Roles of supersymmetry in mathematics?

[arivero]

TL;DR Summary

Beyond the mathematical formalization of superfields, I wonder if supersymmetry generators have direct application in mathematics

Considering the definition of supersymmetry as a set of operators that extend the transformations of Poincare group, I wonder if they are of some value for mathematics, particularly for differential geometry. Most of the formalism I can find relate to "superspace", which does not seem a natural object in geometry. But perhaps I am missing the most mathematical literature.

 

[fresh_42]

Superalgebras are per construction mathematical objects in the first place. Whether they serve some purpose besides physics is another question. Personally, I don't think so. However, they are mathematical objects that can be studied in their own rights. E.g. we can perform analysis on supermanifolds.

 

[lavinia]

Here is a paper by Witten

http://personal.cimat.mx:8181/~gil/...rse/witten_supersymmetry_and_morse_theory.pdf

A classic book explaining Morse theory is Milnor's book unexpectedly titled "Morse Theory". Much of it is applications of Morse theory to Differential Geometry.

 

[martinbn]

The wiki article says that Atiyah-Singer can be given a simpler prove with supersymmetric quantum mechanics, but there is no reference.

https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem#Heat_equation

 

[lavinia]

The wiki article says that Atiyah-Singer can be given a simpler prove with supersymmetric quantum mechanics, but there is no reference.

https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem#Heat_equation

http://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B235_395_1984.pdf

 

[fresh_42]

http://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B235_395_1984.pdf

Friedan, Windey is (independently) from 1984.

The first one was from Ezra Getzler 1983:
Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92, No. 2, 163–178 (1983).

It might be interesting to note that there are analog theorems on supermanifolds for the theorem of implicit functions, and the transformation theorem for integrals. I have a book in which it is noted that the latter can be used to find exact solutions in the theory of dissipation systems in nuclear or condensed matter physics, known as the supersymmetric trick. The authors also published a paper about integral theorems about supersymmetric invariants that do not have a classical correspondence.

Also interesting in the context might be:
https://www.sciencedirect.com/science/article/abs/pii/0003491677903359

 

[arivero]

The idea of producing almost isospectral operators via factorisation seems be a useful one, but it sounds as a trick more for the toolbox. Given H, you factor it as Q Q*, then you check on Q* Q and voila, new operator paired to the original.

What I would expect to be real use of supersymmetry is something that applies Poincare group, or some map acting in the points of a manifold, to transform the functions on this manifold, and then extends it via susy transformations to obtain something more.