⇤Math diary
May 6, 2023: Geometry on the space of functions, and conformal transforms
Geometry of the space of functions
Today I tried to understand geometry on spaces of functions. For example, consider $\mathcal{H}$ the space of $L^2$ functions from $S^1$ to $\mathbb{R}$. What are differential forms on $\mathcal{H}$? What is the exterior derivative? How does this look as a structure on $\mathbb{R}$? This is to build up to the De-Rahm complex on spaces of maps from $S^1$ to $X$, and the chiral De Rahm algebra that this induces on $X$. I’m certain this information exists somewhere. It’s probably standard, in fact I think it’s implicit in some things I’m supposed to know (like Atiyah-Bott’s argument that the curvature is a moment map.) Still, I want to try to figure it out myself.
Suppose you have a functional $S:\mathcal{H} \to \mathbb{R}$. What is its gradient? Well, writing $S = \int_{S^1} \mathcal{L}(f,f’)$, I worked out \(\nabla S = \int_{S^1}\frac{\mathrm{d}l \mathcal{L}}{\mathrm{d}l f} - \frac{\mathrm{d}}{\mathrm{d} x}\frac{\mathrm{d}l \mathcal{L}}{\mathrm{d}l f'}\) Look familiar? Try and find a critical point where $\nabla S = 0$, and these become the Euler legrange equations. In other words, one side of the euler largeangian equations is simply the gradient of the functional! This means we can actually compute stuff.
Moreover, its the first instance I’m really appreciating that geometry on spaces of functions is quantum field theory. I suppose I have been warned. Witten said hodge theory on loop space was just symmetry, etc. Still, its amazing that something as simple as the gradient pops
Conformal tranforms
Also, I read today about the group of conformal transformations, to prepare myself for conformal field theory. The idea is quite cool. You start with $\mathbb{R}^{p,q}$, which is $\mathbb{R}^{n}$ for $n = p+q$ with an indefinite metric of signature ${p,q}$. To find the conformal transforms, you embed $\mathbb{R}^{p,q}$ into $S^p \times S^q \subset \mathbb{R}^{p+1} \times \mathbb{R}^{q+1}$, where $S^p$ is the unit sphere. We endow this with a lorentzian metric. Then, the conformal transforms of $\mathbb{R}^{p,q}$ all come from linear transforms of $\mathbb{R}^{p+1,q+1}$ which preserve the product of unit spheres. This is in turn anything which preserves the metric on $\mathbb{R}^{p+1,q+1}$. So, the group of conformal transforms of $\mathbb{R}^{p,q}$ is isomorphic to $SO(p+1,q+1)$.
The stragity of examining the level sets of the quadratic form preserved by the lie group reminded me of a twitter thread I wrote a while back. The idea is, we can tell if a group is compact or not by wether the “unit sphere” in the preserved metric is compact or not. Moreover, we can find the maximal compact subgroup from the geometry of the unit sphere. This thread was from when I had like 50 followers, so it got no engagement. Looking back on it, I actually used to be funny, and the idea is quite attractive. Couple things I would change, but overall, good little thread.