10
$\begingroup$

In this MO question it is stated that there is a relation between some aspects of condensed matter physics (namely the fractional quantum Hall effect) and the algebraic geometry of Hilbert schemes.

Having made a quick google search without immediate results, I'm curious to know:

  1. How does this interaction between the two topics happen?

Let's have a more general look. I'm aware of the relation between Hilbert schemes -and also other kinds of moduli spaces of sheaves- and instantons. Anyway, here I would be interested to know of other examples more in the vein of the MO question linked above.

  1. Do moduli spaces in the sense of algebraic geometry -of which Hilbert schemes are a special case- have any link with physics (except string theory, high energy and elementary particle physics)?

Last but not least:

  1. What about moduli spaces and applied mathematics?
$\endgroup$
  • $\begingroup$ Moduli spaces come up in quite a variety of cases. For example, you could imagine protein folding to be a differential equation partially defined on a moduli space (you need to keep things embedded in euclidean space, of course...) The surface would be some kind of equi-potential surface around the molecule. $\endgroup$ – Ryan Budney Feb 16 '16 at 4:10
  • $\begingroup$ @RyanBudney: cool, do you have some reference? $\endgroup$ – Qfwfq Feb 16 '16 at 15:16
  • $\begingroup$ This is a topic that Joergen Andersen sometimes talks about but it looks like it hasn't been all put together in one paper. But some of his work flows in this sea of ideas. pure.au.dk/portal/en/andersen@qgm.au.dk $\endgroup$ – Ryan Budney Feb 16 '16 at 19:56
  • 1
    $\begingroup$ Should this question be made community wiki? $\endgroup$ – Qfwfq Feb 25 '16 at 9:25
  • 1
    $\begingroup$ Here's a recent article by Penner along the lines of @RyanBudney's comment ams.org/journals/bull/0000-000-00/S0273-0979-2016-01524-2 $\endgroup$ – j.c. Feb 26 '16 at 17:29
5
$\begingroup$

Regarding question 2: FQHEs are examples of topological phases of matter, whose effective field theories are topological quantum field theories, whose quasi-particle excitations are (essentially) described by modular tensor categories (MTCs). Since what we're usually interested in for constructing models of topological quantum computing (and related things) is the algebraic data of these quasi-particles, the problem of classifying topological phases is very closely related to the problem of classifying modular tensor categories.

Since we're talking about categories what we really want to do is classify them up to something, which in this case is (braided) monoidal equivalence. It turns out though that, for every equivalence class of MTCs, representatives of that equivalence class can be constructed from solutions to certain polynomial equations called the pentagon, hexagon, and pivotal equations. The collection of these solutions define an algebraic set $X$.

For a given $X$ there exists an algebraic group $G$ which acts on $X$ such that for points $F\text{ and }F' \in X$, $F$ and $F'$ give rise to monoidally equivalent categories if and only if there exists $g \in G$ such that $g \cdot F = F'$. Thus the orbits of $G$ in $X$ are in 1-1 correspondence with equivalence classes of categories, and so now we can consider the problem of classifying orbits of $G$.

It turns out that in doing this we have almost the nicest possible situation imaginable - $G$ is reductive, all orbits have the same dimension and are in fact the irreducible components of $X$. This then implies that we can construct another algebraic set $Y$ which is an orbit space for $X$ - that is to say that the points of $Y$ are in 1-1 correspondence with orbits of $G$ in $X$ and the regular functions on $Y$ are those regular functions on $X$ which are invariant under the action of $G$. All of this allows us to classify orbits (i.e. MTCs) by looking at the evaluations of $G$-invariant functions on $X$.

Picking these functions is a really hard problem in general, but for MTCs we have a set of generic candidates: Every MTC gives you a pair of matrices $(S,T)$ which are the so called modular data of the category. They are called this because they specify a representation of the modular group $SL(2,\mathbb Z)$. It is conjectured that MTCs are classified by their modular data.

Bringing things back to physics, the $(S,T)$ matrices have physical meaning - the entries of the $S$-matrix encodes the mutual statistics between particle types and the $T$-matrix encodes the self-statistics. Additionally, the entries of $S$ and $T$ are given by the evaluations of regular functions on $Y$ which is to say that they are given by the evaluation of $G$-invariant regular functions on $X$.

That this has physical meaning can be seen by noting that $G$ is (essentially) the gauge group for our TQFT. Given two quasi-particles $a$ and $b$, the state space $V_{a b}$ for their composite system is finite dimensional and decomposes in to subspaces $V_{a b}^c$, where $c$ is another quasi-particle type (including the vacuum) and whose dimension is the number of fusion channels from $a\otimes b$ to $c$. $G$ is the direct product of the groups of basis transformations on the $V_{ab}^c$ spaces.

The information in paragraphs 1,6, and 7 is pretty standard and can basically be found in these lecture notes. The details for paragraph 2 can be found in arxiv:1305:2229 and for paragraphs 3 and 4 in arxiv:1509.03275.

$\endgroup$
4
$\begingroup$

In response to [2], the classification of topological insulators (or more generally, of any gapped condensed matter, so also superconductors) relies on moduli spaces, see for example this tutorial by Daniel Freed. (For a more formal description, see here.)

$\endgroup$
3
$\begingroup$

Re: q. 3), consider the geometry of the space of phylogenetic trees.

$\endgroup$
3
$\begingroup$

You can consider Nekrasov-Shatashvili for application to Toda chain or Calagero-Moser type systems. That nominally looks like a string or high energy particle link, but those systems are integrable chain systems. Looking for any experiments on these sorts of systems, the setup might be some optical trap like http://www.nature.com/nature/journal/v440/n7086/full/nature04693.html

I don't know if you would call this sort of Atomic, Molecular, Optical system as a cheating answer because it is designed to have that integrable structure.

Same principal for any other $\mathcal{N}=2$ way of getting any other integrable chain that could be engineered with laser traps.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.