Gauge Fields, Knots and Gravity

Wayne LawtonDepartment of Mathematics

National University of Singapore matwml@nus.edu.sg

(65)96314907

Lecture based on book (same title) by

John Baez and Javier Muniain

Objective & Strategy

Manifold, Tangent Space, Bundle, Vector Field

Survey Entire Book – a Vast Landscape

Maxwell, Yang-Mills, Chern-Simons, Knots

Familiar Peak : Basic Math, Linear Algebra, Calculus

View the Landscape from a Peak

Lecture One: Structures, Affine Spaces, Derivatives

Differential Form, Exterior Derivative, DeRham Theory

Affine Connection, Covariant Derivative, Curvature

General Relativity, ADM, New Variables

Lie Groups, Lie Algebras, Flows, Principle Bundles

Structures

Elements – propositional and predicate logic

Products – relations (equivalence, order, functions)

Functions

Sets

Composition – category of sets

Structures – semigroups, groups, rings, fields, modules, vector spaces, algebras, Lie algebras

Algebra

Vector Fields on Sphere

Two Dimensional Sphere – A Manifold

Module of Tangent Vector Fields

},)(|),({)( 2322 SvvvFRSCFSVect

}1|{3

1

232 i ivRvS

Ring of Continuous Real-Valued Functions ),( 2 RSCFree Module of Vector-Valued Continuous Functions

),( 32 RSC

Theorem This module is spanned by three elements, but it is not free (it does NOT have a basis).

TRANSFORMATION GROUPS

GgSSTSSG gT |:

a setis a group

SxxxT ,)(1 GbaTTT baba ,,

,G ,S

Stabilizer Subgroup xxTGgS gx )(|

and a function

define Orbit GgxTO gx |)(For

Sx

Theoremxx SGO / is the set of left cosets

Theorem1

)( gSgS xxTg

Definitions Transitive and Free Transformation Groups

AFFINE SPACE

SSV Tis a vector space

and a transformation group

pqTVvSqp v )(!,,

Example 1

that is both transitive and free, this means that

Example 2

svsvTVS )),((,

V over a field

F

2},,{ ZFVqpS qqTppT ),0(,),0(pqTqpT ),1(,),1(

AFFINE SPACE

in an affine space we can define sub-affine spaces, eglines, planes, etc that correspond to orbits of subspaces of

q

qpv p

r

v

)(rr vTv

For any point VVOSSx x }0/{,however an affine space is NOT a vector space, however

Vwe can also define affine transformations of S by usingtranslations of V and linear transformations of V

Bases and Charts

definea basis

,VBWe can parameterize an affine space S as follows:

and construct a mapping

Choose ,Sx

Bbbbfxy )(

}supportfinite|:{)(0 fFBfBF )(: 0 BFS by

SyBFfy ),()( 0where

If V is finite dimensional and B is an ordered basis then)dim(

0 )( VFBF is a chart and its entries are the coordinate functions on S

EUCLIDEAN SPACEIs an quadruplet

is a real affine space

positive definite :

RVV :bilinear : a linear function of each argument

is a mapping that is

),V,(S, T)V,(S, T

symmetric : ),(),( uvvu 0),(,0 uuu

Definition , Vvu 0),( vuare orthogonal if

Definition , Sqp have distance

),(),( qpqpqpd Question Is this what Euclid had in mind ?

REAL AFFINE SPACE

For finite dimensional V, a canonical topology on S, and a canonical differentiable structure on V

is one where F = R, in such a space we can define

Convex combinations of points in S

SR:

Vttt

))()((lim)( 1

0

If is differentiable (at t) then

Its time to turn our attention to derivatives

A New Look at Derivativesand

)()()()()( ahfLhLaffhL Rac )(then there exists

Ra is linear

and satisfies

)(),)(()()( RCfaDfacfL

Theorem If RRCL )(:

such that

Proof First we observe that 0)constant( LTaylors Theorem implies that there exists )(RCh such that Rxxhaxafxf ),()()()(therefore ))()(()()( aDfxLahxLLf and the result follows by choosing )()( xLac Remark : this is the converse of Leibniz Law

DERIVATIONS

a point

and more generally to any manifold (to come).

RSCL )(:and the concept can be extended to

Definition Tangent Space at

Definition Functions like L are called derivations at

)(STaDefinitions Vector fields as derivations on

Ra

Sa is the set

of derivations at a and denoted by

)(SC

Remarks Why tangent spaces on the sphere are different, Lie algebra of vector fields, Leibniz Law and binomial theorem, exponential of derivations

MANIFOLDS

is Hausdorff, paracompact, and admits of charts

Definition A Manifold is a topological space X that

dRUXU

Implicit Function Theorem If

)()(1

UUUU

nFm RR and

0)(| pFRpX m and there exists

XpkpDF ,))((rankmk such that

then X is a km dimensional manifold.

MANIFOLDS

}1|),,{( 22223 zyxzyxSR

222

43222

41 )(),,( yxzyxzyxF

3114 }0),,(|),,{( RzyxFzyxSSR

}0)(:{),3( 33 MFRMRSO10))det(1,()( RMMMIMF T

}1|),{( 2212 yxyxSR

Tangent SpaceCategory

Contravariant Functor )()(*

XCYC f

),(),( qYpX f

)(),()),(()())(( ** YChXTvhfvhvf pppqp

fhhf )(*

Covariant Functor YX f

Covariant Functor )()( * YTXT qf

p

qpf )(

covcontrcont cont,contcov covcont )()(, )(

*1 dqq RTXTXUq

}{)(

1)( i

d

i

idq xvRT

Tangent Space

such that

)(,),()()()()( XCbaabpvbvpaabv ppp

)(),()),(()())(( ** YChXTvabfvabvf pppqp

pvRecall that })({)( RXCXT pv

p

is linear and satisfies

Continuous functions map tangents to tangents since

))(())(()())( *** bfaffbfafababf

))(())(())(())(())(( ***** pbfafvbfvpafabfv ppp

)()())(()())()(( ** qbavfbvfqa qpqp

Fiber BundlesDefinition

Homeomorphisms (local trivializations)

FUU

)(1

UBBE ,

Fiber

Transition Functionsupu ),(1

BuFu ,)(1

FUUFUU )()(

1

satisfy )))((,(),(1 pugupu where Uuug ,1)(

UUuugug ,))(()( 1

UUUuugugug ),()()(

Tangent Bundle

Charts

yield

dRUX

dRUU

*)(1 )(

Definition XXTXTXp p

)()(

where the fiber is homeomorphic to dR

Problem Show that the transition functions are linear maps on each fiber and derive explicit expressions for them in terms of the standard coordinates on dR

Tangent Bundle

},|{2 RyxiyxCRU kk

Example 212 UUS

where chart 1 is given by stereographic projection

If we identify

)))((,()(,( 11121 vzzbaz dz

dyx

}{\2kk pSU

)()( *1 CTUT kk then

)(},0{\ CTbaibavCz zyx

Remark vzvzvz dzd 21

*1 )(

hence Chern Class = -2

and chart 2 is ster. proj. composed with y-y

Induced BundlesDefinition Given a bundle

we construct the induced (or pullback) bundle

BB f'

BE

and a continuous map

')(' *

BE f

})()(|),({ '' ebfBEbeE and

where

bbef )),)(((*

http://planetmath.org/encyclopedia/InducedBundle.html

SectionsDefinition A section of a bundle BE

is a continuous map Bs idsEB

Remark For each local trivialization of a bundle

and choice of FpBU i

FUU

)(1

we can construct a section

of the bundle that is induced by

})(|),({' euUEueE )),(()( 1 puusp

upu ),(1

ps

since

Vector FieldsDefinition A vector fieldon a manifold M is a section of the tangent bundle, it corresponds to an R-transformation groups (perhaps local) on M.

MR pgRrMMTr ,:

MppTrg rp ),()(

This means that the trajectories

)(MTM v

satisfy

RrMTrgvrg rgprp p

),())(()),(( )(*

defined by

Distributions and ConnectionsDefinition A distribution d on a manifold M is a map that assigns each point p in M to subspace of the tangent space to M at p so the map is smooth. Twodistributions c and d are complementary if

BE

the vertical distribution d on E is defined by

Definition For a smooth bundle (spaces manifolds, maps smooth)

}0)(|)({)( * vMTvpd p Definition A connection on the bundle is a distribution on E that is complementary with the vertical distribution

EppdpcETp ),()()(

Theorem For a connection c the projection induces an induces isomorphism of c(p) onto T_p(B) for all p in E

Holonomy of a ConnectionTheorem Given a bundle BE

Proof Step 1. Show that a connection allows vectors in T(B) be lifted to tangent vectors inT(E) Step 2. Use the induced bundle construction to create a vector field on the total space of the bundle induced by a map from [0,1] into B. Step 3. Use the flow on this total space to lift the map. Use the lifted map to construct the holonomy.

and points p, q in B then every nice path (equivalence set of maps from [0,1] into B) defines a diffeomorphism (holonomy) of the fiber over p onto the fiber over q.

Remark. If p = q then we obtain holonomy groups. Connections can be restricted to satisfy additional (symmetry) properties for special types (vector, principle) of bundles.

Theorem of FrobeniusTheorem A distribution is defined by a foliation iffit is involutive.This means that if v and w are two vector fields subordinate to the distribution then their commutator [v,w] is also subordinate to the distribution. All involutive distributions give trivial holonomy groups if the base manifold is simply connected.