. This product state exists in the doubled Hilbert Ψ 푍퐵 ⊗ 푍��퐵� 푍��퐵� space generated by the quantization of the phase space ( , ), which is itself doubled � to ( , ). By dividing ( , ) into real and imaginary3 parts we can find a 퐻 푋 ℝ representation3 of as a function3 of the real parts of all the periods: (Re , Re ). This 퐻 푋 ℂ 퐻 푋 ℂ once again yields a function on ( , ) which is now independent of the choice of퐼 Ψ Ψ 푋퐼 퐹 symplectic coordinates. The function3 which we have arrived at via the above 퐻 푋 ℝ construction coincides with the “Wigner function” of which expresses the phase-space Ψ density associated with . 푍퐵 Now choose a basis of 푍 퐵and cycles { , } in ( , ) and denote the and cycle periods by and , respectively. can now퐼 be written as a function of the cycle 퐴 퐵 퐴퐼 퐵 퐻3 푋 ℤ 퐴 퐵 periods: = ( 퐼). For = Re and = Re , the Wigner function of is 푋퐼 퐹 푍퐵 퐴 given by 퐼 퐼 푍퐵 푍퐵 푋퐼 푃퐼 푋퐼 푄 퐹 푍퐵 (Re , Re ) = exp( )| ( + )| . (3.41) 퐼 퐼 2 It is this WignerΨ function푋퐼 which퐹 we∫ seek푑Φ 퐼to identify−푄 Φ with퐼 푍 퐵 푃(퐼[ ])푖.Φ This퐼 can be achieved if we identify and as the periods of [ ] ( , ). We will conclude this chapter by 푍퐻 휌 showing the validity 퐼of this identification for large3 which is equivalent to demonstrating 푃퐼 푄 휌 ∈ 퐻 푋 ℝ agreement between and the Wigner function of at the string tree level. In a later 휌 chapter we shall look 퐻deeper into this identification to퐵 see whether these connections persist at one loop. 푍 푍

At the string tree level (large limit) is dominated by the tree level free energy .

Setting = exp ( ) we can rewrite퐵 (3.41) as 0 푖 휌 푍 퐹 푍퐵 − 2 퐹0

(Re , Re ) = exp( ) ( + ) ( + ) 퐼 퐼 퐼 퐼 퐼 퐵 퐼 퐼 ��퐵��퐼��퐼�� Ψ= 푋 exp퐹( ∫ 푑)expΦ ( −푄(Φ +푍 푃))exp푖Φ ( 푍 (푃 + 횤Φ )) 퐼 푖 푖 ∫ 푑Φ퐼 −푄 Φ퐼 − 2 퐹0 푃퐼 푖Φ퐼 2 퐹��0��푃�퐼��횤�Φ��퐼�� = exp( ( + ) + ( + ) ). (3.42) 푖 푖 퐼 퐼 0 퐼 퐼 ��0��퐼��퐼� 퐼 Now the integral∫ over푑Φ is −in2 a퐹 form푃 which푖Φ allows2 퐹 us푃 use횤 Φthe method− 푄 Φ of steepest descent to approximate it. The function ( ) = ( + ) + ( + ) has Φ퐼 푖 푖 퐼 its critical point where = Re푆 Φ= Re− 2 (퐹0 +푃퐼 )푖.Φ At퐼 this2 퐹��0�,� �푃we�퐼�� have:��횤�Φ��퐼� − 푄 Φ퐼 퐼 ∂퐹0 퐼 푄 휕푋퐼 퐹 푃 푖Φ Φ = + (Re )(Im ) 푖 퐼 푖 퐼 퐼 푆 − 4 푋퐼퐹 4 푋�퐼퐹�� − 퐹 푋퐼 = . (3.43) 푖 퐼 푖 퐼 4 푋퐼퐹�� − 4 푋�퐼퐹 We recognize this as the Hitchin functional = evaluated at the value of 푖 for which Re = and Re = . 푉퐻 − 4 ∫푀 Ω ∧ Ω� Ω 퐼 퐼 Finally, we note푋퐼 that푃퐼 the arguments퐹 푄 presented here have only been rigorous at the classical level. In Chapter VIII, we return to the conjectured relation between the topological B-model and the Hitchin functional . There we will consider a suitable reformulation of this connection after incorporating퐻 the one-loop contributions to the partition functions. 푉

IV. BPS Black Holes

In the early 1970s a striking and beautiful correspondence was discovered between the laws of thermodynamics and the laws of black hole dynamics. In particular the Bekenstein-Hawking entropy was found to exhibit behaviour identical to that of thermodynamic entropy. A central question in black hole physics concerns the precise statistical mechanical interpretation of black hole entropy. Derivation of the Bekenstein- Hawking entropy via counting black hole microstates would promote the sharp analogy existing between the laws of thermodynamics and the dynamics of black holes to a deeper and more precise identification of the two. In this chapter we discuss some relevant ideas and in particular the entropy of so-called BPS black holes. In the process we give a brief introduction to N=2 supergravity and towards the end of the chapter we consider an important connection between black holes and Hitchin functionals.

4.1 Black hole entropy

We begin by considering the Reissner-Nordsrom solutions of the classical Einstein- Maxwell field theory of gravity and electromagnetism in four space-time dimensions:

= 1 + + 1 + + . (4.1) 2 2 −1 2 2퐺푀 푄 2 2퐺푀 푄 2 2 2 2 2 These solutions푑푠 are− � static,− 푟 spherically푟 � 푑푡 symmetric� − 푟black푟 holes� 푑푟 with a푟 flat푑Ω space-time geometry at spatial infinity and are characterized by a charge and a mass . For = 0 we see a reduction to Schwarzschild solutions. The black hole horizon has associated with 푄 푀 푄 it an area that is defined by the area of the two-sphere determined by the horizon, and a surface gravity which is interpreted as the force measured at spatial infinity required to 퐴 hold a unit test mass in place. 휅푠 Under the celebrated analogy between the laws of thermodynamics and black hole dynamics, the first law

= , (4.2)

translates into 훿퐸 푇훿푆 − 푝훿푉

= + + . (4.3) 휅푠 훿퐴 2휋 4 The term in (4.3) gives the훿푀 Hawking temperature휇훿푄 Ω which훿퐽 leads us to the identification of 휅푠 the black 2hole휋 entropy with one quarter the area of the event horizon:

= . (4.4) 1 푆 4 퐴

4.2 BPS black holes

The N-extended four-dimensional supersymmetry algebra is given by

, = 2 , , = , (4.5) 퐴 †퐵 휇 퐴퐵 퐴 퐵 퐴퐵 훼 훽 훼훽 휇 훼 훽 훼훽 where , = 1, …�푄, label푄 � the supercharges휎 훿 푃 which�푄 푄 are� taken휖 푍to be Majorana spinors. is a complex antisymmetric matrix of central operators which can be skew-diagonalized퐴퐵 퐴 퐵 푁 푍 using R-symmetry rotations. By Schur’s lemma the eigenvalues are constant on irreducible representations and the same holds true for the mass = . Using 푍푖 (4.5) we can show that the mass is bounded by the eigenvalues , 2which are휇 known as 푀 −푃휇푃 the central charges. We have: 푍푖 | | | | 0. (4.6) 2 2 2 If at least one of the inequalities푀 ≥ for푍1 in≥ (4.6)푍2 is≥ saturated,⋯ ≥ then part of the supercharges operate trivially and consequently the corresponding representation will be smaller than a generic massive representation and will푀 have a different spin content. These representations are known as BPS representations. The special case in which all the central charges are equal so that = | | , for all , is known as a ½-BPS representation. 2 2 푀 푍푖 푖 Now consider once more the Reissner-Nordstrom solution (4.1). We note that for the Reissner-Nordstrom black hole we need to distinguish three different cases based on the relative values of the charge and the mass . For > we find both an interior and an exterior event horizon. When = the solution is said to be an extremal black hole, 푄 푀 푀 푄 for which the surface gravity vanishes and the two horizons coalesce into one. In the 푀 푄 < case we find a physically unacceptable solution where the electromagnetic energy exceeds that of the total energy and the event horizon is replaced by a naked singularity. 푀 푄 Thus extremal black holes saturate the bound for physically acceptable black hole solutions. In the context of = 2 supergravity, the extremal Reissner-Nordstrum black 푀 ≥ 푄 hole is a standard example of a so-called BPS soliton. In fact, as it has four Killing spinors, it is a ½-BPS solution푁

We now discuss an important property of BPS states which is relevant to the significance of these states in the physics literature. There is a natural embedding of Einstein- Maxwell theory into = 2 supergravity. In the next section we present a brief introduction to = 2 supergravity, but here we point out that when performing the 푁 embedding into supergravity theory, the classification of the Reissner-Nordstrom black hole solutions acquires푁 an interpretation in terms of the supersymmetry algebra. This algebra has a central extension proportional to the black hole charge(s) and the unitary representations of the algebra must have masses that are greater than or equal to the

charge(s) in accordance with (4.6). When this bound is saturated, we obtain BPS supermultiplets as was defined earlier. Because of the fact that these BPS supermultiplets are smaller than the generic massive supermultiplets, they are stable under (adiabatic) changes of the coupling constants and the relation between charge and mass remains preserved. Finally we emphasize that as defined above, the BPS black hole solutions saturate a bound implied by the supersymmetry algebra and are characterized by their possession of some residual supersymmetry.

4.3 = supergravity

Supergravity푵 ퟐ is an extension of supersymmetry which arises naturally when one attempts to make supersymmetry transformations local following the approach taken in gauge theories. Starting with global supersymmetry, we find that we necessarily have to introduce a graviton field (and its supersymmetric partner the gravitino) in the process of making supersymmetry transformations local, thus incorporating gravity. Here we briefly introduce the = 2 supermultiplets and then move on to discuss the construction of supersymmetric actions. Some familiarity with the basic concepts of supersymmetry is assumed in what푁 follows.

4.3.1 Supermultiplet Decomposition

Of particular interest are the vector and the Weyl supermultiplets. The tensor supermultiplets and the hypermultiplets will not be considered in detail as they play a subsidiary role. The covariant fields and field strengths of the gauge fields associated with the vector or Weyl supermultiplet make up a chiral multiplet which is described in superspace by a chiral superfield. Scalar chiral fields in = 2 superspace have 16 bosonic and 16 fermionic field components which are reduced to 8 bosonic and 8 푁 fermionic field components by a constraint which expresses higher- components of the superfield in terms of lower- components or their space-time derivatives. The vector 휃 supermultiplet and the Weyl supermultiplet are related to these reduced chiral multiplets. 휃 The vector supermultiplet is described by a scalar reduced chiral superfield, whose lowest- component is a complex field. There is also a doublet of chiral fermions , where is an (2) R-symmetry doublet index. We recall that the R-symmetry group is 휃 푋 Ω푖 by definition the maximal group that rotates the supercharges in a way that commutes with Lorentz푖 symmetry푆푈 and is compatible with the supersymmetry algebra. The R- symmetry group in = 2 supersymmetry is (2) × (1), which acts on the spinors. The chirality of the spinor field is determined by the position of the index , such that 푁 푆푈 푈 has positive chirality and negative. The fields and are lowest- components in 푖 Ω푖 the anti-chiral superfield, related푖 to the chiral superfield via푖 complex conjugation. At the Ω 푋� Ω 휃 -level, we find the field strength of the gauge field and an auxiliary field written as 2 휃 퐹휇휈

39 a symmetric real tensor = = . In this way we have obtained precisely the 8 + 8 independent field components or off-푘푙shell degrees of freedom. 푌푖푗 푌푗푖 휀푖푘휀푗푙푌 For the Weyl supermultiplet, the reduced chiral superfield is an anti-selfdual Lorentz tensor rather than a scalar. In the case of extended supersymmetry the Weyl superfield is assigned to the antisymmetric representation of the R-symmetry group so that its lowest order -component is written as , where the , indices denote the components of space-time tangent space tensors. Its푖푗 complex conjugate corresponds to the anti-chiral 휃 푇푎푏 푎 푏 superfield and has a corresponding selfdual tensor . In light of its tensorial nature, the Weyl supermultiplet has 24 + 24 off-shell degrees of freedom. We now consider the 푇푎푏푖푗 covariant components of the Weyl multiplet. At the linear -level we find that the fermions decompose into the field strength of the gravitini and a doublet spinor . The 휃 gravitini field strengths account for 16 degrees of freedom and combined with the푖 8 휒 independent components corresponding to the spinors , we count 24 off-shell degrees of freedom. At the -level we find the Weyl tensor, field푖 strengths of the gauge fields 휒 associated with R-symmetry2 and a real scalar field . These respectively account for 5, 휃 4 × 3 and 1 off-shell degrees of freedom, which combine with the 6 degrees of freedom 퐷 associated with to give the full 24 bosonic degrees of freedom. 푖푗 The Weyl multiplet푇푎푏 contains the fields of = 2 conformal supergravity and one can write an invariant action that is quadratic in its components. We also note that the 푁 gravitini transform into and in locally supersymmetric Lagrangians of vector 푖푗 multiplets that are at most푎푏 quadratic in space-time derivatives, acts as an auxiliary 푇 푖푗 field and couples to a field-dependent linear combination of the 푎푏vector multiplet field strengths. For this class of Lagrangians, all the fields of the Weyl푇 multiplet with the exception of the graviton and gravitini fields act as auxiliary fields.

As was discussed, our aim is to move towards a theory of local supersymmetry. For this, the vector multiplets must first be formulated in a supergravity background, which leads to extra terms in the supersymmetry transformation rules as well as in the superfield components which are dependent on the supergravity background. Some of these additional terms correspond to the introduction of covariant derivatives. As mentioned before, we consider only vector and Weyl multiplets here. The Weyl multiplet describes the supermultiplet of conformal supergravity and so to achieve consistency we need to formulate the vector supermultiplet in a superconformal background, and indeed the vector supermultiplet is a representation of the full superconformal algebra. Therefore all the superconformal symmetries can be realized as local symmetries.

4.3.2 Supersymmetric Lagrangians

As both the vector and Weyl supermultiplets are described by chiral superfields even beyond their linearized form, we construct supersymmetric actions by taking some function of the vector superfield and the square of the Weyl superfield. Expansion of the superfields in components generates multiple derivatives of this function which depend on the lowest- components, and = ( ) . As the function ( , ) is 휃 holomorphic (i.e. it is independent of the complex푖푗 conjugates2 and ) we take the 휃 푋 퐴 푇푎푏 휀푖푗 퐹 푋 퐴 imaginary part of the resulting expression. 푋� 퐴̅ Here we focus on a few characteristic terms in the supersymmetric action. The scalar kinetic terms are accompanied by a coupling to the Ricci scalar and the scalar field of the Weyl multiplet as follows 퐷 ( ), (4.7) 휇 퐼 휇 퐼 1 퐼 퐼 휇 퐼 � 휇 �퐼 6 퐼 � �퐼 where denotesℒ ∝ 푖the�휕 derivative퐹 휕 푋 − of휕 퐹 휕with푋 �respect− 푖 � to푅 − 퐷. When� 퐹 푋 F −depends퐹 푋 on , this 퐼 푖푗 will generate퐼 interactions between the kinetic term for the vector multiplet scalars푎푏 and the tensor 퐹field of the Weyl multiplet. This퐹 pattern continues푋 for other terms. 푇

The kinetic term of the vector fields is proportional to the second derivative of the function and is given by

퐹 + (4.8) 1 −퐼 1 퐼 푖푗 −푎푏퐽 1 퐽 푎푏푘푙 4 퐼퐽 푎푏 4 � 푎푏 푖푗 4 � 푘푙 Finally weℒ consider∝ 푖퐹 a� term퐹 involving− 푋 푇 the휀 square� �퐹 of the− Weyl푋 푇 tensor,휀 � contained⋯ in the tensor ( ),

ℛ 푀 16 2 ( ) ( ) 16 + , (4.9) 푐푑 −푎푏 푎푏푖푗 푐 퐴 푐푑 푎 푐푏푖푗 where ℒdenotes∝ 푖 퐹a superconformally� ℛ 푀 푎푏ℛ 푀 covariant− derivative푇 퐷 which퐷 푇 also� contains⋯ terms proportional to the Ricci tensor. [25] 퐷푎 4.4 The microscopic description of black holes

As was alluded to at the start of this chapter, a key challenge in the study of black hole thermodynamics has been the development of a statistical mechanical interpretation of the black hole entropy. Ideas from string theory have led to progress in this direction, which have made possible the identification of the black hole entropy with the logarithm of the degeneracy of states ( ) of charge belonging to a certain system of microstates. In string theory these microstates are provided by the states of wrapped brane configurations of given momentum푑 푄 and winding.푄 Calculation of the black hole solutions

41 in the corresponding effective field theory with charges specified by the brane configuration leads us to a remarkable correspondence: the black hole area is equal to the logarithm of the brane state degeneracy, at least in the limit of large charges. [25]

To understand the relation between microscopic and field-theoretic descriptions of black holes we need to recall that strings exist in more dimensions than the familiar four space-time dimensions. In most situations the extra dimensions are compactified on some internal manifold and one has the standard Kaluza-Klein scenario leading to effective field theories in four dimensions, describing low-mass modes of the fields associated with appropriate eigenfunctions푋 on the internal manifold. The original space-time is then locally written as a product × , where is our four-dimensional space-time, so that at each point of we4 have a corresponding4 space . In this picture, the size of 푀 푋 푀 is such that it will not휇 be directly4 observable. We emphasize that the space need not be 푥 푀 푋 푋 the same at every point and in principle the internal spaces defined at various points of 푋 belong to some well-defined class of fixed topology parameterized by certain moduli. These4 moduli appear as fields in the four-dimensional effective field theory. 푀 We now return to look at our black hole through this higher-dimensional lens. Note that in this perspective the internal spaces as well as the four-dimensional space-time fields (such as the metric ) will vary over . In the effective field-theoretic context 푋 only the local degrees of freedom of strings and4 branes are captured. But extended objects 푔휇휈 푀 may also carry global degrees of freedom since they can also wrap themselves around non-trivial cycles of the internal space . This wrapping tends to occur at a particular point in and so in the context of the four-dimensional effective field theory will 푋 manifest itself4 as a pointlike object. This wrapped object is the string theory representation푀 of the black hole.

We have thus found two complementary descriptions of the black hole. One is the usual story based on general relativity where a point mass generates a global space-time solution to Einstein’s equation and is known as the so-called macroscopic description. The alternative picture which we have just discussed is based on the internal space where an extended object is entangled in one of its cycles and is referred to as the microscopic description. We note that the microscopic picture does not immediately involve gravitational fields and so can easily be described in flat space-time.

We will not delve deeper into the structure of this complementary picture by discussing the technical aspects of deriving the black hole entropy by microscopic arguments. We will suffice it to underline the important fact that the microscopic expression for the black hole entropy will depend only on the charges and not on other quantities, such as the values of the moduli fields at the horizon. 푄

4.5 The attractor mechanism

Clearly the microscopic description of black holes would lose much of its appeal if one could not find a way of connecting it to the macroscopic description. It is expected that any description of the black hole microstates would require some version of a quantum theory of gravity. String theory, which is of course one such theory of quantum gravity, has in recent decades been shown to provide a link between these two complementary pictures. In particular, the fact that in string theory gravitons are closed string states which interact with the wrapped branes ensures that such a link does exist. These interactions are governed by the string coupling and we are thus faced with an interpolation in the coupling constant. 푔푠 Unfortunately, relating the two pictures in practice has turned out to be an exceedingly difficult proposition, so much so that a realistic comparison of results between the two is usually not possible. However, in the case of extremal BPS black holes it has been demonstrated that reliable predictions are indeed possible such that the microscopic and macroscopic approaches can be successfully compared to create a deeper understanding of black holes.

When introducing the microscopic description of black holes we noted that the expression for the black hole entropy in this picture depends only on the charges. The corresponding field-theoretic calculation on the other hand may also depend on the values of the moduli fields at the horizon, for instance. Thus, arriving at any reasonable agreement between these two pictures necessitates that the moduli take fixed values at the horizon which may depend only on the charges. This is indeed the case for extremal solutions as was shown in the context of = 2 supersymmetric black holes.[26][27][28] The values taken by the fields at the horizon are independent of their asymptotic values at spatial infinity and this fixed point behaviour푁 is determined by so-called attractor equations.

In the presence of higher-derivative interactions where explicit construction of black hole solutions proves to be a challenge, one can usually find the fixed-point values without the need for interpolation between the horizon and spatial infinity, by focusing on the near-horizon region. Where the symmetry of the near-horizon region is sufficiently strong, the attractor mechanism can be expressed through a variational principle for a suitably defined entropy function. [29]

The = 2 BPS attractor equations can be determined by classifying all possible = 2 supersymmetric solutions through use of supersymmetric variations of the 푁 fermions in an arbitrary bosonic background. Setting these variations equal to zero 푁 implies strong restrictions on this background, as was shown in [28] for = 2 supergravity with an arbitrary number of vector multiplets and hypermultiplets, including 푁

43 higher-order derivative couplings proportional to the square of the Weyl tensor. It was found that the horizon geometry and the values of the fields of interest are completely determined by the charges. In particular, it was found that the extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. Furthermore, supersymmetry also determines the entropy as given by the Noether charge definition of Wald [30]. In the absence of charges we have a flat Minkowski space-time with constant moduli and = 0. 푖푗 It is not feasible to present a detailed and comprehensive 푇explanation푎푏 of the black hole attractor phenomenon in this section. Our aim is only to convey a sense of its crucial role in providing a link between the microscopic and macroscopic descriptions of black holes and hopefully pass on enough of an understanding of the attractor mechanism to make it possible for the reader unfamiliar with this phenomenon to experience some appreciation of the results in the remainder of this chapter.

Earlier we saw how geometric structures can be constructed using -forms defined on a manifold . Here we wish to emphasize a somewhat analogous way of looking at the 푝 attractor mechanism in superstring theory compactified on . In this picture, given a 푀 black hole charge , which can be represented by an integral cohomology class on , the 푀 attractor mechanism fixes certain metric data on at the black hole horizon. That is, 푄 푀 it provides a map 푔휇휈 푀 . (4.10)

We now turn our attention to the link existing푄 ⟼ 푔between휇휈 black holes and Hitchin functionals. Following [9] we suggest that the map (4.10) can be seen as arising from these functionals. In a sense, the metric flow of the internal manifold from spatial infinity to the black hole horizon can be viewed as a geodesic flow as determined by the Hitchin functional. Indeed this new approach is more powerful in the sense that the Calabi-Yau form of the metric follows from the action principle which gives the relation between the charge and the metric, whereas it is an underlying assumption in the other picture. We shall now consider how far we can take these ideas in the case of BPS black holes in dimensions 4 and 5.

4.5 BPS black holes in four dimensions

We consider four-dimensional BPS black holes in Type IIB string theory compactified on a Calabi-Yau 3-fold . Recall that in Chapter II we defined Hitchin’s functional as 푀 푉퐻 ( ) = , (4.11) 1 푉퐻 휌 2 ∫푀 휌� ∧ 휌

44 where is a stable 3-form in six dimensions. We also saw that we could rewrite (4.11) in terms of = + ( ) as 휌 Ω 휌 푖 휌 � 휌 ( ) = . (4.12) 푖 퐻 4 푀 � By varying within a fixed cohomology푉 휌 class− ∫ [ ]Ω, the∧ Ω critical points of generate holomorphic 3-forms on with real part . So by considering variations of , we find 휌 휌 푉퐻 the imaginary part of as a function of its real part. 푀 휌 푉퐻 Now we consider Ωthe effect of the attractor mechanism. For a given fixed charge in the theory on = , the attractor mechanism gives the value of at the black hole 푄 horizon and the real part4 of is equal to , where is the Poincare dual of . So we 푀 ℝ Ω notice the natural identification ∗ ∗ Ω 푄 푄 푄 [ ] = . (4.13) ∗ Here we note that under the above identification,휌 푄 restricting to variations within a fixed cohomology class corresponds precisely to fixing the black hole charge. We have thus 휌 established a connection between and the attractor mechanism at least classically.

The low-energy theory of Type IIB푉퐻 string theory compactified on a Calabi-Yau 3-fold is = 2 supergravity in four dimensions. Let , = 1, 2, , , , be the scalar 푀 components of the , ( ) vector multiplets of 퐼 = 2 supergravity.2 1 The vector multiplet 푁 푋 퐼 ⋯ ℎ part of the effective action2 1 is determined by the holomorphic prepotential ( ) which ℎ 푀 푁 defines a special Kahler geometry of the moduli space of , with Kahler potential . Let ℱ 푋 { , } be a basis of ( , ) and { , } be the dual basis of ( , ). Writing ̈ 푀 ̈ 퐾 (퐼 ) = we have 퐼 3 퐴 퐵퐼 퐻3 푀 ℤ 훼퐼 훽 퐻 푀 ℤ 퐹 퐼 푋 휕 퐼 ℱ = , (4.14)

퐼 퐼 푋 = ∫퐴 Ω, (4.15)

퐼 퐼 =퐹 ∫퐴 Ω ( ) (4.16) 퐼 퐼 and Ω 푋 훼퐼 − 퐹퐼 푋 훽

= log ( ). (4.17) 퐼 퐼 In this context, is related to the퐾 Ka−hler potential푖 푋� 퐹퐼 − 푋 by퐹� 퐼

퐻 푉 ( ) = ̈ = m퐾( ) = . (4.18) 푖 1 퐼 1 −퐾 퐻 4 푀 � 2 �퐼 4 Now let = [ ] (푉 ) and휌 denote− ∫ itsΩ Poincare∧ Ω 퐼 dual푋 by퐹 . Then푒 there exist , such that ∗ 퐼 푄 훾 ∈ 퐻3 푀 푄 푝 푞퐼

= . (4.19) ∗ 퐼 퐼 The central charge field is then given푄 by 푝 훼퐼 − 푞퐼훽

( ) = / = / = / ( ). (4.20) 퐾 2 퐾 2 ∗ 퐾 2 퐼 퐼 퐼 퐼 For a static spherically푍 푄 푒symmetric∫훾Ω extremal푒 ∫푀 Ω BPS∧ 푄 black푒 hole푝, the퐹 −semiclassical푞 푋 black hole entropy can be expressed as

= | | = | | 2 m( ). (4.21) 2 퐼 퐼 퐼 The scalar components푆 of휋 푍the vector휋 푝 multiplets퐹퐼 − 푞퐼푋 describe⁄ 퐼 푋 the퐹�퐼 complex structure moduli of . The equation (4.21)퐼 is valid for values of the moduli fields at the black hole 푋 horizon. The attractor mechanism specifies the values of the moduli fields at the horizon as a푀 function of the charges through the following attractor equations [32]:

Re( ) = , (4.22) 퐼 퐼 Re(퐶푋 ) = 푝, (4.23)

퐼 퐼 =퐶퐹2 푞. 퐾 (4.24) 2 Due to invariance of the entropy with respect퐶 − to푖푍 dilatations,̅푒 we may set = 1 to find that the charges and are simply given by the real parts of and , respectively. 퐶 Substituting these퐼 into (4.21) we obtain 퐼 푝 푞퐼 푋 퐹퐼 = Im( ). (4.25) 휋 퐼 퐼 Combining this result with the expression푆 2 given푋 for퐹� in (4.18), we find that at a critical point of : 푉퐻 휌 푐 푉 퐻 = ( ), [ ] = , (4.26) ∗ where is the semiclassical푆퐵퐻 black휋 hole푉퐻 휌 entropy.푐 휌 푄

We now푆퐵퐻 discuss a conjecture put forth in [9] which would suggest a deep and powerful connection between Hitchin’s theory and black holes. In [15] and [33] the “mixed” partition function is defined by

퐵퐻 푍 ( , ) = ( , ) , (4.27) 퐼 퐼 퐼 퐼 −휙 푞퐼 퐵퐻 푞퐼 퐼 where and are in the forms푍 휙(4.22)푝 - (4.23),∑ Ω 푝 =푞 Im푒 ( ) and the ( , ) are 휋 integer black퐼 hole degeneracies. A key point is that퐼 ( ) performs퐼 a “mixed”퐼 counting 푝 푞퐼 휙 푖 퐶푋 Ω 푝 푞퐼 푍퐵퐻 푄

of the states of the black hole. Now we formally define the partition function by the path integral 푍퐻

( ) = [ ] exp( ( )). (4.28)

퐻 ∗ 퐻 The conjecture can be stated푍 as 푄 ∫휌 =푄 퐷휌 푉 휌

( ) = ( ). (4.29)

A key point supporting this remarkable푍퐵퐻 statement푄 푍퐻 is푄 that provided the path integral defining ( ) converges, it would define a function of the charge which agrees with the black hole entropy on leading asymptotics. 푍퐻 푄 푄 Another interesting observation is that splitting ( ) into , cycles and the charge into electric and magnetic charges, = ( , ), we3 can identify with a Wigner 퐻 푀 퐴 퐵 푄 function derived from the model partition function : 푄 푝 푞 푍퐵퐻 퐵( ) = exp( )| 푍퐵( + )| . (4.30) 퐼 2 This is essentially the same푍퐵퐻 푄as the∫ link푑Φ existing푖푄 betweenΦ퐼 푍퐵 푝 and푖Φ the model which was discussed in Chapter III. There we also discussed how the functional is related to a 푉퐻 퐵 combination of the and models which is consistent with the role that this same and 푉퐻 combination plays in the counting of black hole entropy. [15] 퐵 퐵� 퐵 The퐵� conjecture directly connects Hitchin’s theory to the microscopic description of black holes.

4.5 BPS black holes in five dimensions

In this section we will argue that for a stable 4-form defined on the 6-manifold , the Hitchin functional ( ) = / can be related to black hole entropy in the 5- 1 휎 푀 dimensional theory obtained by compactifying3 2 M-theory on . The BPS black holes in 푉푆 휎 6 ∫푀 휎 this 5-dimensional theory are constructed by wrapping M2-branes over 2-cycles of , 푀 and are characterized by a charge ( , ) = ( , ) and a spin . 4 푀 First we consider the = 0 case. Making푄 ∈ 퐻2 a푀 parallelℤ 퐻identification푀 ℤ to the one푗 made in (4.13) during our analysis of 4-dimensional BPS black holes, we write 푗 [ ] = . (4.31) ∗ The value of the moduli at the horizon, as휎 determi푄 ned via the attractor mechanism ([34], [35]) are then given by a Kahler form satisfying = . We then have the following 1 relation between the black hole entropy and the classical2 value of ( ): ̈ 푘 2 푘 휎 푆퐵퐻 푉푆 휎

~ = / . (4.32) 3 2 퐵퐻 For non-zero black hole spin 푆, we introduc∫푀 푘 ∧e푘 a ∧6-푘form∫ 푀field휎 and identify with the integral cohomology class of : 푗 퐽 푗 퐽 = [ ] ( , ). 6 Noting that the entropy of the spinning푗 black퐽 ∈ 퐻hole푀 inℤ five dimensions scales as , we define the modified functional 3 2 �푄 − 퐽 푆 푉 ( , ) = , (4.33) 3 2 푆 where 푉 휎 퐽 ∫푀 �휎 − 퐽

= . 3 2 푖1푖2푗1푗2푘1푘2 푖3푖4푗3푗4푘3푘4 푖1푖2푖3푖4 푗1푗2푗3푗4 푘1푘2푘3푘4 푖1푖2푖3푖4푘1푘2 푗1푗2푗3푗4푘3푘4 We휎 −emphasize퐽 �휎 that the휎 Kahler휎 form is− invariant퐽 under퐽 the introduction�휖 of the휖 6-form .

In the case of 5-dimensional̈ BPS black푘 holes, the conjecture analogous to (4.29) is that퐽

( ) = [ ] exp( ( , )) (4.34) ∗ 푆 휎 =푄 푆 should count the degeneracies푍 of푄 the BPS∫ black퐷휎 hole characterized푉 휎 퐽 by charge and spin , where = [ ] ( , ). 6 푄 푗 푗 퐽 ∈ 퐻 푀 ℤ

V. The Geometry of Calabi-Yau Moduli Spaces

In this chapter we introduce the notion of Calabi-Yau moduli spaces as parameter spaces describing deformations of Calabi-Yau manifolds. We shall see how the moduli space admits a metric whose form implies a product structure on the moduli space, consisting of two factors corresponding to complex-structure deformations and K hler-structure deformations. We then consider the geometry of these spaces in some detail and find, in 푎̈ particular, that in either case the K hler potentials are given by Hitchin functionals.

5.1 Calabi-Yau moduli spaces 푎̈

The Betti number associated with a manifold is defined as the dimension of the th de Rham cohomology ( ) and is a fundamental topological number. In the case of 푏푟 푀 Kahler manifolds, the Betti numbers푟 decompose as 푟 퐻 푀 ̈ = , , (5.1) 푘 푝 푘−푝 푟 푝=0 where , are Hodge numbers which푏 count∑ theℎ number of harmonic ( , )-forms on the manifold.푝 푞The Hodge numbers do not provide a complete topological characterization of the manifoldℎ and some Calabi-Yau manifolds with the same Hodge numbers푝 푞 can be smoothly related via deformations of the parameters characterizing their size and shape. These parameters are known as the moduli. We consider a Calabi-Yau manifold with fixed Hodge numbers and study the fluctuations arising from deformations of the metric.

In the context of superstring compactifications, the zero modes of the graviton field generate the four-dimensional metric together with a set of massless scalar fields arising from the internal components of the metric . In Calabi-Yau compactifications, 푔휇휈 the metric does not generate any massless vector fields since = 0. Requiring the 푔푚푛 background to remain Calabi-Yau under a small variation of the1 metric on the internal space leads to differential equations whose solutions determine푏 the number of ways that the background metric can be deformed in a manner that preserves supersymmetry and the topology. The moduli can then be interpreted as the coefficients of the independent solutions to these differential equations and are given by the expectation values of massless scalar fields known as moduli fields. The moduli provide a topologically invariant parametrization of the size and shape of the internal Calabi-Yau manifold.

5.2 Deformations of Calabi-Yau three-folds

We perform a small variation

+ , (5.2)

푔푚푛 ⟶ 푔푚푛 훿푔푚푛

and demand that the Calabi-Yau conditions are satisfied both before and after the variation. Thus, in particular, both and + must be Ricci-flat

( )푔푚푛= 0 = 푔푚푛( +훿푔푚푛). (5.3)

Writing = , 푅we푚푛 impose푔 a gauge푅푚푛 fixing푔 훿푔 condition 푚 푚푝 푚 푚푝 훿 푔 푔 훿 푔 = (5.4) 푚 1 푚 푚푛 푛 푚 that removes the metric deformations∇ 훿 푔that correspond2 ∇ 훿푔 simply to coordinate transformations. Now the expansion of ( + ) = 0 yields the following differential equations for : 푅푚푛 푔 훿푔

훿푔 + 2 = 0, (5.5) 푘 푝 푞 푘 푚푛 푝푞 where we have used ( )∇=∇0.훿 It푔 can be shown푅푚 푛 훿that푔 the differential equations for the pure components and the mixed components decouple. We interpret + 푅푚푛 푔 as a Kahler metric and use it to define the Kahler form 훿푔푎푏 훿푔푎푏� 푔 훿푔 ̈ = ̈ . (5.6) 푎 푏� � Each of the ten-dimensional superstring퐽 푖푔 푎theories푏푑푧 ∧ contain푑푧̅ an NS-NS 2-form . Following compactification on a Calabi-Yau 3-fold, the internal (1,1)-form will have , zero 퐵 modes generating massless scalar fields in four dimensions. The complexified K 1hler1 퐵푎푏� ℎ form whose variations give rise to , massless complex scalar fields in four 푎̈ dimensions is defined by combining the1 1 real closed 2-form with the Kahler form : 풥 ℎ = + . 퐵 ̈ 퐽 (5.7)

Although the purely holomorphic and풥 purely퐵 antiholomorphi푖퐽 c metric components vanish, we may consider variations of these to nonzero values producing a variation in the complex structure. Each such variation is associated with a complex (2,1)-form

, (5.8) 푐푑� 푎 푏 푒̅ which is harmonic if the variationsΩ푎푏푐푔 respect훿푔푑�푒̅푑 (5.3).푧 ∧ 푑푧 ∧ 푑푧̅

5.3 The moduli space metric

The moduli space is essentially a parameter space describing complex and Kahler deformations of Calabi-Yau manifolds. This space is endowed with a natural metric [3] consisting of a sum of two pieces corresponding to deformations of the complex structurë and deformations of the complexified Kahler form

= [ + ( )] , (5.9) 2 1 푎푏� 푐푑� 6 푎푐 푏�푑� 푎푑� 푐푏� 푎푑� 푐푏� where 푑is푠 the volume2푉 ∫ 푔 of푔 the 훿Calabi푔 푔-Yau manifold훿푔 훿푔 −. The훿퐵 form훿퐵 of the�푔 moduli푑 푥 space metric implies a local splitting of the moduli space structure into a product 푉 푀 ( ) = , ( ) × , ( ). (5.10) 2 1 1 1 We now take a deeper lookℳ into푀 the geometryℳ 푀 of eachℳ of 푀these two spaces.

5.4 The complex-structure moduli space

First we consider the space of complex-structure metric deformations. Let , = 1, … , , , be local coordinates for the complex-structure moduli space and훼 define a 푡 set of (2,1)-2forms1 = ( ) by 훼 ℎ 1 푎 푏 푐̅ 훼 훼 푎푏푐̅ 휒 2 휒 푑푧 ∧ 푑푧 ∧ 푑푧̅ ( ) = / , (5.11) 1 푑� 훼 훼 푎푏푐̅ 푎푏 푐푑̅ � where we raise and lower indices휒 using the− 2 hermitianΩ 휕푔 metric.휕푡 We may invert these relations to find that

= ( ) , (5.12) 1 푐푑 훼 � 2 � 훿푔푎�푏 − ‖Ω‖ Ω�푎� 휒훼 푐푑푏훿푡 where = . Writing the moduli space metric in the form 2 1 푎푏푐 ‖ ‖ 6 푎푏푐 � Ω Ω Ω = 2 , (5.13) 2 훼 훽� 훼훽� we find that 푑푠 퐺 훿푡 훿푡̅

= � 훼 훽� 푖 ∫ 휒훼 ∧ 휒훽̅ 훼 훽� 퐺훼훽� 훿푡 훿푡̅ − � � 훿푡 훿푡̅ on substituting (5.12) into (5.9). 푖 ∫ Ω ∧ Ω�

Under a change in complex structure, becomes a linear combination of and , and the holomorphic (3,0)-form can be expressed as a linear combination of a (3,0)-form and 푑푧 푑푧 푑푧̅ the (2,1)-forms defined in (5.11). Differentiating = , we 1 find that 푎 푏 푐 휒훼 Ω 6 Ω푎푏푐푑푧 ∧ 푑푧 ∧ 푑푧 = + , (5.14)

where = / . Towards the end휕훼 Ωof this퐾 훼sectionΩ 휒훼 we derive the precise form of , but for now we note that훼 it depends only on the coordinates of the complex structure 휕훼 휕 휕푡 퐾훼 moduli space and not on the coordinates of the Calabi-Yau훼 manfold . 푡 푀

Noting that = , and using = together 푖 ∫ 휒훼∧휒�훽� 2 1 훼 훽� 훼 훽� � � � with (5.14), 퐺we훼훽 find휕 that훼휕훽 풦the metric on the퐺 훼complex훽훿푡 훿푡̅-structure− � 푖 ∫moduliΩ∧Ω� � 훿space푡 훿푡 ̅is Kahler with the Kahler potential given in terms of the Hitchin functional by ̈ ̈ , = log푉퐻( ). (5.15) 2 1 Now choose a basis of three-cycles풦 ,− ( , 푖 ∫=Ω0∧, …Ω�, , ) satisfying 퐼 2 1 퐽 = = 퐴 퐵 and퐼 퐽 ℎ = = 0. (5.16) 퐼 퐼 퐼 퐼 퐽 퐽 퐽 퐽 퐼 퐽 The dual cohomology퐴 ∩ 퐵 basis−퐵 ∩ 퐴, satisfies훿 퐴 ∩ 퐴 퐵 ∩ 퐵 퐽 퐼 = 훼 훽= and = = . (5.17) 퐽 퐽 퐽 퐼 퐼 퐼 퐴 퐼 퐼 퐼 퐵퐽 퐽 퐽 These properties∫ 훼are preserved∫ 훼 ∧ 훽 under훿 the action of∫ the훽 symplectic∫ 훽 ∧ 훼 modular−훿 group (2 , + 2; ). 2 1 푆푝We defineℎ a setℤ of coordinates ( = 1, … , , ) of the complex-structure moduli space by = / where are the훼 periods of 2the1 holomorphic 3-form : 훼 훼 0 퐼 푡 훼 ℎ 푡 푋 푋 푋 = 퐴 , for = 0, … , , . Ω (5.18)

퐼 퐼 2 1 As the moduli space is spanned푋 by∫퐴 theΩ , the 퐼 periods ℎ 퐼 푋 = 퐵 (5.19)

퐼 퐵퐼 are functions = ( ) and we can express퐹 ∫ theΩ holomorphic 3-form as

퐹 퐼 퐹 퐼 푋 = ( ) . Ω (5.20) 퐼 퐼 The relation = + impliesΩ 푋 the훼퐼 condition− 퐹퐼 푋 훽 = 0, which can be used to show that 휕훼Ω 퐾훼Ω 휒훼 ∫ Ω ∧ 휕퐼Ω = = . (5.21) 퐽 퐼 1 퐽 퐼 퐼 퐽 퐽 Thus, we find that the periods퐹 푋 are휕퐹 given⁄휕푋 by 2the휕� derivatives푋 퐹 �⁄휕푋 of the prepotential = . It is clear from (5.21) that the prepotential is homogenous of degree 2. 퐵 1 퐽 퐽 Using퐹 2 푋the퐹 expression of the Kahler potential in terms of the Hitchin functional we have

, , = ̈ = ( ) 2 1 2 1 −풦 ℎ 퐼=0 퐼 퐼 푒 푖 ∫푀 Ω ∧ ,Ω� −푖 ∑ , ∫퐴 Ω ∫퐵퐼 Ω� − ∫퐴 Ω� ∫퐵퐼 Ω = ( ). (5.22) 2 1 2 1 −풦 ℎ 퐼 퐼 ⇒ 푒 −푖 ∑퐼=0 푋 퐹�퐼 − 푋� 퐹퐼

Therefore, the prepotential is seen to completely determine the Kahler potential.

Finally we note that for a function퐹 of the moduli space coordinates̈ that is independent of the Calabi-Yau coordinates, the transformation 퐼 푓 푋 ( ) (5.23) 푓 푋 is a symmetry of the theory since is Ωuniquely⟶ 푒 definedΩ only up a function = ( ). The Kahler metric is also invariant under (5.23). Ω 푓 푓 푋 Differentiating̈ , = log( ) and using = + , we find the explicit form of to be 2 1 = , . We can now define the covariant derivative � 훼 훼 훼 풦 − 2푖1∫ Ω ∧ Ω 휕 Ω 퐾 Ω 휒 퐾 훼 퐾 훼 − 휕 훼 풦 = + , (5.24) 2 1 satisfying 퐷훼 휕훼 휕훼풦

( ) . (5.25) 푓 푋 5.5 The K hler-structure moduli퐷훼 spaceΩ ⟶ 푒 퐷훼Ω

Define a cubic퐚̈ form on the space of real (1,1)-forms by

( , , ) = , (5.26) and note that the volume of the휅 compact휌 휎 휏 6-dimensional∫푀 휌 ∧ 휎 ∧ 휏 manifold can be written as

= = ( , , ). 푀 (5.27) 1 1 The space of (1,1) cohomology푉 classes6 ∫푀 퐽 ∧ is퐽 endowed∧ 퐽 6 휅 with퐽 퐽 퐽a natural inner product

( , ) = , (5.28) 1 where is the Hodge star operator퐺 휌 on휎 the 2Calabi푉 ∫푀 휌-∧Yau⋆ 휎 manifold. Using the identity = + ( , , ) , we can rewrite the metric as ⋆ 1 ⋆ 휎 −퐽 ∧ 휎 4푉 휅 휎 퐽 퐽 퐽 ∧ 퐽 ( , ) = ( , , ) + ( , , ) ( , , ). (5.29) 1 1 2 Choosing a real basis퐺 휌 휎 ( =−12,푉…휅, 휌 ,휎) 퐽of harmonic8푉 휅 휌 (퐽1,퐽1)휅-forms,휎 퐽 퐽 the equation 1 1 푒 훼 훼 =ℎ + = , (5.30) 훼 defines the coordinates . The moduli풥 퐵 space푖퐽 metric푤 푒훼 can now be written as 훼 푤 = , = , , (5.40) 1 휕 휕 1 1 � 훼 훽� 퐺훼훽 2 퐺�푒훼 푒훽� 휕푤 휕푤� 풦

53 where the Kahler potential , is related to the volume of the Calabi-Yau manifold according to 1 1 ̈ 풦 , = = 8 . (5.41) 1 1 −풦 4 The space spanned by is a푒 Kahler manifold.3 ∫푀 퐽 ∧ 퐽 We∧ 퐽 notice,푉 of course, that the Kahler potential is once again given훼 by one of Hitchin’s volume functionals. 푤 ̈ ̈ Finally, we introduce a new coordinate and define a homogeneous function = ( ) of degree 2 which acts as the analogue0 to the complex structure moduli-space 푤 퐺 prepotential: 퐺 푤 ( ) = . 훼 훽 훾 (5.42) 1 푤 푤 푤 0 훼 훽 훾 The Kahler potential , is퐺 then푤 related6 푤 to this∫ 푒function∧ 푒 ∧ as푒 1 1 , , ̈ 풦 = ( / / ), (5.43) 1 1 1 1 −풦 ℎ 퐼 퐼 퐼 퐼 where = 1. 푒 푖 ∑퐼=0 푤 휕퐺̅ 휕푤� − 푤� 휕퐺 휕푤 0 푤

VI. Applications of Hitchin’s Hamiltonian Flow

After a brief review of the relevant mathematical ideas from Chapter II, we consider several applications of Hitchin’s Hamiltonian flow construction. In particular, we consider the implications of the Hamiltonian flow for topological M-theory on × and the canonical quantization of topological M-theory. Towards the end of the chapter6 1 we introduce a particularly interesting class of manifolds known as half-flat manifolds푀 푆 which arise in the context of = 1 string compactifications. We will demonstrate how these manifolds can be identified with the phase space of Hitchin’s Hamiltonian flow. 풩 6.1 Hitchin’s functionals

We begin with푽ퟕ a review of the main result from Chapter II on the geometry of stable 3-forms in 7 dimensions. Specifically, we construct a functional ( ) on a manifold of real dimension 7 which endows with a metric of holonomy , where is a stable 푉7 Φ 푋 3-form. We are aware, of course, that stable 4-forms also exist in 7 dimensions and 푋 퐺2 Φ indeed, as we shall see, we can construct a functional ( ) depending on a 4-form which endows with the same structure. In essence, these functionals are different 푉7 퐺 퐺 sides of the same mathematical coin. 푋 푉7 Let be a stable 3-form. Recall that generates a structure on simply because the exceptional Lie group is the stabilizer of . Define a symmetric tensor by Φ Φ 퐺2 푋 2 퐺 = Φ , 퐵 (6.1) 1 푖1⋯푖7 푗푘 푗푖1푖2 푘푖3푖4 푖5푖6푖7 using the Levi-Civita tensor퐵 in seven− 144 Φdimensions.Φ ΦWe now휖 construct a metric on in terms of using the tensor 푔 푋 Φ 퐵 = det ( ) / . (6.2) −1 9 푗푘 푗푘 The Hitchin functional ( ) now푔 has a 퐵simple interpretation퐵 as the volume of as defined by the metric . That is, 푉7 Φ 푋 푔 ( ) = = det ( ) / . (6.3) We can rewrite (6.3) as 1 9 푉7 Φ ∫푋 �푔Φ ∫푋 퐵 ( ) = , (6.4)

7 where denotes the Hodge star operator푉 Φ associated∫푋 Φ ∧ ∗ Φwith the metric .

We now∗ repeat our familiar analysis by fixing the cohomlogy class 푔of so that

= + where = 0,

and is some 2-form on Φ. PerformingΦ0 푑퐵 the first variation푑Φ0 of ( ) and using homogeneity we find that 퐵 푋 푉7 Φ ( ) = . (6.5) 7 7 Thus, for a critical point we must have훿푉 Φ ⁄훿Φ 3 ∗ Φ

= = 0, (6.6)

which holds if and only if defines 푑aΦ metric푑 ∗ofΦ holonomy on .

We now turn to an interestingΦ construction of ( ) where 퐺2 is a 푋stable 4-form. We begin

by choosing a basis for the 21-dimensional7 space of alternating bilinear vectors , 푖푗 푉 퐺 퐺 so that = for , = 1, , 7 and = 1, , 21. We now define a 21 × 21 matrix2 Q �푒푎 � Λ 푉 by 푖푗 푗푖 푒푎 −푒푎 푖 푗 ⋯ 푎 ⋯ = , (6.7) 1 푖푗 푘푙 푎푏 푎 푏 푖푗푘푙 in terms of which the functional (푄G) is written2 푒 푒 as퐺

푉7 ( ) = det (Q) / . (6.8) 1 12 푉7 퐺 ∫푋 This functional turns out to be equal to the one we would have obtained had we instead performed a construction parallel to (6.3), in effect expressing ( ) as a volume functional based on a metric derived from the 4-form . In either case we end up with the 푉7 퐺 functional 퐺 ( ) = , (6.9)

7 where we emphasize that here is 푉the퐺 Hodge∫푋 star퐺 ∧ operator∗ 퐺 defined by the metric derived from the stable 4-form . ∗ Considering the first variation퐺 of ( ) with restricted to a fixed cohomology class, we once again find 푉7 퐺 퐺 = G = 0, (6.10)

implying that the critical points of 푑퐺( ) are푑 in∗ precise correspondence to the metrics of holonomy on . 푉7 퐺 퐺2 푋

We shall return to this ( ) functional later when we consider the effective action of the

7-dimensional topological7 M-theory. For now we move on to a brief review of the construction of Hitchin’s푉 Hamiltonian퐺 flow.

6.2 ( ) structures and metrics with holonomy

Recall푺푼 thatퟑ for a 6-dimensional manifold we defined푮ퟐ the Hitchin functionals ( ) and ( ) by 푀 푉푆 휎 퐻 푉 휌 ( ) = = / , (6.11) 1 1 3 2 푆 and 푉 휎 6 ∫푀 푘 ∧ 푘 ∧ 푘 6 ∫푀 휎

( ) = = ( ) , (6.12) 푖 1 퐻 4 푀 � 2 푀 where is a stable 3-form 푉and휌 is a− stable∫ Ω 4∧-form.Ω We∫ saw휌� 휌 how∧ 휌 when extremized ( ) and ( ) respectively generate a symplectic structure (with symplectic form ) and a 휌 휎 푉푆 휎 complex structure (with holomorphic 3-form ) on . Provided we can combine these 푉퐻 휌 푘 two structures in a consistent manner, we can think of as the Kahler form on , with Ω 푀 respect to the complex structure defined by . As discussed in Chapter II, the symplectic 푘 ̈ 푀 and complex structures on are compatible in this way provided the following two Ω conditions are satisfied: 푀 = 0, (6.13)

and 푘 ∧ 휌

( ) = 2 ( ). (6.14)

These two consistency equations encode푉퐻 휌 the basic푉푆 휎 compatibility requirements that be of type (1,1) in the complex structure given by and that there be agreement between 푘 the volume forms defined by the symplectic and holomorphic structures. Under these Ω conditions, and combine to generate an (3) structure on . Furthermore, if and satisfy = 0 and = 0, as well as the compatibility conditions, then is a 푘 Ω 푆푈 푀 푘 Calabi-Yau 3-fold. Ω 푑푘 푑Ω 푀 In Theorem 2.6 of Chapter II, we saw how for closed and defined on , we can always uniquely find a metric of holonomy on the manifold × ( , ), where ( , ) 휌 휎 푀 is some interval. This was achieved by allowing and to time evolve with respect to 퐺2 푀 푎 푏 푎 푏 the flow governed by the equations 휌 휎 = , (6.15)

and 휕휌⁄휕푡 푑푘

= = , (6.16)

where is interpreted in as the “time”휕휎⁄휕푡 variable.푘 ∧ 휕푘 As⁄휕푡 was −demonstrated,푑휌� these flow equations are equivalent to the characteristic holonomy equations = = 0 for 푡 the 3-form defined by 퐺2 푑Φ 푑 ∗ Φ Φ = ( ) + ( ) . (6.17)

6.3 Topological M-theory on Φ× 휌 푡 푘 푡 ∧ 푑푡 ퟔ ퟏ Let and be a 3-form and 2-푴form, respectively,푺 on the 6-dimensional manifold and consider the 7-manifold = × . Define a 3-form on by 휌 푘 1 푀 푋 푀 푆 = + . Φ 푋 (6.18)

If we now assume that is stable onΦ , it 휌follows푘 ∧ 푑푡that and are stable on and so generate an (3) structure on , provided they satisfy the compatibility conditions Φ 푋 휌 푘 푀 (6.13) and (6.14). The integrability conditions of this (3) structure are 푆푈 푀 = 0, 푆푈 (6.19)

= 푑푘+ ( ) = 0, (6.20)

and these are precisely equivalent푑 toΩ 푑�휌 푖휌� 휌 �

= = 0. (6.21)

Interpreting (6.21) as the equations of푑Φ motion푑 ∗ ofΦ topological M-theory on , we see a reduction to a 6-dimensional theory whose classical solutions are Calabi-Yau 3-folds. 푋 Recalling the link existing between the functionals , and the topological and + models respectively, it perhaps seems natural to interpret the 6-dimensional 푉푆 푉퐻 퐴 reduction of the topological M-theory on as corresponding in some sense to a 퐵 퐵� combination of the and models. This might provide some insight into the 푋 nonperturbative completion of the topological string, which is expected to involve a 퐴 퐵 combination of the and models. Under the above construction, the compatibility conditions between the symplectic and holomorphic structures on provide the coupling 퐴 퐵 between the and models. 푀 6.4 Canonical퐴 quantization퐵 of topological M-theory

In this section we consider the canonical quantization of the 7-dimensional theory whose action is given by the functional ( ) = , where is a stable 4-form on the manifold = × . We fix the cohomology class [ ] ( , ) so that 푉7 퐺 ∫푋 퐺 ∧ ∗ 퐺 퐺 1 4 푋 푀 푆 퐺 ∈ 퐻 푋 ℝ

= + , where is a 3-form gauge field and = 0. Inspired by the ideas in the previous section, we express in the form 퐺 퐺0 푑Γ Γ 푑퐺0 퐺 = + , (6.22) so that it’s Hodge dual takes the form퐺 휎 휌� ∧ 푑푡

= + . (6.23)

Writing the ( ) action explicitly∗ in퐺 terms휌 of푘 ∧=푑푡 + , we find that after imposing the first compatibility condition = 0, the action can be written simply as a 푉7 퐺 퐺 휎 휌� ∧ 푑푡 linear combination of and 푘 ∧ 휌 퐻 푆 ( ) ( ) ( 푉) = 푉 × (2 + 3 ) = 2 + 3 . (6.24) 7 1 퐻 푆 퐻 푆 We now write the푉 gauge퐺 field∫푋= 푀 in푆 the푑푡 same푉 general푉 form푉 using휌 a 3푉-form휎 and a 2-form whose components lie only along , Γ 훾 훽 푀 = + . (6.25)

Assuming that and take the formsΓ given훾 above,훽 ∧ 푑푡 we find that ( , ( )) and ( , ) are related by 퐺 Γ 휎 휌� 휌 훾 훽 = + + , (6.26)

휌� =휌�0 +푑훽 , 훾 ̇ (6.27) so that ( , ) spans the configuration space.휎 휎 0Variation푑훾 of the action with respect to yields 훾 훽 훽 = = 0, (6.28)

which generates the gauge transformation훿푉7⁄훿훽 푑휌

+ . (5.29)

Thus following canonical quantization 훾of→ the훾 theory푑휆 defined by Hitchin’s ( ) action, the reduced phase space is parameterized by ( , ) for 푉7 퐺 ( ) 훾 휌 ( ), (6.30) 3 3 훾 ∈ Ω 푀 ⁄Ω푒푥푎푐푡( ). 푀 (6.31) 3 Using homogeneity, we calculate the휌 conjugate∈ Ω푐푙표푠푒푑 momentum푀 to be

훾 = , 휋 (6.32) 7 휋훾 4 휌

which immediately suggests the canonical commutation relation

{ , } = , (6.33)

where we interpret the canonical 훿훾variable훿휌 ∫as푀 훿훾“position”∧ 훿휌 and as “momentum”.

This commutation relation suggests an uncertainty훾 principle for휌 the quantum Calabi-Yau, whereby it would be impossible to perform simultaneous measurements of the complex and Kahler structures at the same point in . In this way, the and models will not be independent in the conjectured reduction of topological M-theory to a combination of the ̈ 푀 퐴 퐵 and models.

Now퐴 under퐵 a variation of that satisfies ( ) = 0, a variation is induced in the nonperturbative model partition function via the coupling to Lagrangian branes while 훾 푑 훿훾 the Kahler form is preserved. These closed variations of parameterize an ( , ) in 퐴 the phase space which is canonically conjugate to the cohomology class of 3by (6.33). In ̈ 훾 퐻 푀 ℝ this sense we see a mixing of the and model variables, with the parameter 휌 representing the Lagrangian D-brane tension in the model being conjugate to the 퐴 퐵 훾 model 3-form . 퐴 퐵 It is argued in [9]휌 that this conjugacy of the degrees of freedom of the and models within the context of topological M-theory might be related to the conjectured S-duality 퐴 퐵 relating the and models. The connection of the nonperturbative amplitudes of the model to Lagrangian D-branes and the field, together with the connection of the 퐴 퐵 퐴 nonperturbative amplitudes of the model to the D1-brane and the -field, is viewed as 훾 supporting the idea that the full nonperturbative topological string can be interpreted as a 퐵 퐵 single object consisting of a combination of the and models.

6.5 Half-flat manifolds in = string compactifications퐴 퐵

We now turn to the use 퓝of Hitchin’sퟏ Hamiltonian flow construction in the context of the study of so-called half-flat manifolds. It is known that 6-dimensional manifolds with (3) structure have a significant connection to the space of string vacua with minimal ( = 1) supersymmetry. An (3) structure on is defined by the existence of a non- 푆푈 degenerate almost hermitian structure together with a globally defined 3-form of 풩 푆푈 푀 type (3,0) with respect to . Such a structure푛 determines a Riemannian metric on 퐽푚 Ω , which can in turn be used푛 to define a 2-form of type (1,1) in terms of the almost 퐽푚 푔푚푛 complex structure : 푀 푛 퐽 퐽 푚 = . (6.34) 푝 푚푛 푚 푝푛 A characteristic property of an orientable퐽 6-퐽manifold푔 with (3) structure is that it admits a single globally defined Majorana spinor which is nonzero everywhere. The 푀 푆푈 휂

existence of such a spinor implies a reduction of the structure group of from (6) to (3). We can construct the differential forms and using the spinor as 푀 푆푂 푚푛 푚푛푝 푆푈 = 퐽 Ω 휂

Re 퐽푚푛 =푖휂̅Γ7Γ푚푛휂

푚푛푝 7 푚푛푝 ImΩ =−휂̅Γ Γ , 휂 (6.35)

푚푛푝 푚푛푝 where is the 6-dimensional chiralityΩ operator푖휂Γ and 휂 , are antisymmetrized products of the corresponding pair and triplet of gamma matrices. Γ7 Γ푚푛 Γ푚푛푝 The (3) invariant spinor satisfies

푆푈 휂 = 0, (6.36) ′ for some connection which may be different∇ 휂 from the Levi-Civita connection of the metric . When the ′structure group is reduced to (3), the difference between and ∇ ∇ defines the intrinsic torsion . In the special case when coincides with the Levi′- 푔푚푛 푆푈 ∇ Civita connection, the spinor is covariantly constant ( =′ 0) and the holonomy group ∇ 푊 ∇ ( , ) is thus given by (3). That is, is a Calabi-Yau manifold when . 휂 ∇휂 The manifolds with special holonomy (3) are Kahler with vanishing first Chern class′ 퐻표푙 푀 ∇ 푆푈 푀 ∇ ≡ ∇ and have metrics that are Ricci flat. Hence, we see that the intrinsic torsion is a 푆푈 ̈ measure of deviation from the Calabi-Yau condition. 푊 The intrinsic torsion lies in (3) with (3) defined by the decomposition (6) = (3) (3) . Using1 the decompositions⊥ ⊥(3) = and 푊 Λ ⊕ 햘햚 햘햚 under (3)⊥, it can be shown that the intrinsic torsion⊥ decomposes into 햘풐 햘햚 ⊕ 햘햚 햘햚 ퟏ ⊕ ퟑ ⊕ ퟑ� five1 torsion classes corresponding to the following (3) modules Λ ≅ ퟑ ⊕ ퟑ 푆푈 푊 푊푖 푆푈 = ( ) ( ) ( ) ( ) ( ) 푊1 ⊕ 푊2 ⊕ 푊3 ⊕ 푊4 ⊕ 푊5 ′ ′ Now and are in theퟏ ⊕ andퟏ ⊕ ퟖrepresentations⊕ ퟖ ⊕ ퟔ ⊕ ofퟔ� ⊕(6)ퟑ, ⊕respectively,ퟑ� ⊕ ퟑ ⊕andퟑ� have the following decompositions ([36], [37]) under (3): 푑퐽 푑Ω ퟐퟎ ퟐퟒ 푆푂 = Im( ) + + = ( 푆푈 ) ( ) ( ) = (6.37) 3 1 4 3 and푑퐽 − 2 푊 Ω� 푊 ∧ 퐽 푊 ퟏ ⊕ ퟏ ⊕ ퟑ ⊕ ퟑ� ⊕ ퟔ ⊕ ퟔ� ퟐퟎ

= + + = ( ) ( ) ( ) = , (6.38) ′ ′ where푑Ω 푊1퐽 ∧ 퐽 푊2 ∧ 퐽 �푊��5� ∧ Ω ퟏ ⊕ ퟏ ⊕ ퟖ ⊕ ퟖ� ⊕ ퟑ ⊕ ퟑ� ퟐퟒ

= = = 0. (6.39)

We note that 푊3 ∧ 퐽 푊3 ∧ Ω 푊2 ∧ 퐽 ∧ 퐽

( ), 0 푊1 ∈ Ω (푀), 2 = 푊2 ∈, Ω( 푀) , ( ), 2 1 1 2 3 3 푝푟푖푚 푝푟푖푚 푊 �푊 � �� ∈=Ω 푀 (⊕ )Ω, 푀 1 푊 4 푊��4� ∈, Ω( )푀. (6.40) 1 0 Distinct classes of (3) structure manifolds푊5 ∈ Ω can푀 be defined by requiring various combinations of the torsion classes to vanish. For example, Calabi-Yau manifolds are 푆푈 characterized as (3) structures whose five torsion classes uniformly vanish.

We consider a particularly푆푈 interesting class of manifolds of (3) structure which appear in the study of mirror symmetry on compactifications of string theories on Calabi-Yau manifolds with NS fluxes. These are known as half-flat manifolds푆푈 and play a significant role in the construction of realistic vacua with minimal ( = 1) supersymmetry. Half- flat manifolds can be viewed as mirrors of Calabi-Yau manifolds with a certain kind of NS-NS fluxes [38] and are characterized by the vanishing풩 of the following torsion components:

Re = Re = = = 0. (6.41)

In terms of and , this defining푊1 property푊2 of half푊4 -flatness푊5 is expressed as

퐽 Ω ( ) = 0,

푑(퐽Re∧ 퐽 ) = 0. (6.42)

Defining and by = and 푑 = ReΩ , we can rewrite (5.42) as 1 휎 휌 휎 2 퐽 ∧ 퐽 휌 =Ω0,

푑휎 = 0, (6.43) together with the constraint = 0. We푑휌 notice that this precisely corresponds to the structure induced on a 6-dimensional hypersurface embedded in a manifold. Thus we 휌 ∧ 퐽 see that Hitchin’s Hamiltonian flow allows us to uniquely extend a half-flat (3) 퐺2 manifold to a manifold = × ( , ) with a metric of holonomy . So the half-flat 푆푈 manifolds arising in the context of = 1 string compactifications can be precisely 푋 푀 푎 푏 퐺2 풩

62 identified with the phase space of Hitchin’s Hamiltonian flow. Furthermore, the ground states are in correspondence with Calabi-Yau manifolds, which are the stationary solutions of the flow equations.

It remains an open question whether all = 1 string vacua can be realized in topological M-theory. 풩

VII. Generalized Geometry

In this chapter we introduce the basic mathematical structure of generalized geometry. The key feature of generalized geometry is the replacement of the tangent bundle with the direct sum of the tangent and cotangent bundles , and use of the Courant 푇 bracket on sections of the bundle as the natural generalization∗ of the Lie bracket 푇 ⊕ 푇 defined on sections of . As we shall see,∗ this gives rise to a geometry with the novel 푇 ⊕ 푇 feature that it transforms naturally under both the action of a closed 2-form and the 푇 usual diffeomorphism group. After setting up this basic structure, we proceed to define the more sophisticated concepts of generalized tangent bundles and generalized퐵 metrics.

7.1 The linear algebra of ∗ Let be an -dimensional 푽vector⊕ 푽 space and be its dual space. The 2 -dimensional space admits a non-degenerate symmetric∗ inner product of signature ( , ) 푉 푛 푉 푛 defined by ∗ 푉 ⊕ 푉 푛 푛 + , + = ( ) + ( ) , (7.1) 1 for , and , 〈. 푣The휉 Lie푤 group휂〉 preserving2 �휉 푤 this휂 푣inner� product together with the canonical orientation existing∗ on is ( ) ( , ). The Lie algebra of 푣 푤 ∈ 푉 휉 휂 ∈ 푉 ( ) is given by ( ) =∗{ : , + ∗ , = 0; , } and 푉 ⊕ 푉 푆푂 푉 ⊕ 푉 ≅ 푆푂 푛 푛 based on the∗ structure, we have∗ the decomposition ∗ 푆푂 푉 ⊕ 푉 ∗ 햘햔 푉 ⊕ 푉 푇 〈푇푥 푦〉 〈푥 푇푦〉 ∀푥 푦 ∈ 푉 ⊕ 푉 푉 ⊕ 푉 = , (7.2) 퐴 훽 푇 � ∗� where End( ), : satisfies 퐵 =−퐴 and : satisfies = . Thus via ( ) = we think of ∗ as a 2-form∗ in , which leads∗ us to the decomposition∗ 퐴 ∈ 푉 퐵 푉 → 푉 퐵 −2 퐵∗ 훽 푉 → 푉 훽 −훽 퐵 푣 푖 푣 퐵 (퐵 ) = End(Λ )푉 . (7.3) ∗ 2 ∗ 2 By considering 햘햔 푉 ⊕ 푉 푉 ⊕ Λ 푉 ⊕ Λ 푉

1 exp (7.4) 1 � � for , we find a particular type of orthogonal퐵 symmetry of in the identity component2 of∗ ( ), known as a -transform and given by ∗ 퐵 ∈ Λ 푉 ∗ 푇 ⊕ 푇 푆푂 푉 ⊕ 푉 + 퐵 + + . (7.5)

Now let ( ) be the Clifford푣 algebra휉 ⟼ 푣 defined휉 푖푣 by퐵 ∗ 퐶퐿 푉 ⊕ 푉 , = , , (7.6) 2 ∗ 〈푣 푣〉 푣 ∀푣 ∈ 푉 ⊕ 푉

64 and consider the action of + on the exterior algebra , given by ∗ ∗ ∗ 푣 (휉 ∈+푉 ⊕) 푉 = + . Λ 푉 (7.7)

We find that 푣 휉 ∙ 휑 푖푣휑 휉 ∧ 휑

( + ) = ( + ) + ( + ) = ( ) = + , + , (7.8) 2 thus푣 defining휉 ∙ 휑 an푖 푣algebra푖푣휑 representation휉 ∧ 휑 휉 ∧ on푖푣휑 휉, ∧known휑 as푖푣 the휉 휑 spin〈 representation푣 휉 푣 휉〉휑 ( ). Since in signature ( , ) the∗ volume∗ element of a Clifford algebra Λ 푉 satisfies =∗ 1, the spin representation splits into two irreducible half-spin 푆푝푖푛 푉 ⊕ 푉 푛 푛 휔 representations2 and corresponding to the ±1 eigenspaces of . 휔 + − We note that the푆 group 푆 ( ) is a double cover of ( 휔 ) as seen by the homomorphism ∗ ∗ 푆푝푖푛 푉 ⊕ 푉 푆푂 푉 ⊕ 푉 : ( ) ( ) ∗ ∗ ( )( ) =휌 푆푝푖푛 푉 ⊕ 푉 ⟶ (푆푂 푉 ⊕)푉, . (7.9) −1 ∗ ∗ Now exp , obtained휌 푥 by푣 exponentiating푥푣푥 푥 ∈ 푆푝푖푛 푉 ⊕in 푉the Lie푣 ∈group푉 ⊕ 푉 ( ), acts on spinors as 2 ∗ ∗ 퐵 퐵 ∈ Λ 푉 푆푝푖푛 푉 ⊕ 푉 exp ( ) = 1 + + + . (7.10) 1 For even dimensional , there퐵 휑exists �an invariant퐵 2 퐵 bilinear∧ 퐵 ⋯ form� ∧ 휑, on ± which is symmetric if dim = 4 and skew-symmetric if dim = 4 + 2. Defining 푉 〈∙ ∙〉 푆 : / / by 푒푣 표푑푑 ∗ 푉푒푣 표푑푑푘 ∗ 푉 푘 휎 Λ 푉 ⟶ Λ ( 푉) = ( 1) , ( ) = ( 1) 푚 푚 we can write the bilinear휎 휑2푚 form on− ± as휑 2푚 휎 휑2푚+1 − 휑2푚+1

푆 , = ( ) . (7.11)

The pairing defined in (7.11) is known〈휑 휓 as〉 the 휎휑Mukai∧ 휓 pairing푛 between two spinors.

We end this section by presenting the definition of a pure spinor. First note that for a nonzero spinor , the null space = { : . = 0} depends equivariantly on under the spin representation, in the sense that for∗ all ( ) 휑 퐿휑 푣 ∈ 푉 ⊕ 푉 푣 휑 ∗ 휑 . = ( ) . 푔 ∈ 푆푝푖푛 푉 ⊕ 푉 (7.12)

푔 휑 휑 Now , = ( + ). = 0 for퐿 , 휌 푔 퐿 , so that 1 〈푣 푤〉 2 푣푤 푤푣 휑 푣 푤 ∈ 퐿휑

, = 0 , . (7.13)

That is, null spaces are isotropic.〈 푣 푤〉 ∀푣 푤 ∈ 퐿휑

Definition 7.1: A spinor is said to be pure if is maximally isotropic, i.e. it has dimension equal to . 휑 퐿휑 7.2 The Courant bracket푑푖푚 푉

We consider a bracket acting on pairs ( , ) of a vector field and a -form on a manifold , defined by 푋 휉 푋 푝 휉 푀 [ + , + ] = [ , ] + ( ). (7.14) 1 푋 푌 푋 푌 The skew-symmetric푋 operator휉 푌 휂 [. , . ] is푋 known푌 ℒ as휂 the− ℒCourant휉 − 2 푑bracket,푖 휂 − 푖which휉 unlike the Lie bracket fails to satisfy the Jacobi identity.

A feature of the Lie bracket on sections of is that it only has automorphisms which are trivial, in the sense that they correspond simply to diffeomorphisms of the manifold . 푇 One of the key properties of the Courant bracket is that it unlike the Lie bracket it also possesses non-trivial automorphisms which are defined by forms. To see this, let 푀 be a closed form and consider the vector bundle automorphism of defined푝+ by1 푃 ∗ 훼 ∈ Ω 퐹 푇 ⊕ Λ 푇 ( + ) = + + . (7.15)

A simple calculation confirms th퐹at푋 indeed휉 푋 휉 푖푋훼

([ + , + ]) = [ ( + ), ( + )]. (7.16)

Our interest in the Courant퐹 푋 bracket휉 푌 will휂 be limited퐹 푋 to 휉the퐹 case푌 where휂 = 1.

7.3 The structure of generalized geometries 푝

We now translate the results from our earlier study of the linear algebra of to the general setting of a manifold of dimension . Denote the tangent bundle of by∗ and 푉 ⊕ 푉 consider the direct sum of the tangent and cotangent bundles , which has a natural 푀 푛 푀 푇 inner product of signature ( , ) determined by ∗ 푇 ⊕ 푇 푛 푛 ( + , + ) = , (7.17) where and are tangent and cotangent푋 휉 푋vectors,휉 respectively.푖푋휉 The decomposition (7.3) translates here to the result that the skew-adjoint endomorphisms of are sections 푋 휉 of the bundle End( ) . As before, we focus on the action of∗ a 2-form 푇 ⊕ 푇 by taking the exponential to2 obtain∗ an2 orthogonal transformation on , yielding 푇 ⊕ Λ 푇 ⊕ Λ 푇 ∗ 퐵 푇 ⊕ 푇

+ + + (7.18)

Consider the action of + 푋 휉 ⟶ on 푋spinors휉 푖푋 defined퐵 by ∗ 푋 휉 ∈( 푇+⊕ 푇) = +휑 . (7.19)

Mirroring the ideas in section 6.1,푋 we휉 think∙ 휑 of푖푋 the휑 bundle휉 ∧ 휑 of differential forms as a bundle of Clifford modules over the Clifford algebra defined by this action of ∗ ∗ . Λ 푇 ∗ Thus, in this context we can think of forms as spinors. The 2-form then exponentiates푇 ⊕ 푇 into the spin group to act on a form as . 퐵 퐵 We are now in a position to see how the휑 Courant⟼ 푒 휑 bracket defined in (7.14) arises naturally when we attempt to endow the direct sum with a suitable differential structure. The Lie bracket and the exterior derivative are ∗related by the equation 푇 ⊕ 푇

2 [ , ] = ([ , ] ) + 2 ( 푑) 2 ( ) + [ , ] (7.20)

If we replace푖 푋 the푌 vector푑 푖푋 fields푖푌 훼 , 푖by푋푑 sections푖푌훼 − ,푖 푌푑 of푖 푋훼 푖,푋 and푖푌 the푑훼 interior products by Clifford products, we arrive at a new bracket [ , ] defined∗ on sections of and 푋 푌 푢 푣 푇 ⊕ 푇 satisfying the identities ∗ 푢 푣 푇 ⊕ 푇 [ , ] = [ , ] + ( ( ) ) ( , ) (7.21)

( )( , 푢)푓푣= ([ ,푓 ]푢+푣 ( , 휋),푢 푓) +푣(−, [푢 ,푣 ]푑푓+ ( , )) (7.22)

where is the휋 projection푢 푣 푤 onto 푢the푣 tangent푑 푢 bundle:푣 푤 (푣 +푢 푤) = 푑. When푢 푤 written out explicitly in terms of = + and = + , we find that this new bracket is precisely 휋 휋 푋 휉 푋 the Courant bracket defined in the previous section as 푢 푋 휉 푣 푌 휂 [ + , + ] = [ , ] + ( ). 1 푋 푌 푋 푌 We note that if 푋 is a휉 close푌 d휂 2-form,푋 푌the Courantℒ 휂 − bracketℒ 휉 − 2 is푑 invariant푖 휂 − 푖 휉under the map + + + . 퐵 We푋 have휉 ⟶ now푋 developed휉 푖푋퐵 all the elements necessary to define generalized geometry. Following Hitchin [39] we define “generalized geometries” as consisting of the data on which are compatible with the ( , ) structure and satisfy an integrability condition∗ expressed by the Courant bracket. The symmetry group of the geometry is then 푇 ⊕ 푇 푆푂 푛 푛 provided by the semi-direct product Diff( ) which replaces the usual diffeomorphism group. 2 푀 ⋉ Ω푐푙표푠푒푑

7.4 The generalized tangent bundle

Our definition of the generalized tangent bundle will rely on ideas related to a topological construct known as a gerbe on a manifold . We briefly review these ideas before discussing the generalized tangent bundle. Taking an open cover { } of , we think of a 푀 gerbe on as a collection of functions 푈훼 푀 푀 : (7.23) 1 훼훽훾 훼 훽 훾 satisfying = = and푔 푈= ∩ 푈 ∩ 푈 ⟶ 푆 = 1 on . −1 −1 −1 훼훽훾 훽훼훾 훽훾훿 훼훾훿 훼훽훿 훼훽훾 훼 훽 훾 훿 In the language푔 of 푔homological⋯ algebra,훿푔 this푔 simply푔 푔 means푔 that a gerbe푈 on∩ 푈 is∩ thought푈 ∩ 푈 of as a 2-cocycle. We define a connective structure on a gerbe as a collection of 1-forms 푀 in ( ) satisfying + + = on . 1 −1 훼훽 훼 훽 훼훽 훽훾 훾훼 훼훽훾 훼훽훾 훼 훽 훾 Definition퐴 Ω 푈7.2:∩ Let푈 { } be a connective퐴 퐴 structure퐴 on푔 a gerbe푑푔 on 푈. A ∩curving푈 ∩ 푈 of this connective structure is a collection of 2-forms ( ) satisfying = 퐴훼훽 푀 on . This defines a global closed 3-form = 2 = , known as the 퐵훼 ∈ Ω 푈훼 퐵훽 − 퐵훼 푑퐴훼훽 curvature. 푈훼 ∩ 푈훽 퐻 푑퐵훼 푑퐵훽 Now given a connective structure { }, it follows from + + = that + + = 0 so that we have a cocycle with values in the space−1 of 퐴훼훽 퐴훼훽 퐴훽훾 퐴훾훼 푔훼훽훾푑푔훼훽훾 closed 2-forms. Through identification of on with on via the - 푑퐴훼훽 푑퐴훽훾 푑퐴훾훼 field action + + + , this defines∗ a vector bundle ∗ as an extension 푇 ⊕ 푇 푈훼 푇 ⊕ 푇 푈훽 퐵 푋 훼훽 푋 휉 ⟶ 푋 휉 0푖 푑퐴 0. 퐸 (7.24) ∗ Since the ( , ) structure of ⟶ 푇 ⟶is preserved퐸 ⟶ 푇 ⟶ by the action of the 2-form , the vector bundle is also endowed with this∗ structure. Moreover, the preservation of the 푆푂 푛 푛 푇 ⊕ 푇 푑퐴훼훽 Courant bracket by the action of the closed 2-form implies the existence of an 퐸 induced bracket on sections of satisfying (7.21) and (7.22), where is the bundle 푑퐴훼훽 projection : . This structure defines the generalized tangent bundle . 퐸 휋 7.5 Generalized휋 퐸 ⟶ metric푇 s 퐸

Suppose is a Riemannian metric on the manifold . We can think of as a homomorphism : and consider its graph as a subset of . 푔 푀 푔 is then a subbundle on which∗ the indefinite inner product is positive definite.∗ This ∗ observation motivates푔 푇 ⟶ the푇 following definition of a 푉generalized metric푇 ⊕, which푇 푉 is⊂ not푇 ⊕ 푇 explicitly based on the existence of a Riemannian metric on .

푔 푀

Definition 7.3: Let be the generalized tangent bundle for a gerbe with connective structure. A generalized metric is a subbundle of rank on which the induced 퐸 inner product is positive definite. 푉 ⊂ 퐸 푛 Since the inner product is positive definite on the generalized metric and zero on the cotangent bundle , we must have = 0. Hence, defines a splitting of the exact 푉 sequence 0 ∗ 0. Similarly,∗ a second splitting is defined by the 푇 푉 ∩ 푇 푉 orthogonal complement∗ since the ( , ) signature of the inner product on implies a ⟶ 푇 ⟶ 퐸 ⟶ 푇 ⟶ negative definite inner product⊥ on . 푉 ⊥ 푛 푛 퐸 Locally, the splitting provides a 푉family of sections { } of satisfying = on . A tangent vector is then lifted ∗locally∗ to + . 푉 퐶훼 푇 ⊗ 푇 Define ( , ) = ( ) and note that 퐶훽 − 퐶훼 푑퐴훼훽 푈훼 ∩ 푈훽 푋 푋 푖푋퐶훼 퐶 훼 푋 푌 푖 푋 퐶 훼 푌 ( , ) = ( + , + ) (7.25) is positive definite by the positive퐶훼 푋 푌definiteness푋 푖푋 of퐶훼 the푋 induced푖푋퐶훼 inner product. We have,

( , + ) = ( , ) ( , ) = 0, (7.26) 푇 1 푇 푌 훼 푋 훼 훼 훼 showing that 푌 − 푖 퐶 is orthogonal푋 푖 퐶 to 2 .� 퐶Half푋 the푌 difference− 퐶 푌 푋 of� the two splittings + 푇 and 푌 훼 is the symmetric part of : 푌 − 푖 퐶 푇 푉 푋 훼 푋 훼 훼 푋 푖 퐶 푋 − 푖 퐶 ( + 퐶 ), (7.27) 1 푇 푋 훼 푋 훼 which is positive definite. Therefore,2 we푖 퐶 see that푖 퐶 half the difference of the splittings defined by and yields a metric on . ⊥ Another interesting푉 푉 observation is that푔 the 푀average of the two splittings yields the skew symmetric part of , which is a 2-form defining a curving of the connective structure as can be seen by the relation = satisfied on . This generates a 퐶훼 퐵훼 curvature . 퐶훽 − 퐶훼 푑퐴훼훽 푈훼 ∩ 푈훽 퐻

VIII. Generalized Calabi-Yau Manifolds

Now that we have developed a solid foundation for generalized geometry, we introduce a generalized complex structure as a particular kind of structure that is compatible with this general background. We show how symplectic and complex structures are included as extremal special cases of this higher geometric construct. We also define the notion of generalized Calabi-Yau manifolds and confirm that these manifolds are special cases of generalized complex manifolds.

8.1 Linear generalized complex structures

Consider a real finite-dimensional vector space of dimension . We know that a nondegenerate bilinear form defines a symplectic structure on . Using the 푉 푛 interior product, a symplectic form 2 can be thought of as a mapping : from 휔 ∈ Λ 푉 푉 to its dual space . This gives an equivalent way of defining a symplectic structure on 휔 휔 푣 ⟼ 푖푣휔 as an isomorphism∗ : satisfying 푉 푉 ∗ 푉 휔 푉 ⟶ 푉 = , (8.1) ∗ where is the linear dual of the mapping휔 . −휔 ∗ A complex휔 structure on is an endomorphism휔 : such that

푉 = 퐽1푉. ⟶ 푉 (8.2) 2 We now take advantage of the structure of퐽 the −2 -dimensional direct sum to embed these two very different structures into a higher geometric construct. This ∗higher 푛 푉 ⊕ 푉 structure is known as the generalized complex structure on .

Defininition 8.1: A generalized complex structure on is an푉 endomorphism of satisfying both = 1 and = . ∗ 2 ∗ 푉 풥 푉Now⊕ 푉let be a complex 풥structure− on 풥 and consider−풥 the endomorphism 0 퐽 푉 = , 0 (8.3) −퐽 풥퐽 � ∗� written with respect to . It is easy to see that퐽 = 1 and = , confirming that is a generalized complex∗ structure. Similarly if2 is a symplectic∗ structure on , 푉 ⊕ 푉 풥퐽 − 풥퐽 −풥퐽 the endomorphism 풥퐽 휔 푉 0 = , (8.4) 0−1 풥휔 � −휔 � 휔

is a generalized complex structure since it satisfies = 1 and = . Thus we 2 ∗ see that complex and symplectic structures on a finite휔-dimensional 휔vector space휔 appear as extremal special cases of linear generalized complex풥 structures.− Specifically,풥 −풥 complex and symplectic structures on correspond to diagonal and anti-diagonal linear generalized complex structures, respectively. 푉 8.2 Generalized complex structures on manifolds

In this section we define a generalized complex structure on a smooth manifold of even dimension. The definition that we present here is the one given by Hitchin in [40]. Through a sequence of propositions in Gualtieri’s thesis [41], one can see that this definition is a natural extension of the notion of a generalized complex structure on a finite-dimensional vector space as defined in the previous section.

The following definition is inspired by the observation that in the case of an even dimensional manifold , an almost complex structure : with = 1 has a + -eigenspace in which is a subbundle whose sections are closed2 under the Lie 푀 퐽 푇 ⟶ 푇 퐽 − bracket and satisfies = . The Newlander-Nirenberg Theorem then implies 푖 푇 ⊗ ℂ 퐸 that the subbundle defines an integrable complex structure on . Replacing by 퐸 ⊕ 퐸� 푇 ⊗ ℂ and the Lie bracket by the Courant bracket, we arrive at Hitchin’s definition after 퐸 푀 푇 introducing∗ an isotropic condition. 푇 ⊕ 푇 Definition 8.2: Let be a smooth manifold of dimension 2 and consider the bundle . A generalized complex structure on is a subbundle ( ) such 푀 푛 that ∗ ∗ 푇 ⊕ 푇 푀 퐸 ⊂ 푇 ⊕ 푇 ⊗ ℂ ) = ( ) ∗ 푖 )퐸 the⊕ space퐸� of푇 ⊕sections푇 ⊗ ofℂ is closed under the Courant bracket

푖푖 ) is isotropic. 퐸

The푖푖푖 퐸isotropic condition ensures the compatibility of the almost complex structure : with the indefinite metric on , in the sense that the metric ( + , ∗ + ) = ( ∗ + ) is pseudo-hermitian. Furthermore,∗ the obstruction to 퐽 푇 ⊕ 푇 ⟶ 푇 ⊕1푇 푇 ⊕ 푇 condition ) requiring closure under the action of the Courant bracket is tensorial only if 푋 휉 푌 휂 2 푖푋휂 푖푌휉 is isotropic. 푖푖 This퐸 definition of a generalized complex structure yields a reduction of the structure group of from (2 , 2 ) to ( , ) (2 , 2 ), together with an integrability condition.∗ In the special case where = on and = on , we 푇 ⊕ 푇 푆푂 푛 푛 푈 푛 푛 ⊂ 푆푂 푛 푛 recover an ordinary complex manifold. ∗ 퐽 퐼 푇 퐽 −퐼 푇

8.3 Spinors

We recall from the previous chapter that due to the linear structure of , there exists a decomposition of the spin representation into two irreducible half-spin representations∗ 푇 ⊕ 푇 and . This decomposition corresponds to a splitting of the bundle of formal sums of differential+ − forms on into the bundle of even forms and the bundle 푆 푆 of odd forms. Considering∗ ∗ that the action of 푒푣( ∗, ) ( ) on is표푑푑 ∗ Λ 푇 푀 Λ 푇 ∗ ∗ ∗ Λ 푇 |det 퐺퐿| 푛 ℝ ⊂, 푆푝푖푛 푇 ⊕ 푇 Λ 푇 (8.5) 1⁄2 ∗ where denotes the standard action휑 ⟼ of 푀 (푀, 휑) on , we note the following relation between∗ the spin bundle and the bundle of forms ∗ ∗: 푀 푀 ∈ 퐺퐿 푛 ℝ Λ∗푇∗ 푆= ( ) . Λ 푇 (8.6) ∗ ∗ 푛 1⁄2 Thus, we have 푆 Λ 푇 ⊗ Λ 푇

= ( ) , (8.7) + 푒푣 ∗ 푛 1⁄2 푆 = Λ 푇 ⊗ (Λ 푇 ) . (8.8) − 표푑푑 ∗ 푛 1⁄2 Recall also that the spin representations푆 Λ 푇 and⊗ Λ 푇have an associated invariant bilinear form , which takes values in the one+-dimensional− space . As defined in (8.11), 푆 푆 the bilinear form is equivalent to the exterior product pairing up푛 to∗ a choice of sign. 〈휑 휓〉 Λ 푇 The action of + is determined by Clifford multiplication to be

푋 휉 ( + ) = + , (8.9)

and in this representation the action푋 휉 of∙ 휑 푖푋휑 is휉 ∧ 휑 2 ∗ exp ( ) = 1 +퐵 ∈+Λ 푇 + . (8.10) 1 By removing the trivial line퐵 휑bundle� ( 퐵) 2, 퐵we∧ can퐵 think⋯ � ∧of휑 forms as spinors. It is easy to see that the null space = { + 푛 (1⁄2 ) ( + ). = 0} of a spinor Λ 푇 is an isotropic subspace. A pure spinor was defined∗ earlier by the condition that is 퐿휑 푋 휉 ∈ 푇 ⊕ 푇 ⊗ ℂ ∶ 푋 휉 휑 휑 maximally isotropic. In the case of the pure spinor 1 , its null space is the maximal 퐿휑 isotropic subspace = . Transforming by 0 ∗ we obtain the pure ∈ Λ 푇 spinor exp (1) = 1 + + + ∗ , whose maximal isotropic2 ∗ subspace is 푈 푇 ⊂ 1푇 ⊕ 푇 퐵 ∈ Λ 푇 = exp ( ). 2 퐵 퐵 2 퐵 ⋯ 푈Therefore퐵 a 푇generalized complex structure is defined in the neighbourhood of each point by a complex pure spinor field, up to multiplication by a scalar. In general these locally defined pure spinors are local trivializations of a complex line bundle. However, for a

particular type of generalized complex structure known as a generalized Calabi-Yau manifold, the pure spinor is global. We now turn to the study of these structures.

8.4 Generalized Calabi-Yau manifolds

Definition 8.3: A generalized Calabi-Yau manifold is an even-dimensional manifold with a closed form / which is a complex pure spinor for the orthogonal 푀 vector bundle satisfying푒푣 표푑푑 , 0 at each point. 휑∗∈ Ω ⊗ ℂ The pure spinor푇 ⊕ associated푇 with〈휑 a 휑genera�〉 ≠ lized Calabi-Yau manifold ( , ) defines a maximal isotropic subbundle ( ) where 휑 ∗ 푀 휑 휑 = = { 퐸+⊂ 푇(⊕ 푇 ⊗) ℂ ( + ) = 0} (8.11) ∗ 휑 휑 is the annihilator퐸 of 퐿. The condition푋 휉 ∈ 푇,⊕ 푇 0⊗ impliesℂ ∶ 푋 휉 ∙ 휑

휑 〈휑 =휑�〉 ≠ = 0 (8.12)

� and thus 퐸휑 ∩ 퐸휑 퐸휑 ∩ 퐸�휑

= ( ) . (8.13) ∗ Now all that remains to show that퐸휑 ⊕a generalized퐸�휑 푇 ⊕ Calabi푇 ⊗-Yauℂ manifold is indeed a special case of a generalized complex manifold is to confirm that sections of are closed under the Courant bracket. To see this, assume that + and + lie in so that 퐸휑 휑 ( + ) = 푋 +휉 푌= 0 휂 퐸 (8.14)

and 푋 휉 ∙ 휑 푖푋휑 휉 ∧ 휑

( + ) = + = 0. (8.15)

Recalling that the Lie derivative푌 is휂 defined∙ 휑 by푖푌휑 =휂 ∧ 휑 + and using = 0, = and = , we find that ℒ푋 푑푖푋 푖푋푑 푑휑 푋 푌 푖 휑 − 휉 ∧ 휑 푖[ ,휑] =−휂 ∧(휑 )

푋 푌 푋 푌 푌 푋 푖 휑 = ℒ 푖 휑 − 푖 ℒ 휑 ( )

= −ℒ푋휂 ∧ 휑 − 휂 ∧ ℒ푋휑 −+푖푌푑( 푖푋휑 )

= −ℒ푋휂 ∧ 휑 −+ 휂 ∧ 푑푖푋휑 +푖푌 푑휉( ∧)휑

푋 푌 [ , ] = −ℒ 휂 ∧ 휑 + 휂 ∧( 푑휉) ∧ 휑 푖 푑휉 ∧ 휑 − 푑휉 ∧ 휂 ∧ 휑 (8.16)

푋 푌 푋 푌 By skew-symmetry,⇒ 푖 휑 −ℒ 휂 ∧ 휑 푖 푑휉 ∧ 휑

[ , ] = ( [ , ] [ , ] ) 1 푋 푌 휑 2 푖 푋 푌 휑 − 푖 푌 푋 휑 = ( + + ) 1 2 −ℒ푋휂 푖푌푑휉 ℒ푌휉 − 푖푋푑휂 ∧ 휑 = ( + + + ) 1 2 −푖푋푑휂 − 푑푖푋휂 푖푌푑휉 푖푌푑휉 푑푖푌휉 − 푖푋푑휂 ∧ 휑 = [2( + ) 2( + ) ( )] 1 2 푑푖푌 푖푌푑 휉 − 푑푖푋 푖푋푑 휂 − 푑 푖푌휉 − 푖푋휂 ∧ 휑 = [ ( )] . 1 푌 푋 푌 푋 That is, ℒ 휉 − ℒ 휂 − 2 푑 푖 휉 − 푖 휂 ∧ 휑

[[ , ] + ( )]. = 0 1 푋 푌 푋 푌 푋 푌 ℒ 휂[ −+ℒ 휉, −+2 푑]푖 휂 =− 푖0, 휉 휑 (8.17) so that [ + , + ] as⇒ claimed.푋 휉 푌 휂 ∙ 휑

We will 푋now휉 show푌 how휂 ∈ complex퐸휑 and symplectic manifolds appear as special cases of generalized Calabi-Yau manifolds. First consider an -dimensional complex manifold with a non-vanishing holomorphic form of degree . For any of type (0,1) (1,0) 푚 퐶 and of type , we have 푀 ∗ Ω 푚 푣 ∈ 푇 ⊗ ℂ 휉 ∈ 푇 ⊗ ℂ ( + ) = + = 0. (8.18)

Hence, is pure and we have푣 a maximal휉 ∙ Ω annihilator푖푣Ω 휉 ∧ subspaceΩ of real dimension 2 . Since is closed and satisfies Ω 푚 Ω , = ( 1) 0, (8.19) 푚 the pair ( , ) defines a generalized〈Ω Ω�〉 Calabi− -YauΩ ∧ manifold.Ω� ≠

Now let 푀퐶 beΩ a symplectic manifold of dimension with symplectic form and recall 1 that 푆 is a pure spinor. We obtain another pure spinor 푀0 ∗ 푚 휔 ∈ Λ 푇 = exp (8.20) 푒푣 by exponentiating the 2-form 휑. Note that푖휔 ∈ Ω= 0 ⊗andℂ

푖휔 , ~푑휑 0 (8.21) 푚 since is closed and non-degenerate.〈휑 Thus,휑�〉 휔( ,≠ ) is a generalized Calabi-Yau manifold. 휔 푀푆 휑

8.5 B-field transforms

Let ( , ) be a generalized Calabi-Yau manifold and consider the action of a real closed 2-form on given by 푀 휑 2 ∗ 퐵 ∈ Λ 푇 휑 = . (8.22) ′ 퐵 As noted earlier, is both closed and 휑pure. 푒Since휑 the -field acts via the Spin group, we have ′ 휑 퐵 , = , = , 0, (8.23) ′ ′ 퐵 퐵 implying that we obtain another〈휑 휑��〉 generalized〈푒 휑 푒 Calabi휑�〉 〈-휑Yau휑�〉 manifold≠ ( , ) through the action of the B-field on . The B-transform provides a way of interpolating′ between 푀 휑 symplectic and Calabi-Yau structures as we shall see. 휑 For a symplectic manifold with symplectic form , the pure spinor is = and the transform takes the form 푖휔 휔 휑 푒 = , (8.24) ′ 퐵+푖휔 where is a nonzero complex constant.휑 푐푒

Let 푐= + be the holomorphic non-degenerate (2,0)-form of a holomorphic symplectic푐 manifold of complex dimension 2 and note that each of and 휔 휔1 푖휔2 defines a real symplectic structure on . For 0, the action of = / on the 푀 푘 휔1 휔2 generalized Calabi-Yau manifold ( , ) yields the generalized Calabi-Yau manifold 푀 푡 ≠ 퐵 휔1 푡 ( , ( )), where 푖휔2⁄푡 푀 푒 푀 휑 푡 ( ) = ( ) . (8.25) 휔1+푖휔2 ⁄푡 The limit 휑 푡 푒

= lim ( ) = ( + ) (8.26) ! 푘 1 푘 푡⟶0 1 2 gives the (2 , 0)-form of 휑a Calabi-Yau푡 structure휑 푡 푘via휔 the B푖-휔field transform of the symplectic structure defined by / . 푘 8.6 Structure group reductions휔 2 푡

As was mentioned earlier, a generalized complex structure is a reduction of the structure group of from (2 , 2 ) to ( , , ) together with an integrability condition. Here we consider∗ the reduction of the structure group defined by a Calabi-Yau manifold 푇 ⊕ 푇 푆푂 푛 푛 푈 푛 푛 ( , ). As , 0, / | , | defines a non-vanishing section

푀 휑 〈휑 휑�〉 ≠ 휑 � 〈휑 휑�〉

( ) / (8.27) + 푒푣 ∗ 푛 1 2 of the spinor bundle . We푆 can⊗ rewriteℂ ≅ Λ this푇 as⊗ Λ 푇 ⊗ ℂ

푆 ( ) / , (8.28) + 푒푣 ∗ 푛 1 2 휑 휑 since ( ) 푆 is⊗ a maximallyℂ ≅ Λ 퐸 isotropic⊗ Λ 퐸subbundle⊗ ℂ. Thus defines a non- vanishing section of∗ ( ) / and so determines a complex volume form on , 퐸휑 ⊂ 푇 ⊕ 푇 ⊗ ℂ 휑 implying a reduction of푛 the structure1 2 group to ( , ) ( , ) (2 , 2 ). Λ 퐸휑 퐸 Although we will note pursue this line of thought푆푈 much푛 푛 ⊂deeper,푈 푛 푛we ⊂would푆푂 like푛 to푛 note that by considering further reductions of the structure group to even smaller subgroups, one is led to other generalized geometries. For example, fixing 2, 3 or 4 spinors by subgroups of (4, 4) in 4 dimensions yields the special double covers

푆푝푖푛1) (2, 2) (4, 2)

2) 푆푈(1, 1) ⟶ 푆푂(4, 1)

3) 푆푝(1) × ⟶(푆푂1) (4) corresponding푆푝 푆푝 to a ⟶generalized푆푂 Calabi-Yau stucture, a generalized hyperk hler structure and a generalized hyperk hler metric, respectively. [12] [41] 푎̈ 푎̈

IX. The Generalized Hitchin Functional

In this chapter we present a variational approach to generalized Calabi-Yau manifolds in the 6-dimensional case, thereby reconnecting with many of the key ideas we started with. After setting up the necessary algebraic background, we use a certain invariant quartic form to construct a volume functional which yields generalized Calabi-Yau manifolds as its critical points. Equipped with this generalized Hitchin functional, we take a deeper look into the connection between the topological -model and the Hitchin action discussed in Chapter III. 퐵 9.1 An invariant quartic function

To set up the necessary algebra, we work with the group (12, ) instead of (6, 6) and focus on a complex 6-dimensional vector space . The spin spaces ± are 푆푝푖푛 ℂ of real dimension 32 and constitute symplectic representations due to the skew symmetry 푆푝푖푛 푉 푆 of the spinor bilinear form , in this dimension. Set = ( ) / and note that a symplectic action of a Lie group on a vector space+ defines푒푣 ∗ a moment6 1 map2 〈휑 휓〉 푆 푆 ≅ Λ 푉 ⊗ Λ 푉 : by ∗ 퐺 푆 휇 푆 ⟶ 햌 ( )( ) = ( ) , , (9.1) 1 where , and : 휇End휌 푎 gives2 〈 휎the푎 representation.휌 휌〉 By identifying the Lie algebra (12, ) with its dual via the bi-invariant form ( ), ( ) can be seen as 휌 ∈ 푆 푎 ∈ 햌 휎 햌 ⟶ 푆 taking values in the Lie algebra. 햘햔 ℂ 푡푟 푋푌 휇 휌 Let ( ) / be a basis vector and write = + + with respect to the decomposition6 1 2 ( ) = End( ) . We find that 휈 ∈ Λ 푉 ∗ 2 푎∗ 퐴 2 퐵 훽 햘햔( )푉(⊕) 푉= ( ) , 푉 =⊕ Λ (푉 ⊕/2Λ) 푉+ , = 0, (9.2) 1 so that : 휇0 휈for 푎any 2 〈휎( 푎 휈) 휈/〉. Hence,〈− 푡푟퐴 the moment휈 퐵 ∧map휈 ν 〉vanishes on any pure spinor. We now construct an invariant6 1 2 quartic function which will provide a key link to pure spinors.휇 휈 ⟼ 휈 ∈ Λ 푉

Definition 9.1: Using the moment map of the spin representation of (12, ), we define an invariant quartic function on by 휇 푆 푆푝푖푛 ℂ 푞( )푆= ( ) . (9.3) 2 A crucial property of this quartic form푞 휌is that푡푟 (휇 )휌 0 if and only if there exists a unique pair of pure spinors { , } satisfying , 0 such that = + . [40] 푞 휌 ≠ Furthermore, it can be훼 show훽 n that a pair〈훼 훽of〉 pure≠ spinors ,휌 satisfy훼 훽

훼 훽

( + ) = 3 , . (9.4) 2 If = + is real, { , } forms푞 a 훼complex훽 conjugate〈훼 훽〉 pair. In the case that is real, we have , = , = , so that ( ) > 0 by (8.4). If is not real, then 휌 훼 훽 훼 훽 훼 , = , is imaginary and ( ) < 0. We thus arrive at the following lemma which 〈훼 훽〉 〈훼 훼�〉 〈훼 훼〉 ∈ ℝ 푞 휌 훼 connects the quartic form to the pure spinors satisfying , 0. 〈훼 훽〉 〈훼 훼�〉 푞 휌 Lemma 9.1: If is a pure푞 spinor with ( )휑< 0, then 〈 is휑 the휑�〉 real≠ part of a complex pure spinor satisfying , 0. 휌 ∈ 푆 푞 휌 휌 9.2 Generalized휑 geometry〈휑 from휑�〉 ≠ a variational perspective

Consider the real group × (6, 6) and note that it acts transitively on the open set ∗ ℝ 푆푝푖푛 = { : ( ) < 0}, (9.5)

when we take the vector space 푈 to be휌 real.∈ 푆 We푞 휌 now use the quartic form to construct an (6,6)-invariant homogeneous function of degree 2 on . 푉 Definition푆푂 9.2: Let = { : ( ) < 0}휙 and define : 푈 by + 푈 휌 ∈ 푆 푞 휌( ) = ( )/3. 휙 푈 ⟶ ℝ (9.6)

According to Lemma 8.1, can휙 휌be written�−푞 as휌 = + in terms of a pure spinor that satisfies , 0. From (8.4) we see that , = ( )/3 < 0, so that the 휌 ∈ 푈 휌 휑 휑� 휑 function is related to the purely imaginary number 2 , by 〈휑 휑�〉 ≠ 〈휑 휑�〉 푞 휌 휙 ( ) = , . 〈 휑 휑 � 〉 (9.7)

Although for definiteness we fixed 푖휙= 휌 〈휑 휑�〉 ( ) / in our analysis, we could just as easily have worked with += 푒푣 ∗ ( 6 ) 1 2. Thus, we have 푆 푆 ≅ Λ 푉 ⊗ Λ 푉 effectively defined an invariant homogeneous− 표푑푑 function∗ :푛 1⁄2 of degree 2 on an 푆 Λ 푉 ⊗ Λ 푉 open set in / . 6 ∗ 푒푣 표푑푑 ∗ 휙 푈 ⟶ Λ 푉 Now let 푈 be Λa compact푉 oriented 6-dimensional manifold and note that a form / ( ) is stable if it lies in the open set at each point of . We now construct 푀 the generalized푒푣 표푑푑 Hitchin functional as a volume functional 휌 ∈ Ω 푀 푈 푀 풢 ℍ ( ) = ( ), (9.8)

풢 defined on the space of even or oddℍ differential휌 ∫푀 휙 forms.휌 Note that is invariant under both diffeomorphisms and the action of the -field . ℍ풢 퐵 퐵 휑 ⟼ 푒 휑

Before considering the critical points of this functional, we note that the derivative of at is related to the bilinear symplectic form according to 휙 휌 ( ) = , . (9.9)

With as above, we state and prove the퐷휙 key휌̇ result〈휌� regarding휌̇〉 the generalized Hitchin functional . 휌� 풢 Theorem 9ℍ.1: A closed stable form / ( ) is a critical point of ( ) in its cohomology class if and only if + defines푒푣 표푑푑 a generalized Calabi-Yau structure on . 휌 ∈ Ω 푀 ℍ풢 휌 Proof: The first variation yields휌 푖휌� 푀

( ) = ( ) = , . (9.10)

풢 As the variation is restricted훿 ℍto a휌 fixeḋ ∫ cohomology푀 퐷휙 휌̇ ∫class,푀〈휌� 휌 wė〉 have = . That is,

( ) = , . 휌 ̇ 푑훼 (9.11)

풢 Recall that the bilinear form . , . 훿 ℍis equivalent휌̇ ∫푀〈 휌to� 푑훼an 〉exterior product pairing together with a choice of sign determined by the function : / / , where 〈 〉 ( ) = ( 1) and ( ) = ( 1) 푒푣 .표푑푑 Thus,∗ we have푒푣 표푑푑 ∗ 푚 푚 휎 Λ 푉 ⟶ Λ 푉 휎 휓 2 푚 − 휓 2 푚 휎 휓2푚+(1 ) = − ( 휓) 2푚+1 (9.12)

풢 and by an application of Stokes’훿ℍ theorem휌̇ we∫푀 휎find휌� that∧ 푑훼

( ) = ± ( ) = ± ( ) = ± , . (9.13)

풢 The variation훿ℍ vanishes휌̇ for∫푀 all푑휎 휌�pre∧cisely훼 when∫푀 휎 푑 휌=� 0∧. 푑훼That is, the∫푀〈 푑closed휌� 훼〉 form is a critical point if and only if = 0, where 2 = + . Since , 0, we see that 훼 푑휌� 휌 ( , ) is a Calabi-Yau manifold. 푑휑 휑 휌 푖휌� 〈휑 휑�〉 ≠ 푀 휑

9.3 The topological -model at one loop ∎

In Chapter III, we퐁 noted that the Hitchin functional ( ) defined for a real stable 3- form on a 6-dimensional manifold is related to the topological B-model with target 푉퐻 휌 space . The relation that we suggested was an identification of the partition function of 휌 푀 the Hitchin functional with the Wigner transform of the partition functions of the B and B 푀 topological strings on , which we supported with classical arguments. In this section, � we present a modification to that conjecture after considering the quantization of the Hitchin functional at the푀 quadratic or one-loop order. [42]

First we consider the one-loop contribution to the B-model partition function . The holomorphic Ray-Singer torsion for the -complex of the vector bundle of holomorphic 푍퐵 -forms , ( ) on is defined as ̅ 푝 0 휕 푝 Ω 푀 푀 ( ) , = (det ) , (9.14) 푞+1 푅푆 푛 ′ −1 푞 퐼휕� 푝 �∏푞=0 ∆푝푞 where det denotes the zeta function regularized product of nonzero eigenvalues of the Laplacian′ acting on ( , )-forms. ∆푝푞 푝푞 It is shown in [21]∆ that the one푝-loop푞 contribution to the B-model partition function is 퐵 given by the product of the holomorphic Ray-Singer1 -torsions , as 퐹 푅푆 � (휕̅ ) 퐼휕 푝 = log ( , ) . (9.15) 푝+1 퐵 푛 푅푆 −1 푝 1 푝=0 휕� 푝 Expansion in terms of the determinants퐹 det∏ 퐼 yields ′ 푝푞 = log ∆ (det )( ) . (9.16) 푝+푞 퐵 1 푛 푛 ′ −1 푝푞 1 푝=0 푞=0 푝푞 Using the Hodge duality퐹 relation2 ∏= ∏, we can write∆ the one-loop factor in the B-model partition function as 퐼푝 퐼3−푝

, = exp( ) = . (9.17) 퐵 3 퐵 1−푙표표푝 1 Now recall that can be expressed푍 as a volume− 퐹 functional1 퐼 ⁄퐼 0

푉 퐻 ( ) = ( ), (9.18)

퐻 where ( ) defines a measure on 푉. We휌 saw∫ 푀how휙 휌a critical point of this functional defines a complex structure on together with a holomorphic (3,0)-form + . We 휙 휌 푀 refer to this structure as a Calabi-Yau 3-fold. We fix the cohomology class of by 퐼 푀 휌 푖휌� writing = + for a 2-form which needs to be quantized in order to test the 휌 conjecture at one-loop. The partition function of the Hitchin functional can be written as a 휌 휌0 푑훽 훽 path integral over the 2-form :

([ 훽 ]) = exp ( + ) . (9.19)

To perform the quantization푍퐻 at휌0 one-loop,∫ 퐷훽 we need�−푉 to퐻 consider휌0 푑훽 the� second variation of . This second variation is a non-degenerate function of modulo the action of 푉퐻 diffeomorphisms Diff( ) combined with gauge transformations + . If the 훽 cohomology class [ ] lies in a certain open set in ( , ), then has a unique 푀 훽 ⟶ 훽 푑휆 critical point modulo gauge transformations and thus3 generates a local mapping from 휌0 퐻 푀 ℝ 푉퐻 ( , ) into the moduli space of Calabi-Yau structures on . [8] 3 퐻 푀 ℝ 푀

Recall that the first variation of is given by

푉 퐻 ( ) = . (9.20)

퐻 Let be a critical point of defining훿푉 휌 a complex∫푀 훿휌 structu∧ 휌� re on . The second variation evaluated at is 휌푐 푉퐻 퐼 푀 푐 ( ) 휌 ( , ) = ( ) = , (9.21) 2 2 퐻 푐 휌 퐻 where is a certain complex푉 휌structure훿휌 on훿 푉the 휌space ∫of푀 훿휌3-forms∧ 퐽훿휌 = ( ) which commutes with the Hodge decomposition of with respect to and has3 eigenvalues6 퐽 푊 Λ ℝ ⊗ ℂ = + on , , , and = on , , , . [8] 3 0 2 1 1 2 푊0 3 퐼 퐽The one푖 -loop푊 quantum푊 partition퐽 − function푖 푊 for푊 the Hitchin theory can now be written as

( ) , ([ ]) = exp ( , ) . (9.22) 2 퐻 1−푙표표푝 푐 퐻 푐 Writing the 2-form 푍 as = 휌+ ∫+퐷훽 in terms�−푉 of 휌the 푑훽Hodge� decomposition with respect to the complex structure on , we find that 훽 훽 훽20 훽11 훽02 ( ) ( , ) = 퐼 푀 2 퐻 푐 푉 휌 훿휌 = ∫푀 푑훽( ∧ 퐽푑훽+ + ) ( + + )

20 11 02 20 11 02 = ∫푀 푑(훽 + 훽 + 훽 ) ∧ (퐽푑 훽 + 훽 훽 )

( ) 푀 20 11 02 20 11 02 ( , ) = ∫ 푑 훽 훽 , 훽 ∧ 푖푑훽 퐽푑훽 − 푖 푑훽 (9.23) 2 푐 11 11 where we ⇒have푉퐻 used휌 the훿휌 fact that∫푀 휕 the훽 form∧ 휕̅훽 lies entirely in the -eigenspace of to write = and then = 0 by Stokes’ theorem. The term in 푑훽20 푖 퐽 the second variation vanishes in a similar manner. The and terms do not appear in 퐽푑훽20 푖푑훽20 ∫푀 푑훽 ∧ 푑훽20 푑훽02 the intergral as they can both be removed by a diffeomorphism which leaves the action 훽20 훽02 invariant. Indeed, for each there exists a unique vector field which generates a diffeomorphism that simultaneously removes and . We are thus left with a path 훽 integral over with the remaining gauge invariance given by = + . 훽20 훽02 We will not go훽11 through the technical details of the quantization of훿훽 the11 휕̅휆10 휕휆01 term which is skilfully presented by V.Pestun and E.Witten in [42]. Here we note that the ∫푀 휕훽11 ∧ 휕̅훽11 one-loop path integral of the -theory was calculated in [42] to be given by a product of Ray-Singer torsions as 푉퐻 , = / , (9.24)

푍퐻 1−푙표표푝 퐼1 퐼0

which disagrees with the B-model partition function , = . That is, the genus one free energy of the topological B-model disagrees with the one-loop3 free energy of the 푍퐵 1−푙표표푝 퐼1⁄퐼0 Hitchin functional .

Fortunately, we can푉 퐻escape this discrepancy by replacing with the generalized Hitchin

functional ( ), where this time ( ) is a formal퐻 sum 표푑푑 푉 풢 ℍ 휌 휌=∈ Ω + 푀+ (9.25)

of odd forms ( ). We saw휌 that휌 if1 lies휌3 in휌 the5 open set defined in the previous section, then it generates푖 a generalized almost complex structure on determined by a 휌푖 ∈ Ω 푀 휌 푈 pure spinor satisfying 2 = + . Furthermore, if = = 0 then this generalized 푀 almost complex structure is integrable and we have a critical point of the functional 휑 휑 휌 푖휌� 푑휌 푑휌� ( ). If ( ) = 0, then the generalized complex structure coincides with an ordinary complex structure and the critical points of the generalized Hitchin functional correspond ℍ풢 휌 푏1 푀 to Calabi-Yau structures on . Therefore, we see that in this context the generalized Hitchin functional ( ) is an extension of the functional and the classical agreement 푀 between the B-model partition function and that of Hitchin theory will continue to hold ℍ풢 휌 푉퐻 when we replace by . However, since the fluctuating degrees of freedom are different, these two theories are not equivalent on a quantum level. 푉퐻 ℍ풢 For trivial ( ), the quadratic expansion of is given by

1 풢 푏 푋 ℍ , (9.26)

where is a complex structure on ∫푀(훿휌).∧ With퐽훿휌 respect to Hodge decomposition, has + eigenvalues on the spaces , , 표푑푑, , , , , and the one-loop expansion of the 퐽 Ω 푀 퐽 classical generalized Hitchin functional1 0 3 0 near2 1 a critical3 2 point is 푖 Ω Ω Ω Ω ( ) 푐 ( , ) = + 휌 . (9.27) 2 풢 푐 푀 11 ̅ 11 00 ̅ 22 Note the additionalℍ 휌 훿휌 ∫ term�훽 compared∧ 휕휕훽 to 훽(9.2∧3).휕 휕Carrying훽 � out the quantization of (9.27) as in [42], we find that the partition function of the generalized ∫푀 훽00 ∧ 휕휕̅훽22 Hitchin model at one loop is given by 푍퐺퐻

, = , (9.28) 3 퐺퐻 1−푙표표푝 1 in agreement with the partition function푍 of the topological퐼 ⁄퐼0 B-model (9.17).

Thus we see that the one-loop contribution to the partition function of the B-model is equal to the one-loop contribution to the partition function of generalized Hitchin theory:

, = , . (9.29)

푍퐺퐻 1−푙표표푝 푍퐵 1−푙표표푝

Finally, we note that the precise relation of the Hitchin functional to the full partition function of the topological B-model, if indeed such a link exists, has yet to be determined.

X. Supergravity in the language of Generalizeed Geometry

We present a reformulation of ten-dimensional type II supergravity in the language of generalized geometry by considering an (9,1) × (1,9) (10,10) × structure on the generalized tangent space. Following the construction of torsion-free, compatible+ generalized connections, we are led to the푂 definition푂 of the⊂ generalized푂 Ricciℝ tensor and generalized Ricci scalar. This brings us to a position from which we can express the supersymmetry variations and equations of motion in an elegant and manifestly (9,1) × (1,9)-covariant form.

푆푝푖푛10.1 Type II 푆푝푖푛supergravity

We begin with a brief introduction to the structure of = 10 type II supergravity in the democratic formalism [44]. The type II fields are 푑 ( ) ± ± , , … , , , (10.1) 푛 휇휈 휇휈 휇1 휇푛 휇 where is the metric, is�푔 the 2퐵-form퐴 potential휓 and휆 � is the dilaton. The RR ( ) potentials푔휇휈 are given by 퐵휇휈… where is odd for type IIA휙 and even for type IIB. In type ± ± 푛 IIA, and are chiral 1components푛 of the gravitino and dilatino 퐴휇 휇 푛 휇 휓 휆 = + and = + , (10.2) + − + − 휇 휇 휇 where 휓( ) ±휓= 휓 ± and 휆 ( )휆 ± =휆± ±. (10.3) 10 10 In type IIB, we have훾 휓휇 ∓휓휇 훾 휆 휆

= = , + and (10.4) 휇 + 휇 휓− 휆− 휓 � 휇 � 휆 � � where ( ) ±휓= ± and ( ) 휆± = ±. (10.5) 10 10 휇 휇 As we shall see, it is natural훾 휓 to replace휓 ± with the combination훾 휆 −휆

± =휆 ± ±. (10.6) 휇 휇 휌 훾 휓 − 휆 ( ) ( ) We write the field strengths as a sum of even or odd forms = ( ) , where the - ( ) 퐵 퐵 form RR field strength is = ( ). Letting =퐹 , the∑ 푛bosonic퐹 푛 “pseudo-푛 action” [43] can be written as퐵 퐵 퐹푛 푒 ∧ 푑퐴 푛−1 퐻 푑퐵 ( ) = + 4( ) ( ) . ! 2 1 −2휙 2 1 2 1 1 퐵 2 푆퐵 2휅 ∫ �−푔 �푒 �ℛ 휕휙 − 12 퐻 � − 4 ∑푛 푛 �퐹 푛 � �

The term “pseudo-action” refers to the failure of to imply the self-duality relation ( ) [ / ] ( ) ( ) = ( 1) ( ) satisfied by the RR fields.퐵 Here denotes the Hodge star 퐵 퐵 푆 operator and [푛 ]2 is the integer part of . 퐹 푛 − ∗ 퐹 10−푛 ∗ Using the Levi푛-Civita connection , the푛 fermionic action takes the form

1 ∇ 1 = [ 2 4 2 ∇ H 2 2 −2휙 +휇 휈 + +휇 + + + +휇 + 퐹 2 � 1 휈 휇 � 휇1 � 휇 푆 − � �−푔 푒 � 휓 훾 ∇ 휓 − 휓 +∇ 휌 −H 휌 ̅ 휌 − 휓 휓 푘 2 2 + 휇휈휆 + + 휇휈휆 + + + 휇 휈 휆 휇휈 휆 1 +− 휓� 퐻 2훾 휓 − 휌 퐻 4 훾 휓 휌2 ∇휌 + H 2 −2휙 −휇 휈 − −휇 − − − −휇 − � 휈1 휇 � 휇 1 � 휇 + 푒 � 휓 훾 +∇ 휓 − 휓 ∇ 휌 − 휌̅ H 휌 휓 휓 2 2 − 휇휈휆 − − 휇휈휆 − − − 1 휇 휈 휆 휇휈 휆 휓� 퐻 훾 휓 F ()B 휌 퐻 +훾 휓F−()B 휌 , 휌 � 4 −휙 + 휈 휇 − + − − 푒 �휓�휇 훾 훾 휓휈 휌 휌 �� up to and including quadratic fermionic terms. [43]

Now setting the fermions to zero, the bosonic equations of motion can be written in the form

( ) ( ) … + 2 = 0, (10.7) ( )! … 1 휆휌 1 2휙 1 퐵 퐵 휆1 휆푛−1 휇휈 휇휆휌 휈 휇 휈 푛 휇휆1 휆푛−1 휈 ℛ − 4 퐻 퐻 ∇ ∇ 휙 − 4 푒 ∑ ( 푛) −1 퐹 퐹 ( ) … = 0, (10.8) ( )! … 휇 −2휙 1 1 퐵 퐵 휆1 휆푛−2 ∇ �푒 퐻휇휈휆� − 2 ∑푛 푛−2 퐹휇휈휆1 휆푛−2퐹 ( ) + = 0, (10.9) 2 2 1 1 2 with a Bianchi identity for ∇: 휙 − ∇휙 4 ℛ − 48 퐻

퐹 ( ) ( ) = 0. (10.10) 퐵 퐵 Variation of the fermionic action푑 퐹given− above퐻 ∧ 퐹 yields the fermionic equations of motion with leading-order fermionic terms:

± ± ± ± 휈 1 휆휌 1 휆 1 휈휆 휈 24 휈휆휌 휈 휇 2 휈휇 휆 휇 8 휇휈휆 훾 ��∇ ∓ 퐻 훾 − 휕 휙� 휓[( ) 퐻] 휓()B� − �∇ ∓ 퐻 훾 � 휌 = (±1) F()n , (10.11) 1 휙 푛+1 ⁄2 휈 ∓ 16 푒 ∑푛 훾 훾휇 휓휈 2 ± ( ) ± 1 휈휆 휇 휇 1 휈휆 휇 8 휇휈휆 휇 휇 24 휇휈휆 휇 �∇ ∓ 퐻 훾 − 휕 휙� 휓 [(− 훾) ∇] ∓()B 퐻 훾 − 휕 휙 휌 = (±1) F()n . (10.12) 1 휙 푛+1 ⁄2 ∓ 16 푒 ∑푛 휌

We now consider the supersymmetry transformations in type II supergravity. In both type IIA and type IIB, supersymmetry variations can be parameterized by a pair of chiral spinors ±. [43] Choosing an orthonormal frame with respect to the metric , the bosonic supersymmetry transformations in type IIA can be expressed as 휖 푒휇 푔휇휈 = + , 푎 + 푎 + − 푎 − 휇 휇 휇 훿푒= 2 휖̅ 훾[ 휓 ] 휖2̅ 훾 [휓 ], + + − − 휇휈 휇 휈 휇 휈 훿퐵 log( 휖̅ )훾=휓 − 휖̅ 훾 휓 , 1 1 + + 1 − − ( ) 훿휙 − 4 훿 −푔 − 2 휖̅ 휌 − 2 휖̅ 휌 ( ) … = … … 퐵 푛 1 −휙 + 휈 − −휙 + − 1 푛 1 푛 1 푛 푒 ∧ 훿퐴 휇 휇 2 �푒 휓�휈 훾휇 휇 훾 휖 − 푒 휖̅ 훾휇 휇 휌 � … + … . (10.13) 1 −휙 + 휈 − −휙 + − 휇1 휇푛 휈 휇1 휇푛 The type IIB bosonic supersymmetry− 2 �푒 휖̅ transformations훾 훾 휓 푒are 휌also̅ 훾 given휖 by� the above after a change in sign in the last line, so that the final equation becomes

( ) ( ) … = … … 퐵 푛 1 −휙 + 휈 − −휙 + − 1 푛 1 푛 1 푛 푒 ∧ 훿퐴 휇 휇 2 �푒 휓�휈 훾휇 휇 훾 휖 − 푒 휖̅ 훾휇 휇 휌 � + … + … . (10.14) 1 −휙 + 휈 − −휙 + − 휇1 휇푛 휈 휇1 휇푛 The fermionic supersymmetry transformations2 �푒 휖̅ 훾 훾 are 휓 푒 휌̅ 훾 휖 �

± ± [( ) ] ()B = + (±1) F()n , 1 휈휆 1 휙 푛+1 ⁄2 ∓ 훿휓휇 �∇휇 ∓ 8 퐻휇휈휆훾 � 휖 16 푒 ∑푛 훾휇휖 ± = ±. (10.15) 휇 1 휈휆 휇 휇휈휆 휇 Finally, we note that the훿휌 symmetry훾 �∇ group∓ 24 of퐻 the훾 NSNS− 휕 bosonic휙� 휖 sector is given by the semi-direct product Diff( ) . [43] This is, of course, precisely the group of transformations that preserve generalized2 geometries. 푀 ⋉ Ω푐푙표푠푒푑 10.2 Conformal split frames

Let be a 10-dimensional spin manifold. The potential is a locally defined 2-form patched as 푀 퐵 ( ) = ( ) ( ) (10.16)

풊 풋 푖푗 across the overlap of coordinate퐵 patches,퐵 − 푑whereΛ the ( ) satisfy

푖 푗 푖푗 푈 ∩ 푈 ( ) + ( ) + ( ) = ( 푑).Λ (10.17)

Λ 푖푗 Λ 푗푘 Λ 푘푖 푑Λ 푖푗푘

Thus, according to our definition in Chapter VII, defines a connective structure on a grebe on . Recall that the generalized tangent bundle was defined as an extension 퐵 푀 0 0퐸, (10.18) ∗ where and denote the tangent⟶ and푇 cotangent⟶ 퐸 ⟶ 푇bundles,⟶ respectively. Consider the bundle ∗ 푇 푇 = det (10.19) ∗ which has a natural (10,10) × structure퐸� .푇 Now⊗ 퐸choose a basis { } for and let { } be the dual basis on . Define the+ split frame for by the construction given in 푎 푂 ℝ 푒̂푎 푇 푒 [43] as ∗ 푇 �퐸�퐴� 퐸� = = (det )( + ) for = ,

퐴 푎 푎 푒푎̂ 퐸� = 퐸� = (det푒) 푒̂ 푖 퐵 for 퐴 = 푎 + . (10.20) 푎 푎 A split frame is one퐸�퐴 that퐸 is defined푒 by푒 a splitting 퐴 of the푎 generalized푑 tangent bundle. Using (10.20), we have 푇 ⟶ 퐸 , = (det ) , 2 �퐴 �퐵 0 퐴퐵 where is the (10,10) metric〈퐸 =퐸 〉 . 푒Since휂 = det det is a frame- 1 0 핀 ∗ dependent휂 conformal푂 factor, we 휂see that2 � :� = 1, …Φ,20 defines푒 ∈ what푇 is known as a 핀 conformal basis on . It is useful to conformally rescale (10.20) by a function so that �퐸�퐴 퐴 � =퐸�푥 = (det )( + ) for = , 휙 −2휙 퐴 푎 푎 푒̂푎 퐸� = 퐸� = 푒 (det푒) 푒̂ 푖 퐵 for 퐴 = 푎 + . (10.21) 푎 −2휙 푎 10.3 Generalized퐸�퐴 Connections퐸 푒 푒 푒 퐴 푎 푑

Following [46], we define the Dorfman derivative for = + ( ) and = + ( ) by 퐿푉푊 푉 푣 휆 ∈ Γ 퐸 푊 푤 휁 ∈ Γ 퐸 � = + , (10.22)

as a generalization of the Lie퐿 derivative푉푊 ℒ푣 푤whichℒ푣 captures휁 − 푖푤푑휆 the symmetries of the bosonic NSNS sector. A generalized connection is defined as a first-order linear operator that is compatible with the (10,10) × structure and satisfies + 퐷

푂 =ℝ + , (10.23) 퐴 퐴 퐴 퐵 퐴 퐷푀푊 휕푀푊 Ω푀 퐵푊 − Λ푀푊

using frame indices. We can construct such a generalised connection using the split frame defined in (10.21) by lifting a connection to an action on :

= +∇ ( ) for퐸 � = , ∇ 퐴 푎 푎 푀 퐴 휇 푎 휇 푎 �퐷 푊 �퐸� = 0�∇ 푤 � 퐸 � ∇ 휁 퐸 for 푀 = 휇 + 10. (10.24) ∇ 퐴 A generalized connection�퐷푀푊 �퐸� 퐴generates a corresponding generalized푀 torsion휇 T as a linear map : ( ) ( ) defined by 2 푇 Γ 퐸 ⟶ Γ Λ 퐸 ⊕ ℝ ( ). = , (10.25) 퐷 where is the Dorfman derivative푇 푉 with푊 퐿 replaced푉푊 − 퐿 푉by푊 . 퐷 10.4 The퐿푉 ( , ) × ( , ) structure 휕 퐷

Following푶 [41]ퟗ ퟏwe consider푶 ퟏ ퟗ an (9,1) × (1,9) principal subbundle of the (10,10) × bundle. This is equivalent to specifying a metric of signature (9,1), a -field and a 푂 푂 푂 dilaton+ . Thus it provides the appropriate generalised structure to describe the NSNS ℝ 푔 퐵 supergravity fields. Geometrically, the (9,1) × (1,9) structure defines a splitting of 휙 into two 10-dimensional subbundles 푂 푂 퐸 = , (10.26)

such that the natural (10,10) metric퐸 restricts퐶+ ⊕ to퐶 −give a metric of signature (9,1) on and a metric of signature (1,9) on . Following [43] we construct a frame 푂 퐶+ where defines an orthonormal basis for and defines an orthonormal+ basis− 퐶− �퐸�푎 � ∪ �퐸�푎� � for . That+ is, − �퐸�푎 � 퐶+ �퐸�푎� � 퐶 − , = , + + 2 〈퐸�푎 , 퐸�푏 〉 = Φ 휂푎푏 , − − 2 푎� � 푎�푏� 〈퐸� , 퐸�푏 〉 = 0−,Φ 휂 (10.27) + − where (det ) is a fixed density〈퐸�푎 퐸� which푎� 〉 gives an isomorphism between and . Note that in (10.27),∗ and are flat metrics of signature (9,1). Φ ∈ Γ 푇 퐸� 퐸 � We can explicitly construct휂푎푏 a 휂generic푎�푏 (9,1) × (1,9) structure as

= 푂 +푂 + , + −2휙 + + + �푎 푎 푎 푒̂푎 퐸 = 푒 �−푔�푒̂ 푒 + 푖 퐵�, (10.28) − −2휙 − − − � 퐸�푎� 푒 �−푔�푒̂푎� − 푒푎� 푖푒̂푎 퐵�

where the fixed conformal factor in (10.27) is = and { }, { }, and their duals { }, { }, are two independent orthonormal frames−2휙 for the metric+ . −That is, Φ 푒 −푔 푒̂푎 푒̂푎� + − � 푎 푎� 푒 푒 = = , 푔 +푎 +푏 −푎� −푏� � 푔푎푏( , 휂푎푏) 푒= ⊗, 푒 휂푎�푏,푒 ⊗= 푒 . (10.29) + + − − � � One can also construct the푔 푒 invariant̂푎 푒̂푏 generalised휂푎푏 푔 �metric푒̂푎� 푒̂푏 � 휂푎�푏

= + , (10.30) −2 푎푏 + + 푎�푏� − − 푎 푏 푎� 푏� where = is퐺 theΦ fixed�휂 conformal퐸� ⊗ 퐸� density.휂 퐸In� the⊗ 퐸coordinate� � frame, we have −2휙 Φ 푒 �−푔 = . (10.31) −1 −1 1 푔 − 퐵푔 퐵 −퐵푔 퐺푀푁 2 � −1 −1 � The pair ( , ) parameterizes the coset−푔 ( 퐵(10,10)푔× )푀푁/ (9,1) × (1,9). + 10.5 Torsion퐺 Φ-free, compatible generalized푂 connectionsℝ 푂 푂

A generalized connection is said to be compatible with the (9,1) × (1,9) structure if = 0. Any generalised connection can be written in the form 퐷 푂 푂

퐷퐺 =퐷 + (10.32) 퐴 ∇ 퐴 퐴 퐵 in terms of . Imposing the퐷 compatibility푀푊 퐷푀푊 conditionΣ푀 and퐵푊 setting the torsion of to zero, we find that ∇ 퐷 퐷

= ( ) + 푏 푏 1 푏 푐 2 푏 푏 푐 +푏 푐 푎 + 푎 + 푎 푐 + 푎 푐 푎푐 + 푎 푐 + 퐷 푤 ∇ 푤 − 6 퐻 푤 − 9 훿 휕 휙 − 휂 휕 휙 푤 퐴 푤 = , 푏 푏 1 푏 푐 푎� + 푎� + 푎� 푐 + 퐷 푤 ∇ 푤 − 2 퐻 푤 = + , 푏� 푏� 1 �푏 푐̅ 푎 − 푎 − 푎 푐̅ − 퐷 푤 ∇ 푤 2 퐻 푤 = + + , (10.33) 푏� 푏� 1 푏� 푐̅ 2 푏� 푏� 푐̅ −푏� 푐̅ 푎� − 푎� + 푎� 푐̅ − 푎� 푐̅ 푎�푐̅ − 푎� 푐̅ − where the퐷 푤± are u∇ndetermined푤 6 퐻 tensors푤 − 9 that�훿 do휕 not휙 − contribute휂 휕 휙� 푤to the퐴 torsion.푤 [43]

Thus, we can퐴 always construct torsion-free compatible connections . We also note that in contrast to the Levi-Civita connection in ordinary Riemannian geometry, the conditions defining a torsion-free, compatible generalized connection are not strong퐷 enough to uniquely determine .

퐷

Since is assumed to be a spin manifold, we can promote the local structure to (9,1) × (1,9). Let ( ±) be the spinor bundles associated with the subbundles 푀 ± and denote the corresponding gamma matrices by , . By definition, a generalized 푆푝푖푛 푆푝푖푛 푆 퐶 connection acts on spinors as 푎 푎� 퐶 훾 훾 = + , + + 1 푎푏 + 퐷푀휖 휕푀휖 4 Ω푀 훾푎푏휖 = + , (10.34) − − 1 푎�푏� − � ± 퐷푀휖 휕푀휖 4 Ω푀 훾푎�푏휖 where ( ( ±)). We can uniquely build a set of four operators from these derivatives: 휖 ∈ Γ 푆 퐶 = , + 1 푏푐 + 퐷푎�휖 �∇푎� − 8 퐻푎�푏푐훾 � 휖 = + , − 1 푏�푐̅ − � 퐷푎휖 �∇푎 8 퐻푎푏푐̅훾 � 휖 = , 푎 + 푎 1 푎푏푐 푎 + 훾 퐷푎휖 �훾 ∇푎 − 24 퐻푎푏푐훾 − 훾 휕푎휙� 휖 = + . (10.35) 푎� − 푎� 1 푎�푏�푐̅ 푎� − 푎� 푎� 푎�푏�푐̅ 푎� We now define a traceless훾 퐷 휖 generalized�훾 ∇ Ricci24 퐻 tensor훾 − by훾 휕either휙� 휖of 0 푎푏� = [ , 푅] , 1 0 푎 + 푎 + � 2 푅푎푏�훾 휖 훾 퐷푎 퐷푏 휖 = [ , ] , (10.36) 1 0 푎� − 푎� − 푎�푏 푎� 푏 and a generalized curvature scalar2 푅 훾 휖by 훾 퐷 퐷 휖

=푅( ) , (10.37) 1 + 푎 푏 푎� + 푎 푏 푎� or alternatively, − 4 푅휖 훾 퐷 훾 퐷 − 퐷 퐷 휖

= . (10.38) 1 − 푎� 푏� 푎 − 푎� 푏� 푎 10.6 Fermionic degrees −of4 freedom푅휖 � 훾 퐷 훾 퐷 − 퐷 퐷 �휖

The type II fermionic degrees of freedom correspond to spinor and vector-spinor representations of (9,1) × (1,9). The spinor bundles ( ±) decompose into spinor bundles ±( ) and ±( ) of definite chirality under ( ). For the gravitino 푆푝푖푛 푆푝푖푛 푆 퐶 degrees of freedom we have 10 푆 퐶+ 푆 퐶− 훾

( ( )), ( ) , (10.39) + ∓ − + and the dilatino휓푎 �degrees∈ Γ 퐶− of⊗ freedom푆 퐶+ correspond to휓 푎 ∈ Γ�퐶+ ⊗ 푆 퐶− �

±( ) , ( ) . (10.40) + + + Finally, for the supersymmetry휌 ∈ Γ�푆 parameter퐶+ � s, we have휌 ∈ Γ�푆 퐶− �

±( ) , ( ) . (10.41) + − + Note that in (10.39)-(10.4휖 ∈1Γ),� 푆the 퐶upper+ � signs correspond휖 ∈ Γ to�푆 type퐶− IIA� and the lower signs to type IIB supergravity.

10.7 RR fields

± ( ) Let ( / ) be the spin bundles of definite chirality under , where denotes the Cliff(10,10; ) gamma matrices. The RR field strengths 20= transform as 푆 1 2 Γ ( )Γ spinors of positive (negative) chirality for type IIA (type IIB). The generalized metric ℝ 퐹 ∑푛 퐹 푛 structure leads to the decomposition of Cliff(10,10; ) into Cliff(9,1; ) Cliff(9,1; ), and thus the identification of ( / ) with ( ) ( ). Using the spinor norm on ℝ ℝ ⊗ ℝ ( ), we interpret as a map from sections of ( ) to sections of ( ). We ( / 푆) 1 2 푆 퐶+ ⊗ 푆 퐶− denote the image under this isomorphism by 푆 퐶− 퐹 ∈ 푆 1 2 푆 퐶− 푆 퐶+

#: ( ) ( ), (10.42)

and its conjugate map by 퐹 푆 퐶− ⟶ 푆 퐶+

# : ( ) ( ). (10.43) 푇 Finally, we note that the (9,1) ×퐹 (1푆,9퐶)+ structure⟶ 푆 퐶 provides− two chirality operators (±) on (10,10) × spinors given by + 푂 푂 Γ 푆푝푖푛 ( ) = ℝ … … , and ( ) = … … . (10.44) ! ! + 1 푎1 푎10 − 1 푎�1 푎�10 푎1 푎10 푎�1 푎�10 Γ 10 휖 Γ( ) Γ [ / ] ( Γ) 10 휖 Γ Γ The self-duality condition ( ) = ( 1) ( ) of the RR field strengths can now 퐵 푛 2 ( ) 퐵 be expressed as a chirality 퐹condition푛 − under ∗ 퐹:10 −푛 − ( ) Γ= , (10.45) − ± where ( / ). Γ 퐹 −퐹

1 2 퐹 ∈ 푆

10.8 The equations of motion

The supersymmetry variations can be expressed in an elegant, locally (9,1) × (1,9) covariant form using the torsion-free compatible generalized connection . 푆푝푖푛 Consider the fermionic variations (10.15). Using the uniquely determined spin operators (10.35),푆푝푖푛 these can be written as 퐷

= + # , + + 1 − 훿휓푎� 퐷푎�휖 16 퐹 훾푎�휖 = + # , − − 1 푇 + 푎 푎 푎 훿 휓 퐷 =휖 16 퐹 ,훾 휖 + 푎 + 훿휌 = 훾 퐷푎휖 . (10.46) − 푎� − For the NSNS bosonic fields, the supersymmetry훿휌 훾 퐷푎� variations휖 can be express in terms of the generalized metric as

= = 2( + )vol + 2( + ) , (10.47) + + − − 2 + + − − 푎푎� 푎�푎 푎 푎� 푎� 푎 퐺 푎푎� where vol훿퐺 = 훿퐺 =휖̅ 훾. Finally,휓 휖 ̅the훾 variation휓 of the휖 ̅RR휌 potential휖̅ 휌 퐺 can be written as a bispinor −2휙 퐺 푒 �−푔 Φ 퐴

( #) = ( ) ( + ), (10.48) 1 푎 + − + − + − 푎� + − �푎 푎� where the upper16 (lower)훿퐴 sign훾 corresponds휖 휓 − 휌 to휖̅ type∓ IIA휓 휖(Type̅ 훾 IIB).휖 휌 ̅

We now write the supergravity equations of motion in the language of generalized geometry. In terms of the generalized Ricci tensor , the generalized curvature scalar , and the Mukai pairing , introduced in Chapter VII,0 the equations of motion for , 푅푎푏� and become 푅 〈∙ ∙〉 푔 퐵 휙 = vol , , 0 1 −1 푎푏� 퐺 푎푏� 푅 − 16 = 0, 〈 퐹 Γ 퐹 〉 (10.49)

in direct analogy to the Einstein equations.푅 The RR supergravity fields equations of motion take the compact form

= = 0. (10.50) 1 퐴 퐴 Finally, the bosonic pseudo-action2 Γ and퐷 퐹 the fermionic푑퐹 action are written as

푆퐵 푆퐹

= (vol + , ( ) ). (10.51) 1 1 − 퐵 2 퐺 and 푆 2휅 ∫ 푅 4 〈퐹 Γ 퐹〉

= 2vol [ + + 2 + 2 1 +푎� 푏 + −푎 푏� − + 푎� − −푎 2 � 푆퐹 − 2휅 ∫ 퐺 휓� 훾 퐷푏휓푎� 휓� 훾 퐷푏휓푎 휌̅ 퐷푎�휓 휌̅ 퐷푎휓 ( # + # )]. (10.52) + 푎 + − 푎� − 1 + − + 푎 푎� − 푎 푎� �푎� 푎 −휌̅ 훾 퐷 휌 − 휌̅ 훾 퐷 휌 − 8 휌̅ 퐹 휌 휓 훾 퐹 훾 휓

XI. Final Remarks

11.1 Related topics in the existing physics literature

In this section we draw attention to some parts of the existing physics literature that are related to the ideas covered in this dissertation. In [31] a connection is established between the Hitchin functionals used to define the actions in form theories of gravity and entanglement measures for systems with a small number of constituents. It is then argued that this connection lies behind the Black Hole/Qubit Correspondence (BHQC) relating certain black hole entropy formulas in supergravity to multipartite entanglement measures of quantum information theory.

In our development of generalized geometry, we defined a generalized complex structure as a particular kind of geometrical structure that is compatible with the basic set up of generalized geometry. We could also consider generalized versions of other geometries such as generalized structures. In [47] the quantization of the effective target space description of topological M-theory is considered in terms of the Hitchin 퐺2 functional whose critical points yield 7-dimensional manifolds with holonomy. The one-loop partition function for this theory is calculated and a generalized version of it is 퐺2 shown to be related to generalized geometry. This is then compared to the topological string. The Hitchin functional description of the topological string in six dimensions is 퐺2 then related to the reduction of the effective action for the generalized theory. 퐺2 2 In [48] Pestun extends the OSV conjecture = [33] to topological퐺 strings on 2 generalized Calabi-Yau manifolds. It is shown that the classical black hole entropy is 푍퐵퐻 �푍푡표푝� given by the generalized Hitchin functional introduced in Chapter IX. The geometry differs from an ordinary geometry if ( ) 0, where is the generalized Calabi-Yau manifold. 푏1 푋 ≠ 푋 In [49] the structure of “exceptional generalized geometry” (EGG) is explained and shown to provide a unified geometrical description of 11-dimensional supergravity backgrounds. On a -dimensional background, the action of the ( , ) symmetry group of generalised geometry is replaced in EGG by the exceptional -duality group ( ) as 푑 푂 푑 푑 originally described in [50]. The metric and form-field degrees of freedom combine into a 푈 퐸푑 푑 single geometrical object and the differential structure of EGG is given by an analogue of the Courant bracket. Exceptional generalized geometry is ideally suited for a systematic description of supergravity backgounds.

In [52] a unified description of bosonic 11-dimensional supergravity restricted to a -dimensional manifold is presented for all 7. The theory is based on a natural ( ) × action on an extended tangent space. The bosonic degrees of freedom as well 푑 + 푑 ≤ 퐸푑 푑 ℝ

94 as the diffeomorphism and gauge symmetries are unified as a “generalized metric”. The analogues of the Levi-Civita connection and Ricci tensor are also introduced and the bosonic action is simply given by the generalized Ricci scalar. The equations of motion are then elegantly encoded by the vanishing of the generalized Ricci tensor. The significance of the formulation for M-theory is also considered. Finally, the completion of the description of supergravity by inclusion of the fermionic degrees of freedom and supersymmetry transformations (at least to leading order) will be presented as the main result of [53].

11.2 Directions for future research

Deeper investigations into many of the bold claims made in “Topological M-theory as Unification of Form Theories of Gravity” [9] should provide fertile ground for original research. Certainly a more rigorous treatment of some of the key conjectures contained in that paper would be desirable. Some of these include the conjectured relations of Hitchin functionals to the models of topological string theory and BPS black holes. In particular, the relation of the topological model to Hitchin’s functional was seen to break down at one loop. Although we saw that we could achieve agreement between the topological 퐵 푉퐻 model and Hitchin theory by replacing with the generalized Hitchin functional, the 퐵 precise form of the full relationship between퐻 the two remains an open question. Attempts to test this link at higher loops might prove푉 to be worthwhile.

In the context of the use of generalized geometry in the description of supergravity backgrounds, one could consider the description of higher-derivative correction terms to the theory, assuming that the generalized structure is preserved. One could also consider explicit constructions of supergravity backgrounds with particular special structures on the generalised tangent space. A deeper question is whether there can be any extension of the generalized geometrical description for exotic string backgrounds. Such a description would have to retain concepts such as connections and curvature as some sensible limit of the full string theory while moving away from the usual 10-dimensional manifold.

References

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