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Selfduality in geometry : Yang-Mills connections and selfdual lagrangians Donaldson, Jason Roderick 2007

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SELFDUALITY IN GEOMETRY: YANG-MILLS CONNECTIONS AND SELFDUAL LAGRANGIANS by JASON RODERICK DONALDSON B.Sc, Simon Fraser University, 2005 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 2 007 © Jason Roderick Donaldson, 2007 A b s t r a c t The convex theory of selfdual Lagrangians recently developed by Ghoussoub analyses junctionals rooted i n an expanse of p a r t i a l d i f f e r e n t i a l equations and finds t h e i r minima not v a r i a t i o n a l l y but rather by r e a l i z i n g that they assume a prescribed lower bound. This i s exactly the circumstance i n the selfdual and a n t i - s e l f d u a l Yang-Mills equations that arise i n the physical f i e l d theory and the study of the geometric and topological structure of four-dimensional manifolds. I expose the Yang-Mills equations, building up the geometry from student-level and subsequently outline the setting of self d u a l Lagrangians. The theories are c l e a r l y analogous and the l a s t section f e i n t s at the exact l i n k . T a b l e o f C o n t e n t s Abstract . i i Table of Contents i i i CHAPTER 1 Introduction 1 CHAPTER 2 Geometric Buildup 3 The Covariant Derivative and Curvature Form 3 Li e Structure 8 Connections 13 The Hodge Star and Volume 18 A Generalized Domain for Covariant Derivatives and the Second Bianchi Identity 27 CHAPTER 3 The Yang-Mills F u n c t i o n a l 1 The Functional Defined and the Yang-Mills Connections 31 A Representation of Yang-Mills Connections 34 The Example of Two Dimensions 36 A Preliminary Fact 4 0 Yang-Mills i n Four Dimensions: The Selfdual Equations .... 45 Cohomology and Chern Classes 4 8 Computation of Chern Classes 52 Examples and the Topological Charge 56 A F i r s t Lower Bound for the Yang-Mills Functional 58 Instantons as Absolute Minima ••• 60 CHAPTER 4 The Convex Selfdual Framework 64 The Convex Setup: The Legendre Transform, S u b d i f f e r e n t i a l , and Fenchel Inequality 64 The Basic Example for our Application 69 Lagrangians 71 An Example from P a r t i a l D i f f e r e n t i a l Equations 72 The Link with Instantons 76 The Future 81 References 83 Chapter 1: Introduction What follows are two narrative introductions to loosely connected fields i n modern analysis intended to be fully self-contained for a graduate student ap-proaching the subjects after first graduate course sequences in differential ge-ometry and real analysis. Given the basic understanding of finite-dimensional abstract manifolds—atlases, vector bundles, and differentiation—a student should experience the first sections as a solidification of notation and a re-minder of definitions before learning the next batch of geometry—Lie and exterior algebras and integration—with the express purpose of understanding the Yang-Mil l s functional and the so-called selfdual and anti-selfdual equa-tions for connections in four dimensions. Throughout I try to take special care to clarify the domains of operators and state clearly their alternative representations, since i n learning geometry I have been and am sti l l stumped particularly often by formulations switch and many spaces present themselves together here. The exposition of the Yang-Mil l s theory should illustrate some geometric concepts, namely cohomology, while building up to some nice fundamental results. I compute the expression for the functional's crit ical points varia-t iona l^ , but the point is the four-dimensional case where the attainment of the lower bound, not the first order condition, gives the essential equations. This is the essential analogy wi th the convex theory of chapter 3 . Jurgen Jost's book Riemannian Geometry and Geomemtric Analysis was my main 1 reference and I largely follow its exposition. Fina l ly the thi rd chapter outlines Ghoussoub's theory of selfdual L a -grangians. It strongly resembles the four-dimensional Yang-Mil l s theory in that it proposes to reformulate partial differential equations as minimization problems whose solutions determined depending on the realization that they attain their lower bounds. The key tool here is the Fenchel inequality of convex analysis. The framework's main strengths are its simplici ty—it re-lies only on basic functional analysis and very basic convex analysis—and its overarchingness—it applies to an expanse of partial differential equations. I sketch some of the theory wi th a story-telling, pedagogical bent and formally align it wi th the Yang-Mil ls case i n the final section. Whi le Yang-Mil ls fits effortlessly into this framework the formal computations carried out at the end have yet to be fully justified mathematically. I hope (and suspect) they wi l l be soon, especially because I believe too that they wi l l extend to sti l l richer circumstances. 2 Chapter 2: Geometric Buildup To begin we clarify the definitions and notations, that are often malleable in the literature, for the objects essential to differential geometry that con-stitute the basal elements of what follows. The covariant derivative and its induced curvature are the essential objects of Yang-Mil ls theory. They sit on top of gauge theory and relate algebra and topology to real analysis and especially, as is our true subject, to modern partial differential equations which we view here through the application convex analysis in the calculus of variations. The Covariant Derivative and Curvature Form A covariant derivative 1 on a smooth, compact, finite dimensional manifold A4 is a map D : X(M) x X(M) -+ X(M), D : (V, X) ^ DVX where X(M) denotes the space of smooth vector fields on the manifold, X(M) = {X G C ° ° ( M ; TM) ; Xp e TPM for each p e M). 1 A covariant derivative is often called a connection. I will employ another convention and refer to the connection as the operator A in the decomposition D = d+A. Note though that while they are not equal in this formulation, there is still a one-to-one correspondence between covariant derivatives and connections. 3 In general we assume that the manifold has no boundary, dM = 0. Since in fact D depends on the first argument only pointwise—as opposed to local ly— we may replace V wi th its image, VP = u, and say instead, for each p G M , D : TPM x X(M) - » £ ( A 4 ) , D : ( i t , X) i - > DUX, or, finally but importantly, equivalently exchange the tangent space i n the domain for its dual in the range, as D : X(M) X(M) x T;(M) D : X ^ DX. The covariant derivative is a derivative i n the sense of parallel transport: Given a curve a G C 1 ( M ; M) and a linear isomorphism Pt,c • TA(0)M -> T C T ( t ) X satisfying (PtAu)>PtAv))a(t) = (u>v)*(o) for al l u, v G T ^ ^ A ^ and £ G K (where (•, •) denotes the chosen metric on 4 TPM), the equation dt t=o (1) defines the action of covariant derivative in the direction of any u = <r(0) := da/dt £ TPA4, p = a(0). Bu t only up to the choice of the parallel transport map P—although this doesn't depend on the curve a so long as it satisfies the ini t ia l condition. This definition clearly depends on the specification of the metric on TA4, but a more abstract equivalent formulation does not. The selection of covariant derivative to employ, however, typically depends on the metric. For example, the canonical Lev i -C iv i t a covariant derivative is the unique covariant derivative V satisfying the torsion-free condition for every X, Y, Z e X(M), where the Lie Bracket by the commutator [•,•]: X(M) x X(M) X{M) acts as VxY - VYX - [X, Y} = 0 (2) and the defining formula X((Y,Z)) = (VXY,Z) + (Y,VXZ) [X, Y) := XY - YX 5 which means for 1 ' J dxidx* d& dx*' [ ' X - XI 8 Y -YI 3 dx1' 9x z The definition of Yang-Mil l s covariant derivatives depends explicitly on the manifold's metric, but relies further on the global minimization of the man-ifold's curvature. The curvature of a covariant derivative is its composition wi th itself in the second argument: FD:=DoD: TPM x TPM x X(M) -* X(M). (4) Here, again consider the alternative formulation: FD : x{M) -> x(M) x T;M x T;M, and introduce the notation VtP(TM) := X(M) x ClP(M) (5) understanding that tt°(TM) = X(M). Here £lS(A4) is space of sections of the total space of p-differential forms. To 6 be explicit (and perhaps, wi th apologies, verbose), the background notation is as follows: For x G M, p-forms at x are p l\T*xM := {UJ : TXM x ... x TXM - » R ; co is linear} and the corresponding total space is ,\T*M := I J /\T;M. A s wi th any vector bundle, there is a projection ir : f\p T*M. —> A4 such that for each x E M., v K-\X) = F\T*xM which allows the definition of the sections of the p-forms (and by analogy sections generally 2) as Return to our examination of the curvature: In this language it is the 2 O f course the vector fields are simply the sections of the tangent bundle, £ ( A t ) := r ( T A t ) . A later section addresses this definition as necessary for the expansion of the theory, but for now I attempt to sick as much as possible to the simplest case and spell out the required basics, because, as is already apparent, there is a lot of theory and notation to bog us down despite that my intended focus is examples. 7' mapping FD : Q°{TM) -> n2{TM). In the case of the Lev i -C iv i t a covariant derivative the curvature, R := i*V, takes a simple form: For u, v G TPM. R(u,v) : X(M) 3£(A4) as # ( u , v ) X = V U V W X - V „ V U X - V[UMX (here it suffices to consider the bracket [u,v] = [Y, Z]p where the vector fields Y and Z are such that Yp = u and Zp = v). Whi le the Lev i -C iv i t a co-variant derivatives come easily and give us a curvature form, the formulation required for the Yang-Mil ls functional and hence Yang-Mil ls covariant deriva-tives (which, in a manner opposite to the Lev i -C iv i t a case, result inversely from the curvature) wi l l remain elusive since they depend on the development of further algebraic and analytic—geometric—structures on the manifold. Lie Structure Consider a group G which is itself a manifold (G = M) where the group actions GxG-^G, (x,y)~xy (7) 8 and G^G, x ^ x ' 1 (8) are smooth. G is called a Lie group. The diffeomorphic left translation Lx : G —> G, Lxy := xy, (9) produces the essential mapping L x * : X(G) —> 3L(G), LX*X = dLxX = X o Lx, (10) for exterior (total) derivative d, that defines the subspace Q of X(G) called the Lie algebra of G by 0 := {X e X(G) ; Lx*Xy = Xxy for al l x, y G G } . (11) (In this restricted context the above equation for ( L x ) * suffices to define the push forward operator denoted by subscript star.) Observe computationally that for X, Y e g and x e G, L>x*[X, Y] = [LX*X, LX*Y], (12) so [X, Y] is itself left-invarient and hence belongs to g. A n d note that the 9 existence of such a bracket map [•,•]: g x g —> g satisfying [X,X} = 0 . (13) and [X, [Y, Z\\ + [Y, [Z, X}} + [Z, [X, Y}} = 0 (14) whenever X, Y, Z £ g defines the Lie algebra in a general setting. Thus the vector fields on M. themselves are a Lie algebra. For I G G , the inner automorphism Ax : G —» G defined by Axy = xyx'1, (15) induces an automorphism on g v ia differentiation. For identity element e of G, Axe = e, so the exterior derivative at e maps the tangent space at e into itself, Ax*{e) : TeG <—> TeG, and the so-called adjoint representation of G, the map ad : g —• g given as ad : x i — • A x *(e) , (16) yields an automorphism on g. The first example of the imposition of Lie structure on a manifold comes in the form of the general linear group over the real numbers, the totality of 10 nonsingular, real matrices of a given dimension, G L ( n , R ) := {X G M(ra, R ) ; det X ^ 0}, (17) where, of course, M(n, R ) denotes the set o f n x n real matrices. The group actions are i n fact C°°. A n d the associated Lie algebra gl(n, R) comes from firstly attaching a bracket with the usual form, [X, Y] — XY — YX for X, Y G M ( n , R ) , and then equating the matrices wi th left-invariant vector fields through a mapping, M ( n , R ) —» X, X i — • X bijectively v ia the triple sum, Xa '— __. ciikXkj f——J , (18) i,j,k=l \ a x i j J a for each a = (oy) G G L ( n , R) and where {(d/dxij)a ;1 < i,j < n} is a, basis for TaA^ which has dimension n23 The next example is the orthogonal group, O(n) = {X G M ( n , R ) ; XTX = I}, (19) which has Lie algebra o(n):={Xegl(n,R);XT + X = 0}, (20) 3 Commonly the Lie algebra is studied separately from its sections, above. Following, for example, Urakawa, here we term these sections themselves the Lie algebra as it is illustrative in the present case, but since the definitions differ only by an isomorphism they are flexible in the literature and we will sometimes change conventions, assuming the context will make the definition clear. 11 for zero matr ix 0 = (0);j and the special orthogonal group, SO(n) := {X £ 0(n); det X = 1}, (21) which is equipped wi th the same algebra as the orthogonal group, so(n) := o(n) (22) But the structures which wi l l provide the neatest examples in the follow-ing context of self-dual Yang-Mil l s theory are the unitary group, U ( n ) : = { Z e M ( n , C ) ; n = I} , (23) where as usual star denotes the adjoint matrix, and the special unitary group, SU(n) := {Z e U(n) ; d e t Z = 1}. (24) Their Lie algebras are u(n) :={Z eM(n,C)]Z* + Z = 0}, (25) and su(n) := {Z e u(n) ; trzT = 0}, (26) respectively. Here the association of matrices to left-invariant vector fields 12 must be carried out as in the above case of the general linear group. The adjoint representation assists in selecting a metric for on the manifold. C a l l the inner product, (- ,•): g x g —> E , is ad(C7)-invariant (and hence further bi-invariant) if whenever X, Y G g and x G G (ad(x)X, ad(x)Y) = (X, Y). (27) Now for X, Y G g = o(n),g = u(n), or g = su(n) the negative trace of the product defines ad(G)-invariant metric (X, Y) = -tvXY. (28) C o n n e c t i o n s Having established a metric demand that the covariant derivative respect it v ia the relationship d {X, Y) = (DX, Y) + (X, DY) (29) for vector fields X and Y. C a l l such covariant derivatives metric. Henceforth metrics wi l l be assumed Ad(G)-invariant and connections assumed metric. The covariant derivative restricted from the vector fields to a Lie subalgebra 13 has image contained in the same algebra, : 0 —>• 0 x Q}(M), since it commutes wi th the commutator. Immediately we have the same quality for the curvature, FD : 0 -+ 0 x Q2(M), or FD:Q°(TM;Q)^Q2(TM;g). To investigate the action of the connection further recall its local represen-tation i n terms of the Christoffel symbols, r ! ; f (30) J OXk d*i OXj (summation convention applied here on the left and throughout), DmX = (i\t) + T)k(a(t))^(t)t;k(t)) (31) for X = £l d/dxl and a = ald/dx\ V iew the Christoffel symbols as a map from the tangent space into the general linear group, A := (Tljk)ijtk, A : TM - * 0 l ( n , K ) , 14 A:a^(r)k(a)^)ik, (32) or, perhaps more clearly rendered on basis vectors, A (^) = Fti**' ( 3 3 ) where n = dim Ml (i, k = l,...,n) and A(a) now multiplies X G X as a matrix, ( A ^ A ^ ^ a ) ) ^ . (34) But , moreover, interpret the operator A is itself (locally) a gi-valued one-form, A e Q\TM\gt). In the decomposition of the covariant derivative the first term is independent of the direction a and in fact is just the exterior derivative of J at i . A s such write simply, D = d + A, (35) and call the operator A a connection on the tangent bundle TMl. In this formulation, the expression for the covariant derivative applied to a section of the tangent bundle, again parameterized via a local curve in M. so that X = £,ld/dx% G TXA4, as expressed i n terms of the Christoffel symbols becomes 15 The connection represents the curvature v ia the decomposition Fd+A(X) = (d + A)o(d + A)X = {d + A)(dX + AX) = d2X + d(AX)+AdX+ A(AX) = (dA)X - AdX + AdX + A A AX, since the first term vanishes because d2 = 0 and the minus sign comes from distributing the derivative v i a the product rule because A is a one-form. Thus Fd+A = dA + A A A. (37) The covariant derivative's being metric implies further that the corresponding connection is skew-symmetric, A(X) E o{n) (38) for every X G TM. To see the result consider an orthonormal basis { E \ , E n } for the fibres TpM. for each p 6 A4 generated by inverting the bundle's projection and charts from an orthonomal basis of R n 4 , so (El(p),Ej(p))=5ij. 4 For details see Jost page 38. 16 Observe now that since a vector field applied to any constant is zero for X G TPM X(Ei,Ej) = 0. Realize that the definition of a metric connection says that X {Ei, Ej) = (DxEi, Ej) + (Eu DxEj) or here X {Eu Ej) = (A(X)El, Ej) + (Et, A(X)Ej) having used the fact that wi th in a bundle chart the basis vectors are constant so for the exterior derivatived defined there dEi = 0 for i = l , . . . , n , to eliminate the exterior derivative parts of the covariant derivative. Understanding that A(X) = (A{X)Y'j : TXM - TXM is just and n x n matrix, formalize matrix multiplication in the current no-tation as A(X)Ei = (A(X))^Ej, 17 and now just compute 0 = X(Ei,Ej) = ((A(X))i'kEk,Ej) + (Ei,(A(X)y'kEk) = (A(X)y>k (Ek, Ej) + {A(X)Y<k (Eu Ek) = (A(X)Y'k 5kj + (A(X)Y'k 5ik = [A{X)Y'j + {A(X))» or, for every X e TM, A(X) = -AT{X) with the superscript denoting transposition, ( A r ) ! J = (A)- 7 ' 1 which is of course the definition of skew symmetry is the group structure is real and the transpose of a matr ix thus corresponds wi th its adjoint, {Au, v) = (u, ATv) , Thus we write i n general Aen\TM;o). (39) The Hodge Star and Volume Given a ci-dimensional Riemannian manifold M, define an innerproduct 18 on the p-fold exterior product f\pT*M: For / i , u € T*M, (fii A ... A fjLp, wi A ...A up) := det ( (^ ,0^)) . . . , (40) where the metric (-,-)x = ('•>') '• TXM —> E on the tangent fibre at each x E A4, induces the innerproduct on the cotangent space, T*M. For the local-coordinate basis {dl/dxl}™=1 of TXM the equations d < D = 6 - ' < 4 i ) define the corresponding basis {dx1}^ for T*Ml and hence an innerproduct by (/J,,LU) = gVfjLiUj (42) for fi = fiidx1, u> = u>idxz, and where glj are the entries in the matr ix inverse to that locally representing the metric, g = Qij dx1 <g> dxj. (43) The basis {tixij A ... A dxip ; 1 < ii < i2 <•••• < ip < n] defines /\PT*M. 19 The Hodge "star" operater n—p * : /\T:M - l\T*XM for 0 < p < d is defined uniquely by the requirement that it be linear and that *(ei A ... A ep) = ep+i A ... A en, (44) whenever { e i , e n } is a positive orthonormal basis of T*M and *(ei A ... A ep) = - e p + i A ... A en, (45) whenever { e i , e d } is a negative orthonormal basis of T*M. The definition of a positive basis comes from prescribing that a basis B be positive and defining that a basis B' be positive exactly if the change of basis matrix, A : B ^ B ' has positive determinant, d e t A > 0. The Hodge operator provides a reformulation of the inner product on p-forms: v (•,•) : l\T*M —> E {v, w) = *(w A *v) = *(v A *w). (46) To see that it truly is an innerproduct, realize first that it is bi-linear be-20 cause both the wedge and star are linear. Now, given an orthornomal basis {ei , ...,en} for the cotangent bundle, Bp := {eh A ... A eip ; 1 < i < n, ix < ... < ip} is a basis for f\pT*A4. Realize that product vanishes for distinct basis vec-tors, {eh A ... Aeip,ejl A ... Aejp) = (eh A ... A eip) A *(eh A ... A ejp) = eilA...AeipAejp+1A...Aejn = Sij * 1 since unless the vectors coincide the last wedge product wi l l have a repeated element and thus be zero. This fact and bi-linearity show reflexivity, posi-tive definiteness and symmetry and thus demonstrate that the given formula defines an innerproduct on the exterior algebra. From multilinear algebra we know that for a p-by-p matrix A Avi A ... A Avp = det A(vx A .... A vp) for any vectors u s l y u v £ T*M.. Consider in particular the case where p — n and {ui,ujn} constitutes a basis for the cotangent space (so p = d) and A is the change-of-basis matrix relating it to an orthonormal basis {ei , ...,ep} 21 as Au>i = e{. We can relate the n-fold exterior products of the bases vectors i n terms of a k ind of volume measure: eiA...Aed = Aco1 A .... A Acjn = (det A) ui A ... A un det(((Ji,u>j)) u>i A ... A uin since AAT = (uuUj). Observe the immediate consequence of the definition of the operator *: *(ei A .... A en) = 1 to write the above as Hence if uji = dx1 then _ u)xA...Au}n *1 = v /det((w i ,a ; i )) dx1 A ...Adxn V d e t ( 9 i j ) Recall that glj = g~} and again mind the rules for determinants to uncover 22 the so-called volume form *1 = y/gdx1 A ... Adxn, (47) defining the standard shorthand, := ^detgij. (48) A s such, for n c M vol(fi) := / ^Jgdx1 A ... A dxd = / *1, Jo. Jn where the integral of any continuous function / : Ml —> M. on the manifold is naturally defined by summing over the coordinate neighbourhoods of the system {(Ua,a);a € A } as / f*l:= f {foa-l)^dxa. (49) JUa Ja(UQ) The right-hand integral is well defined over the neighbourhood a(Ua) C MJ1 with volume element dxa = dx*...dx%. The extension to an arbitrary subset O on the manifold follows from the existence of a parti t ion of unity <j>a G C°°(M;R) by which is well defined since we insist that supp(0 a ) C Ua. 23 Note here the formula for the autocomposition of the Hodge operator, **: APT*M-+ APT*M, * * = ( _ I ) P ( " - P ) . (51) We know that for an orthonormal vectors { e i , e p } in T*M. *(ei A ... A ep) = ep+x A ... A en. such that the orthonormal basis { e i , e p , ep+i, en} is (defined to be) positive. A n d for the orthonormal vectors { e p + i , . . . , e n }, * ( e p + i A ... A en) — (det A) ex A ... A ep where A : T*M —> T ^ A 4 is the change-of-basis matr ix that reorders the basis as A : { e i , e p , e p + i , e n } i—> {ep+i, en> eii ••••> e p}-The sign of the determinant by definition determines whether { e p + 1 , e n , ex, is positive (with respect to the convention established by asserting { e i , e p , e. to be positive). If {e[, ...,e'n} is a negative basis—the image of the positive basis under a transformation wi th negative determinant—then i n general we have *(ei A ... A e'p) = - e p + 1 A ... A e n . 24 But reordering the wedge product exchanging the place of the p-form and the (n — p)-form gives ei A ... A ep A e p _ i A ... A en = ( - l ) p ( n ~ p ) e p + 1 A ...en Ae1A ... A ep, (52) by the standard formula for such rearrangements. A n d since *(ei A ... A ep A e p _ i A ... A e n ) = *(Aep+1 A ...Aen A Ae1 A ... A Aep) = det A (e p+i A ... A en A e\... A ep), we conclude that det A = (-l)P(n~p) and thus * ( e i A . . . A e p ) = (det A) e1 A ... A ep = ( - l f ( n - p ) e 1 A . . . A e which says ** = (—l)p(n p) as hoped. Now define the global L 2 - innerproduct of u e T*M as (yu,w) := / (fj,fu) * 1 J M and use the star formulation of the innerproduct to note that if p = n/2—so 25 * : /\PT*M -> f\pT*M— then {*(eh A ... A eip), *(eh A ... A ejp)) = *( * (e^ A ... A e i p ) A *(*( e i l A ... A ejf = *( * ( e i l A ... A eip) A (e^ A ... A e j p ) ) , having exploited the fact that ** is the identity i n this case, here again observe that if the basis vectors are distinct then the product vanishes. Contrariwise if vectors above are the same then *(eh A ... Ae i p ) A (eh A ... A eip) so (*{eh A ... Aeip),*(eh A . . . A e J ) Linearity extends this to (*v, *w) = (v, w). Likewise ( « , « , ) - / < ™ , ™ > . l = / < » , » > . ! = ( « . » ) , J M J M = ehA...Aein = (ehA...Aeip)A*(eilA...Aei) = *((e i l A ... A eip) A *(e< 1 A ... A e i p )) = (eh A ... Aeip,eh A ... Aeip). 26 so we can add as a corollary that * is an L 2 - isometry: II * V\\L2 := (*v, *v) = (v, v) = |H|z ,2. A Gernalized Domain for Covariant Derivatives and the Second Bianchi Identity While here we wi l l stick mostly to covariant derivatives oporating on vec-tor fields—sections of the tangent bundle—the operator as defined functions identically on the space of sections of any vector bundle (E, IT, Ai), where E is a vector space and ix its projection onto A4, T(E) := {s eC\M;E) ; 7ros = ldM}. (53) A n d hereafter we write QP(E) := T(E) x np(M), (54) (p < d im/vf) which agrees wi th our current definition of Vtp(TM) since Y(TM) = X{M). Now we wish to extend the covariant derivative to this space flp(E) of sections crossed wi th forms. Motivated by the requirement the the covariant derivative satisfy a Liebnitz product and expoiting the established exeritor 27 derivative for forms, take for X G T(E) and UJ G QP(M) that D : T(E) x W(M) -> x QP+1(M), D(X ®u) := DX ALU + X ®duj (55) wi th the understanding that (X <g> A u; 2 := AT <g) (tvi A u; 2), whenever X eF(E) and a>i, o;2 G flp(M). Furthermore, for distinct bundles Ei and E2 wi th assosiated covariant derivatives D\ and D 2 respectively, define the covariant derivative on E\ x E2 v ia D(X®Y) := DiX®Y + X®D2Y (56) whenever X E E1 and Y E E2. In particular this defines a covariant deriva-tive Z? on the space E®E*\ For X := ^ <g> w-7' G r(£ <g> £7*) = (d + A ) AT = dX + AiQviQxJ) = dX + ( A } ^ ^ <8> u>* - vk ® J). But the last term is just the Lie bracket of the connection wi th th section, so DX = dX + [A,X]. (57) 28 C a l l E <g> E* =: End-E, motivated by the fact that each u e E* is an endomorphism on the bundle E. Define further AdE := {T : E ^ E ; T is linear, T* = - T } (58) or A d i ? is the subset of End i? for which the endomorphism for each fibre is skew symmetric. From the previous section we know a connection A is skew symmetric in the sense that A(X) £ o so for D = d + A we have A e Q.\AdE). (59) Now view the curvature tensor, FD : T(E) -+ tf(E) = (T(E)y x tf(M), . as a two-form assuming values in E <g> J51*, F D 6 r(£)<8> (r(£))*®ft2(A4), 29 and apply the result to commute the connection of the curvarture: DFD = DFD + [A,F] = d2 + dA A A - A A dA + [A, dA + A A A] = dA A A - A A dA + A A dA - dA A A + [A, A A A] = [A, A A A] = [Aidx\ Ajdxj A Akdxk] = AiAjAk(dxi A dxj A dxk - dxj A dxk A dx*) = 0. Revealing the formula DFD = 0, (60) termed the Second Bianchi Identity, which wi l l prove usuful in our ini t ia l demonstration of the selfduality of the Yang-Mi l l s equations in four dimen-sions. 30 Chapter 3: The Yang-Mills Functional Define the here our central subject, the Yang-Mil ls functional, which is the norm of curvature over a manifold viewed as a function of connections wi th a given Lie structure. Physically, the Yang-Mil ls connections are stationary points of the field strength. This second chapter examines the functional and its cri t ical points, even-tually examining our main focus of the functional's absolute minimizers in four dimensions and the corresponding equations for the, the selfdual and anti-self dual equations, Fr> = *FD and Fp — — * Ft,-The Functional Defined and the Yang-Mills Connections Define an inner product for A, B G W(TM. \ g), where A = X ® u and B = Y ® v for X, Y e g and u, v G QP{M), v ia (A,B) (X®u,Y ®v) {X,Y)s(u,v)KPTiM (X, Y) * (u A *V). So in the cases outlined (the algebras o, u, and su), ( A B) txXY * (cu A *u). 31 A n d analogously to above the L2 scalar product is (A,B)L, := f (A,B)*1. JM Thus finally define the Yang-Mil ls functional as the L 2 - n o r m of the curvature— I -I2 == <-,->-as yM : Vtl(T*M\o) —>• E , yM(A) = [ \Fd+A\2 * 1 = / (Fd+A, Fd+A) * 1. (61) Sometimes (as below) it wi l l be easier to view this as a function of the covariant derivative rather than the connection, and when no ambiguity wi l l result we wi l l employ the same notation—yM(d + A) := yM{A). The objective is to choose a connection that is stationary wi th respect to this square energy. The traditional approach is variational. For covariant derivatives D and D G Q}(TM.;Q) and a vector field l e g consider FD+Wx = (D + tD)o(D + tD)X = D2X + tD(DX) +W + W ADX + t2D ADX = (FD + tDD + t2D AD)X, having employed the fact that D(DX) = (DD)X -DA DX. Now take the variational derivative of the functional in order to find the conditions of the 32 stationary points: 5yM(D) = -£| yM(D + W) dt d_ dl t = 0 J M 'M - 2JM(Db,F0).l, since (FD,tDD) and (tDD, FD) are the only first-order terms in the expan-sion of the scalar product. Thus setting 8yM.(D) = 0 yields the equation (DD, FD)L2 = 0, for all covariant derivatives D, for the functional's crit ical points. Given an arbitrary covariant derivative D : —> f ^ T ^ / v f ) introduce the operator D* : Q}(TXM) —> Q 0 , termed the dual covariant derivative to D, defined by the L2 relationship (D*X,Y)L2m = (X,DY)L2m, (62) for every X G ^(TM) and Y € ^ ( T T W ) , in order to rewrite the character-ization of the functional's stationary points as (D, D*FD)L2 = 0 33 for every D, or better, D*FD = 0. (63) Covariant derivatives satisfying this equation are called Yang-Mil l s covariant derivatives. (Likewise A is a Yang-Mil ls connection if d+A solves the above.) A Representation of Yang-Mills Connections Returning to the decomposed representation of the covariant derivative, D = d + A, express the connection in components as A — Aidx1, where A(X) = AidxyX), Ai e Ql(n). The connection acts on a vector field Y v i a exterior differentiation and exte-rior product wi th the connection, DY = (d + A)Y = dY + A ^ A Y. Utilise the skew symmetry of the connection to write the rewrite the dual covariant derivative simply i n terms of the adjoint to the exterior derivative: {X, DY) = ( X , dY + Aidx1 AY) = (d*X, Y) - (AX, dx* A Y). (64) 34 Now represent the curvature as F = Fijdx* A dx3' i n normal coordinates, i.e. the Kronecker delta represents the metric and the Ohristoffel symbols vanish, 9i j ~ ^ij > A n d here the one-form d*F is 8F-d*F = d^Fijdx1 A dx3) = -—^dx3, (65) ax 1 understanding that summation is taken over i as well as j despite the indices residing "on the same level"—i.e. both being formally contravariant. Substitute this formula into the representation of the 1? innerproduct involving the covariant derivative in terms of its canonical decomposition to get (F, DY) = (d*F, Y) — (AiF, dx* A Y) 8F = ( - -g^dxj, Y) - (AkFijdx1 A dxj, dxk A Y) = ( - ^ d x i , Y) - (MFy - F^dx3, y ) = (- ^dx>\ Y) - {[A, dx3, y ) , 35 where having executed the summation in normal coordinates eliminated the fc-index and reduced the right-hand innerproduct to agree wi th the right-hand one i n domain and here the Lie bracket denotes exactly the symmetricness of the curvature as \A-i, F%j ] = A i Fij Fij Aj. So and A is a Yang-Mil l s connection if ^ + [ ^ , 1 = 0 (66) for each j — 1 , n (of course F^ corresponds to A as FJ+A = F^dx1 A dx1*). The Example of Two Dimensions If n = dim M. = 2 then the orthogonal group is Abel ian, if A, Be o(2, E ) then AB = BA. The determinant of an orthogonal matr ix is always plus or minus one, I det 4^1 = 1 36 for al l A 6 o(n) since, by definition, AA = L so and det AAT = det A det AT = det I = 1 det A = det A implies (det A)2 = 1 which gives the desired fact. Thus, in two dimensions, where, A'1 = 1 det A Cl22 — t l i2 - a 2 1 a n the equation A 1 = AT implies V a22 —^12 - o 2 i a n an a 2 i = ± —a 1 2 a 2 2 37 which says every A 6 o(2) has either the form \ s t -t s or V t -s Having established this there just for cases to compute directly to see that the group is Abel ian. Of course that the Lie bracket vanishes identically is an immediate and t r iv ia l consequence, AB - BA = AB - AB = 0. We call such a Lie algebra t r ivial . A n d thus the skew-symmetric bundle Ad(TM) C TM. x T*M is also tr ivial , which means i t is isomorphic to the direct product of the manifold wi th the real numbers, we say (writing equality for short) A d ( T A l ) = M x R . In this representation the covariant derivative coincides wi th the exterior derivative, D = d (67) or the connection vanishes locally—its derivatives do not vanish, since the representation Da = d + Aa depends on the coordinate system {(Ua,a);a £ A} (mind that this set of 38 charts A is unrelated to the connection A), and this decomposition is not i n general global. In this setting the fundamental equations al l simplify greatly. A A A = 0 reduces the curvature to Fd+A = dA + A/\A = dA (68) and thus here the Bianchi identity follows immediately from the fact that the autocomposition of the exterior derivative vanishes identically DFD = dFD = d(dA) = d2A = 0 (69) Lastly, the local absence the connection reduces dual covariant derivative reduces to the adjoint exterior derivative, D* = d* so the Yang-Mil ls equations read <TF = 0. Employing above identity for the curvature expands this to d*dA = 0. This low-dimensional context unifies Yang-Mil l s theory wi th the study of 39 harmonic forms since the Laplacian on forms A : Clp(M) —» QP(M) is defined as A := dd* + d*d. (70) So tr ivial ly the curvature is harmonic in this context, AFD = 0. Furthermore, if, without motivation, we assume the so-called gauge condi-tion, d*A = 0, (71) and we have immediately dd*A = 0. Hence the the above sequence of equations shows that the connection A is also automatically harmonic A A = (dd* + d*d)A = dd*A + d*dA = 0. A Preliminary Fact Recall Stokes's Theorem for forms, for any smooth (n — l )-form u> wi th 40 compact support, / du * 1 = (p u * 1, JM JdM (the Hodge operators of course correspond the respective cotangent bundles, T*M and T*dM, to generate the appropriate volume elements) and, more-over, since we are considering a manifold without boundary, dM — 0, du * 1 = 0. M Now, for a € A P 1 T*M a n d P e /\P T*M, apply the formula to the (n - 1)-form that comes from taking the wedge product of a (p — l)-form a and (n — p)-form */3: / d(a A *B) * 1 = 0. J M Mind ing the product rule for forms and keeping i n mind that since d * 3 e **(d*p) = (-l)(p-1){n-p+1)d*p, we can compute d (aA*/3 ) = d a A * / 3 + ( - l ) p _ 1 a A d * / ? = d a A * / ? + ( - l ) p - 1 ( - l ) ( p - 1 ) ( n - p + 1 ) a A * * d * / ? . The exponent expands to n(p — l)—p2 + 3p — 2 and because p and p2 always 41 have the same parity, we can cancel most of the terms and write ( _-gp- l ( _1^ (p- l ) ( r c-p+l ) _ The autocomposition of the Hodge star on the space of n-forms is the identity (assuming a positive basis as throughout), so carry on by employing this along wi th just linearity and the definition of the innerproduct daA*(3 + (-l)n(p-1)aA**d*P = * * (da A */3 + ( - l ) n ( p - 1 } a A * * d * (3) = *(*(daA*p) + (-l)n(p-1)*(aA**d* = *((da,(3) + ( - l ) ' l ( p - 1 } (a, *d * /? ) ) . Bu t by Stokes's Theorem we can say that this integrates to zero, f d(aA*(3)*l= [ *((da,/3) - (a,(-l)n(p-1)+1*d*f3)) * 1 = 0. JM JM Since this holds for any (p — l )-form a and p-form (3, the integrand must be zero and the above is in fact a statement about the relationship between the exterior derivative and the Hodge star, (da, (3) + ( - l ) ^ - 1 ) (a, *d * (3) = 0 or (da, (3) = (a, ( - 1 ) " ( P - 1 ) + 1 * d * (3) , 42 which reads like the definition of the adjoint exterior derivative and says exactly that d* = ( _ I ) » ( P - I ) + I * d * . (72) The result extends to covariant derivatives. The Hodge operator acts on elements of X <g> u G flp(TM), wi th X G X and u £ f\P(T*M), as *(X'<g)a;) = X <2> (73) which is to say it is defined to act normally on the form but leave the vector field alone. In contrast recall that the image of connection A belongs to the orthogonal group, A(X) G o(n) for each X G TM, in the sense that for A = Aidx1 the map Ai : TXM TXM is skewTsymeetric. Thus the matrix Ai acts only on the vector field but leaves the form alone, Ai(X ®u) = AiX ®u. (74) A s such the operators commute, *Ai = Ai folloing from the simple compu-43 tation, (*Ai)(X®u) = *(AiX®u}) = AiX <S> *OJ = Ai(X <g> *UJ) = (Ai*)(X®oj) and i n particular, again since * * = (—1)P(™-P) ) o r (_I)P("-P)+I * * = Ai = * *Ai = *Ai*. Return to the formula for the dual covariant derivative D*, (D*X, Y) = (d*X, Y) - (AiX, dxi A Y) (75) and manipulate to get (D*X,Y) = (-l)n(?-V+1(*d*X,Y) - ( - l ) p ( " - p ) + 1 ( *A * X,dxi AY). Now suppose that the manifold is of even dimension and look for the dual covariant derivative of a form of even order—that is n and p above are even. In the next, especially pertinent section, we wi l l be narrow our focus to the situation when n = 4 and want to apply D* to the curvature, which is a 44 two-form. So, to resume, write (D*X, Y) = (-*d*X,Y) + (-*Ai*X, dx1 A Y) (76) and we can say D*X = - *(d + A)* = - * £ > * , (77) as long as we understand that for X <g> u G VLP(TM) such that X G X and u) G JT2P(A4) here the connection A applies to the "form part" u> v ia contraction, not multiplication, that is to say exactly that (A(X ®u),Y) = (AiX ® u, dx* A Y) for every Y G flp~1(TA4); that the dual covariant derivative must decrease the order of the form part by one motivates this view. Yang-Mills in Four Dimensions: The Selfdual Equations Recall that in four dimensions the Hodge star is an L2 isometry on the space of two-forms, 2 2 * : /\T*M - » f\T*M. Given normal coodinates about x such that {dx1, dx2, dx3, dx4} describes 45 a basis for the fibre, define A + := span-fob;1 A dx2 + dx3 A dx4, dx1 A dx4 + dx2 A dx3, dx1 A dx3 - dx2 A dx4} and A " := spanjdx 1 A d x 3 + dx2 A dx 4 , d x 1 A dx2 - dx3 A dx4, dx1 A dx4 - dx2 A dx3}, and realize that since the six vectors described above are independent and /\2 T*M. has dimension six, A^™)-(-r>).(3-, in our case of two-forms. Thus: 2 A + © A - = /\T*M. (78) The division of the fibre into the prescribed subspaces speaks to the Hodge star. Computat ion minding the behaviour of the operator reveals *a = ±a (79) for a € A + or a E A~—that is to say that the decomposition is into the eigenspaces of * corresponding to the eigenvalues 1 and —1. Interestingly, this splitting corresponds to the splitt ing of the special or-46 thogonal group in the sense of isomophism A + = A ~ = so(3) and 2 f\T*M= 50(4)= 5o (3 )©so (3 ) . Here we wi l l turn to covariant derivatives D wi th curvature tensors FJJ = *FD G A + called selfdual and FD — — * FD G A - called anti-self dual. Fur-thermore, call a connection inducing selfdual curvature an instanton and one inducing anti-selfdual curvature an anti-instanton. The Bianchi formula DFD = 0 implies that D*F, D 0 (80) for FD G A + or FD G A " . A n d hence *D*FD = 0. (81) Bu t the main result of the previous section says that * D * = -D* 47 in this case, and we can now say D*FD = 0. (82) Or all instantons and anti-instantons are Yang-Mills connections. Cohomology and Chern Classes Our nicest examples of Yang-Mi l l s theory w i l l pertain to the covariant derivatives on the compact subgroups U(m) and SU(ra) of the complex gen-eral linear group G L ( r a , C ) , so return to the abstract setting define it on an arbitrary vector bundle E wi th rank m. In this context we wi l l be able to rewrite the Yang-Mil l s functional elegantly and read off its minimizers. To begin we need the concept of Chern classes and thus of cohomology. We say that two p-forms, a, P G Clp(M), are cohomologeous if their difference is exact, that is, there exists a (p — l)-form, 7 G ilp~1(A4), such that a — (3 = 0 7 . This cohomology relation is equivalence relation that partitions the space {a G QP(A4) ; da = 0} of closed forms in QP(A4). The set of all equivalence classes, [a] := {P G ttp(M) ; a - R is exact, dp = 0}, 48 itself defines a vector space, HP(M) := {[a] ; a £ Q, da = 0}, (83) called the p-th de Rham cohomology group. The Chern classes are such equivalence classes belonging to such a group that depend, for our definition, on the elementary symmetric polynomials, ^ ( A x , . . . , A m ) : = Yl K-Kj, (84) l<a\<...<aj<m or, more precisely, on the matr ix polynomials, Pj : M(m, C ) -> C where for B £ M(m, C ) Pj(B):=pl(X1,...,Xm) (85) and A i , A m £ C are the (ordered) eigenvalues of B. We have the essential property that the polynomials represent i n the expansion over the product of a first-order monomials, for t £ C , m n(*-^ )=p**m"j'. 3 = 1 49 having employed the shorthand p> := p*(\i, A m ) . The identity looks the same carried over to the matr ix case, Y[(t- Xj) = Pj(B)tm~j. (86) 3=1 But here realize that of course m H(t - Xj) = 0 exactly when t is an eigenvalue of B. But for t to be an eigenvalue by definition Ba = ta, for each a E Cm which reformulates into the familiar equation de t (B - tld) = 0. (87) Since these polynomials have the same roots and both have leading coefficient one they must be equal, Pj(B)tm-j = det(B - tld), (88) and we have a tool, the determinant, wi th which to compute elementary symmetric polynomials. These polynomials are homogenous wi th degree j and thus take map 50 p-forms to jp-iorms. In particular we here consider the curvature as a homo-morphism, FD G AdE = EndE = Romc{E; E) = M(m, C ) , so, Pj : Q2(M) tt2j{M), Pj{FD) e fl2i(M). (89) These polynomials are i n fact exact, dPj{FD) = 0 (90) and further are independent of the covariant derivative 5 , P>(FDl) = P>(FD2) (91) for any covariant derivatives Dx, D2 : 0 —> £lJ(E). So, independently of the covariant derivative—F := FD for arbitrary D—we can define the elements of the 2j?-th cohomology group, G /Y^ ' (A4) , (92) 5See Jost, pages 125-126 for a proof. ct(E) := 51 called the Chern classes of the bundle E. Computation of Chern Classes Now, to compute the classes, exploit the formula d e t ( ^ F D + fId)=|>(^)r-or, i n terms the curvature of an arbitrary covariant derivative, ] T Cj {E)tm~i det \T^FD + tld) . (93) 3=0 To simplify, divide by tm and remember that the determinant is m-homogeneous, -• m m - _ > ( £ ) t " ^ = 5>,(£)r^ 3=0 3=0 1 / i — det — FD + tld tm V 2 T T = det ( ^ - F D + Id Realize that the curvature's eigenvalues are two-forms since FD &° —> fl2(E). Aj is an eigenvalue of FD when for X £ T(E), FD(X) = AjX G n2{E). (94) 52 This extends naturally since the roots t £ C and leading coefficient of the equation n tid-iM=° ( 9 5 ) again coincide wi th those of the determinant above. Thus, replacing Id with 1 i n the product to agree wi th the formalism, det(^+M)=n((-iA0- (96) Moreover we w i l l exploit the fact from linear algebra that the trace is the sum of the eigenvalues, m tvB = £ A , to compute the Chern classes v ia the expression E^=n ( i -^ A i ) . <w 3=0 3=1 V where we have substituted r := 1/t. V i z , for m = rank(E) = 1, = cQ(E)+c1(E)r 3=0 5 3 But the zeroth-order symmetric polynomial is one, here c0(E) = 1. Thus since al l other terms above cancel. Furthermore, since wi th rank(.E') = 1, B : E —> E, and hence likewise FD : Q° —> tt2(E), has only (exactly) one eigenvalue must be equal to its trace, tiFD = Ai, or, moreover, Ci(E) = ±trF. ( 98 ) To find c2(E) consider the case in which m = 2 so Co(E) + Cl(E)r + c2(E)r2 = (l + ^ r ^ l + ^ A 2 r i 1 = 1 + ; r - ( A i + A 2 ) r — — A i A A2r2. LIT 4 7 T A n d simply matching coefficients gives and C l ( £ ) = ^ ( A i + A 2 ) = ^ t r F , c2(E) = —~Ai A A 2 . 4 7 T 54 Jost gives the general formula for the second Chern class of tangent bundle E wi th rank m as 6 nrn — 1 1 c2(E) - — Cl(E) A Cl(E) = — t r F 0 A F0, (99) Am, 07TZ for the "trace free part" of F, FQ := F — —trF • ldE. m But for our discussion focus on two examples, the cases when D is a u ( l ) -covariant derivative for i l lustration and, more importantly, when D is a su(2)-covariant derivative. 6 See page 127. 55 Examples and the Topological Charge Firstly, i n the case when E is a complex line bundle wi th structure group U ( l ) the curvature is just a two-form, Fd+A = dA=:f, for an arbitrary u(l)-connection A. This is analogous wi th the two-dimentional real case discussed already. Thus the trace-eigenvalue approach is here very simple, or alternatively, we get the first Chern class tr ivial ly from the deter-minant definition, c0(E) + Cl(E)r = det ( £-fr + Id j = 1 + ^-fr and thus c i ( £ ) = (ioo) There are only two Chern classes here (the above and c0(E) = 1) so this wholly defines the topological structure of the bundle E. Moreover, secondly let E have structure group U(2) so that the curvature, FD : n°(E;su(2)) tf(E;su(2)), 56 is represented by a matr ix of two-forms FD = (ft n ^ ft fi e su(2) (101) where, for j, k = 1, 2, fi e W(M), and trFD = ft + f\ = 0 as required in the definition of S U . Thus right away the first Chern class vanishes, ci(£7) = trF = 0. A n d now in this formulation we can compute the second Chern class from the determinant, CQ(E) + CL(E)r + c2(E)r2 = det ( Id + — FDr 2"7T = det / rJtr + 1 i-J\r \ 2ir fir Sh + 1 4TT2 4TT2 (ft^f22-f^f?)r2 + ^(ft + f22)T + l ( / i A ^ - z j A / y + i , 57 since trFD = 0. Clearly now we can write the second Chern class, = " ^ ( / l A / l - ^ A / 2 ) (102) 4TT 2 ^tr(FDAFD). A n d we have defined the topological structure of E7. Integrate over the second Chern class over manifold to obtain the second Chern number known as the topological charge and written, —k = -k(M,su(2)) := -c2(E)[M] = ---- f t r ( F A F) * 1 (103) (In fact this is a constant over the fundamental class [Jvi] of oriented, four-dimensional, compact manifolds.) A First Lower Bound for the Yang-Mills Functional i Now recall that on su the ad-invariant innerproduct is given by minus the trace, and the specified innerproduct on Q2(E;su) is, for X <&u>, Y <g> v € tt2(E;su), wi th X, Y e V(E) and u, v £ Q.2(M), (X <8) Y ® ^ ) n 2 ( E ; s u ) = - t r ( X y ) w A *v, 7Freed and Uhlenbeck say, "The characteristic class [c2{E)\ classifies SU(2) bundles over compact 4-manifolds, but this classification fails in higher dimensions." See page 33 for references. 58 so that / (F, *F)Q2(E;SU) = - [ t r ( F A F) * 1 = -8n2k. (104) JM JM Recall that the Hodge operator is an L 2 -isometry, so (F,F) = {*F1*F), and (dropping the implied su-subscript) employ this to reformulate the Yang-Mi l l s functional v ia yM(D) = [ (FD,FD}*1 JM = [{FD,FD) + (*FD,*FD)>j*l = I f ({FD ~ *FD, FD - *FD)) *1+ f (FD, *FD) * 1 2 JM V ' JM 1 / \ t D - *rD z ->M FD - *FD\2 * 1 - 8 T T 2 A ; . Since the square is positive we can bound the functional from below by the topological charge, yM(D) > -8n2k for every su(2)-covariant derivative D. Remember we have said nothing about the sign of —A; and since the functional is a norm write yM > max{-87r 2A;, 0}. (105) 59 Instantons as Absolute Minima No matter the covariant derivative write the curvature into its selfdual and anti-selfdual components as Fd+A = F = F+ + F~ for F+ G A + and F~ G A " . From here express the Yang-Mil ls functional as yM(A)= [ \F\2*1 = f ( F , F ) * 1 JM JM = / (F++ F~,F++ F~) *l JM = f ((F+,F+) + 2(F+,F-) + (F-,F-))*l JM V ' - f ( ( F \ F + ) + (F-,F-))*l= f ( | F + | 2 + | F -JM v ' JM since (F+,F~) — 0 as a result of the orthogonality of A + and A ~ . M i n d that, by assumption, F+ = *F+ and F~ = - * F~, 60 and break down the second Chern class: 8n2c2(E) = t r ( F A F ) = t r ( ( F + + F~) A (F+ + F~)) = t r ( F + A F+ + F~ A F _ ) = t r ( F + A F + ) + t r ( F _ A F~) = t r ( F + A * F + ) - t r ( F ~ A *F~) = - | F + | 2 + | F - | 2 , since the cross terms F+ A F _ and F~ A F+ cancel each other out and (of course) the trace is linear. Integrating returns us to the topological charge and looks like 8 7 r 2 f c = f ( | F + | 2 - | F - | 2 ) * 1 , (106) JM which looks remarkably like our current expression for the Yang-Mil l s functional-it differs only i n the sign of one term. Comparing the two we see / ( | F + | 2 + | F - | 2 ) * 1 > / ( | F + | 2 - | F - | 2 ) * 1 JM JM which implies yM > 8ir2\k\ (107) and strengthens our bound form the previous section. We see that equality is attained, yM(D) = 87i2\k\, 61 i.e. the Yang-Mil l s functional has an absolute minimum, exactly when either the self dual or the anti-self dual part of the curvature vanishes, F+ = 0 or F~ = 0, which is to say that the covariant derivative D is an instanton or anti-instanton. A n d which one depends on the sign of (minus) the Chern number k. Minimize the difference yM — 8ir2k, firstly for k > 0: yM - I ( | F + | 2 - | F - | 2 ) * 1 = / ( | F + | 2 + | F - | 2 ) * 1 - / [\F+\2-\F JM JM JM = 2 / | F - | 2 * 1 , IM which is minimized at zero when FD — F+ £ A+ is selfdual, D is an instan-ton, by definition. Likewise, if k < 0 corresponds to anti-selfdual curvature, yM - f ( | F + | 2 - | F - | 2 ) * 1 = / (\F+\2 + \F-\2)*l+ [ ( | F + | 2 - | F JM JM JM = 2 / | F + | 2 * 1 , ' M which again is minimized at zero, this time when FD = F~ £ A ~ and D is an anti-instanton. We have not, remember, discussed the existence of such solutions to the self-dual equations. Whi le Cliff Taubes has constructed examples of solu-tions, there is no theoretical guarantee that the topological minimum wi l l be 62 attained for any covariant derivative. O n the other hand, in some settings, for example in the case of line bundles over the two-spheres S 2 x S 2 8 there exist covariant derivatives that are neither instantons or anti-instantons but minimize the Yang-Mil ls functional by assuming the bound. According to Freed and Uhlenbeck, page 37. 63 Chapter 4 : The Convex Selfdual Framework I hope that the resurgence of convex methods recently incited by Ghous-soub wi th lead to facile investigation of the absolute minima of the Yang-Mil ls functional in four dimensions because (in some contexts) basic analysis wi l l displace geometric, topological, and algebraic methods and alleviate the chal-lenge of working at the intersection of these fields. The selfdaul Yang-Mil ls equations are poised to become among the canonical of examples of classical equations reinterpreted as representatives from more general classes that wi l l provide diverse extensions of and offshoots from known work. The Convex Setup: The Legendre Transform, Subdifferential, and Fenchel Inequality Firs t ly define the Legendre transform (p* : X* -> R U {+00} of convex, lower semicontinuous functional op : X —• MUJ+oo} defined on Banach space X, ip\y) := sup{(x, y) - <p(x) ; x G X}. (108) Directly from the definition of the supremum, write simply <p(x) + <P*(y)>(x,y) (109) for every x G X and y G X*, called the Fenchel Inequality. Note here 64 that (p* w i l l itself always be convex, which direct manipulation of convex combinations reveals. Now define independently the subdifferential of ip, do? : X —> 2X , as the set-valued map dcp : x {y G X* ; ip(z) > <p(x) + (z - x, y) for al l z G X}. (110) Here realize that if <p = f is a smooth function on X = X* — W1 then the subdifferential reduces to the singleton of the gradient, df = { V / } , by supposing for every x G K n , and in particular x = x 0 + /ie, for h G R and standard orthonormal basis vectors ei, ...,en, so /(x) > / ( X Q ) + (x - X Q ) • y / (x 0 + hei) > /(x 0) + he{ • y and thus l im /(x 0 + hei) - / ( X Q ) h < ej • y < l im / i -»0+ / (x 0 + fee.) - / ( X Q ) or, component-wise 65 So the subdifferential contains at most one element. y = ® ( X o ) " - l : ( x o ) ) = v / ( X o ) ' which is of course in fact in the set as the tangent plane at any point x0 lies below the a convex function: /(x0) + V/(xo).(x-x 0)</(x). This intuit ion from finite-dimensional calculus contextualizes the subd-ifferential as a derivative that does not depend explicitly on a l imi t and as such freely applies to functionals at their points of nonsmoothness, taking on multiple values i n this situation. The connection between the derivative wi th the Legendre Transform is the crux of the elementary selfdual theory, and comes when the Fenchel Inequality is attained: ip(x) + tp*(y) = (x,y) if and only if y G dtp(x). ( H I ) Tr iv ia l manipulation of the supremum proves the implication, (x, V) - <p(x) = sup{(x, y) - <p(x)}. > (z, y) - <p(z) for every z G X, which reads y G dip(x), and the converse is only one line 66 longer: If for every z € X • <f(z) > <p(x) + (z- x,y) then (x,y) - (p(x) > (z,y) - <p(z). Since the inequality holds for all z, take the supremum of the right-hand side (x,y) - (p(x) = sup{(z,y) - <p(z) ; z e X} = tp*(y). Now suppose further that the space is reflexive, X** = X, so that </?** X -> R , and (p**(x) = sup {(x,y)-<p*(y)} yex* = sup { (x,y) - sup{(z,y) - <p(z)}} yex* zex < <p(x), since (x,y) - sup{(z,y) - ip(z) ; z £ X} < <p(x). That is, ip** < (p. When the functional is convex in fact the reverse inequality holds, ip > <p** 67 so we have equality: ^ = tp. (112) A p p l y this identity in the Fenchel inequality to recover its dual equivalent in terms of the subdifferential: tp**(x) + <p'(y) = (p(x) + <p*(y) = (x, y), automatically gives the thi rd equivalent proposition, thus y G dip(x) if and only if x G dip*(y). (113) remembering of course that this is also equivalent to attainment in the Fenchel inequality. 68 The Basic Example for our Application Now, poised to address the theory of Langragians on X x X* that con-stitutes the essence of the chapter, turn firstly to a concrete example that is illustrative here and to be fruitful in the sequel. If ip(x) = ^\x\p then <p*(y) = sup j (a;, j/) - ^\x\p ; x G A"| is calculable variationally. A t maximizing x, (x,y) - -\x\p < (x + tw,y) - -\x + tw\p P P for any w £ X and every i e R . Thus one-dimensional calculus gives d_ dt x + tw, y) - -\x + tw\p } = 0, which implies {w, y) — \x + tw\p 2 (x + tw, w) = 0 t=o or Thus (w, y) — \x\p 2 (x, w) = 0. x\p 2x = y 69 and \x\ = provided the found functional does belong to the dual space. Perhaps note quickly that for a function / E X = L p ( R n ) , the maximizer in fact is a member of the dual, \f\p~2f E X* = Lq(Rn), where q is the exponent dual to p, i.e. i + I = l , as i / r 2 / | / | p < o o . Returning to the general case, plugging i n for x yields the Legendre transform <p* to be (\\• f)\y) = M ^ + 1 --\v\& = (i - -) \v\& = -\p J P \ pj q y\ having noted that here the inner product does not complicate computations and reduces simply by first inserting x for y, (x,y) = (x, \x\p~2x) = \x\p = \y\p-5. In this context the Fenchel Inequality reads ^\x\p + ±\y\i>{x,y), whenever x E X, y E X*. 70 Lagrangians C a l l a functional L : X x X* —> R U {+00} a Lagrangian i f it is lower semicontinuous and convex in both variables and not identically +00. Star denotes the Legendre transform in both variables, L*(q,y) = sup{(q,x) + (y,p) - L(x,p) ; x £ X,p £ X*}. (114) If L*(p,x) = L(x,p) for all (p,x) £ X* x X (115) then the Lagragian L is called anti-selfdual. The first (general) example of such a functional is the sum of a convex functional on X and its Legendre transform on X*: L{x,p) = <p{x) + <p*(p). (116) Convexity and lower-semicontinuity follow immediately from the same as-sumptions on <p and consequently <p*, namely for every fixed p, <p(x) > (x,p) and that the sum of lower-semicontinuous functions is lower-semicontinuous. Computat ion of the Legendre transform is simple since the variables separate: L*(q,y) = sup{(q,x) + (y,p) - <p(x) - cp*(p) ; x £ X,p £ X*} = sup{(g,x) - tp(x) ; x £ X} + sup{(?/,p) - <p*(p) ; p £ X*} = <p\q) + H?\v) = f(y) + = L(y,q), 71 where the linearity of X, X*, and (•, •) allowed us to reposition the minus signs in . the arguments of the supremum operators. Now, remarkably, the functional I : X —> M , defined by I(x) := L(x, Ax) - {x, Ax) = ip(x) + tp*(Ax) - (x, Ax), (117) for any linear A : X —• X* (which cleary preserves the necessary convexity), satisfies in f{ / (x ) ;x G X} = 0 = I(x), (118) again supposing that (p is coercive to ensure the existence of such a x G X. Recognize this minimum from the main inequality—I(x) — 0 when <p{x) + ip*(Ax) = (x,Ax) (119) or, moreover, Ax G d<p(x). (120) Thus, replacing the second-order Euler-Lagrange equations for crit ical points, self-duality gives a first-order (in a convex sense) equation to find the mini-mizer. An Example from Partial Differential Equations To reify the overarching setting and demonstrate the power of this case in 72 particular, consider one of the simplest examples from Ghoussoub's series of papers on the subject, the non-symmetric Dirichlet problem: Given functions / : Q ->• R and smooth a : Q R n , for 0 C R n bounded, = - Au + \u\p-2u + f on fi, (121) M = 0 on dfl, for u G i fo (^ ) - To reformulate this equation as a selfdual, convex minimiza-t ion problem construct the functional, :=4 / IV^I2 + - / \u\p + f fu, (122) 1 Jn P Jn Jn whose form is motivated by the fact that its Frechet derivative is equal to the right hand side of the above sample application integrated. Where for u G X, the Frechet derivative at u is the solution of |*(» + to)-»(») + OT(u)| 3 IMU i f the l imit exists and agrees for every v G X. Since the norm in the numerator is just absolute value, we can rewrite this explicitly as, r t T / , ,. V(u + tv) - + DV(u) , DV(u) = hm —- '- ^-L (124) 73 where v 6 X is any vector wi th norm one. For completeness write explicitly the Lagrangian L : H£(Q) x # - 1 ( f 2 ) -+ R , as L(u,-u) = * (u) + and realize the interesting and essential quality that throughout the following the Legendre conjugate <3/* remains uncalculated. B u i l d further the operator A : HQ(Q) —• LQ(Q,), A(u) = a • V u = aiQ~T' i=l 1 and now finally the functional I : HQ(Q) —> R , /(it) = Ait) — (it, Au), or, replacing A and L by their definitions and integrating the product by parts, I(u) := #(u) + * * ( a « V i i ) + / div(a) |u | 2 . Convexity and lower-semicontinuity follow immediately from their definitions and restrict the exponent so that p > 1 to ensure a growth condition sufficient for the attainment of the infimum. Hence look for u such that I(u) = i n f { / ( « ) ; u E H^(0)} = 0, (125) or, moreover, a • \7u G d^(u). 74 Relying on the established proof of the analogous fact i n finite dimensions, realize that if a functional I has Frechet derivative DI at u 6 X, then dl(u) = {DI(u)}. Thus the calculation of the one-dimensional l imit , bearing i n mind the zero boundary condition imposed by HQ (Q) when integrating by parts, yields the desired equation: a • Vu = Au + \u\p~2u + /. 75 The Link with the Instantons This final section, which indicates how the first and second chapters should be unified, wi l l unfortunately be both cursory and formal. Contrari-wise it be viewed positively how easy and powerful the newer, analytic for-malism is and also how, once resolved in the present context it should extend to other st i l l richer ones. Namely the generalization of Yang-Mil ls theory to address p-energies. Ghoussoub has extended the theory as outlined to minimize functionals I : X —>R of the form I(x) := L(Ax, Tx) - (Ax, Tx) (126) for selfdual Lagrangian L : X x X* —>• R where either one of the two operators A : X —> X or r : X -»• X*, is nonlinear. Here, / > 0 and the functional attains its minimum. We care about the basic case when, 76 for a convex functional \& : X —• R , L ( x , p ) = * ( x ) + tf*(p). In this case we obtain / ( x ) = * ( A x ) + ^*(Tx) - (Ax, l x ) and the minimizer x such that I(x) = 0 satisfies Tx E 9 * (Ax) (127) as a result of the key relationship between the subdifferential and the Fenchel inequality. Now reassign notation to match the geometric setting and (formally) apply this result. O n the second special unitary group, the Yang-Mil ls func-tional aligns wi th this setting from completing the square as explained at the end of the second chapter. yM(A) = [ \Fd+A\2*l (128) JM = \ f \Fd+A\\l + \ [ \*Fd+A\\l (129) 2 J M ZJM 77 Now define replace the general Banach space wi th the two-forms which con-stitute a Hilbert space and thus their own dual space by the Riesz theorem. Label the elements of the convex theory explicitly for a vector bundle E over the manifold Ml: Begin wi th the convex functional, which here takes the form of a second power, * : ft2(Ad£;su(2)) -> E , 1 \ I2 2 1 1 and we already know it is its own Legendre transform, = = I| • | 2 : Q 2 (Ad£;su (2)) R . Now, on the underlying algebra, define the operators A , T : su(2) -»• tt2 (AdE; su(2)) whose actions are defined by generating the curvature and its Hodge star, A : A H > F d + A = dA + A A A and r : A n . 78 Thus yM{A) = [ tf(AA) * 1 + / (IM) * 1. (130) JM JM But since the inner product of the curvature wi th its Hodge star is constant, (FD, *FD) = -8ir2k so the Yang-Mil l s functional has the same minimizers as the functional IyM(A) := yM(A) + 87i2k (131) = f * ( A A ) * 1 + f y*(TA)*l- f < A A , r A ) * l , JM JM JM which fits in wi th the theory of self dual Lagrangians. Thus we have trans-ported the Yang-Mil ls functional into a partial differential equations context where hopefully if can be addressed wi th simple convex techniques and the more direct language of real analysis. Since 0 t t ( x ) = 0 Q | a ; | 2 ) ={X}, the equation TA e d^(AA) for the minimizer A such that IyM(A) = 0, 79 becomes TA € {AA}, or, rather, replacing the operators by their values, *Fd+A e {Fd+A}. This is t r ivial ly equivalent to the pair of equation, Fd+A = *Fd+A, which, of course, is the selfdual Yang-Mil l s equation requiring that the curva-ture Fd+A be selfdual thus that the covariant derivative d+A be an instanton. To recover the anti-selfdual equation, simply exploit that | * Fd+A\2 = | — *Fd+A\2 and write the Yang-Mil l s functional as yM(A) = lf \ F d + A \ 2 * l + I f \ - * F d + A \ 2 * l . (132) Z JM Z JM Thus the above programme applies identically wi th the operator T replaced by r' : A ^ - * F d + A and al l else identical. This gives way to F'A e {AA} 80 or - * Fd+A G {Fd+A}, which, exactly in parallel wi th the selfdual case above, is exactly the equation Fd+A = — * Fd+A, the anit-self dual equation for an anti-instanton d + A. The Future There is a dense literature surrounding instantons and their role in gauge theory—we did not talk about gauge groups, but know that al l Yang-Mil ls connections have gauge equivalent curvatures—and the insights they provide into topology. Alas, we have hardly touched on the real geometry of four-dimensional manifolds. We have seen, though, the facile application of a new and surprisingly easy theory i n our complex setting. As a suggestion for further investigation then, focus on exploiting this framework and realize that here we can naturally extend the theory to p-energies: yMp(A):=- f \Fd+A\p*l + - [ \ * F d + A \ q * l - [ (Fd+A,*Fd+A)*l, VJM QJM JM (133) where 1/p + 1/q = 1. B y the same argument as in the p = 2 case (yjvi2 = 81 yM), but wi th the functionals defined as * = -|-r, and * * = -|-T we get the subdifferential, 3*(s) = aQ|*r) = { I IP" x\x\ Thus the minimum of yMp is attained at A when *F<I+A — FD+A\FD+A P . 82 References Freed, D . , and K . Uhlenbeck, 1991, Instantons and Four-Manifolds, Second Ed., Springer-Verlag, New York. N . Ghoussoub: "Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions", AIHP-Ana lyse non linaire (In press) 35 pp N . Ghoussoub: 2005, " A class of self-dual differential equations and its vari-ational principles". Jaffe, A . , and C . Taubes, 1980, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhaiiser, Boston. Jost, J . , 2005, Riemannian Geometry and Geometric Maps, Fourth Ed., Springer, Heidelberg. Urakawa, H . , 1990, The Calculus of Variations and Harmonic Maps, A M S Translations of Mathematical Mongraphs, Providence. K . Wehrheim: "Anti-self-dual instantons wi th Lagrangian boundary con-ditions I: El l ip t ic theory", Comm.Math .Phys . 254 (2005), n o . l , 45-89. 83 Wehrheim, K . , 2004, Uhlenbeck Compactness, E M S Series of Lectures Mathematics. 84 

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