UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-ubc_2007-0084a.pdf [ 1.97MB ]
JSON: 831-1.0080450.json
JSON-LD: 831-1.0080450-ld.json
RDF/XML (Pretty): 831-1.0080450-rdf.xml
RDF/JSON: 831-1.0080450-rdf.json
Turtle: 831-1.0080450-turtle.txt
N-Triples: 831-1.0080450-rdf-ntriples.txt
Original Record: 831-1.0080450-source.json
Full Text

Full Text

SELFDUALITY IN GEOMETRY:  YANG-MILLS CONNECTIONS  AND SELFDUAL LAGRANGIANS  by JASON RODERICK DONALDSON B.Sc,  Simon F r a s e r  U n i v e r s i t y , 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  Abstract  The convex theory of s e l f d u a l Lagrangians r e c e n t l y developed by Ghoussoub analyses j u n c t i o n a l s r o o t e d 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 f i n d s t h e i r minima not v a r i a t i o n a l l y but r a t h e r by r e a l i z i n g t h a t they assume a p r e s c r i b e d bound.  lower  T h i s i s e x a c t l y the circumstance i n the s e l f d u a l and  a n t i - s e l f d u a l Y a n g - M i l l s equations t h a t a r i s e i n the p h y s i c a l f i e l d theory and the study of the geometric and  topological  s t r u c t u r e of f o u r - d i m e n s i o n a l m a n i f o l d s . I expose the Y a n g - M i l l s equations, b u i l d i n g up the geometry from s t u d e n t - l e v e l  and subsequently o u t l i n e the s e t t i n g of  s e l f d u a l Lagrangians.  The t h e o r i e s  are c l e a r l y analogous and the  l a s t s e c t i o n f e i n t s at the exact l i n k .  Table  Abstract  of  Contents  .  Table of Contents CHAPTER 1  i i i  Introduction  1  CHAPTER 2 Geometric B u i l d u p The C o v a r i a n t D e r i v a t i v e and Curvature Form Lie Structure Connections The Hodge S t a r and Volume A G e n e r a l i z e d Domain f o r C o v a r i a n t D e r i v a t i v e s and the Second B i a n c h i I d e n t i t y CHAPTER 3  The Y a n g - M i l l s  i i  3 3 8 13 18 27  Functional 1  The F u n c t i o n a l D e f i n e d and the Y a n g - M i l l s Connections A R e p r e s e n t a t i o n of Y a n g - M i l l s Connections The Example of Two Dimensions A P r e l i m i n a r y Fact Y a n g - M i l l s i n Four Dimensions: The S e l f d u a l Equations Cohomology and Chern C l a s s e s Computation of Chern C l a s s e s Examples and the T o p o l o g i c a l Charge A F i r s t Lower Bound f o r the Y a n g - M i l l s F u n c t i o n a l Instantons as Absolute Minima  31 34 36 40 .... 45 48 52 56 58 ••• 60  CHAPTER 4 The Convex S e l f d u a l Framework The Convex Setup: The Legendre Transform, S u b d i f f e r e n t i a l , and Fenchel I n e q u a l i t y The B a s i c Example f o r our A p p l i c a t i o n Lagrangians An Example from P a r t i a l D i f f e r e n t i a l Equations The L i n k w i t h Instantons The Future  64  References  83  64 69 71 72 76 81  Chapter 1: Introduction W h a t follows are two narrative introductions to loosely connected fields i n modern analysis intended to be fully self-contained for a graduate student approaching the subjects after first graduate course sequences i n differential geometry and real analysis. G i v e n 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 reminder of definitions before learning the next batch of geometry—Lie and exterior algebras and integration—with the express purpose of understanding the Y a n g - M i l l s functional and the so-called selfdual and anti-selfdual equations for connections i n 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 still stumped particularly often by formulations switch and many spaces present themselves together here. T h e exposition of the Y a n g - M i l 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 critical points variat i o n a 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. T h i s is the essential analogy w i t h the convex theory of chapter 3 . Jurgen Jost's book Riemannian  Geometry  and Geomemtric  1  Analysis  was m y m a i n  reference and I largely follow its exposition. F i n a l l y the t h i r d chapter outlines Ghoussoub's theory of selfdual L a grangians. It strongly resembles the four-dimensional Y a n g - M i l l s theory i n that it proposes to reformulate partial differential equations as m i n i m i z a t i o n problems whose solutions determined depending on the realization that they attain their lower bounds.  T h e key tool here is the Fenchel inequality of  convex analysis. T h e framework's main strengths are its simplicity—it relies only on basic functional analysis and very basic convex analysis—and its overarchingness—it applies to a n expanse of partial differential equations. I sketch some of the theory w i t h a story-telling, pedagogical bent and formally align it w i t h the Y a n g - M i l l s case i n the final section. W h i l e Y a n g - M i l l s 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 w i l l be soon, especially because I believe too that they w i l l extend to still 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 constitute the basal elements of what follows. T h e covariant derivative and its induced curvature are the essential objects of Y a n g - M i l l s theory. T h e y 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 i n the calculus of variations.  The Covariant Derivative and Curvature Form A covariant derivative on a smooth, compact, finite dimensional manifold 1  A 4 is a m a p  D : X(M) x X(M) -+ D : (V, X) ^ where X(M)  X(M)  X(M),  DX V  denotes the space of smooth vector fields on the manifold,  = {X G C ° ° ( M ; TM)  ; X  p  e TM P  for each p e  M).  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. 1  3  In general we assume that the manifold has no boundary, dM  = 0. Since i n  fact D depends on the first argument only pointwise—as opposed to locally— we may replace V w i t h its image, V = u, and say instead, for each p G M , P  D :TM P  D  x X(M)  : (it,  X)  -» £(A4),  D X,  i->  U  or, finally but importantly, equivalently exchange the tangent space i n the domain for its dual i n the range, as  D : X(M)  x  X(M)  D : X ^  T;(M)  DX.  T h e covariant derivative is a derivative i n the sense of parallel transport: G i v e n a curve a G C ( M ; M)  and a linear isomorphism  1  Pt,c • T )M  -> T  A(0  C T ( t )  X  satisfying (PtA )> tA ))a(t) u  P  =  v  ( > )*(o) u  v  for all u, v G T ^ ^ A ^ and £ G K (where (•, •) denotes the chosen metric on  4  T M), P  the equation  (1)  dt t=o  defines the action of covariant derivative i n the direction of any u = <r(0) := da/dt  £ T A4, p = a(0). B u t only up to the choice of the parallel transport P  map P—although  this doesn't depend on the curve a so long as it satisfies  the initial condition. T h i s definition clearly depends on the specification of the metric on TA4,  but a more abstract equivalent formulation does not.  T h e selection of covariant derivative to employ, however, typically depends on the metric. For example, the canonical L e v i - C i v i t a covariant derivative is the unique covariant derivative V satisfying the torsion-free condition  VxY  - VX  - [X, Y} = 0  Y  (2)  and the defining formula  X((Y,Z))  for every X, [•,•]: X(M)  = (V Y,Z)  Y , Z e X(M), x X(M)  +  X  X{M)  (Y,V Z) X  where the Lie Bracket by the acts as  [X, Y) := XY  5  -  YX  commutator  which means 1  '  dxidx*  J  d& dx*'  [  '  for X -  X  I  Y -Y  8  3  I  dx ' 1  9x  z  T h e definition of Y a n g - M i l l s covariant derivatives depends explicitly on the manifold's metric, but relies further on the global m i n i m i z a t i o n of the manifold's curvature. T h e curvature of a covariant derivative is its composition w i t h itself i n the second argument:  F :=DoD:  TM  D  P  x TM P  x X(M)  -* X(M).  (4)  Here, again consider the alternative formulation:  F  D  : x{M)  -> x(M)  x T;M  x  T;M,  and introduce the notation  Vt (TM) P  := X(M)  x Cl (M) P  (5)  understanding that tt°(TM)  =  X(M).  Here £l (A4) is space of sections of the total space of p-differential forms. To S  6  be explicit (and perhaps, w i t h apologies, verbose), the background notation is as follows: For x G M, p-forms at x are p l\T* M  := {UJ : T M  x  X  x ... x T M  - » R ; co is linear}  X  and the corresponding total space is ,\T*M  := I J  /\T;M.  A s w i t h 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* M x  which allows the definition of the sections of the p-forms (and by analogy sections generally ) as 2  R e t u r n to our examination of the curvature: In this language it is the 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. 2  7'  mapping F  : Q°{TM)  D  ->  n {TM). 2  In the case of the L e v i - C i v i t a covariant derivative the curvature, R := i*V, takes a simple form: For u, v G T M. P  R(u,v)  : X(M)  3£(A4)  as #(u,v)X = V V X - V „ V X U  W  V X  U  [UM  (here it suffices to consider the bracket [u,v] = [Y, Z] where the vector fields p  Y  and Z are such that Y  p  = u and Z  p  = v).  W h i l e the L e v i - C i v i t a co-  variant derivatives come easily and give us a curvature form, the formulation required for the Y a n g - M i l l s functional and hence Y a n g - M i l l s covariant derivatives (which, i n a manner opposite to the L e v i - C i v i t a case, result inversely from the curvature) w i 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  8  (7)  and G^G, are smooth. L  x  x^x'  G is called a L i e group.  (8)  1  T h e diffeomorphic left translation  : G —> G, Ly  := xy,  x  (9)  produces the essential mapping L * : X(G) —> 3L(G), x  L *X X  = dL X x  = X oL,  (10)  x  for exterior (total) derivative d, that defines the subspace Q of X(G)  called  the L i e algebra of G by  0 := {X e X(G)  ; L *X x  = X  y  xy  for a l l x, y G G } .  (11)  (In this restricted context the above equation for ( L ) * suffices to define the x  push forward operator denoted by subscript star.) Observe computationally that for X, Y e g and x e G,  L>x*[X, Y] = [L *X, L *Y], X  X  (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 L i e algebra i n a general setting. Thus the vector fields on M. themselves are a L i e algebra. For I G G , the inner automorphism A  : G —» G defined by  x  Ay x  = xyx' , 1  (15)  induces an automorphism on g v i a differentiation. For identity element e of G, A e = e, so the exterior derivative at e maps the tangent space at e into x  itself, A *{e) : T G x  e  <—> T G, e  and the so-called adjoint representation of G,  the map ad : g —• g given as  ad : x i — • A * ( e ) , x  (16)  yields an automorphism on g. T h e first example of the imposition of L i e structure on a manifold comes i n 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. T h e 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 w i t h left-invariant vector fields through a mapping, M ( n , R ) —» X, X i — • X bijectively v i a the triple sum,  X  a  '—  __.  ciikX  kj  i,j,k=l  f——J \  a x  ,  (18)  ijJa  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 n  23  The next example is the orthogonal group,  O(n) = {X  G  M(n, R); X X T  = I},  (19)  which has L i e algebra  o(n):={Xegl(n,R);X  T  + X = 0},  (20)  C o m m o n l y 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. 3  11  for zero m a t r i x 0 = (0);j and the special orthogonal group,  SO(n) := {X £ 0(n);  det X = 1},  (21)  which is equipped w i t h the same algebra as the orthogonal group,  so(n) := o(n)  (22)  B u t the structures which w i l l provide the neatest examples i n the following context of self-dual Y a n g - M i l 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 L i e 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 i n the above case of the general linear group.  The  adjoint representation assists i n 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)  Now for X,  = (X, Y).  (27)  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)  Connections Having established a metric demand that the covariant derivative respect it v i a 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 w i l l be assumed Ad(G)-invariant and connections assumed metric. T h e covariant derivative restricted from the vector fields to a Lie subalgebra  13  has image contained i n the same algebra,  : 0 —>• 0 x  Q}(M),  since it commutes w i t h the commutator.  Immediately we have the same  quality for the curvature,  F  :  D  -+ 0 x  0  Q (M), 2  or F :Q°(TM;Q)^Q (TM;g). 2  D  To investigate the action of the connection further recall its local representation i n terms of the Christoffel symbols,  r!;f J  (30)  OX  d*i OXj  k  (summation convention applied here o n the left and throughout),  DX m  for X = £ d/dx l  l  = (i\t)  + T) (a(t))^(t)t; (t))  and a = a d/dx\ l  V i e w the Christoffel symbols as a map  from the tangent space into the general linear group, A :=  A : TM  (31)  k  k  -*0l(n,K),  14  (T j )ij k, l  k  t  A:a^(r) (a)^) , k  (32)  ik  or, perhaps more clearly rendered on basis vectors,  A  (^)  where n = dim Ml (i, k = l,...,n)  Fti**'  =  and A(a)  ( 3 3 )  now multiplies X  G X as a  matrix, ( A ^ A ^ ^ a ) ) ^ .  (34)  B u t , 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 i n fact is just the exterior derivative of J  at i . A s  such write simply, D = d + A, and call the operator A a connection  on the tangent bundle TMl.  (35) In this  formulation, the expression for the covariant derivative applied to a section of the tangent bundle, again parameterized v i a a local curve i n M. so that X = £, d/dx l  %  G T A4, as expressed i n terms of the Christoffel symbols becomes X  15  T h e connection represents the curvature v i a the decomposition  F (X) d+A  =  (d + A)o(d  + A)X  =  {d + A)(dX  +  =  dX  =  (dA)X  2  AX)  + d(AX)+AdX+ - AdX  A(AX)  + AdX + A A  AX,  since the first term vanishes because d = 0 and the minus sign comes from 2  distributing the derivative v i a the product rule because A is a one-form. Thus F  d+A  = dA + A A A.  (37)  T h e covariant derivative's being metric implies further that the corresponding connection is skew-symmetric,  A(X)  for every X G  E o{n)  (38)  TM.  To see the result consider an orthonormal basis { E \ , E } for the fibres n  TpM. for each p 6 A4 generated by inverting the bundle's projection and charts from an orthonomal basis of R  n 4  , so  (E (p),E (p))=5 . l  4  j  ij  For details see Jost page 38.  16  Observe now that since a vector field applied to any constant is zero for  X G TM P  X(E ,E ) i  = 0.  j  Realize that the definition of a metric connection says that  X {Ei, Ej) = (D Ei, x  Ej) + (E  D Ej)  Ej) + (E ,  A(X)Ej)  u  x  or here  X {E Ej) = (A(X)E , u  l  t  having used the fact that w i t h i n 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  :TM X  -  TM X  is just and n x n matrix, formalize m a t r i x multiplication i n the current notation as  A(X)Ei  =  (A(X))^Ej,  17  and now just compute  0 = X(E ,E ) i  or, for every X e  j  =  ((A(X)) ' E ,E )  +  =  (A(X)y>  (E , Ej) + {A(X)Y<  =  (A(X)Y'  5  + (A(X)Y'  =  [A{X)Y'  +  {A(X))»  i k  k  k  j  (E ,(A(X)y' E ) k  i  k  k  k  k  j  u  E) k  5  k  kj  (E  ik  TM, A(X)  =  -A {X) T  w i t h the superscript denoting transposition, ( A ) r  ! J  =  (A)- ' which is of 7 1  course the definition of skew symmetry is the group structure is real and the transpose of a m a t r i x thus corresponds w i t h its adjoint,  {Au, v) = (u, A v) T  ,  Thus we write i n general Aen\TM;o).  (39)  The Hodge Star and Volume G i v e n a ci-dimensional Riemannian manifold M, define an innerproduct  18  on the p-fold exterior product f\ T*M:  For / i , u €  p  (fii A ... A fjLp, wi A ...A u ) p  where the metric (-,-)  ('•>') '• T M  = x  T*M,  := det ( ( ^ , 0 ^ ) ) . . . ,  —> E on the tangent fibre at each  X  x E A4, induces the innerproduct on the cotangent space, T*M. local-coordinate basis {d /dx }™ l  of T M  l  =1  d  X  < D  define the corresponding basis {dx }^ 1  =  (40)  6  for  For the  the equations  - '  < 4 i )  T*Ml and hence an innerproduct  by (/J,,LU) = gVfjLiUj  for fi = fiidx , u> = u>idx , and where g 1  z  lj  (42)  are the entries i n the m a t r i x inverse  to that locally representing the metric,  g = Qij dx <g> dx . 1  (43)  j  T h e basis {tixij A ... A dx  ip  defines  ; 1 < ii < i <•••• < i < n] 2  /\ T*M. P  19  p  T h e Hodge "star" operater n—p * : /\T:M  -  l\T* M X  for 0 < p < d is defined uniquely by the requirement that i t be linear a n d that *(ei A ... A e ) = e i A ... A e , p  p+  n  (44)  whenever { e i , e } is a positive orthonormal basis of T*M a n d n  *(ei A ... A e ) = - e p  p +  i A ... A e , n  (45)  whenever { e i , e } is a negative orthonormal basis of T*M. T h e definition d  of a positive basis comes from prescribing that a basis B be positive a n d 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. T h e 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 be20  cause b o t h the wedge and star are linear. Now, given an orthornomal basis { e i , ...,e } for the cotangent bundle, n  Bp := {e A ... A e h  is a basis for f\ T*A4.  ip  ; 1 < i < n, i  x  < ... < i } p  Realize that product vanishes for distinct basis vec-  p  tors,  {e  h  A ... Ae ,e ip  A ... Ae )  jl  =  jp  (e A ... A e ) A *(e A ... A e ) h  ip  =  h  jp  e A...Ae Ae A...Ae il  =  ip  jp+1  jn  Sij * 1  since unless the vectors coincide the last wedge product will have a repeated element and thus be zero. T h i s fact and bi-linearity show reflexivity, positive definiteness and symmetry and thus demonstrate that the given formula defines an innerproduct on the exterior algebra. F r o m multilinear algebra we know that for a p-by-p m a t r i x A  Avi A ... A Av = det A(v A .... A v ) p  for any vectors u s and {ui,uj } n  l  y  u  v  x  p  £ T*M.. Consider i n particular the case where p — n  constitutes a basis for the cotangent space (so p = d) and A  is the change-of-basis m a t r i x relating it to a n orthonormal basis { e i , ...,e } p  21  as Au>i = e . {  We can relate the n-fold exterior products of the bases vectors i n terms of a k i n d of volume measure:  eiA...Ae  d  =  Aco A .... A Acj  =  (det A) ui A ... A u  1  n  n  det(((Ji,u>j)) u>i A ... A ui  n  since AA  =  T  (uuUj).  Observe the immediate consequence of the definition of the operator *:  *(ei A .... A e ) = 1 n  to write the above as _  u) A...Au} x  v  Hence if uji = dx  1  /det((w ,a; )) i  dx A 1  V lj  i  then *1 =  Recall that g  n  d e t  ...Adx  n  (9  i j  )  = g~} and again m i n d the rules for determinants to uncover  22  the so-called volume form  *1 = y/gdx  A ... Adx ,  1  (47)  n  defining the standard shorthand,  := ^det .  (48)  gij  A s such, for n c M  vol(fi) := / ^Jgdx Jo.  A ... A dx  1  = / *1, Jn  d  where the integral of any continuous function / : Ml —> M. on the manifold is naturally defined by summing over the coordinate neighbourhoods of the system {(U ,a);a a  € A } as  /  f*l:=  JU  f  {foa- )^dx . l  (49)  a  Ja(U )  a  Q  T h e right-hand integral is well defined over the neighbourhood a(U ) a  w i t h volume element dx  a  C MJ  1  = dx*...dx%. T h e extension to an arbitrary subset  O on the manifold follows from the existence of a partition of unity <j> G a  C°°(M;R)  by  which is well defined since we insist that s u p p ( 0 ) C U . a  23  a  Note here the formula for the autocomposition of the Hodge operator,  **: A T*M-+ P  A T*M, P  * * = (_I)P("-P).  (51)  We know that for an orthonormal vectors { e i , e } i n T*M. p  *(ei A ... A e ) = e p  A ... A e .  p+x  n  such that the orthonormal basis { e i , e , e i, e } is (defined to be) positive. p  p+  n  A n d for the orthonormal vectors { e i , . . . , e } , p +  n  * ( e i A ... A e ) — (det A) e A ... A e p +  n  x  p  where A : T*M —> T ^ A 4 is the change-of-basis m a t r i x that reorders the basis as A : { e i , e , e + i , e } i—> {ep+i, p  p  n  e > ii ••••> p}e  e  n  T h e sign of the determinant by definition determines whether { e  p + 1  , e , e, n  x  is positive (with respect to the convention established by asserting { e i , e , e. p  to be positive). If {e[, ...,e' } is a negative basis—the image of the positive n  basis under a transformation w i t h negative determinant—then i n general we have *(ei A ... A e' ) = - e p  24  p + 1  A ... A e . n  B u t reordering the wedge product exchanging the place of the p-form and the (n — p)-form gives  ei A ... A e A e _ i A ... A e p  p  n  = (-l)  p ( n  ~  e  p )  p + 1  A ...e  by the standard formula for such rearrangements.  *(ei A ... A e A e _ i A ... A e ) p  p  n  p  n  p  A n d since  A ...Ae  =  det A (e +i A ... A e A e\... A e ),  p+1  A Ae  n  1  p  A ... A Ae )  n  p  p  and thus  =  (det A) e A ... A e 1  =  which says ** = (—l) (  (52)  p  *(Ae  n  p  ... A e ,  1  =  we conclude that det A = (-l)P( ~p)  *(eiA...Ae )  Ae A  n  (-lf  ( n  -  p )  p  e A...Ae 1  ) as hoped.  Now define the global L - i n n e r p r o d u c t of 2  u e T*M  as  (yu,w) : = / (fj, u) * 1 f  J M  and use the star formulation of the innerproduct to note that if p =  25  n/2—so  * : /\ T*M P  {*(e  -> f\ T*M—  A ... A e ), *(e  h  then  p  ip  h  A ... A e )) jp  =  *( * (e^ A ... A e ) A *(*(  =  *( * (  ip  eil  A ... A e  jf  A ... A e ) A (e^ A ... A e ) ) ,  eil  ip  jp  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  *(e  A ... A e ) A (e  h  ip  A ... A e )  h  =  ip  e A...Ae h  in  =  (e A...Ae )A*(e A...Ae ) h  ip  il  i  so  (*{e  h  A ... Ae ),*(e ip  h  =  *((e  =  (e  il  A ... A e ) A *(e ip  <1  h  A ... Ae ,e ip  h  A ...  Linearity extends this to (*v, *w) = (v, w).  Likewise  ( « , « , ) -  A ... A e )) ip  A ...AeJ)  / < ™ , ™ > . l  JM  26  = / < » , » > . JM  ! =  ( « . » ) ,  Ae ). ip  so we can add as a corollary that * is an L - i s o m e t r y : 2  II * V\\L2  := (*v,  *v)  = (v, v) =  |H|z,2.  A Gernalized Domain for Covariant Derivatives and the Second Bianchi Identity W h i l e here we w i l l stick mostly to covariant derivatives oporating on vector 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 = ld }.  (53)  M  A n d hereafter we write  Q (E) := T(E) x n (M), P  (54)  p  (p < d i m / v f ) which agrees w i t h our current definition of Vt (TM) p  Y(TM)  =  since  X{M).  Now we wish to extend the covariant derivative to this space fl (E) p  of  sections crossed w i t h forms. M o t i v a t e d 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)  D : T(E) x W(M)  D(X  ®u)  and UJ G Q (M) P  ->  x  that  Q (M), P+1  := DX ALU + X ®duj  (55)  w i t h the understanding that  (X <g>  whenever X eF(E)  A u; := AT <g) (tvi A u; ), 2  and a>i, o; G 2  2  fl (M). p  Furthermore, for distinct bundles Ei and E2 w i t h assosiated covariant derivatives D\ and D respectively, define the covariant derivative on E\ x E 2  2  via D(X®Y) whenever X E E  1  := DiX®Y  and Y E E . 2  tive Z? on the space E®E*\  =  (56)  2  In particular this defines a covariant deriva-  For X := ^  = (d + A ) AT =  + X®D Y  dX +  <g> w- ' G r(£ <g> £7*) 7  AiQviQxJ)  dX + ( A } ^ ^ <8> u>* -  v ® k  J).  B u t the last term is just the Lie bracket of the connection w i t h t h section, so DX  = dX + [A,X].  28  (57)  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 E n d i ? for which the endomorphism for each fibre is skew symmetric. F r o m the previous section we know a connection A is skew symmetric i n the sense that A(X)  £ o so for D = d + A we have  A e Q.\AdE).  (59)  Now view the curvature tensor,  F  D  : T(E) -+  tf(E)  = (T(E)y  x  tf(M),  as a two-form assuming values i n E <g> J5*, 1  F  D  6 r(£)<8>  (r(£))*®ft (A4), 2  29  .  and apply the result to commute the connection of the curvarture:  DF  D  =  DF  + [A,F]  =  d + 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\ Ajdx  =  A A A (dx  D  2  j  i  i  j  k  A  A dx ] k  k  A dx A dx j  k  - dx A dx j  k  A dx*) = 0.  Revealing the formula DF  D  = 0,  (60)  termed the Second Bianchi Identity, which w i l l prove usuful i n our initial demonstration of the selfduality of the Y a n g - M i l l s equations i n four dimensions.  30  Chapter 3: The Yang-Mills Functional Define the here our central subject, the Y a n g - M i l l s functional, which is the norm of curvature over a manifold viewed as a function of connections w i t h a given Lie structure. Physically, the Y a n g - M i l l s connections are stationary points of the field strength. T h i s second chapter examines the functional and its critical points, eventually examining our m a i n focus of the functional's absolute minimizers i n four dimensions and the corresponding equations for the, the selfdual and anti-selfdual equations, Fr> = *F  D  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 i a  (X®u,Y  (A,B)  ®v)  {X,Y) (u,v) s  (X, Y)  KPTiM  * (u A *V).  So i n the cases outlined (the algebras o, u, and su),  (A  B)  txXY  31  * (cu A *u).  A n d analogously to above the L  scalar product is  2  (A,B) ,  :=  L  f JM  (A,B)*1.  Thus finally define the Y a n g - M i l l s functional as the L - n o r m of the curvature— 2  I -I  2  == <-,->-as  yM yM(A)  = [  : Vt (T*M\o)  —>• E ,  l  \F \  *1= /  2  d+A  (F , d+A  F )  * 1.  d+A  (61)  Sometimes (as below) it w i l l be easier to view this as a function of the covariant derivative rather than the connection, and when no ambiguity w i l l result we w i l l employ the same notation—yM(d  + A) :=  yM{A).  T h e objective is to choose a connection that is stationary w i t h respect to this square energy. T h e traditional approach is variational. For covariant derivatives D and D G Q}(TM.;Q)  D+W  F  x  =  (D + tD)o(D  =  DX  =  (F  2  and a vector field l e g  +  + tD(DX)  tD)X +W  + tDD + t D  + W ADX  2  AD)X,  having employed the fact that D(DX)  = (DD)X  D  consider  -DA  + tD 2  DX.  ADX  Now take the  variational derivative of the functional i n order to find the conditions of the  32  stationary points:  5yM(D)  =  -£| dt d_ dl  yM(D  t  =  0  J'M M  since (F ,tDD) D  + W)  2J (Db,F ).l, M  and (tDD, F ) D  0  are the only first-order terms i n the expan-  sion of the scalar product. T h u s setting 8yM.(D)  (DD,  F)2 D  L  = 0 yields the equation  = 0,  for all covariant derivatives D, for the functional's critical points. G i v e n an arbitrary covariant derivative D : D* : Q}(T M) X  the L  2  —> f ^ T ^ / v f ) introduce the operator  —> Q , termed the dual covariant derivative to D, defined by 0  relationship  (D*X,Y) 2 L  for every X G ^(TM)  m  = (X,DY) 2 , L  m  (62)  and Y € ^ ( T T W ) , i n order to rewrite the character-  ization of the functional's stationary points as  (D, D*FD)L  2  33  =  0  for every D, or better, D*F  = 0.  D  (63)  Covariant derivatives satisfying this equation are called Y a n g - M i l l s covariant derivatives. (Likewise A is a Y a n g - M i l l s 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 i n components as A — Aidx , 1  A(X)  =  where  AidxyX),  Ai e Ql(n). T h e connection acts on a vector field Y v i a exterior differentiation and exterior product w i t h 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 + Aidx  1  AY)  = (d*X, Y) - (AX,  34  dx* A Y).  (64)  Now represent the curvature as  F = Fijdx* A dx ' 3  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  d*F = d^Fijdx  1  8FA dx ) = -—^dx , ax 3  (65)  3  1  understanding that summation is taken over i as well as j despite the indices residing "on the same level"—i.e. b o t h being formally contravariant. Substitute this formula into the representation of the 1? innerproduct involving the covariant derivative i n terms of its canonical decomposition to get  (F, DY)  =  (d*F, Y) — (AiF, dx* A Y) 8F  =  ( - -g^dx ,  Y) - (AkFijdx  =  ( - ^dxi,  Y)  - (MFy  =  (-  Y)  - {[A,  j  ^dx>\  1  35  A dx , dx j  k  - F^dx , 3  dx , 3  y),  A Y) y)  where having executed the summation i n normal coordinates eliminated the fc-index and reduced the right-hand innerproduct to agree w i t h the right-hand one i n domain and here the L i e bracket denotes exactly the symmetricness of the curvature as \A-i, F%j ]  =  A i Fij  Fij Aj.  So  and A is a Y a n g - M i l l s connection if  ^  for each j — 1 , n  + [^,1  = 0  (of course F^ corresponds to A as FJ+A = F^dx  (66)  1  A dx *). 1  The Example of Two Dimensions If n = d i m M. = 2 then the orthogonal group is Abelian, if A, Be  o(2, E )  then AB =  BA.  T h e determinant of an orthogonal m a t r i x is always plus or minus one,  I det ^41 = 1  36  for a l l A 6 o(n) since, by definition,  AA  = L  so det AA  = det A det A  T  T  = det I = 1  and det A = det A implies (det A)  = 1  2  which gives the desired fact. Thus, i n two dimensions, where,  A'  =  1  the equation A  1  = A  1 det A  Cl22  -a  2 1  — tli2  an  implies  T  22  a  V- o i 2  —^12  a  = ±  an —a  n  37  12  a i 2  a  2 2  which says every A 6 o(2) has either the form  \  s  t  -t  s  or  t  -s  V  Having established this there just for cases to compute directly to see that the group is A b e l i a n . O f course that the Lie bracket vanishes identically is an immediate and t r i v i a l consequence,  AB - BA = AB - AB = 0.  We call such a L i e algebra trivial. Ad(TM)  C TM.  x T*M  A n d thus the skew-symmetric bundle  is also trivial, which means it is isomorphic to  the direct product of the manifold w i t h the real numbers, we say (writing equality for short) A d ( T A l ) = M x R. In this representation the covariant derivative coincides w i t h the exterior derivative, D = d  (67)  or the connection vanishes locally—its derivatives do not vanish, since the representation D  a  = d+ A  a  depends on the coordinate system {(U ,a);a a  38  £ A} (mind that this set of  charts A is unrelated to the connection A), and this decomposition is not i n general global. In this setting the fundamental equations a l l simplify greatly. A A A = 0 reduces the curvature to  F  d+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  DF  D  = dF  = d(dA) = d A = 0 2  D  (69)  Lastly, the local absence the connection reduces dual covariant derivative reduces to the adjoint exterior derivative,  D* = d*  so the Y a n g - M i l l s equations read  <TF = 0.  E m p l o y i n g above identity for the curvature expands this to  d*dA = 0.  T h i s low-dimensional context unifies Y a n g - M i l l s theory w i t h the study of 39  harmonic forms since the Laplacian on forms A : Cl (M) p  —» Q (M) P  is defined  as A := dd* + d*d.  (70)  So trivially the curvature is harmonic i n this context,  AF  D  = 0.  Furthermore, if, without motivation, we assume the so-called gauge condition, 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> w i t h  40  compact support, /  du * 1 = (p  JM  u * 1,  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  P e /\  d  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: / J  d(a A *B) * 1 = 0. M  M i n d i n g the product rule for forms and keeping i n m i n d that since d * 3 e  **(d*p)  =  (-l) - - d*p, (p  1){n  p+1)  we can compute  d(aA*/3)  =  daA*/3 + ( - l )  =  daA*/?+(-l) - (-l)  p _ 1  p  T h e exponent expands to n(p — l)—p  2  41  1  aAd*/? ( p  -  1 ) ( n  -  p + 1 )  aA**d*/?.  + 3p — 2 and because p and p  2  always  have the same parity, we can cancel most of the terms and write  (_-gp-l(_1^(p-l)(rc-p+l)  _  T h e 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 w i t h just linearity and the definition of the innerproduct  daA*(3 + (-l) - aA**d*P n(p  1)  =  * * (da A */3 + ( - l )  =  *(*(daA*p)  =  *((da,(3) + ( - l ) '  n ( p  -  1 }  a A * * d * (3)  +  (-l) - *(aA**d* n(p  l ( p  -  1 }  1)  (a, *d * / ? ) ) .  B u t by Stokes's Theorem we can say that this integrates to zero,  f JM  d(aA*(3)*l=  [  *((da,/3)  - (a,(-l) - *d*f3)) n(p  1)+1  * 1 = 0.  JM  Since this holds for any (p — l)-form a and p-form (3, the integrand must be zero and the above is i n fact a statement about the relationship between the exterior derivative and the Hodge star,  (da, (3) + ( - l ) ^ - ) (a, *d * (3) = 0 1  or (da,  (3) = (a,  (-1)"(P- )+ 1  42  1  * d * (3) ,  which reads like the definition of the adjoint exterior derivative and says exactly that d* = ( _ I ) » ( P - I ) + I *  d  * .  (72)  T h e result extends to covariant derivatives. T h e Hodge operator acts on elements of X <g> u G fl (TM), p  w i t h X G X and u £ f\ ( *M), P  *(X'<g)a;) = X <2>  T  as  (73)  which is to say it is defined to act normally on the form but leave the vector field alone. I n contrast recall that the image of connection A belongs to the orthogonal group, A(X)  G o(n) for each X G TM,  i n the sense that for  A = Aidx the map 1  Ai : T M X  TM X  is skewTsymeetric. Thus the m a t r i x 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)  Ai =  ) o  r  (_I)P("-P)+I * * =  * *Ai =  *Ai*.  R e t u r n to the formula for the dual covariant derivative D*,  (D*X, Y) = (d*X, Y) - (AiX, dx A Y)  (75)  i  and manipulate to get  (D*X,Y)  = (-l) ?-V (*d*X,Y) n(  +1  - (-l) "p (  p ) + 1  ( * A * X,dx  i  AY).  Now suppose that the manifold is of even dimension and look for the d u a l covariant derivative of a form of even order—that is n and p above are even. In the next, especially pertinent section, we w i l l be narrow our focus to the situation when n = 4 a n d 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,  dx A Y)  (76)  1  and we can say D*X  = - *(d + A)* = - * £ > * ,  (77)  as long as we understand that for X <g> u G VL (TM)  such that X  P  G X  and u) G JT2 (A4) here the connection A applies to the "form part" u> v i a P  contraction, not multiplication, that is to say exactly that  (A(X  for every Y G fl ~ (TA4); p  1  ®u),Y)  = (AiX  ® u, dx* A Y)  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 i n four dimensions the Hodge star is an L  isometry on the  2  space of two-forms, 2  * : /\T*M  2  -»  f\T*M.  G i v e n normal coodinates about x such that {dx , 1  45  dx , 2  dx , 3  dx } 4  describes  a basis for the fibre, define  A  := span-fob; A dx  +  1  + dx  2  3  A dx ,  A dx  4  + dx  2  A dx ,  dx  1  A dx  2  A d x , d x A dx  2  - dx  3  A dx ,  dx  1  A dx  4  dx  1  3  3  - dx  2  A dx } 4  and  A " := s p a n j d x A d x + dx 1  3  4  1  4  4  - dx  2  A  and realize that since the six vectors described above are independent and /\  2  T*M. has dimension six,  ^A™)-(-r>).(3-, i n our case of two-forms. Thus: 2  A + © A - = /\T*M.  (78)  T h e division of the fibre into the prescribed subspaces speaks to the Hodge star. C o m p u t a t i o n minding the behaviour of the operator reveals  *a = ±a  for a € A  +  (79)  or a E A ~ — t h a t is to say that the decomposition is into the  eigenspaces of * corresponding to the eigenvalues 1 and —1. Interestingly, this splitting corresponds to the splitting of the special or46  dx }, 3  thogonal group i n the sense of isomophism  A  = A ~ = so(3)  +  and 2  f\T*M=  50(4)=  5o(3)©so(3).  Here we w i l l t u r n to covariant derivatives D w i t h curvature tensors FJJ = *F  GA  D  +  called selfdual and F  D  — —* F  D  GA  -  called anti-selfdual. Fur-  thermore, call a connection inducing selfdual curvature an instanton and one inducing anti-selfdual curvature an anti-instanton. T h e B i a n c h i formula = 0  DF  D  implies that D*F,  for F  D  GA  +  or F  D  D  0  (80)  G A " . A n d hence  *D*F  = 0.  D  B u t the m a i n result of the previous section says that  *D* =  47  -D*  (81)  in this case, and we can now say  D*F  = 0.  D  Or all instantons  and anti-instantons  (82)  are Yang-Mills  connections.  Cohomology and Chern Classes Our nicest examples of Y a n g - M i 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 general linear group G L ( r a , C ) , so return to the abstract setting define it on an arbitrary vector bundle E w i t h rank m.  In this context we w i l l be able to  rewrite the Y a n g - M i l l s functional elegantly and read off its minimizers. To begin we need the concept of C h e r n classes and thus of cohomology. We say that two p-forms, a, P G Cl (M), p  are cohomologeous if their difference  is exact, that is, there exists a (p — l)-form, 7 G il ~ (A4), p  a  — (3 =  1  such that  07.  T h i s cohomology relation is equivalence relation that partitions the space {a G Q (A4) P  ; da = 0} of closed forms i n Q (A4). P  T h e set of all equivalence  classes, [a] := {P G tt (M) ; a - R is exact, dp = 0}, p  48  itself defines a vector space,  H (M)  := {[a] ; a £ Q, da = 0},  P  (83)  called the p-th de R h a m cohomology group. The C h e r n classes are such equivalence classes belonging to such a group that depend, for our definition, on the elementary symmetric polynomials,  ^(A ,...,A ):= x  Yl  m  K-Kj,  (84)  l<a\<...<aj<m  or, more precisely, o n the m a t r i x polynomials,  P  j  : M(m, C ) -> C  where for B £ M(m, C )  P (B):=pl(X ,...,X )  (85)  j  1  and A i , A  m  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  3=1  49  j  having employed the shorthand p> := p*(\i,  A ) . T h e identity looks the m  same carried over to the m a t r i x case,  Y[(t-  Xj) = P (B)t ~ . j  m  j  (86)  3=1  B u t here realize that of course m  H(t  - Xj) = 0  exactly when t is an eigenvalue of B.  B u t for t to be an eigenvalue by  definition Ba = ta, for each a E C  m  which reformulates into the familiar equation  d e t ( B - tld) = 0.  (87)  Since these polynomials have the same roots and b o t h have leading coefficient one they must be equal,  P (B)t j  m  j  = det(B - tld),  (88)  and we have a tool, the determinant, w i t h which to compute elementary symmetric polynomials. These polynomials are homogenous w i t h degree j and thus take map 50  p-forms to jp-iorms.  In particular we here consider the curvature as a homo-  morphism,  F  D  G AdE = EndE = Rom {E;  E) = M(m, C ) ,  c  so,  P  j  : Q (M)  tt {M),  2  2j  P {F )  e  j  D  fl i(M).  (89)  2  These polynomials are i n fact exact,  dP {F ) j  D  = 0  (90)  and further are independent of the covariant derivative , 5  P>(F )  = P>(F )  Dl  for any covariant derivatives D , D x  (91)  D2  2  : 0 —>  £l (E). J  So, independently of the covariant derivative—F := F  D  for arbitrary  D — w e can define the elements of the 2j?-th cohomology group,  c (E) t  5  :=  G /Y^'(A4),  See Jost, pages 125-126 for a proof.  51  (92)  called the C h e r n classes of the bundle E.  Computation of Chern Classes Now, to compute the classes, exploit the formula  det(^F  D  +  f  Id)=|>(^)r-  or, i n terms the curvature of an arbitrary covariant derivative,  ]T  {E)t ~i m  Cj  det \T^FD  + tld)  .  (93)  3=0  To simplify, divide by t  m  -•  and remember that the determinant is  m  m  3=0  3=0  m-homogeneous,  - _ > ( £ ) t " ^ = 5>,(£)r^ 1 / i — det — F t  V2TT  m  =  det ( ^ - F  D  D  + tld  + Id  Realize that the curvature's eigenvalues are two-forms since FD fl (E). 2  Aj is an eigenvalue of F  D  F (X) D  when for X £  = AjX  52  G n {E). 2  &° —>  T(E),  (94)  T h i s extends naturally since the roots t £ C and leading coefficient of the equation  n -iM=°  ( 9 5 )  tid  again coincide w i t h those of the determinant above. Thus, replacing Id w i t h 1 i n the product to agree w i t h the formalism,  (^ )=n( -i 0-  det  (  +M  A  (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 C h e r n classes v i a the expression  E^=n( -^ i). i  3=0  3=1  A  V  where we have substituted r := 1/t. V i z , for m = rank(E)  = 1,  =  c (E)+c (E)r Q  3=0  53  1  <w  B u t the zeroth-order symmetric polynomial is one, here c (E)  = 1. Thus  0  since a l l other terms above cancel. Furthermore, since w i t h rank(.E') = 1, B : E —> E, and hence likewise FD : Q° —> tt (E), has only (exactly) one 2  eigenvalue must be equal to its trace,  = Ai,  tiF  D  or, moreover, Ci(E)  To find c (E) 2  Co(E) + (E)r Cl  = ±trF.  (98)  consider the case i n which m = 2 so  + c (E)r  2  2  + ^  r  =  (l  ^  l  +  =  i 1 1 + ;r-(Ai + A ) r —— A i A 2  LIT  47T  A n d simply matching coefficients gives  C l  (£) = ^ ( A i + A ) = ^trF, 2  and c (E) 2  = —~Ai 47T  54  ^ A  A A . 2  2  r  Ar. 2  2  Jost gives the general formula for the second C h e r n class of tangent bundle E w i t h rank m as  6  c (E) 2  nrn — 1  - —  (E)  Cl  1  A (E) Cl  = — trF A F, 07T  Am,  0  0  (99)  Z  for the "trace free part" of F,  FQ := F — —trF m  •  ld . E  B u t for our discussion focus on two examples, the cases when D is a u ( l ) covariant derivative for illustration 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 w i t h structure group U ( l ) the curvature is just a two-form,  Fd+A =  dA=:f,  for an arbitrary u(l)-connection A. T h i s is analogous w i t h the two-dimentional real case discussed already. Thus the trace-eigenvalue approach is here very simple, or alternatively, we get the first C h e r n class trivially from the determinant definition,  c (E) 0  + (E)r Cl  = det ( £-fr  + Id j = 1 +  ^-fr  and thus  (ioo)  ci(£) = There are only two C h e r n classes here (the above and c (E) 0  = 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,  F  D  : n°(E;su(2))  tf(E;su(2)),  56  is represented by a m a t r i x of two-forms  F  D  (ft n ^e su(2)  =  (101)  ft fi  where, for j, k = 1, 2,  fi e W(M), and trF  D  = ft + f\ = 0  as required i n the definition of S U . Thus right away the first C h e r n class vanishes, ci(£7) = trF = 0. A n d now i n this formulation we can compute the second C h e r n class from the determinant,  CQ(E) + (E)r CL  + c (E)r  2  2  =  det ( Id + —  Fr D  2"7T  rJtr + 1  /  =  det  \  2ir  2  4TT  2  Sh  fir  +1  (ft^f 2-f^f?)r  + ^(ft  ( / i A ^ - z j A / y  + i,  2  4TT  i-J\r  57  2  + f )T + l 2  2  since trF  D  = 0. Clearly now we can write the second C h e r n class,  =  " ^ ( / l A / l - ^ A / 4TT  2  2  )  (102)  ^tr(F AF ). D  D  A n d we have defined the topological structure of E . 7  Integrate over the second C h e r n class over manifold to obtain the second C h e r n number known as the topological  —k = -k(M,su(2))  charge and written,  := -c (E)[M]  = ----  2  f  t r ( F A F) * 1  (103)  (In fact this is a constant over the fundamental class [Jvi] of oriented, fourdimensional, 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 Q (E;su) 2  tt (E;su), 2  w i t h X, Y e V(E) and u, v £  (X <8)  Y ® ^)  n 2 ( E ; s u )  is, for X <&u>, Y <g> v €  Q. (M), 2  = - t r ( X y ) w A *v,  Freed and Uhlenbeck say, "The characteristic class [c {E)\ classifies SU(2) bundles over compact 4-manifolds, but this classification fails in higher dimensions." See page 33 for references. 7  2  58  so that /  (F, *F)  = -  Q2(E;SU)  [  JM  t r ( F A F) * 1 = -8n k.  (104)  2  JM  Recall that the Hodge operator is an L -isometry, so 2  (F,F)  =  {*F *F), 1  and (dropping the implied su-subscript) employ this to reformulate the YangM i l l s functional v i a  yM(D)  =  [  (F ,F }*1 D  D  JM  =  [{F ,F ) D  =If  2 JM 1  >  D  ({FD ~ *F , F D  D  D  - *F )) D  '  V  / \F t D->M D  z  (*F ,*F ) j*l  +  D  - *F *r \ 2  D  D  *1-  *1+  f  (F , *F ) D  D  *1  JM  8TT A;. 2  Since the square is positive we can bound the functional from below by the topological charge, yM(D)  >  for every su(2)-covariant derivative D.  -8n k 2  Remember we have said nothing  about the sign of —A; and since the functional is a n o r m write  yM  > max{-87r A;, 0}. 2  59  (105)  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  +  GA  +  and F~ G A " .  F r o m here express the Y a n g - M i l l s functional as  yM(A)=  [ \F\ *1 JM 2  =  f (F,F)*1 JM  =  / (F + JM  =  f JM  -  since (F ,F~) +  F~,F +  +  f JM  ((F ,F ) +  F~)  +  *l  + 2(F ,F-)  +  +  +  ( ( F \ F  +  )  +  '  that, by assumption,  +  f (|F | JM  + (F-,F-))*l=  v  — 0 as a result of the orthogonality of A  F  (F-,F-))*l '  V  = * F + and F~ = - *  60  F~,  +  2  + |F-  and A ~ . M i n d  and break down the second C h e r n class:  8n c (E) 2  2  = tr(FAF)  =  tr((F  =  tr(F  +  AF  =  tr(F  +  A F ) + tr(F  =  tr(F  +  A * F ) - t r ( F ~ A *F~)  =  since the cross terms F  AF  +  + F~) A (F  +  F~))  + F~ A F )  +  _  +  A F~)  _  +  -|F | +|F-| , +  2  2  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 87r fc= 2  f (|F+| -|F-| )*1, JM 2  (106)  2  which looks remarkably like our current expression for the Y a n g - M i l l s functionalit differs only i n the sign of one term. Comparing the two we see  /  (|F | +  2  + |F-| )*1>  /  2  JM  (|F | -|F-| )*1 +  2  2  JM  which implies yM  >  8ir \k\ 2  (107)  and strengthens our bound form the previous section. We see that equality is attained, yM(D)  =  61  87i \k\, 2  i.e. the Y a n g - M i l l s functional has an absolute m i n i m u m , exactly when either the selfdual or the anti-selfdual part of the curvature vanishes,  F  +  = 0 or F~ = 0,  which is to say that the covariant derivative D is an instanton or antiinstanton. A n d which one depends on the sign of (minus) the C h e r n number k. M i n i m i z e the difference yM  yM  -  I  (|F | -|F-| )*1 +  2  2  — 8ir k,  firstly  2  =  JM  /  for k > 0:  (|F+| + | F - | ) * 1 2  2  JM  =  /  [\F \ -\F + 2  JM  2 /  |F-| *1, 2  IM  which is minimized at zero when F  —F  £ A+ is selfdual, D is an instan-  +  D  ton, by definition. Likewise, if k < 0 corresponds to anti-selfdual curvature,  yM  -  f  (|F | -|F-| )*1 +  2  2  =  JM  /  (\ \ + \F-\ )*l+ +  2  2  F  JM  =  2 /  [  (|F | -|F +  2  JM  |F | *1, +  2  '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.  W h i l e Cliff Taubes has constructed examples of solu-  tions, there is no theoretical guarantee that the topological m i n i m u m will be  62  attained for any covariant derivative. O n the other hand, i n some settings, for example i n the case of line bundles over the two-spheres S x S 2  2  8  there  exist covariant derivatives that are neither instantons or anti-instantons but minimize the Y a n g - M i l l s 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 Ghoussoub w i t h lead to facile investigation of the absolute m i n i m a of the Y a n g - M i l l s functional i n four dimensions because (in some contexts) basic analysis will displace geometric, topological, and algebraic methods and alleviate the challenge of working at the intersection of these fields. T h e selfdaul Y a n g - M i l l s equations are poised to become among the canonical of examples of classical equations reinterpreted as representatives from more general classes that w i l l provide diverse extensions of and offshoots from known work.  The Convex Setup: The Legendre Transform, Subdifferential, and Fenchel Inequality F i r s t l y define the Legendre transform (p* : X* -> R U {+00}  of convex,  lower semicontinuous functional op : X —• M U J + o o } 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)  for every x G X and y G X*, called the Fenchel Inequality.  64  (109)  Note here  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 —> 2  X  , as the  set-valued map  dcp : x  {y G X*  ; ip(z) > <p(x) + (z - x, y)  for all z  Here realize that if <p = f is a smooth function on X  (110)  G X}.  = X* — W  1  then  the subdifferential reduces to the singleton of the gradient, df = { V / } , by supposing /(x) >  / ( X Q )  + (x -  • y  XQ)  for every x G K , and i n particular x = x + /ie, for h G R and standard n  0  orthonormal basis vectors ei, ...,e , so n  /(x  0  + hei) > /(x ) + he 0  {  • y  and thus  lim  /(x  0  + hei) h  / ( X Q )  < ej • y < l i m /i-»0+  or, component-wise  65  /(x  0  + fee.) -  / ( X Q )  So the subdifferential contains at most one element.  y  =®  ( X o )  "-l:  ( x o )  )  =  v/(Xo)  '  which is of course i n fact i n the set as the tangent plane at any point x lies 0  below the a convex function:  /(x ) + V / ( x o ) . ( x - x ) < / ( x ) . 0  T h i s intuition from  0  finite-dimensional  calculus contextualizes the subd-  ifferential as a derivative that does not depend explicitly on a l i m i t and as such freely applies to functionals at their points of nonsmoothness,  taking  on multiple values i n this situation. T h e connection between the derivative w i t h 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)  i f and only if y G dtp(x).  (HI)  T r i v i a 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( ) > <p(x) + (zz  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 , a n d  (p**(x) = =  sup yex*  {(x,y)-<p*(y)}  sup { (x,y) - sup{(z,y) - <p(z)}} yex* zex  < <p(x),  since (x,y)  - sup{(z,y) - ip(z) ; z £ X} < <p(x).  T h a t is, ip** < (p. W h e n the functional is convex i n fact the reverse inequality holds, ip > <p**  67  so we have equality: ^  =  tp.  (112)  A p p l y this identity i n the Fenchel inequality to recover its dual equivalent i n terms of the subdifferential:  tp**(x) + <p'(y) = (p(x) + <p*(y) = (x, y),  automatically gives the t h i r d 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 i n the Fenchel inequality.  68  The Basic Example for our Application Now, poised to address the theory of Langragians on X x X* that constitutes the essence of the chapter, t u r n firstly to a concrete example that is illustrative here and to be fruitful i n the sequel. If ip(x) = ^\x\  p  <p*(y) = sup j (a;, j/) - ^\x\  p  then  ; x G A"|  is calculable variationally. A t m a x i m i z i n g x,  (x,y)  -  -\x\ P  p  < (x + tw,y)  - -\x P  +  tw\  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  (x + tw,  2  w)  or (w, y) — \x\  p  2  (x, w) = 0.  Thus x\  p  x  2  69  = y  t=o  = 0  and \x\ = provided the found functional does belong to the dual space. Perhaps note quickly that for a function / member of the dual, \f\ ~ f p  E X  = L ( R ) , the maximizer i n fact is a p  n  E X* = L (R ),  2  q  where q is the exponent dual  n  to p, i.e. i + I = l , as  i/r / 2  |/| <oo. p  Returning to the general case, plugging i n for x yields the Legendre transform <p* to be  (\\• f)\y) \p J  = M ^  +  1  - -\v\& = (i - -) \v\& = - y\ \  P  pj  q  having noted that here the inner product does not complicate computations and reduces simply by first inserting x for y,  (x,y)  = (x, \x\ ~ x) p  2  = \x\  p  =  In this context the Fenchel Inequality reads  ^\x\  p  whenever x E X,  +  ±\y\i>{x,y),  y E X*.  70  \y\p-5.  Lagrangians C a l l a functional L : X x X* —> R U {+00}  a Lagrangian i f it is lower  semicontinuous and convex i n b o t h variables and not identically +00.  Star  denotes the Legendre transform i n b o t h 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. T h e 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 assumptions o n <p and consequently <p*, namely for every fixed p, <p(x) > (x,p) and that the sum of lower-semicontinuous functions is lower-semicontinuous. C o m p u t a t i o n of the Legendre transform is simple since the variables separate:  L*(q,y)  =  sup{(q,x)  =  sup{(g,x) - tp(x) ; x £ X} + sup{(?/,p) - <p*(p) ; p £ X*}  =  <p\q) +  =  f(y)  +  + (y,p)  - <p(x) - cp*(p) ; x £ X,p  H?\v) =  L(y,q),  71  £  X*}  where the linearity of X, X*, and (•, •) allowed us to reposition the minus signs i n . 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 i n 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 m i n i m u m from the m a i n 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 critical points, self-duality gives a first-order  (in a convex sense) equation to find the m i n i -  mizer.  An Example from Partial Differential Equations To reify the overarching setting and demonstrate the power of this case i n  72  particular, consider one of the simplest examples from Ghoussoub's series of papers on the subject, the non-symmetric Dirichlet problem: G i v e n functions / : Q ->• R and smooth a : Q  M  R , for 0 C R n  =  - Au + \u\ - u  =  0 on  p  2  n  bounded,  + f on  fi,  (121)  dfl,  for u G i f o ( ^ ) - To reformulate this equation as a selfdual, convex minimization problem construct the functional,  =4  / IV^I + - / \u\ + f fu, 2  :  1  Jn  p  P Jn  Jn  whose form is motivated by the fact that its Frechet derivative  (122) 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)-»(») IMU +  +  OT(u)|  3  if the limit exists and agrees for every v G X. Since the norm i n the numerator is just absolute value, we can rewrite this explicitly as,  r  t  T  /  DV(u)  ,  ,. V(u + tv) = h m —'-  73  + DV(u)  ^-L  ,  (124)  where v 6 X is any vector w i t h norm one. For completeness write explicitly the Lagrangian L : H£(Q) x # ( f 2 ) -+ R , as L(u,-u) = * ( u ) + - 1  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 • Vu =  Q~T'  ai  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)}  or, moreover, a • \7u G  74  d^(u).  = 0,  (125)  Relying on the established proof of the analogous fact i n finite dimensions, realize that if a functional I has Frechet derivative DI dl(u)  = {DI(u)}.  at u 6 X,  then  Thus the calculation of the one-dimensional limit, bearing  i n m i n d the zero boundary condition imposed by HQ (Q) when integrating by parts, yields the desired equation:  a • Vu = Au + \u\ ~ u + /. p  75  2  The Link with the Instantons T h i s final section, which indicates how the first and second chapters should be unified, w i l l unfortunately be b o t h cursory and formal. Contrariwise it be viewed positively how easy and powerful the newer, analytic formalism is and also how, once resolved i n the present context it should extend to other still richer ones. Namely the generalization of Y a n g - M i l l s 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 m i n i m u m . 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  = 0  I(x)  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 Y a n g - M i l l s functional aligns w i t h this setting from completing the square as explained at the end of the second chapter.  yM(A)  =  [  \F \ *l  (128)  2  d+A  JM  =  \f  2JM  \F \\l d+A  +\ [  JM  Z  77  \*F \\l d+A  (129)  Now define replace the general Banach space w i t h the two-forms which constitute a Hilbert space and thus their own dual space by the Riesz theorem. L a b e l the elements of the convex theory explicitly for a vector bundle E over the manifold Ml: Begin w i t h the convex functional, which here takes the form of a second power,  * : ft (Ad£;su(2)) -> E , 2  1  \  2  1  I  2  1  a n d we already know it is its own Legendre transform,  =  = I| • | : Q ( A d £ ; s u ( 2 ) ) 2  2  R.  Now, on the underlying algebra, define the operators  A , T : su(2) -»• tt (AdE; su(2)) 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 + / JM  (IM) * 1.  (130)  JM  B u t since the inner product of the curvature w i t h its Hodge star is constant,  (F ,  *F )  D  =  D  -8ir k 2  so the Y a n g - M i l l s functional has the same minimizers as the functional  I (A) yM  :=  yM(A)  + 87i k  (131)  =  f * ( A A ) * 1 + f y*(TA)*lJM JM  2  f JM  <AA,rA)*l,  which fits i n w i t h the theory of selfdual Lagrangians. Thus we have transported the Y a n g - M i l l s functional into a partial differential equations context where hopefully if can be addressed w i t h simple convex techniques and the more direct language of real analysis. Since  0tt(x) = 0 Q | a ; | ) 2  the equation TA e  d^(AA)  for the minimizer A such that  IyM(A)  79  = 0,  ={X},  becomes TA €  {AA},  or, rather, replacing the operators by their values,  *F  e  d+A  {F }. d+A  T h i s is trivially equivalent to the pair of equation,  Fd+A =  *F , d+A  which, of course, is the selfdual Y a n g - M i l l s equation requiring that the curvature F  be selfdual thus that the covariant derivative d+A be an instanton.  d+A  To recover the anti-selfdual equation, simply exploit that | * F \  2  d+A  | — *F \  2  d+A  =  and write the Y a n g - M i l l s functional as  yM(A)  = lf Z  \F  \ *l + I 2  d + A  JM  Z  f \-*F JM  \ *l. 2  d  +  A  (132)  Thus the above programme applies identically w i t h the operator T replaced by r' : A ^  - * F  and all else identical. T h i s gives way to  F'A  e  80  {AA}  d + A  or - *F  G  d+A  {F }, d+A  which, exactly i n parallel w i t h the selfdual case above, is exactly the equation  F  d+A  = —*  F , d+A  the anit-self dual equation for an anti-instanton d + A.  The Future There is a dense literature surrounding instantons and their role i n gauge theory—we d i d not talk about gauge groups, but know that a l l Y a n g - M i l l s connections have gauge equivalent curvatures—and the insights they provide into topology. Alas, we have hardly touched on the real geometry of fourdimensional manifolds. We have seen, though, the facile application of a new and surprisingly easy theory i n our complex setting. A s a suggestion for further investigation then, focus on exploiting this framework and realize that here we can naturally extend the theory to p-energies:  yM (A):=p  f VJM  \F \ *l p  d+A  + - [  \*F  QJM  \ *l- [ q  d  +  A  (F ,*F )*l, d+A  d+A  JM  (133) where 1/p + 1/q = 1. B y the same argument as i n the p = 2 case (yjvi  2  81  =  yM),  but w i t h the functionals defined as  * = -|-r,  and * *  = -|-T  we get the subdifferential,  3*(s) = aQ|*r) = {x\x\ I  Thus the m i n i m u m of yM  p  IP"  is attained at A when  *F<I+A — F \F D+A  P D+A  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", A I H P - A n a l y s e non linaire (In press) 35 pp  N . Ghoussoub: 2005, " A class of self-dual differential equations and its variational 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 M a t h e m a t i c a l Mongraphs, Providence.  K . Wehrheim:  "Anti-self-dual instantons w i t h Lagrangian boundary con-  ditions I: E l l i p t i c theory", C o m m . M a t h . P h y s . 254 (2005), n o . l , 45-89.  83  Wehrheim, K . , 2004, Uhlenbeck  Compactness,  Mathematics.  84  E M S Series of Lectures  


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items