Mysterious triality
Abstract
Mysterious duality was discovered by Iqbal, Neitzke, and Vafa in 2001 as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of Mtheory and del Pezzo surfaces, both governed by the root system series $E_k$. It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics which gives rise to the same $E_k$ symmetry pattern. We present a sequence of topological spaces, starting with the foursphere $S^4$, and then forming its iterated cyclic loop spaces $\mathcal{L}_c^k S^4$, within which we discover the $E_k$ symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its $E_k$ symmetry pattern and that of toroidal compactifications of Mtheory is no longer a mystery, as each space $\mathcal{L}_c^k S^4$ is naturally related to the compactification of Mtheory on the $k$torus via identification of the equations of motion of $(11k)$dimensional supergravity as the defining equations of the Sullivan minimal model of $\mathcal{L}_c^k S^4$. This gives an explicit duality between algebraic topology and physics. Thereby, we extend IqbalNeitzkeVafa's mysterious duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: is there an explicit relation between the del Pezzo surfaces $\mathbb{B}_k$ and iterated cyclic loop spaces of $S^4$ which would explain the common $E_k$ symmetry pattern?
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.14810
 Bibcode:
 2021arXiv211114810S
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Quantum Algebra;
 83E50 (Primary) 55P62;
 14J26;
 14J45;
 14J81;
 81T30 (Secondary)
 EPrint:
 70 pages