Supermetric

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

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

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

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

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