Supersymmetry gauge theory including supergravity is mainly developed as a Yang - Mills gauge theory with spontaneous breakdown of supersymmetries. There are various superextensions of pseudo-orthogonal Lie algebras and the Poincaré Lie algebra. The nonlinear realization of some Lie superalgebras have been studied. However, supergravity introduced in SUSY gauge theory has no geometric feature as a supermetric.
In gauge theory on a principal bundle
P
→
M
with a structure group
K
, spontaneous symmetry breaking is characterized as a reduction of
K
to some closed subgroup
H
. By the well-known theorem, such a reduction takes place if and only if there exists a global section
h
of the quotient bundle
P
/
H
→
M
. This section is treated as a classical Higgs field.
In particular, this is the case of gauge gravitation theory where
P
=
F
M
is a principal frame bundle of linear frames in the tangent bundle
T
M
of a world manifold
M
. In accordance with the geometric equivalence principle, its structure group
G
L
(
n
,
R
)
is reduced to the Lorentz group
O
(
1
,
3
)
, and the associated global section of the quotient bundle
F
M
/
O
(
1
,
3
)
is a pseudo-Riemannian metric on
M
, i.e., a gravitational field in General Relativity.
Similarly, a supermetric can be defined as a global section of a certain quotient superbundle.
It should be emphasized that there are different notions of a supermanifold. Lie supergroups and principal superbundles are considered in the category of
G
-supermanifolds. Let
P
^
→
M
^
be a principal superbundle with a structure Lie supergroup
K
^
, and let
H
^
be a closed Lie supersubgroup of
K
^
such that
K
^
→
K
^
/
H
^
is a principal superbundle. There is one-to-one correspondence between the principal supersubbundles of
P
^
with the structure Lie supergroup
H
^
and the global sections of the quotient superbundle
P
^
/
H
^
→
M
^
with a typical fiber
K
^
/
H
^
.
A key point is that underlying spaces of
G
-supermanifolds are smooth real manifolds, but possessing very particular transition functions. Therefore, the condition of local triviality of the quotient
K
^
→
K
^
/
H
^
is rather restrictive. It is satisfied in the most interesting case for applications when
K
^
is a supermatrix group and
H
^
is its Cartan supersubgroup. For instance, let
P
^
=
F
M
^
be a principal superbundle of graded frames in the tangent superspaces over a supermanifold
M
^
of even-odd dimensione
(
n
,
2
m
)
. If its structure general linear supergroup
K
^
=
G
L
^
(
n
|
2
m
;
Λ
)
is reduced to the orthogonal-symplectic supersubgroup
H
^
=
O
S
^
p
(
n
|
m
;
Λ
)
, one can think of the corresponding global section of the quotient superbundle
F
M
^
/
H
^
→
M
^
as being a supermetric on a supermanifold
M
^
.
In particular, this is the case of a super-Euclidean metric on a superspace
B
n
|
2
m
.
