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}.

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