What is the usefulness in mathematics, I know it can provide topological invariants, any other things?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
One can prove many things (enumerative results in algebraic geometry, results in the geometric Langlands program, mirror symmetry) by making physics-style computations in various quantum field theories. Currently, one is not able to make such arguments rigorous in general, and hence they become a method for producing conjectures in mathematics, not theorems. One would like to upgrade their status to being proved theorems (!), and one approach would be to find a framework in which the field-theoretic computations become valid mathematical deductions.
One current framework in which one can do something like this is that supplied by the bordism hypothesis of Baez and Dolan, recently proved by Jacob Lurie. This provides a rigorous means of constructing certain topological field theories, which is already beginning to find applications. But (for example) this framework is not encompassing enough to explain mathematically the results of Kapustin and Witten in the direction of deducing geometric Langlands from mirror symmetry.