Bundles, Loops and Symmetries: an Introduction to String Compactifications

Bundles, loops and symmetries: an introduction to string compactications

Paul Dempster November 26, 2012

Abstract The following are a mixture of introductory notes on bundles and holonomy, with emphasis on the physical applications, geared towards those who want to understand the ideas behind gauge group breaking by Wilson loops, in the style of the classic string compactication techniques of the 80s. They should be taken as preparatory reading for the author's journal club talk.

1 Motivation

The Green-Schwarz anomaly cancellation paper [1] which kick-started the rst superstring revolution opened up the possibility that one could obtain phe- nomenologically realistic models from string theory. The main idea was, and still is, to consider the spectrum of string vacua of the form M4 × K, where M4 is Minkowski and K some compact manifold. By restricting the four-dimensional physics to be something which we want, the hope is to try to narrow down the possible string vacua. In these notes, we will investigate one of the earliest attempts along these lines [2, 3], which motivated the following 20-odd years of research, and as such should be something that every student of strings encounters at least once in their lifetime. We begin by reviewing some mathematical concepts that we will need further on, namely bundles (vector, principal, and associated), holonomy and homotopy. We try to relate these at all times to familiar physics-y situations which you should have encountered before, though not necessarily using the same language. Once we've built our foundations we go on to investigate, and hopefully elucidate, some of the claims made in [3].

2 Bundles

Where we introduce the mathematical ideas behind bundles.

1 Heuristically speaking, bundles are to direct products of manifolds what manifolds are to at space. Locally, they look the same, but globally there is a much richer structure. As far as references go for this section, I would strongly recommend chapter 9 of [4], upon which most of this material is based, and should be your rst port of call if you want to add some rigour to the description we give. Some advice from personal experience is to always try to picture what's going on geometrically. It is an unfortunate consequence of my typesetting skills that these notes will not contain any diagrams.

2.1 Fibre bundles General denition. Sections.

2.1.1 Denition and properties We quickly dene a bre bundle, and mention some of the most useful properties of it, which we will use later in our main discussions. The bundle consists of a manifold, E, called the total space; a manifold, M, the base space; a manifold F called the (typical) bre; and a Lie group G called the structure group, which acts on F (on the left). Alongside these elements, we have a number of maps, which build up the structure of the bundle. The most basic of these is the projection π : E → M. The important point here is that, given some point p ∈ M, the preimage1 −1 π (p) denes the bre over p, which we call Fp. This bre is isomorphic to the typical bre F mentioned earlier. Our base manifold M, being a manifold, comes with a structure of its own. Specically, we have some open covering {Ui} of M. We dene the map φi : −1 Ui × F → π (Ui), which we call the local trivialization. The reason for this terminology is that −1 maps the collection of bres −1 to the direct φi π (Ui) product Ui × F . The important point here is that, if we take some point u ∈ −1 in the bre over , −1 , for some element . π (p) p φi (u) = (p, fi) fi ∈ F Now, take this open cover of M and consider a point p ∈ Ui ∩ Uj. Then we have two local trivializations φi and φj to choose from on Ui ∩ Uj. Take the point u ∈ π−1(p) living in the bre over p. Then we have

−1 −1 φi (u) = (p, fi), φj (u) = (p, fj), i.e. we can choose a dierent element of the bre F depending on our choice of local trivialization.

Now, we have a map tij(p): F → F relating these two choices. Recall that we said before that the structure group G acts in this manner, so the tij(p) should be elements of G. In particular, we have

fi = tij(p)fj. (1)

1i.e. the set of points which project down to p ∈ M

2 The transition functions tij(p) should satisfy certain condtitions so that the patches covering M can be glued together nicely. Specically, we should have , −1, and the so-called ech cocycle condition on triple- tii(p) = I tij(p) = tji(p) intersections . tij(p)tjk(p)tki(p) = I All this denes the bre bundle.

2.1.2 Sections

Now that we have a bre bundle, it would be nice if, for each point p ∈ M, we had a way of picking out a particular point u ∈ E. We call this a choice of section. The importance of sections for physics will become clear later. Mathematically, a section of the bundle E is a map s : M → E satisfying . This simply says that the section lies in the bre above . π ◦ s = IM s(p) p We denote the set of sections on M by Γ(M,F ). There are many dierent examples of bre bundles in mathematics, most of which have found some use in physics in one way or another. We now give an overview of three of the most important types.

2.2 Vector bundles

General denition. Tangent bundle of S2.

2.2.1 Denition and properties

A vector bundle is just a bre bundle where the bre is a vector space V . Note that, for vector bundles, the structure group becomes the automorphism group Aut(V ), which for V nite-dimensional is simply the group GL(V ). For example, if we take V = Rk, the structure group is just GL(k, R). Some terminology: A vector bundle whose bre is 1-dimensional is called a line bundle. An important example of a vector bundle which you're probably familiar with is the tangent bundle. Given an m-dimensional base manifold M, the tangent bundle TM is a vector bundle with typical bre Rm.

2.2.2 Example: tangent bundle of S2 It's probably about time for an example now, so let's try to describe the tangent bundle TS2 to the 2-sphere. We know that we can't cover S2 with a single coordinate patch, but we can 2 2 with two. Take the charts {UN ,US} as a covering of S , where UN(S) is S without the South (North) pole. We take projective coordinates (X,Y ) on UN and (U, V ) on US. For completeness, these are given by x y x y (X,Y ) = , , (U, V ) = , − , 1 − z 1 − z 1 + z 1 + z where x2 + y2 + z2 = 1 are the usual coordinates on S2 ⊂ R3. The coordinates (X,Y ) and (U, V ) are related (on UN ∩ US) by

3 X Y U = ,V = − . X2 + Y 2 X2 + Y 2 We note that a vector V ∈ TS2 can be written as

µ ∂ µ ∂ V = VN µ = VS µ , ∂X p ∂U p with µ 2. {VN(S)} ∈ R Now, to nd the transition functions we rst dene the local trivializations. 2 We take u ∈ TS with p = π(u) ∈ UN ∩ US. Then,

−1 µ −1 µ φN (u) = (p, {VN }), φS (u) = (p, {VS }). Using (1), and noting that

µ µ ∂U ν VS = ν VN , ∂X p we nd the transition function tSN (p) to be given by

∂(U, V ) 1 − cos 2θ − sin 2θ t (p) = = , SN ∂(X,Y ) r2 sin 2θ − cos 2θ where (X,Y ) = (r cos θ, r sin θ). Before we move on to consider principal bundles, we should comment on the space of sections of a vector bundle. Given sections s, s0 of a vector bundle over M, we can dene vector addition and multiplication in a point-wise manner, e.g. (s + s0)(p) = s(p) + s0(p), (fs)(p) = f(p)s(p) (for f ∈ F(M)). This has important applications in physics, as we will see later, where elds Φ(x) are considered to be sections of some vector bundle over a spacetime M.

2.3 Principal bundles General denition. Canonical local trivialization. Homogeneous space example.

2.3.1 Denition and properties

A principal bundle is a bre bundle whose bre F is identical with the structure group G. We denote a principal G-bundle over M as P (M,G). As in (1), the transition functions (which in this case are just elements of G) act on F on the left. We can also dene a right-action of G on F . Take a point u ∈ π−1(p) and −1 dene a local trivialization by φ (u) = (p, gi). Then the right action of G is −1 dened by φ (ua) = (p, gia) for a ∈ G. What does this do? Well, we know that π(ua) = π ◦ φi(p, gia) = p = π(u), so this right action moves us around on the bre over p.

4 We note moreover that the action is transitive and free2, see section 9.4.1 of [4].

2.3.2 Canonical local trivialization: a choice of section

For principal bundles, given some section s, there is a preferred choice of local −1 trivialization we can make. In particular, take some p ∈ Ui and u ∈ π (p). Then there is a unique element gu ∈ G such that u = si(p)gu. We dene the canonical local trivialization by −1 . φi φi (u) = (p, gu) If p ∈ Ui ∩ Uj, and we have sections si over Ui and sj over Uj, we see that si(p) = φi(p, e) = φj(p, tji(p)e) = φj(p, etji(p)) from the denition of the transition function. Then, from the denition of the right action, we have

φj(p, etji(p)) = φj(p, e)tji(p) = sj(p)tji(p), so the sections are related by

si(p) = sj(p)tji(p). This has some relevance in physics, as we shall see later. Loosely, when one considers a gauge theory, a section si(x) is considered a choice of gauge for the gauge connection Ai(x). Then, the transition functions let us go between dierent gauge choices. This is exactly what a gauge transformation is.

2.3.3 Example: homogeneous spaces Before we move on to our nal type of bundle, let's go through another example by describing a homogeneous space M = G/H as a principal bundle. Homoge- neous spaces come up in many dierent areas of physics. For example, the way you probably learnt about superspace was to consider it as a quotient SuperPoincaré Superspace = Lorentz . They also appear all over the place in supergravity as the target manifolds of various actions. We now show that, given a closed Lie subgroup H of G, G can be thought of as a principal H-bundle over M = G/H. We dene the right action of H on G as (g, h) 7→ gh for g ∈ G, h ∈ H. Elements of the base space M = G/H are given by equivalence classes [g]. We dene the projection π : G → G/H by π(g) = [g]. We note that π(gh) = π(g), as we would expect since right action by the structure group shouldn't change the bre we're in.

For a local trivialization, we rst dene a local section si over Ui, and make use of the canonical local trivialization as dened above. That is, we take −1 −1 . φi (g) = ([g], si([g]) g) In terms of the usual discussion of homogeneous spaces, the condition π(gh) = π(g) = [g] ∈ M tells us that h lies in the isotropy group of [g]. The section s([g]) ∈ G is what is sometimes called the coset representative.

2That is, if ua = u for some u ∈ P , then a must be the unit element e ∈ G.

5 2.4 Associated vector bundles

General denition. Fields in physics. Complex scalar elds as U(1) line bundles.

2.4.1 Denition and general properties Okay, we've nally got down to the where the physics begins. It turns out that by far and away the most natural way to describe the elds that we see in physics is in terms of associated vector bundles, so it would be a good idea to explain what they are. The idea here is that we have some principal bundle P (M,G) and some vector space V , on which G acts on the left by automorphisms. To be more specic, let's take V = Rk, and let ρ be a k-dimensional representation of G, so that G acts on V by v 7→ ρ(g)v. Dene an action of G on the product space P × V by

(u, v) 7→ (ug, ρ(g)−1v), for u ∈ P and f ∈ F . Then we form a coset space E = (P × V )/G by identifying elements of P × V under the action of G above. That is, (u, v) ∼ (ug, ρ(g)−1v) for any u ∈ P , v ∈ V and g ∈ G. An important point to note here is that we have [ug, v] = [u, ρ(g)v], so the right action on P `induces' a left action in the bre V . If we then dene a projection πE : E → M by πE([u, v]) = π(u) (where π : P → M was the original projection), we see that we have a bre bundle E with base space M, bre V , and structure group G. We note that, in this case, the transition functions on E are given by ρ(tij(p)), where tij(p) are the transition functions on P .

2.4.2 Sections: elds

So, physics. In eld theory, we consider objects Φa(x), which we call elds. They are considered to be functions of some spacetime coordinate (x ∈ M) and they transform under a k-dimensional representation ρ of a gauge group G, a a b Φ (x) → ρ(g) bΦ (x). In the bundle language, then, they are sections of the associated vector bundle k corresponding to the principal -bundle . P ×ρ R G P (M,G) Let's look at a simple example.

2.4.3 Example: scalar elds transforming under U(1) A scalar eld charged under a U(1) gauge group, as can be found in QED for example, can be thought of as a section of the complex line bundle associated with a U(1)-bundle P (M,U(1)), where M is some spacetime. What is this saying? Well, a complex line bundle means that the bre of the associated vector bundle is just C. So elements of the bre look like φ(x) ∈ C.

6 On the principal bundle side, we have the right action of g ∈ U(1) on P . In terms of the vector bundle, the construction says that [ug, v] = [u, ρ(g)v], so the action of an element g ∈ U(1) `induces' an action whereby the vector transforms under a 1-dimensional U(1) representation, i.e. φ(x) 7→ eiα(x)φ(x).

3 Connections on a bundle: Wilson lines

Where we introduce the idea of a connection on a principal bundle. Parallel transport. Wilson lines. We said above that a section of an associated vector bundle corresponds to what we know of as a gauge eld. We know from our study of gauge eld theories that there should be a notion of gauge choice, which doesn't change the physics. It is natural, therefore, to ask how this is encoded mathematically. The main reference for this part of the discussion is chapter 10 of [4]. We will skimp on many of the details, but the ideas are the same.

3.1 Connections on principal bundles General denition. Compatibility condition. Relation with gauge potential. The technical denition of connections on principal bundles, which was talked about by Owen in his rst string seminar, and which can be found in both his accompanying notes and chapter 10 of [4], involves splitting a principal bundle P (M,G) into `vertical' and `horizontal' subspaces, which is achieved through introducing a connection 1-form ω ∈ g ⊗ T ∗P valued in the Lie algebra g of G. If we have a local section σi dened on some open set Ui ⊂ M, then we can dene a local connection 1-form Ai by

−1 −1 (2) ωi = gi dgi + gi Aigi, where −1 for is the canonical local trivialization men- φi (u) = (p, gi) u = σi(p)gi tioned above. The advantage of this denition is the following: we want the splitting of P (M,G) into horizontal and vertical subspaces to be independent of any choice of section, which means that ωi = ωj should match on all open patches Ui ∩Uj 6= ∅. This then tells us how the local connection 1-form `transforms' when we move from one section to another, i.e. when we change gauge. In particular, take sections σ1, σ2 over U ⊂ M such that σ2(p) = σ1(p)g(p). Then, in the language of (2), we have g2 = e, g1 = g. Hence, the condition ω1 = ω2 implies

−1 −1 A2 = g A1g + g dg. But this is precisely the transformation law we'd expect for a gauge eld transforming under G!

7 3.2 Parallel transport Horizontal lift of a curve. Parallel transport as a dierential equation. Wilson lines. The splitting of P (M,G) into vertical and horizontal subspaces by the connection 1-form ω described in the last section allows us to dene a notion of the horizontal lift of a curve γ : [0, 1] → M, that is, a curve γ˜ : [0, 1] → P such that πP ◦ γ˜ = γ and the tangent vector to γ˜(t) always lies in the horizontal subspace. Theorem 10.2 of [4] tells us that, given a curve γ : [0, 1] → M and a point −1 u0 ∈ π (γ(0)) in the bre over γ(0), there exists a unique horizontal lift γ˜(t) in P such that γ˜(0) = u0. If we take γ(t) ∈ Ui a chart on M and σi a section over Ui such that γ˜(t) = σi(γ(t))gi(t), then the proof of this statement can be reduced to the ODE

d g (t) = −A (X)g (t), dt i i i where X is the tangent vector to γ(t) at γ(0). Since gi(0) = e, the formal solution to this is

γ(t) ! gi(γ(t)) = P exp − Ai , (3) ˆγ(0) where P denotes path-ordering. We note that, since Ai is an element of the Lie algebra g of G, gi(γ(t)) ∈ G. So, we have deduced that the unique horizontal lift of the curve γ which passes through u0 ∈ P is given by γ˜(t) = σi(γ(t))gi(γ(t)) for gi(γ(t)) as in (3). Using this idea of the horizontal lift of a curve, we can dene what we mean −1 by parallel transport. Take γ : [0, 1] → M with u0 ∈ π (γ(0)) and let γ˜ be −1 its horizontal lift. This denes a unique point u1 =γ ˜(1) ∈ π (γ(1)), which is called the parallel transport of u0 along γ. In the local form, u1 is given by u1 = σi(1)P exp − Ai . (4) ˆγ 3 The object on the right is exactly what we would call a Wilson line Wγ in the physics literature. The idea, then, is that, given a curve γ : [0, 1] → M with γ(0) = p0, γ1 = p1, and a section σi with σi(p0) = u0 ∈ P , the Wilson line tells us how we deviate from the section as we move along γ. So far we have just been working on the principal bundle, but we will extend everything to an associated bundle when we get to the symmetry breaking chapter.

4 Holonomy of a connection: Wilson loops

Where we think about parallel transport around closed loops. Wilson loops.

3Well, up to factors of i.

8 4.1 Wilson loops Parallel transport around a closed curve. Let's consider a loop (closed curve) γ in the base manifold M, i.e. a map γ : [0, 1] → M with γ(0) = γ(1) = p ∈ M. Considering the horizontal lift γ˜ of this loop, we nd that, in general, γ˜(0) 6=γ ˜(1). In fact, γ can be thought of as −1 −1 dening a transformation τγ : π (p) → π (p) in the bre over p ∈ M. Continuing our discussion from the previous section, we see that taking −1 u =γ ˜(0) ∈ π (p) and γ˜(1) = τγ (u) := ugγ , (4) tells us that gγ = P exp − Ai . (5) ˛γ This is exactly the famous Wilson loop that appears so often in physics! We'll need it again later.

4.2 Holonomy group The set of loops at a point. General denition Now that we know how to describe parallel transport around a closed loop in our base manifold M we can go on to dene the notion of holonomy. This is an important notion in classifying simply connected Riemannian manifolds, as was done by Berger. Consider a principal bundle P (M,G), take a point u ∈ π−1(p) and consider the set of loops γ that are `based' at p ∈ M, i.e. with γ(0) = γ(1) = p. Call this set Cp(M). Then the set of group elements

Φu = {g ∈ G|τγ (u) = ug, γ ∈ Cp(M)} ⊂ G, denes the holonomy group at u. Clearly this is just the set of possible values which the Wilson loop (5) can take. There is a great deal more that can be said with regards to holonomy, in- cluding a mention of the Ambrose-Singer theorem, which relates the holonomy to the curvature of the connection 1-form. However, since we have no use of the curvature in this current discussion, we leave this for a later date.

5 Homotopy

Where we ask questions of the simple-connectedness of a manifold. General denition. Homotopy of K/Γ. The nal ingredient that we will need before embarking on our analysis of symmetry breaking is that of homotopy.

Let M be some manifold, with x0 ∈ M. The set of loops at x0 can be partitioned into classes wherein loops are homotopically equivalent, that is there exists some smooth deformation of each loop in the class into any other loop in the class.

9 The set of all such homotopy classes is the fundamental group π1(M, x0) of M at x0. A few important properties of the fundamental group are

• If M is arcwise connected4, then we can ignore the base point and just speak of π1(M).

• For a simply connected space, the fundamental group is trivial π1(M) = {e}.

Finding the fundamental group for a given manifold M is non-trivial. How- ever, one can show that

1 ∼ (6) π1(S ) = Z, which can be seen intuitively by saying that any loop on S1 can be wrapped some integer (positive or negative depending on the direction) number of times around the space, and that one cannot smoothly deform a loop wrapped m times into one wrapped n times for m 6= n. Using this result, we also nd that, e.g. 2 ∼ , etc. π1(T ) = Z ⊕ Z An important result that we will use later is that, if K0 is a simply connected 5 manifold admitting a discrete symmetry group Γ which acts freely on K0, then π1(K0/Γ) = Γ. A sketch proof of this involves the following: A loop in K0/Γ can be described by the group element g ∈ Γ which takes x0 ∈ K0 to g(x0) ∈ K0. Take two such loops, described by g1,2 ∈ Γ. Then they are homotopic i going around one and then backwards around the other leaves you with something homotopic to the identity loop, which is described by x0 7→ x0. However, we would then have −1 . But, from the denition of a freely-acting g1 g2(x0) = x0 group, this implies g1 = g2, so there is an isomorphism between π1(K0/Γ) and Γ.

6 Symmetry breaking

Where we get to the meat of the talk. Breaking gauge groups. Realistic phenomenology. In this section, we'll make use of the tools that we've seen thus far to try to understand the contents of [3]. In particular, we aim to show how, by performing compactications with Wilson lines (a vague wordy statement which we'll make precise later) we can break the gauge group of the heterotic string to something with which we can hope to do realistic four-dimensional phenomenology.

4 A topological space X is arcwise connected if, for any x0, x1 ∈ X, there exists a path α(s) with α(0) = x0 and α(1) = x1. 5That is, if g(x) = x for any x ∈ K and g ∈ Γ, g must be the identity.

10 6.1 Setup Heterotic string compactications. Holonomy and SUSY breaking. Euler characteristic and number of generations. In the classic paper by Candelas, Horowitz, Strominger and Witten [2] it was shown how, by dening the E8 × E8 heterotic string on a product manifold M4 × K, where M4 is maximally-symmetric (Minkowski, dS, or AdS) and K compact, one can hope to get realistic 4-dimensional physics by changing the properties of the compact manifold K. Let's try to get a feeling of how this is done.

We want to start by considering the low-energy eective action of the E8 ×E8 heterotic string. At the end of [2] the authors also consider a similar construction from a purely stringy point of view, but we won't discuss that here. An excellent overview of what it means to have a low-energy eective action of a certain string theory is given in Chapter 7 of David Tong's string theory notes [5]. All that we will use here is that the low-energy eective action of the E8 × E8 heterotic string is ten-dimensional N = 1 supergravity coupled to super Yang-Mills with gauge group G = E8 × E8. The eld content of the supergravity part of this theory is

SUGRA = {gMN , ψM ,BMN , λ, φ}, where ψM is the gravitino, λ is the dilatino, and the bosonic elds gMN ,BMN , φ are the metric, a 2-form potential, and the dilaton respectively. On the super Yang-Mills side we have

SYM a a = {FMN , χ }, a where F is the eld strength associated to the E8 × E8 gauge potential (connection 1-form) Aa, and χa is a gaugino. The index a labels elds in the adjoint representation of G = E8 × E8.

6.2 The 3-form eld strength and anomaly cancellation In order to cancel the gravitational anomalies present in ten dimensions6, Green and Schwarz [1] had to modify the standard gauge transformation of the 2-form potential B. One then nds that the naive eld strength H = dB of the 2-form potential is not gauge-invariant, and should be replaced instead by the gauge invariant combination

H = dB + ω3L − ω3Y , where ω3Y and ω3L are the Chern-Simons 3-forms associated with the Yang- Mills connection Aa and the SO(1, 9) spin connection ω. Due to this modica- tion, the Bianchi identity for H reads 6Recall that chiral anomalies are generally present in even dimensions, and gravitational anomalies are present if d = 4k + 2.

11 1 dH = trR ∧ R − TrF ∧ F, (7) 30 where R and F are the curvature 2-forms for the spin connection and Yang-Mills connections, and the traces are taken over the vector rep of SO(1, 9) and the adjoint rep of E8 × E8. This is an important equation in what will follow as it relates the geometry (encoded in the spin connection) to the possible gauge elds that we can have. In particular, we will see that restricting the geometry of the compact manifold will break the gauge group G.

6.3 Vacuum solutions and supersymmetry Let's move on and think about particular solutions to our low-energy eective action. We're interested in vacuum solutions of the form M4 × K, where K is compact and M4 non-compact and maximally symmetric. Moreover, we want our solution to preserve some supersymmetry. For various phenomenological reasons we would like to have N = 1 SUSY in four dimensions. Leaving SUSY unbroken in our vacuum solution amounts to the requirement that, for every eld α, we have δα = 0, where is the parameter of our SUSY transformations (since we're dealing with supergravity, should be a function of the spacetime). Now, we know that

δ(bosons) = fermions, δ(fermions) = bosons. We also know that, if we want our vacuum to be Lorentz invariant, the expectation (background) values of the fermionic elds should all vanish. This means that the SUSY variation of the bosonic terms vanishes identically, so we only need to worry about the variations of the fermions (the gravitino ψM , dilatino λ, and gaugino χa). The requisite variations are given in [2]. We make a simplifying assumption at this point (which is not made at this stage in [2] but they use it later on), namely that

H = 0, dφ = 0, (8) so that the ux of the 2-form potential vanishes and the dilaton is constant. One can also consider so-called ux compactications, where H 6= 0, but these will not be discussed here. They are mentioned briey in the nal section. We note that, with these simplications, the variation of the dilatino vanishes identically.

6.4 The geometry of M4

If we look at the variation δψµ of the components of the gravitino along the non- compact directions, we nd that if M4 is maximally symmetric then it should be Minkowski. This is our rst indication that imposing phenomenological constraints aects the possible geometries of our spacetime. The fact that we end up with our non-compact space being Minkowski is non-trivial, since the

12 condition of unbroken supersymmetry in four dimensions is also consistent with AdS spacetimes.

6.5 The geometry of K

We next turn to the condition δψm = 0, which tells us that there exists a covariantly constant real7 spinor η dened on K. The existence of such a spinor, it turns out, severely restricts the form of K. Here, again, we see that phenomenological constraints aect the geometry. We can formulate these restrictions in two equally admissible ways. First, using η we can construct [2] an almost-complex structure J on K, which turns out to be integrable. Hence, K is a complex manifold. Moreover, the metric gmn on K is hermitian with respect to J. Since J was constructed from η, which is covariantly constant, it should itself be covariantly constant. Hence K should be Kähler. Moreover, an integrability condition for η ensures that Rmn = 0, so K is Ricci at. So our compact manifold K is required to be Kähler and Ricci at (which is equivalent to having vanishing rst Chern class [4]) or, in other words, it is a Calabi-Yau manifold. The second way to see that K should be Calabi-Yau is to consider the holonomy group of K [2]. The idea rests on the fact that O(6) is locally isomorphic to SU(4). A priori, the spin connection on K is an O(6) gauge eld. However, the existence of a non-zero η which is covariantly constant means that the spin connection should take values in the little group SU(3) of η. The Ambrose-Singer theorem then tells us that K has holonomy a subgroup of SU(3). Hence, K is Calabi-Yau by Berger's classication of simply connected Riemannian manifolds. So, to recap, the condition that we wanted unbroken supersymmetry and the assumptions (8) restricted our non-compact manifold to be Minkowski, and our compact manifold to be Calabi-Yau. This isn't the whole story yet though. We also need to take into account the condition (7), with H = 0, which relates the curvature of the spin connection (which we've now determined) to the curvature (eld strength) of the sYM gauge eld. One can show [2] that a consistent solution to (7) is obtained if we take the gauge eld for one of the E8 factors to be zero and break the other to E6. The reason, loosely speaking, is that E8 has a maximal subgroup SU(3) × E6. If we embed the spin connection into the SU(3) factor then it turns out that (7) is satised and the gauge eld lives in the E6.

6.6 Number of generations

We've seen now that imposing that our theory retains N = 1 SUSY restricts the form of both the compact manifold K and the noncompact manifold M4. Moreover, we have been led to a gauge group, E6, which is semi-realistic for doing phenomenology with. However, these aren't the only ingredients we would

7Real here means that it lives in the 4 ⊕ 4¯ representation of SU(4).

13 like to have: we would also like to have a fairly realistic (i.e. small) number of chiral generations for our fermions. Before we're able to count how many such generations we're left with after compactication, we want to know what we're really looking for, so let's begin. We start with our ten-dimensional fermions, which live in a spinor representation of SO(1, 9). Under the decomposition of the SO(1, 9) to SO(1, 3)×SO(6) our fermions will split simply into spinor reps of SO(1, 3) and SO(6). Consider a massless fermion, Ψ, in ten dimensions, which satises

iD/ 10Ψ = i(D/ 4 + D/ K )Ψ = 0, where we are using a covariant Dirac operator which includes the spin connection. Naively, we might then deduce that, to the four-dimensional observer, the operator D/ K looks like a mass operator, whose eigenstates give the masses of the four-dimensional fermions. Unfortunately, this doesn't quite work, since we can't simultaneously diagonalise D/ 4 and D/ K : they don't commute. For a detailed explanation of the resolution to this, see Section 14.1.2 of [6]. For our purposes however, we not really interested in the masses of the fermions, since they should all be of order the Planck mass. We only really care about the massless four-dimensional fermions, which are given by the zero modes of the Dirac operator D/ K . Going back to ten dimensions for the minute, we can dene a chirality operator

(10) Γ = Γ1 ... Γ10, which squares to the identity. We also introduce the operators

(4) (K) Γ = iΓ1 ... Γ4, Γ = −iΓ5 ... Γ10, which are chirality operators in four dimensions and on the internal space K, respectively. We have, therefore, Γ(10) = Γ(4)Γ(K). The fermions that we get in ten-dimensional supergravity are required to satisfy a chiral constraint, which we take to be

Γ(10)Ψ = Ψ, from which we can deduce

Γ(4)Ψ = Γ(K)Ψ, so that left/right chiral fermions in four dimensions are left/right chiral fermions on the internal manifold. But why are we now going on about chirality? Well, it turns out that an important object associated with any dierential operator (more generally, any operator that acts as a map between sections of bre bundles, see Chapter 12 of [4]) is its index. For the example we care about (the internal Dirac operator), this index is given by the number of left-chiral zero modes minus the number

14 of right-chiral zero modes. This is what one would usually call the `number of chiral generations', and is exactly the quantity we'd like to compute. Now, there is an important mathematical theorem called the Atiyah-Singer index theorem [4] which relates the index of an operator to some topological invariant on the bundle on which it is dened. In the case we're interested in this is a certain spin bundle over K, for which it turns out that the topological invariant in question is just half the Euler characteristic of K, 1 Ngen = |χ(K)|. 2

However, if we take K = K0 simply connected and of SU(3) holonomy, the number of generations obtained is rather unrealistically high...

6.7 Changing the homotopy class Quotient by a discrete group. Non-contractable loops.

A simple solution to the large number of generations obtained when K0 is simply connected is to quotient out by the action of a discrete symmetry group

Γ acting freely on K0, that is, consider the manifold K = K0/Γ. In this instance, one can show that χ(K) = χ(K0)/|Γ|, so we have hope to be able to reduce the number of chiral generations to a more realistic value by suitably choosing Γ. In [2] the authors construct a model with 4 generations in exactly this way. Moreover, quotienting out by such a discrete symmetry group does not further break supersymmetry.

6.8 Breaking the gauge group

Wilson loops on K/Γ. Gauge transformations. Restrictions. As we showed earlier, in the case of K = K0 simply connected, one of the E8 factors in the gauge group of the heterotic string is broken to E6, which is a realistic8 GUT group in four dimensions.

When K = K0/Γ, however, there is a mechanism whereby this E6 gauge group can be broken further. This fact relies on the observation earlier that the fundamental group of K0/Γ is Γ, and so there exists non-contractable cycles in the manifold. We can then consider non-trivial Wilson loops around such cycles, which, as we will see, break the original gauge group. We follow the discussion in [3]. To start with, let's consider the picture we want. Let's take our gauge group to be G (e.g. G = E6). Then we're interested in a principal G-bundle over K0/Γ. We'll then want to consider an associated k-bundle (i.e. a vector bundle with bre k) to the principal bundle . R P (K0/Γ,G) Let's take a non-contractable loop γ in K0/Γ, and consider its (unique) horizontal lift in P (K0/Γ,G). This induces a map u 7→ ugγ . Now, take a section Φ(x) of the associated k-bundle over x ∈ K0/Γ. The right action u 7→ ugγ in the bres of P (K0/Γ,G) `induces' a left action Φ(x) 7→ 8 In that a single generation of quarks and leptons (plus other things) ts into the 27 of E6.

15 ρ(gγ )Φ(x), where ρ is a k-dimensional representation of G. In fact, the matrix ρ(gγ ) is exactly the Wilson loop Wγ associated with the loop γ, i.e. (5) but where A is now a matrix in a k-dimensional representation of g. In particular, if we consider the loop as dening a map x 7→ g(x) (which is the same point in our identication), on K0/Γ, then we have

Φ(g(x)) = Wγ Φ(x). (9) Now, what happens under a gauge transformation? For Φ(x) living in Rk, the eld transforms under a k-dimensional representation as Φ(x) 7→ V (x)Φ(x). Now, we're interested here in vacuum congurations, so we want to en- sure that we don't accidentally switch on any background gauge elds when performing the gauge transformation. We know that, under a transformation, B(x) 7→ V (x)−1B(x)V (x)+V (x)−1dV (x). So, if we start with B = 0, we'll only stay in a vacuum conguration provided dV = 0, i.e. V (x) = V is constant. As such, we see that, applying the transformation to (9), the left-hand side transforms as

Φ(g(x)) 7→ V Φ(g(x)) = VWγ Φ(x), whereas the right-hand side transforms as

Wγ Φ(x) 7→ Wγ V Φ(x), where we have used the fact that Wγ is gauge invariant. Hence, a gauge transformation is compatible with (9) i [V,Wγ ] = 0. Put another way, our initial gauge group G is broken into the group which commutes with the Wilson loop.

7 Extras

In the discussion of [2] it was explicitly assumed that the various `uxes' (non- trivial vacuum expectation values for the various p-form elds that occur in string theory) are switched o. One can also consider backgrounds with these uxes switched on, which gives a more realistic spectrum for phenomenology. For some good introductory lectures along these lines I'd recommend S.Kachru's notes (and videos) from the Graduate Summer School on String Phenomenology 2012 held at the Simons Center, Stony Brook.

References

[1] Michael B. Green and John H. Schwarz. Anomaly Cancellation in Supersym- metric D=10 Gauge Theory and Superstring Theory. Phys.Lett., B149:117 122, 1984.

16 [2] P. Candelas, Gary T. Horowitz, Andrew Strominger, and Edward Witten. Vacuum Congurations for Superstrings. Nucl.Phys., B258:4674, 1985. [3] Edward Witten. Symmetry Breaking Patterns in Superstring Models. Nucl.Phys., B258:75, 1985.

[4] M. Nakahara. Geometry, Topology and Physics, Second Edition. Graduate Student Series in Physics Series. Taylor & Francis Group, 2003. [5] David Tong. String Theory. 2009.

[6] M.B. Green, J.H. Schwarz, and E. Witten. Superstring Theory: Volume 2, Loop Amplitudes, Anomalies and Phenomenology. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1988.