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I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.
added reference to A convenient category of supermanifolds to supermanifold
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for discussion of the functorial geometry-perspective
So what’s the precise differente/relationship between using supermanifolds and manifolds with spin bundles to describe fermionic fields? I understand that, strictly speaking, the former is about fermionic fields (i.e. anti-commuting fields) whereas the latter is about spinors (of a particular representation), and that these are related by the spin-statistics theorem, but what is the concrete connection between these two approaches? Given a spinor bundle one can take exterior powers and construct a supermanifold in the spirit of Batchelor’s theorem (well, it should be a supermanifold with Spin group action), but what about the converse? The latter theorem is only about vector bundles, it doesn’t care about these being associated vector bundles to a Spin principal bundle.
In particular, say in the NSR string when we speak of requiring a manifold to admit a spin structure, does this mean in the GS string to require the supermanifold to be isomorphic to the exterior power of some spinor bundle?
On the first part: Yes, supermanifolds as such are about having graded-commutative algebras of smooth functions. That these have a Spin-action is a separate requirement.
In general, the math-literature tends to use “super” for anything “$\mathbb{Z}/2$-graded-commutative”, while the physics literature uses it much more specifically for “graded-skew extension of a Poincaré-symmetry group (or of a conformal group).
On the last question: Not sure I understand exactly what you are asking. The point to notice is that for the NSR-string the worldsheet is a supermanifold, while target spacetime is an ordinary manifold, while for the GS-string it is the other way around. (In the “super-embedding formalism” both worldvolume and target-space are supermanifolds.)
So is the Spin-action requirement on a supermanifold essentially that it is modeled not on some random R^{n|m} but on some R^{n|N} for N some Spin representation?
Also, thanks for the super-embedding formalism pointer, clarified a lot.
Yes!
Okay, that makes sense. But then what’s the analogous statement for the other elements in the SO tower? I’m guessing not a lot can be said about this yet since even the definition of stringor bundle is recent, but maybe I’m wrong. I imagine one would construct spaces whose probes are not vector bundles over Cartesian spaces (i.e. super Cartesian spaces) but algebra bundles over Cartesian spaces for the String analog?
By the way, the super-embedding approach indeed seems to get at the heart of the matter.
In Prop. 6.10 on p. 63 here we observe that the GS-action functional for the super $p$-branes without worldvolume gauge fields (hence including the M2 but not the M5) are just the super-volume forms on the super-embedded worldvolume. So this makes them exactly the super-analog of the Nambu-Goto action that one may have hoped for all along.
Regarding pairing higher spin geometry with supergeometry: I see what you have in mind here. Some kind of higher super-geometry with rings of functions given by higher analogs of superalgebras.
There is nothing known about this, and next to nobody has considered this. The only exception is Kapranov 2013 suggesting that the higher gradings of higher supergeometry ought to be controlled by the sphere spectrum, and then my proposal for a definition of “spectral super-scheme” here, along the lines of an observation earlier made by Rezk 2009.
Are stringors not what would make a natural formulation of super string field theory?
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Pierre Deligne, John Morgan, Ch 2 in: Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, DeligneMorgan-NotesOnSusy.pdf:file]
Ivan Mircović, Sec 2 in: Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, Mirkovic-NotesOnSupermathematics.pdf:file]
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