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 $P\to M$ 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 $P/H\to M$. This section is treated as a classical Higgs field.

In particular, this is the case of gauge gravitation theory where $P=FM$ is a principal frame bundle of linear frames in the tangent bundle $TM$ of a world manifold $M$. In accordance with the geometric equivalence principle, its structure group $GL(n,\mathbb R)$ is reduced to the Lorentz group $O(1,3)$, and the associated global section of the quotient bundle $FM/O(1,3)$ 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 $\widehat P\to \widehat M$ be a principal superbundle with a structure Lie supergroup $\widehat K$, and let $\widehat H$ be a closed Lie supersubgroup of $\widehat K$ such that $\widehat K\to \widehat K/\widehat H$ is a principal superbundle. There is one-to-one correspondence between the principal supersubbundles of $\widehat P$ with the structure Lie supergroup $\widehat H$ and the global sections of the quotient superbundle $\widehat P/\widehat H\to \widehat M$ with a typical fiber $\widehat K/\widehat H$.

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 $\widehat K\to \widehat K/\widehat H$ is rather restrictive. It is satisfied in the most interesting case for applications when $\widehat K$ is a supermatrix group and $\widehat H$ is its Cartan supersubgroup. For instance, let $\widehat P=F\widehat M$ be a principal superbundle of graded frames in the tangent superspaces over a supermanifold $\widehat M$ of even-odd dimensione $(n,2m)$. If its structure general linear supergroup $\widehat K=\widehat{GL}(n|2m; \Lambda)$ is reduced to the orthogonal-symplectic supersubgroup $\widehat H=\widehat{OS}p(n|m;\Lambda)$, one can think of the corresponding global section of the quotient superbundle $F\widehat M/\widehat H\to \widehat M$ as being a supermetric on a supermanifold $\widehat M$.

In particular, this is the case of a super-Euclidean metric on a superspace $B^{n|2m}$.

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.

See also

• Supergeometry
• Supergravity
• Super Minkowski space
• Gauge gravitation theory