# Physical effects in supersymmetric theories of the underlying supermanifold being split or non-split?

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Given an $m$-dimensional $C^{\infty}$ manifold $M$ with an $n$-dimensional vector bundle $E$ over $M$, one can use the transition functions of the manifold and the bundle to construct an ($m,n$)-dimensional $(\mathbb{R}^{m,n}_S,DeWitt,H^{\infty})$ supermanifold denoted $S(M,E)$ with body $M$, see Theorem 8.1.1 of Rogers. $\mathbb{H}^{\infty}$ are those functions whose coefficients functions in the Grassmann analytic expansion are real valued, see Definition 4.4.3 of the same book. Supermanifolds that can be constructed in this from a vector bundle in this way are called split supermanifolds.

However, there exist also so-called non-split supermanifolds that can not be constructed in this way. Complex analytic supermanifolds are an example of this.

My question now is what are the physical effects in supersymmetric theories of the underlying supermanifold being  spit or non-split?

What (kind of) supersymmetric theories in physics are based on (non-) split supermanifolds and why? How is it decided which type of supermanifold is needed?

For example which kind of (split or non-split) supermanifolds are applied in the superspace-formalism for supersymmetric quantum mechanics, the MSSM, super Yang-Mills theories, super-gravities, string-theory, etc.?

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