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Gauge Theory Gauge Theory José Figueroa-O’Farrill* http://www.maths.ed.ac.uk/empg/Activities/GT Version of 6th October 2019 These are the notes accompanying the first few lectures on Gauge Theory, a PG course taught in Edinburgh in the Spring of 2006. The only requirement is a working familiarity with differential geometry and Lie groups; al- though scholia on the necessary definitions will be scattered throughout the notes. Any statement which is not proved to your satisfaction is to be thought of as an exercise, even if not explicitly labelled as such! These notes are still in a state of flux and I am happy to receive comments and suggestions either by email or in person. *) j.m.figueroa(at)ed.ac.uk 1 GT 2006 (jmf) 2 Contents 1 Connections on principal fibre bundles3 1.1 Motivation: the Dirac monopole...................................3 1.2 Principal fibre bundles.........................................4 1.3 Connections...............................................6 1.3.1 Connections as horizontal distributions..........................6 1.3.2 The connection one-form...................................8 1.3.3 Gauge fields...........................................8 1.4 The space of connections....................................... 10 1.5 Gauge transformations......................................... 11 1.6 The action of G on A ......................................... 11 2 Curvature 13 2.1 The curvature of a connection.................................... 13 2.1.1 The horizontal projection................................... 13 2.1.2 The curvature 2-form..................................... 13 2.1.3 Gauge field-strengths..................................... 15 2.2 The covariant derivative........................................ 15 2.2.1 Basic forms........................................... 15 2.2.2 The covariant derivative.................................... 16 3 The Yang–Mills equations 19 3.1 Some geometry............................................. 19 3.1.1 The volume form........................................ 19 3.1.2 The Hodge ? operator..................................... 19 3.1.3 Inner product on bundle-valued forms........................... 20 3.2 The variational problem........................................ 21 3.2.1 The action functional..................................... 21 3.2.2 The field equations....................................... 22 3.3 Coupling to matter........................................... 23 4 Instantons 24 4.1 (Anti-)self-duality............................................ 24 4.2 What is c?................................................. 25 4.3 The Chern–Simons form........................................ 26 4.4 The BPST instanton........................................... 26 5 Instanton moduli space 28 5.1 Irreducible connections........................................ 28 5.2 The deformation complex....................................... 29 GT 2006 (jmf) 3 Lecture 1: Connections on principal fibre bundles The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years. — CN Yang, 1979 In this lecture we introduce the notion of a principal fibre bundle and of a connection on it, but we start with some motivation. 1.1 Motivation: the Dirac monopole It is only fitting to start this course, which takes place at the JCMB, with a solution of Maxwell’s equa- tions. The magnetic field B of a magnetic monopole sitting at the origin in R3 is given by x B(x) , Æ 4¼r 3 where r x . This satisfies divB 0 and hence is a solution of Maxwell’s equations in R3 \{0}. Æ j j Æ Maxwell’s equations are linear and hence this solution requires a source of magnetic field, namely the monopole which sits at the origin. We will see later in this course that there are other monopoles which do not require sources and which extend smoothly to the ‘origin’. In modern language, the vector field B is understood as the 2-form F ­2(R3 \{0}) given by 2 1 ¡ ¢ F x1dx2 dx3 cyclic Æ 2¼r 3 ^ Å in the cartesian coordinates of R3. Maxwell’s equations say that dF 0. This is perhaps more evident in Æ spherical coordinates (x r sinθcosÁ,...), where 1 Æ 1 F sinθdθ dÁ . Æ 4¼ ^ Since dF 0 we may hope to find a one-form A such that F dA. For example, Æ Æ 1 A cosθdÁ . Æ¡4¼ This is not regular over all of R3\{0}, however. This should not come as a surprise, since after all spherical coordinates are singular on the x3-axis. Rewriting A in cartesian coordinates x3 x2dx1 x1dx2 A ¡ , Æ 4¼r x2 x2 1 Å 2 clearly displays the singularity at x x 0. 1 Æ 2 Æ In the old Physics literature this singularity is known as the “Dirac string,” a language we shall distance ourselves from in this course. This singularity will afflict any A we come up with. Indeed, notice that F restricts to the (normalised) area form on the unit sphere S R3, whence ½ Z F 1 . S Æ If F dA for a smooth one-form A, then Stokes’s theorem would have forced R F 0. Æ S Æ The principal aim of the first couple of lectures is to develop the geometric framework to which F (and A) belong: the theory of connections on principal fibre bundles, to which we now turn. GT 2006 (jmf) 4 1.2 Principal fibre bundles A principal fibre bundle consists of the following data: • a manifold P, called the total space; • a Lie group G acting freely on P on the right: P G P £ ! (p,g) pg (or sometimes R p) 7! g where by a free action we mean that the stabiliser of every point is trivial, or paraphrasing, that every element G (except the identity) moves every point in P. We will also assume that the space of orbits M P/G is a manifold (called the base) and the natural map ¼ :P M taking a point to Æ 1 ! its orbit is a smooth surjection. For every m M, the submanifold ¼¡ (m) P is called the fibre 2 ½ over m. Further, this data will be subject to the condition of local triviality: that M admits an open cover {U®} 1 and G-equivariant diffeomorphisms à : ¼¡ U U G such that the following diagram commutes ® ® ! ® £ à 1 ® ¼¡ U / U G ® ® £ ¼ pr1 " { U® 1 This means that à (p) (¼(p),g (p)), for some G-equivariant map g : ¼¡ U G which is a fibrewise ® Æ ® ® ® ! diffeomorphism. Equivariance means that g (pg) g (p)g . ® Æ ® One often abbreviates the above data by saying that P G P ¡¡¡¡¡! ? ? ? ? y¼ or y¼ is a principal G-bundle, M M but be aware that abuses of language are rife in this topic. Don’t be surprised by statements such as “Let P be a principal bundle...” We say that the bundle is trivial if there exists a diffeomorphism ª :P M G such that ª(p) ! £ Æ (¼(p),Â(p)) and such that Â(pg) Â(p)g. This last condition is simply the G-equivariance of ª. Æ A section is a (smooth) map s :M P such that ¼ s id. In other words, it is a smooth assignment ! ± Æ to each point in the base of a point in the fibre over it. Sections are rare. Indeed, one has Done? q Exercise 1.1. Show that a principal fibre bundle admits a section if and only if it is trivial. (This is in sharp contrast with, say, vector bundles, which always admit sections.) Solution. If s :M P is a section, define ª :P M G by sending ª(p) (¼(p),Â(p)), where Â(p) G ! ! £ Æ 2 is uniquely specified by p s(¼(p))Â(p). Conversely, given ª :P M G, define a section by s(m) 1 Æ ! £ Æ ª¡ (m,e). u Nevertheless, since P is locally trivial, local sections do exist. In fact, there are local sections s® :U® 1 ! ¼¡ U canonically associated to the trivialisation, defined so that for every m U , à (s (m)) (m,e), ® 2 ® ® ® Æ where e G is the identity element. In other words, g s :U G is the constant function sending 2 ® ± ® ® ! every point to the identity. Conversely, a local section s® allows us to identify the fibre over m with G. 1 Indeed, given any p ¼¡ (m), there is a unique group element g (p) G such that p s (m)g (p). 2 ® 2 Æ ® ® GT 2006 (jmf) 5 On nonempty overlaps U : U U , we have two ways of trivialising the bundle: ®¯ Æ ® \ ¯ ï à 1 ® U G o ¼¡ U / U G ®¯ £ ®¯ ®¯ £ ¼ pr1 pr1 $ z U®¯ 1 For m U and p ¼¡ (m), we have à (p) (m,g (p)) and à (p) (m,g (p)), whence there is 2 ®¯ 2 ® Æ ® ¯ Æ ¯ g¯ (p) G such that g (p) g¯ (p)g (p). In other words, ®¯ 2 ® Æ ®¯ ¯ 1 (1) g¯ (p) g (p)g (p)¡ . ®¯ Æ ® ¯ In fact, g¯®¯(p) is constant on each fibre: 1 g¯ (pg) g (pg)g (pg)¡ ®¯ Æ ® ¯ 1 (by equivariance of g , g ) g (p)g g ¡ g (p) ® ¯ Æ ® ¯ g¯ (p). Æ ®¯ In other words, g¯ (p) g (¼(p)) for some function ®¯ Æ ®¯ g :U G. ®¯ ®¯ ! From equation (1), it follows that these transition functions obey the following cocycle conditions: (2) g (m)g (m) e for every m U , and ®¯ ¯® Æ 2 ®¯ (3) g (m)g (m)g (m) e for every m U , ®¯ βγ γα Æ 2 αβγ where U U U U . αβγ Æ ® \ ¯ \ γ Done? q Exercise 1.2. Show that on double overlaps, the canonical sections s® are related by s (m) s (m)g (m) for every m U . ¯ Æ ® ®¯ 2 ®¯ 1 Solution. Simply observe that g s g g ¡ g s g . u ® ± ¯ Æ ® ± ¯ ± ¯ ± ¯ Æ ®¯ One can reconstruct the bundle from an open cover {U®} and transition functions {g®¯} obeying the cocycle conditions (2) as follows: G . P (U® G) , Æ ® £ » where (m,g) (m,g (m)g) for all m U and g G.
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