# Roles of supersymmetry in mathematics?

• A
• arivero
In summary, the conversation discusses the value of supersymmetry in mathematics, specifically in relation to differential geometry. It is noted that supersymmetry is a mathematical object that can be studied on its own, and there are applications of supersymmetry in fields such as Morse theory and quantum mechanics. The Atiyah-Singer index theorem can also be simplified using supersymmetry. However, it is mentioned that supersymmetry may be seen as more of a tool rather than a fundamental concept in mathematics. Other interesting papers and concepts related to supersymmetry are also referenced in the conversation.
arivero
Gold Member
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.

martinbn
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 said:
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

quasar987
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.

## 1. What is supersymmetry and how does it relate to mathematics?

Supersymmetry is a theoretical concept in physics that proposes a symmetry between particles with different spin values. In mathematics, it has been studied as a way to unify different theories and provide a deeper understanding of mathematical structures.

## 2. What are the main applications of supersymmetry in mathematics?

Supersymmetry has been applied in various areas of mathematics, such as algebraic geometry, representation theory, and topology. It has also been used to study integrable systems and quantum field theory.

## 3. How does supersymmetry impact our understanding of symmetries in mathematics?

Supersymmetry has provided new insights into the concept of symmetry in mathematics, particularly in the study of Lie groups and algebras. It has also led to the discovery of new symmetries in certain mathematical structures.

## 4. What are the current developments in the study of supersymmetry in mathematics?

There is ongoing research in the use of supersymmetry to solve problems in various branches of mathematics, such as number theory and differential geometry. Additionally, there are efforts to develop a mathematical framework for supersymmetric quantum field theories.

## 5. How does the presence or absence of supersymmetry affect the mathematical models used in physics?

The presence of supersymmetry in a physical system can greatly simplify the mathematical models used to describe it, making calculations more manageable. On the other hand, the absence of supersymmetry can also lead to new and interesting mathematical structures and techniques being developed.

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