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 fur- ther on, namely bundles (vector, principal, and associated), holonomy and ho- motopy. 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 lan- guage. 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 2 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 fUig 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 2 −1 in the bre over , −1 , for some element . π (p) p φi (u) = (p; fi) fi 2 F Now, take this open cover of M and consider a point p 2 Ui \ Uj. Then we have two local trivializations φi and φj to choose from on Ui \ Uj. Take the point u 2 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 2 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 2 M, we had a way of picking out a particular point u 2 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 fUN ;USg 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 2 TS2 can be written as µ @ µ @ V = VN µ = VS µ ; @X p @U p with µ 2. fVN(S)g 2 R Now, to nd the transition functions we rst dene the local trivializations. 2 We take u 2 TS with p = π(u) 2 UN \ US. Then, −1 µ −1 µ φN (u) = (p; fVN g); φS (u) = (p; fVS g): 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 2 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 2 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 2 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.