Why can't we take three loops?

Question

Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:

• No loops: here we study a simple Lie algebra $\mathfrak{g}$, a Weyl group, a braid group, or a Hecke algebra, all of which have something to do with a Lie group $G$.
• One loop: here we study an affine Lie algebra $\widehat{\mathfrak{g}}$, a quantum group $U_q(\mathfrak{g})$, an affine Weyl group, an affine braid group, or an affine Hecke algebra, all of which I think have something to do with the loop group $LG$ of $G$.
• Two loops: here we study a double affine Lie algebra $\widehat{\widehat{\mathfrak{g}}}$, an affine quantum group $U_q(\widehat{\mathfrak{g}})$, an elliptic quantum group (whatever that means), a double affine or elliptic Weyl group, a double affine or elliptic braid group, or a double affine or elliptic Hecke algebra, all of which I think have something to do with the double loop group of $G$, or maybe more precisely the space of maps of some sort from an elliptic curve $E$ to $G$.

The suggestion is further that this pattern doesn't continue.

Why doesn't this pattern continue?

I asked around and got an answer that I interpreted as follows. The trichotomy above can be matched up to the trichotomy additive group $\mathbb{C}$ : multiplicative group $\mathbb{C}^{\times}$ : elliptic curve $E$. Here is one story about the match as I understand it, which is not very well.

• One-dimensional algebraic groups give rise to equivariant cohomology theories. The above theories give rise to equivariant cohomology, equivariant K-theory, and equivariant elliptic cohomology respectively.
• Roughly, $\text{Spec } H^{\bullet}_G(\text{pt}) \cong \mathfrak{g}/G \cong \text{Bun}_G(\mathbb{C})$, while $\text{Spec } K^{\bullet}_G(\text{pt}) \cong G/G \cong \text{Bun}_G(\mathbb{C}^{\times})$, and $\text{Spec } E^{\bullet}_G(\text{pt}) \cong \text{Bun}_G(E)$, where by $E^{\bullet}_G$ I mean the equivariant elliptic cohomology theory associated to the elliptic curve $E$.
• There is some yoga in geometric representation theory which I'm not all that familiar with involving building interesting algebras like group algebras of Weyl groups and Hecke algebras by computing the equivariant (co)homology or equivariant K-(co)homology of some varieties of interest, which has something to do with the construction of the affine and double affine Hecke algebras mentioned above.

Since we've run out of one-dimensional algebraic groups, that would be some reason to believe that the pattern doesn't continue. But nevertheless I don't have a good sense of what, if anything, prevents us from studying and saying interesting things about "triple affine Lie algebras," "triple affine Weyl groups," "triple affine Hecke algebras," etc. at least insofar as the triple loop group of a group seems perfectly well-defined. On the geometric side it seems like there's nothing stopping us studying $G$-bundles on higher dimensional varieties. On the cohomological side, cohomology, K-theory, and elliptic cohomology should optimistically just be the first three terms of an entire sequence of cohomology theories at higher chromatic levels, or from the perspective of the Stolz-Teichner program, defined in terms of higher-dimensional field theories...

Answer 1

To elaborate on Kevin's excellent answer, one can account for the current absence of "higher loop" representation theory using physics. Namely, all of the representation theoretic structures you mention fit in very naturally into the study of gauge theory, specifically 4-dimensional $\mathcal N=2$ gauge theories. These come in two main classes (with some intersection) - the quiver gauge theories, which are the natural homes for algebras like Yangians, quantum loop algebras, and elliptic quantum groups; and the class S theories (reductions of the 6d "theory $\mathfrak X$" -- the (2,0) superconformal field theory labeled by a Dynikin diagram - on Riemann surfaces), which are the natural home for geometric Langlands, double affine Hecke algebras, Khovanov homology etc. (the theory Kevin describes associated to $U_q(\mathfrak g)$ is $\mathcal N=4$ super Yang Mills, which is the case when the Riemann surface is the two-torus).

So why should this be relevant? the question of attaching interesting representation theory to maps into Lie groups is very closely linked to the question of finding interesting gauge theories in higher dimensions (the latter is strictly stronger but seems like the most natural framework we have for such questions). Specifically, we want supersymmetric gauge theories, if we want them to have any relation to topological field theory or algebraic geometry etc.

However there are no-go theorems for finding gauge theories in higher dimensions. Even at the classical level it is impossible (thansk to Lie theory, namely the structure of spin representations) to have a supersymmetric gauge theory in more than 10 dimensions ---- any SUSY theory in dimensions above ten also includes fields of spin two and above (so physically is a theory of gravity), while above dimension 11 we have to have higher spin fields still (which physicists tell us doesn't make sense -- regardless it won't be a gauge theory). In any case theories with gravity and other stuff are a very far stretch to be called representation theories!

At the quantum level (which is what we need for representation theory) it's much harder still -- I believe there are no UV complete quantum gauge theories above dimension 4 (in other words higher dimensional theories have to have "other nonperturbative stuff in them"). All of the representation theoretic structures you mention naturally fit into theories that come from six dimensions at best (reduced to 4 dimensions along a plane, cylinder or torus in the quiver gauge theory case to see Yangians, quantum affine algebras and elliptic quantum groups, or along a Riemann surface in the class S case). Studying in particular theory $\mathfrak X$ on various reductions gives a huge amount of structure, and includes things like ``triple affine Hecke algebras" presumably when reduced on a three-torus, but there's a clear upper bound to the complexity you'll get from these considerations.

Now of course you might ask what about theories that don't come from supersymmetric gauge theory? the only interesting source I've heard of for higher dimensional topological field theories is (as you hint) chromatic homotopy theory, in particular the fascianting work of Hopkins and Lurie on ambidexterity in the$K(n)$-local category. This is a natural place to look for "higher representation theory", which is touched on I believe in lectures of Lurie -- but my naive impression is these theories will have a very different flavor than the representation theory you refer to (in particular a fixed prime will be involved, and these theories certainly don't feel like traditional quantum field theory!). But it's a fascinating future direction. For a hint of what kind of representation theory this leads to we have the theorem of Hopkins-Kuhn-Ravenel describing the $n$-th Morava K-theory of BG in terms of n-tuples of commuting elements in G --- i.e. the kind of characters you might expect for G-actions on $(n-1)$-categories.

Answer 2

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra (by which I really mean the HOMFYPT completion thereof) or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a 0-category, i.e. a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi, Adrien Brochier, and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.