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Gauge Fields, Knots and Gravity
Wayne LawtonDepartment of Mathematics
National University of Singapore [email protected]
(65)96314907
Lecture based on book (same title) by
John Baez and Javier Muniain
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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
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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
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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).
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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
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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(
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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
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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
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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 ?
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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
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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
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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
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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.
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MANIFOLDS
}1|),,{( 22223 zyxzyxSR
222
43222
41 )(),,( yxzyxzyxF
3114 }0),,(|),,{( RzyxFzyxSSR
}0)(:{),3( 33 MFRMRSO10))det(1,()( RMMMIMF T
}1|),{( 2212 yxyxSR
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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
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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
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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 ),()()(
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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
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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
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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
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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
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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
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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
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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.
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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.