Theory of connections
1 What is De Rham differential?
A PDF on 
is a function on 
.
Therefore a vector field 
on 
induces a differentiation of PDFs. This is the De Rham differential:
is a function on .
Therefore a vector field on
induces a differentiation of PDFs. This is the De Rham differential:
De Rham differential is induced by
2 Action of wavy Lie algebra on PDF
Suppose that 
acts on a manifold . Then, naturally,
acts on .
This implies that 
acts on PDF on .
3 Definition of connection
3.1 Algebraic interpretation
Connection on a -bundle 
is the same as homomorphism of 
-modules:
-bundle is the same as homomorphism of -modules:
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3.2 Geometrical interpretation as a lift
A connection 
on :
on :
Eq. (7) can be interpreted as the condition that the action of on the
vector fields commutes with the action of 
:
on the
vector fields commutes with the action of :
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In other words, having a connection is equivalent having a “lift”:
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such that vor any 
: 
. In other words, it
is the same as the split of the exact sequence of algebroids over :
: . In other words, it
is the same as the split of the exact sequence of algebroids over :
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usually
is called “Atiyah algebroid”; the projection
is its anchor
The relation between 
and is:
and is:
(This is the subtraction of the vertical component; 
is called the
“vertical component” of 
.) Notice that 
is an ideal in the Lie algebra of vector fields.
The curvature 
is defined as follows:
is called the
“vertical component” of .) Notice that is an ideal in the Lie algebra of vector fields.
The curvature is defined as follows:
It is straightforward to verify that 
where 
is
a function on (but not on , because 
should commute with the action of 
!);
this implies that the type of is indeed (9). Also notice that is automatically
horizonthal. The Jacobi identity implies for any triple of vector fields 
:
where is
a function on (but not on , because should commute with the action of !);
this implies that the type of is indeed (9). Also notice that is automatically
horizonthal. The Jacobi identity implies for any triple of vector fields :
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This is the same as:
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we identified
as
; it was important that
is
-invariant; if it were not invariant, then we would not have been able to identify
with the Lie derivative of
along
-valued function but a
-valued
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where the commutator is the commutator of the vector fields in 
. This
concludes our short review of the connections in the principal bundle.
. This
concludes our short review of the connections in the principal bundle.