- Source: Connection (algebraic framework)
Geometry of quantum systems (e.g.,
noncommutative geometry and supergeometry) is mainly
phrased in algebraic terms of modules and
algebras. Connections on modules are
generalization of a linear connection on a smooth vector bundle
E
→
X
{\displaystyle E\to X}
written as a Koszul connection on the
C
∞
(
X
)
{\displaystyle C^{\infty }(X)}
-module of sections of
E
→
X
{\displaystyle E\to X}
.
Commutative algebra
Let
A
{\displaystyle A}
be a commutative ring
and
M
{\displaystyle M}
an A-module. There are different equivalent definitions
of a connection on
M
{\displaystyle M}
.
= First definition
=If
k
→
A
{\displaystyle k\to A}
is a ring homomorphism, a
k
{\displaystyle k}
-linear connection is a
k
{\displaystyle k}
-linear morphism
∇
:
M
→
Ω
A
/
k
1
⊗
A
M
{\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}
which satisfies the identity
∇
(
a
m
)
=
d
a
⊗
m
+
a
∇
m
{\displaystyle \nabla (am)=da\otimes m+a\nabla m}
A connection extends, for all
p
≥
0
{\displaystyle p\geq 0}
to a unique map
∇
:
Ω
A
/
k
p
⊗
A
M
→
Ω
A
/
k
p
+
1
⊗
A
M
{\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}
satisfying
∇
(
ω
⊗
f
)
=
d
ω
⊗
f
+
(
−
1
)
p
ω
∧
∇
f
{\displaystyle \nabla (\omega \otimes f)=d\omega \otimes f+(-1)^{p}\omega \wedge \nabla f}
. A connection is said to be integrable if
∇
∘
∇
=
0
{\displaystyle \nabla \circ \nabla =0}
, or equivalently, if the curvature
∇
2
:
M
→
Ω
A
/
k
2
⊗
M
{\displaystyle \nabla ^{2}:M\to \Omega _{A/k}^{2}\otimes M}
vanishes.
= Second definition
=Let
D
(
A
)
{\displaystyle D(A)}
be the module of derivations of a ring
A
{\displaystyle A}
. A
connection on an A-module
M
{\displaystyle M}
is defined
as an A-module morphism
∇
:
D
(
A
)
→
D
i
f
f
1
(
M
,
M
)
;
u
↦
∇
u
{\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}
such that the first order differential operators
∇
u
{\displaystyle \nabla _{u}}
on
M
{\displaystyle M}
obey the Leibniz rule
∇
u
(
a
p
)
=
u
(
a
)
p
+
a
∇
u
(
p
)
,
a
∈
A
,
p
∈
M
.
{\displaystyle \nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in M.}
Connections on a module over a commutative ring always exist.
The curvature of the connection
∇
{\displaystyle \nabla }
is defined as
the zero-order differential operator
R
(
u
,
u
′
)
=
[
∇
u
,
∇
u
′
]
−
∇
[
u
,
u
′
]
{\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}
on the module
M
{\displaystyle M}
for all
u
,
u
′
∈
D
(
A
)
{\displaystyle u,u'\in D(A)}
.
If
E
→
X
{\displaystyle E\to X}
is a vector bundle, there is one-to-one
correspondence between linear
connections
Γ
{\displaystyle \Gamma }
on
E
→
X
{\displaystyle E\to X}
and the
connections
∇
{\displaystyle \nabla }
on the
C
∞
(
X
)
{\displaystyle C^{\infty }(X)}
-module of sections of
E
→
X
{\displaystyle E\to X}
. Strictly speaking,
∇
{\displaystyle \nabla }
corresponds to
the covariant differential of a
connection on
E
→
X
{\displaystyle E\to X}
.
Graded commutative algebra
The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a graded
commutative algebra. This is the case of
superconnections in supergeometry of
graded manifolds and supervector bundles.
Superconnections always exist.
Noncommutative algebra
If
A
{\displaystyle A}
is a noncommutative ring, connections on left
and right A-modules are defined similarly to those on
modules over commutative rings. However
these connections need not exist.
In contrast with connections on left and right modules, there is a
problem how to define a connection on an
R-S-bimodule over noncommutative rings
R and S. There are different definitions
of such a connection. Let us mention one of them. A connection on an
R-S-bimodule
P
{\displaystyle P}
is defined as a bimodule
morphism
∇
:
D
(
A
)
∋
u
→
∇
u
∈
D
i
f
f
1
(
P
,
P
)
{\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}
which obeys the Leibniz rule
∇
u
(
a
p
b
)
=
u
(
a
)
p
b
+
a
∇
u
(
p
)
b
+
a
p
u
(
b
)
,
a
∈
R
,
b
∈
S
,
p
∈
P
.
{\displaystyle \nabla _{u}(apb)=u(a)pb+a\nabla _{u}(p)b+apu(b),\quad a\in R,\quad b\in S,\quad p\in P.}
See also
Connection (vector bundle)
Connection (mathematics)
Noncommutative geometry
Supergeometry
Differential calculus over commutative algebras
Notes
References
External links
Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv:0910.1515 [math-ph].