Post #3736714

The exceptional groups E₈, E₇ and E₆ are famous - but in fact we can define Eₙ groups for smaller n too. E₃ is the gauge group of the Standard Model! Some larger ones are gauge groups of famous grand unified theories. I used to think this was cute - but now I think it could be a clue. I'm starting to see how this series of groups is connected to the foundations of quantum physics. Here's part of the story that remains puzzling to me - it's about things called del Pezzo surfaces. A del Pezzo surface is a special kind of 2-dimensional complex manifold, so it's a 4d manifold in the usual sense. Any 4d manifolds gives a lattice, called its 2nd cohomology. A del Pezzo surface gives a lattice with a special vector in it called its 'canonical class'. And if we look at all the lattice vectors orthogonal to this, we get a special sort of lattice called an Eₙ lattice! From this there's a way to get the Eₙ group - that's something you have to learn to understand the Eₙ groups. But what the heck is a del Pezzo surface? In algebraic geometry you can 'blow up' a surface by removing a point p and sticking in a bunch of new points, one for each direction in which you could approach p. But be careful: we're working with complex numbers, and we count two vectors as giving the same 'direction' if is one is some complex number times the other. You get a del Pezzo surface if you take the complex projective plane ℂℙ² and blow it up at a bunch of points in 'general position'. That roughly means that they're random, nothing special about them. And here's the shocking part: if you blow up at n points, you get a del Pezzo surface that gives the Eₙ lattice! What's going on? https://en.wikipedia.org/wiki/Del_Pezzo_surface