Supermetric
Supersymmetry gauge theory including supergravity is mainly developed as a Yang - Mills type 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 with a structure group
, spontaneous symmetry breaking is characterized as a reduction of
to some closed subgroup
. By the well-known theorem, such a reduction takes place if and only if there exists a global section
of the quotient bundle
. This section is treated as a classical Higgs field.
In particular, this is the case of gauge gravitation theory where is a principal frame bundle of linear frames in the tangent bundle
of a world manifold
. In accordance with the geometric equivalence principle, its structure group
is reduced to the Lorentz group
, and the associated global section of the quotient bundle
is a pseudo-Riemannian metric on
, 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 -supermanifolds. Let
be a principal superbundle with a structure Lie supergroup
, and let
be a closed Lie supersubgroup of
such that
is a principal superbundle. There is one-to-one correspondence between the principal supersubbundles of
with the structure Lie supergroup
and the global sections of the quotient superbundle
with a typical fiber
.
A key point is that underlying spaces of -supermanifolds are smooth real manifolds, but possessing very particular transition functions. Therefore, the condition of local triviality of the quotient
is rather restrictive. It is satisfied in the most interesting case for applications when
is a supermatrix group and
is its Cartan supersubgroup. For instance, let
be a principal superbundle of graded frames in the tangent superspaces over a supermanifold
of even-odd dimensione
. If its structure general linear
supergroup
is reduced to the
orthogonal-symplectic supersubgroup
, one can think of the corresponding global section of the quotient superbundle
as being a supermetric on a supermanifold
.
In particular, this is the case of a super-Euclidean metric on a superspace .
References
- Deligne, P. and Morgan, J. (1999) Notes on supersymmetry (following Joseph Bernstein). In: Quantum Field Theory and Strings: A Course for Mathematicians, Vol. 1 (Providence, RI: Amer. Math. Soc.) pp. 41-97 ISBN 978-0-8218-1198-6.
- Sardanashvily, G. (2008) Supermetrics on supermanifolds, Int. J. Geom. Methods Mod. Phys. 5, 271.
External links
- G. Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092.