Transpose Symmetry Groups of Noninvertible Polynomials

by

Print Friendly, PDF & Email

Nathan Cordner and Dr. Tyler Jarvis, Department of Mathematics

Introduction
Mirror symmetry is an area of mathematical research that stems from theoretical physics, particularly from string theory. Solutions of problems in mirror symmetry yield not only interesting mathematical results, but also have important theoretical implications for high energy particle physics.

In Landau-Ginzburg mirror symmetry, there are A and B models which describe certain aspects of the physical world. The mirror symmetry conjecture states that each A-model is isomorphic to, or the same as, a corresponding B-model. Each side is built using a polynomial (W) with its group of symmetries (G). Constructing the isomorphism requires building a transpose polynomial (WT ) and a transpose group (GT ) (see Equation 1). So far this conjectured isomorphism has been proven for a small class of cases, but it has not been proven in general.

 

7

For a certain class of polynomials called invertible, the transpose operation is understood well for both the polynomial and its group of symmetries. However, an open question at the time I began this research was how to define the transpose operation for noninvertible polynomials and their symmetry groups. The purpose of this project was to investigate this question in order to extend the mirror symmetry conjecture to noninvertible polynomials.

Methodology
My initial approach was to focus on studying the symmetry groups. I had recently developed a way to characterize these groups in terms of integer lattices. My idea was to use this characterization, as well as some of the theory developed by Artebani et. al. in [1], to come up with a generalized definition for the transpose operation. It became apparent after a few months that this approach was intractable.

My second approach was to study the polynomial and determine when its dual should exist. Equation 2 is a special case of mirror symmetry that has been proven for invertible polynomials by Krawitz in [2]. My initial assumption was that noninvertible polynomials and their symmetry groups ought to behave in similar fashion as the invertible ones.

 

6

Results
11Discussion

12

 

Conclusion

The results of this project were surprising. They show that our current understanding of mirror sym- metry for invertible polynomials does not extend easily to noninvertible polynomials and symmetry groups. Further research on this question will either require new insight on the behavior of noninvertible polynomials, or will eventually show that such duality does not exist.

References

[1] Michela Artebani, Samuel Boissiere, and Alessandra Sarti, The Berglund-Hubsch-Chiodo-Ruan mir- ror symmetry for K3 surfaces, arXiv:1108.2780v2 [math.AG].

[2] Marc Krawitz, FJRW Rings and Landau-Ginzburg Mirror Symmetry, arXiv:0906.0796v1 [math.AG].