Friday, January 11, 2013

Motivic Mirror Symmetry




















Rolf Schimmrigk's new paper on the arxiv demonstrates that the Omega-motives of exactly solvable mirror pairs of Calabi-Yau threefolds are isomorphic, as expected by mirror symmetry since the L-functions of the Omega-motives of such pairs are predicted to be identical.

String theory suggests a relation between two-dimensional conformal field theory and the geometry of Calabi-Yau varieties.  This comes in the form of relating automorphic forms derived from pure or mixed motives that arise in Calabi-Yau varieties to modular forms that come from Kac-Moody algebras on the worldsheet.  Such relations allow an arithmetic link between the spacetime and worldsheet theory, giving a means to pass from the extra-dimensional geometry to the worldsheet and back.

In the framework of M-theory, it would be of interest to generalize the Calabi-Yau results to G2 manifolds and other higher dimensional compactification spaces.

3 comments:

ThePeSla said...

Kneemo,

I found the paper on motives most enlightening and reassuring at a time when I was exploring some old methods again. I am not a fan of the Calabi-you geometry of compactification but it has it place in the general scheme.

I thought you might like to see the recent posts that I have long held of many of the suggestions or proved methods of this sort of algebra.

www. pesla. blogspot. com the last is the A-brane idea which beyond the M framework theory goes far into more abstract spaces than your closing sentence here on higher generalization.

The PeSla L. Edgar Otto

Mitchell said...

And another from Schimmrigk today.

kneemo said...

@ThePeSla A-brane as in adelic brane?

@Mitchell Yes, Rolf is on a roll.