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Stream: learning: questions

Topic: Quantization of symplectic stacks

John Baez (Oct 28 2024 at 05:07):

Has anyone done geometric quantization for some nice class of stacks? For example if we start with a smooth complex algebraic variety $X$ equipped with a complex line bundle $L$ with a hermitian metric having a positive curvature form, and $G$ is a finite group acting on the whole situation, there should be a 'stacky quotient' $X//G$ that we can still geometrically quantize.

John Baez (Oct 28 2024 at 05:13):

People might tend to think of $X//G$ as an orbifold, but treating it as a stack keeps the higher-categorical structure:

• Eugene Lerman, Orbifolds as stacks?

John Baez (Oct 28 2024 at 05:19):

Urs Schreiber has written on prequantization for stacks, but I don't know if he's done quantization:

• Urs Schreiber, Geometric quantization on moduli ∞-stacks.

John Baez (Oct 28 2024 at 05:28):

This might be relevant but I'm not seeing it emphasize that the result of stacky geometric quantization should be more than just a Hilbert space: it should be something more like a 'groupoid internal to Hilbert spaces', though what exactly it should be I don't know.

John Baez (Oct 28 2024 at 05:30):

(I have a very concrete reason for wanting to know this stuff: an example I'd like to play with.)

David Michael Roberts (Oct 28 2024 at 05:46):

Maybe something like this: a VB-groupoid where the bundles are have Hermitian metrics. There would need to be some kind of compatibility, namely the source and target maps playing well with the metrics.

Then some kind of "2-Hilbert space" of sections? Presumably there's relevant work by the Brazillian school of Lie groupoid geometry.

John Baez (Oct 28 2024 at 06:00):

It would be great if there were a 2-Hilbert space of sections, but I don't see how it works! I don't know that Brazilian school - or at least I don't know them by that name. Who are you talking about? But I can't believe nobody has done this yet, at least for the relatively easy case I mentioned: the quotient by a finite group action.

David Michael Roberts (Oct 28 2024 at 07:10):

Sure. I think I was vaguely remembering this paper, for the 2-vector space of sections of a VB-groupoid https://arxiv.org/abs/1703.09791

David Michael Roberts (Oct 28 2024 at 07:13):

I'm not sure all the people are in Brazil, but there's a non-trivial overlap.

David Corfield (Oct 28 2024 at 07:28):

Anything useful in

• Luigi Alfonsi, Higher geometry in physics?

It has a bunch of further references.

David Michael Roberts (Oct 28 2024 at 07:39):

I agree, though, that in principle, equivariant symplectic geometry for a finite group action should be possible to do without any fancy tech at all, and surely someone's done it!

Chris Grossack (they/them) (Oct 28 2024 at 17:20):

It's my understanding that deformation quantization for (shifted) symplectic stacks is somewhat developed, but geometric quantization still needs a lot of work. For some progress in this direction, you might be interested in a talk Bertrand Toën gave earlier this year, Geometric Quantization for Shifted Symplectic Structures.

Pavel Safranov also has some fairly recent work, which I unfortunately haven't read, in a paper Shifted Geometric Quantization.

John Baez (Oct 29 2024 at 00:21):

Thanks, @David Michael Roberts, @David Corfield and @Chris Grossack (they/them)! I have a specific example in mind, which should not require extremely hard machinery. It's the moduli stack of flat Riemannian metrics on the 2-sphere with 12 points removed, where the holonomy around each point is fixed to be $\exp(\pi i / 3)$ (a sixth root of unity). This particular moduli stack has some amazing properties, discovered by Thurston, who however treated it as an orbifold. But I think it'll be even more exciting if we treat it as a stack and geometrically quantize it.

John Baez (Oct 29 2024 at 00:21):

I wouldn't be surprised if people have already handled examples like this in some way or other. But unless they've studied Thurston's paper and loop quantum gravity they may have missed out on some of the fun to be had!

David Michael Roberts (Oct 29 2024 at 01:29):

One could imagine that the symplectic form $\omega$ on the complex hyperbolic space isn't just equivariant, but for each group element $g$ in the discrete group $$\Gamma$ that gives this moduli stack there is a specific family of 1-form that measures the difference between $\omega$ and $g^*\omega$, in the sense that $\omega - g^*\omega = d\alpha_g$. With this data I can imagine being able to construct a line bundle with connection on the quotient stack. This will be, in particular, a groupoid over the action groupoid, whose object component is a line bundle with connection. Then one can consider sections etc and try to construct the Hilbert space blah blah. It occurs to me that instead of just a Hilbert space, one should probably also get a representation of $\Gamma$ on that Hilbert space.

John Baez (Oct 29 2024 at 04:31):

I believe the symplectic structure on complex hyperbolic space is invariant under the group I'm interested in.

I've thought more about ordinary real hyperbolic space, which can be identified with the hyperboloid

$\{(x_0, \dots, x_n) \; \vert \; x_0^2 = x_1^2 + \cdots + x_n^2 + 1, \quad x_0 > 0 \}$

in $(n+1)$-dimensional Minkowski spacetime $\mathbb{R}^{n+1}$, and it inherits a Riemannian metric from the Lorentzian metric

$- dx_0^2 + dx_1^2 + \cdots + dx_n^2$

on Minkowski spacetime. This Riemannian metric is thus invariant under $\mathrm{SO}_0(n,1)$ (the connected component of the isometry group of Minkowski spacetime).

John Baez (Oct 29 2024 at 04:40):

Now I'm thinking about complex hyperbolic space, which can be identified with

$\{(z_0, \dots, z_n) \; \vert \; |z_0|^2 = |z_1|^2 + \cdots + |z_n|^2 + 1 \}/\mathrm{U}(1)$

in $\mathbb{C}^{n+1}$. Note that now we're first imposing the constraint $|z_0|^2 = |z_1|^2 + \cdots + |z_n|^2 + 1$, which cuts down one dimension, and then modding out by the obvious $\mathrm{U}(1)$ action , which cuts down another.

Complex hyperbolic space has a Kähler structure, and I think it inherits it from the indefinite hermitian form

$- dz_0 d\overline{z}_0 + dz_1 d\overline{z}_1 + \cdots + dz_n d\overline{z}_n$

on $\mathbb{C}^{n+1}$... in a certain sense, which is more complicated than the real case.

I've read that this Kähler structure is invariant under $\mathrm{PU}(n,1)$ (the projectivization of the group of linear transformations of $\mathbb{C}^{n+1}$ preserving the hermitian form).

The imaginary part of the Kähler structure is the symplectic structure $\omega$ I'm interested in. The group I'm interested in is a discrete subgroup of $\mathrm{U}(n,1)$.

John Baez (Oct 29 2024 at 04:54):

I may be confused, but I think one-dimensional complex hyperbolic space is our usual friend the hyperbolic plane.

John Baez (Oct 29 2024 at 06:01):

When we take the quotient of a hyperbolic plane by a well-chosen discrete group of isometries, we get a modular curve. A modular curve is a moduli space of elliptic curves with extra structure. But modular curves typically have 'stacky points' corresponding to elliptic curves with extra symmetries; if we take this into account we treat the modular curve as a moduli stack.

To geometrically quantize a modular curve, we take a line bundle over that curve and form its space of holomorphic sections - which are (ignoring some technical conditions) modular forms.

I think maybe I see how the stacky aspect affects the theory of modular forms, at least in the most vanilla case of the moduli space of plain old, undecorated elliptic curves. But enough for now!

John Baez (Oct 30 2024 at 23:43):

Okay, @David Michael Roberts, I found a nice paper that implicitly studies geometric quantization of symplectic stacks in a very simple but very important special case:

• The canonical ring of a stacky curve, John Voight and David Zureick-Brown.

John Baez (Oct 30 2024 at 23:46):

So here we have a curve with some stacky points, we consider all tensor powers $K^{\otimes n}$ of the canonical line bundle $K$ over this, we form the spaces $\Gamma(K^{\otimes n})$ of sections of these line bundles, and we note there's a multiplication

$\Gamma(K^{\otimes m}) \otimes \Gamma(K^{\otimes n} \to \Gamma(K^{\otimes (m+n)})$

This multiplication is usually treated as the multiplication in a ring that people call the canonical ring or pluricanonical ring.

John Baez (Oct 30 2024 at 23:48):

I suppose a more esthetic viewpoint is to say that this multiplication makes $\Gamma$ into a lax monoidal functor from

• the monoidal category of line bundles that are tensor powers of $K$

to

• the monoidal category of vector spaces!

John Baez (Oct 30 2024 at 23:48):

But anyway, this paper grinds out in great detail what we get for some of the famous stacky curves considered in algebraic geometry - for example, modular curves.

John Baez (Oct 30 2024 at 23:53):

It does not explicitly mention geometric quantization, but that's secretly what the subject of 'canonical rings' is all about!