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INTRODUCTION to GAUGE THEORY 1. Bundles, Connections

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INTRODUCTION to GAUGE THEORY 1. Bundles, Connections INTRODUCTION TO GAUGE THEORY LORENZO FOSCOLO 1. Bundles, connections and curvature M smooth manifold + G Lie group with Lie algebra g • principal G–bundle P over M • – smooth manifold P – free right G–action R: P G P × → – smooth projection to orbit space π : P M = P/G → – local G-equivariant trivialisations π 1(U) U G − ≃ × gauge group = Aut(P ) • G principal bundles and vector bundles • – representation ρ: G GL(V ) ❀ vector bundle E = P V → ×ρ e.g. ρ = Ad ❀ adjoint bundle ad P – G = stabiliser of φ r V s V , (E, φ) modelled on (V, φ ) ❀ principal G–bundle 0 ∈ ⊗ ∗ 0 of φ–adapted frames � � Examples: Hopf circle bundle S3 S1, quaternionic Hopf bundle S7 S4, frame bundle • → → Exercise 5.1 A connection on π : P M is a G–invariant splitting T P = ker π H of • → ∗ ⊕ 0 ker π T P π∗T M 0 → ∗ → → → H is called the horizontal subspace 1 Since ker π P g, a connection is a 1-form A: T P g such that Rg∗A = Ad(g− )A • ∗ ≃ × → – Product/trivial connection on M G induced from identification T G G g × ≃ × – loc. trivialisation P U G: A = trivial + a for a g-valued 1-form a on U |U ≃ × – change trivialisation given by u: U G ❀ A = trivial + u 1du + u 1au → − − infinite-dimensional affine space of connections on P modelled on Ω1(M; ad P ) • A Note: action of gauge group on by pull-back G A Exercise 5.2 horizontal lift XH of a vector field X on M: the unique G–invariant vector field on P such • that XH H T P at every point of P and π XH = X ∈ ⊂ ∗ parallel transport of A • – hor. lift of path γ in M: path γ in P such that π γ = γ and γ (t) H t ◦ ′ ∈ γ(t) ∀ – h π 1(γ(0)) ! hor. lift ❀ parallel transport Π : π 1(γ(0)) π 1(γ(1)) ∀ ∈ − ∃ γ − → − – γ loop ❀ Π = R for some g � G � � γ g ∈ curvature F Ω2(M; ad P ) of A • A ∈ – π F (X, Y ) = [X, Y ]H [XH , Y H ] ∗ A − – F = d d A A ◦ A – = d + a ❀ F = da + a a ∇A A ∧ A , a Ω1(M; ad P ): F = F + d a + a a • ∈ A ∈ A+a A A ∧ Bianchi identity: d F = 0 • A A Date: Feb 5, 2021. 1 2 LORENZO FOSCOLO – G–invariant polynomial p: g R ❀ closed (2 deg p)–form p(FA) on M →deg p – characteristic class [p(FA)] HdR (M; R) independent of A ∈ i e.g. G = U(k): pi(X) = Tr(X ) ❀ Chern classes ci(E) of complex vector bundle E M complex manifold, E M complex vector bundle with Hermitian metric h → Cauchy–Riemann operator on E: C-linear map ∂ : Ω0(M; E) Ω0,1(M; E) such that • E → ∂ (fs) = ∂f s + f ∂ s E ⊗ E – space of Cauchy–Riemann operators: affine space modelled on Ω0,1(M; End E) connection ❀ Cauchy–Riemann operator ∂ = 0,1 • ∇ E ∇ Chern connection: h + ∂ = ! unitary connection with ∂ = 0,1 • E ⇒ ∃ ∇ E ∇ M holomorphic vector bundle with underlying smooth complex vector bundle E • E → – Cauchy–Riemann operator ∂ : in local holomorphic trivialisation ∂ = ∂ – Cauchy–Riemann operator ∂E ❀ sheaf of “holomorphic” sections E( ) = ker ∂ E O E E – ∂ ∂ = 0 E has the structure of a holomorphic bundle E ◦ E ⇐⇒ E Exercise 5.3 2. Flat connections G compact Lie group, e.g. G = U(k) • P M principal G–bundle • → moduli space of flat connections = A F = 0 / • M { ∈ A | A } G as sets = Hom(π (M), G)/G • M 1 Exercise 5.4 fix A and understand local structure of near A • ∈ M M introduce Banach spaces, e.g. Hölder spaces: k,α, k+1,α • A G deformation complex of a flat connection A • d d d 0 Ω0(M; ad P ) A Ω1(M; ad P ) A Ω2(M; ad P ) A . → −→ −→ −→ 0 1 2 and cohomology groups HA, HA, HA fix metric on M ❀ Ω1(M; ad P )k,α = im d ker d • A ⊕ A∗ – im dA: tangent space to the orbit A ,α G · – H0 : Lie algebra of stabiliser 2 of A A GA e.g. G = U(k), SU(k): k+1,α = 1 iff H0 = 0 GA { } A – Slice Theorem ❀ local structure of k,α/ 2,α near A A G k,α k+1,α 1 k,α k+1,α / = A + a a Ω (M; ad P ) , a k,α < �, d∗ a = 0 / SA,� GA { | ∈ � �C A } GA H1 = a Ω1(M; ad P )k,α d a = 0 = d a : gauge-fided infinitesimal deformations of A • A { ∈ | A A∗ } as a flat connection A is irreducible + H2 = 0: k+1,α is a smooth manifold near A with tangent space H1 • A M A (smooth structure independent of α (0, 1), k 1) ∈ ≥ Exercises 5.5, 5.6 3. The Yang–Mills functional (M, g, vol) oriented Riemannian manifold, P M principal G–bundle with G compact Lie group → Yang–Mills functional on : (A) = F 2 vol • A YM M | A| critical points: d F = 0 (where d = ´ d ) • A∗ A A∗ ± ∗ A∗ Exercises 5.9, 5.10 Compactness: INTRODUCTION TO GAUGE THEORY 3 Strong Uhlenbeck Compactness: M n closed, n < p < (p 4 if n = 2), A sequence • 2 ∞ ≥ 3 i p 2,p of YM connections w/ uniformly bounded FAi L after passing subsequence W C∞ � � ⇒ ∃ gauge transformations gi s.t. gi∗Ai A smooth YM connection −→ c∞r2 4 n 2 Monotonicity Formula: c = c(M) s.t. e r − FA is increasing in r • ∃ Br(po) | | ❀ smooth convergence of YM connections with ´unformly bounded energy away from a set • of finite (n 4)–dimensional Hausdorff measure (Uhlenbeck) − – in 4d convergence outside finitely many points Exercise 5.11 4. Instantons Exercise 5.12 4 2 2 2 (M , g) ❀ Λ T ∗M = Λ+T ∗M Λ T ∗M (eingenspaces of ) • ⊕ +− ∗ A instanton/ASD connection if FA = 0 • + 2 (A) = FA FA g + 2 FA instantons are absolute minima of • YM − 〈 ∧ 〉 | | ⇒ + YM ´ ´ d d moduli space : 0 Ω0(M; ad P ) A Ω1(M; ad P ) A Ω2 (M; ad P ) 0 • M → −→ −→ + → Exercises 5.13, 5.14 Donaldson’s Diagonalisation Theorem: M 4 closed, smooth, oriented, simply connected w/ negative definite intersection form q • 2m = � α H2(M; Z) q(α, α) = 1 { ∈ | } P M 4 SU(2)–bundle with c (P ) = 1 • → 2 moduli space of instantons on P • M , – for generic metric g, H2 + = 0 for all A and irr smooth A ∈ M M – if = then dim = 5 M ∕ ∅ M 2 – reducibles ❀ singularities p1, . , pm modelled on cones CP over ∈ M – irr non-empty: p M and � 1 instanton A “concentrated” in B (p) M ∀ ∈ ≪ ∃ p,� � irr non-empty manifold with boundary M CP2 • M ⊔ m M smoothly cobordant to CP2 = q = id • m ⇒ − � Exercises 5.15, 5.16 � 5. Exercises Exercise 5.1. In this exercise we study principal bundles on spheres. (i) Show/convince yourself that isomorphism classes of principal G–bundles on the sphere Sn are in 1:1 correspondence with πn 1(G). (Hint: trivialise the bundle over two hemispheres and consider the gluing map over −the equator.) (ii) Describe principal U(1)–bundles on S2. (iii) Show that there are only two non-trivial SO(3)–bundles on S2. Can you describe the non- trivial one? (Hint: think about rank 3 real vector bundles of the form R L, where R is ⊕ the trivial real line bundle and L is a complex line bundle.) (iv) Show that every SU(2)–bundle on S3 is trivial and that SU(2)–bundles on S4 are classified by an integer. (v) Show that there are only two principal SU(2)–bundles on S5. Can you describe the non- trivial SU(2)–bundle on S5? (Hint: use the homogeneous space presentation S5 = SU(3)/SU(2).) Exercise 5.2. Let π : P M be a principal G–bundle with connection A and let E = P V be → ×ρ an associated vector bundle of rank k. (i) Show that there is a 1:1 correspondence between sections s C (M; E) of E and G– ∈ ∞ equivariant V –valued functions s C (P ; V ) on P . ∈ ∞ � 4 LORENZO FOSCOLO (ii) Show that the formula s ds H defines a covariant derivative on E, i.e. an R–linear map �→ | : C (M; E) C (M; T M E) satisfying the Leibniz rule (fs) = df s + f s ∇A ∞ → ∞ ∗ ⊗ ∇A ⊗ ∇A for all f C (M) and s C� (M; E). ∈ ∞ ∈ ∞ (iii) Work in a local trivialisation P U G where A is given in terms of a Lie algebra |U ≃ × valued 1-form a on U. Note that since G is a subgroup of GL(k), a is simply a matrix k of 1-forms. Similarly, a section s of E U is simply a smooth map s: U R . Show that | → s = ds+as, where d denotes the usual differential and as simply denotes the point-wise ∇A action of the matrix a on the vector s. (iv) In the same way one extends the differential d: C (M) Ω1(M) of functions to the exter- ∞ → ior differential d: Ωp(M) Ωp+1(M) of differential forms, show that the covariant derivat- → ive extends to a covariant exterior differential operator d : Ωp(M; E) Ωp+1(M; E) ∇A A → for any p. Here Ωp(M; E) denotes the space of smooth section of the vector bundle ΛpT M E.
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