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 $P\to M$ 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.

Supermetric

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