# (Super)integrable systems on quiver varieties

In recent papers

https://arxiv.org/abs/2101.05520

https://arxiv.org/abs/2001.06911

(super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed.

My question: are there heuristics and/or conjectures about the existence of (super)integrable systems on more general quiver varieties (i.e. quiver varieties associated to more general quivers)?

Note: in https://arxiv.org/abs/2001.06911 the authors state that "It is generally expected that Nakajima quiver varieties ought to be algebraically completely integrable Hamiltonian systems with a Hitchin-like fibration" but I am wondering what is the status of such expectations and how precisely they have been formulated. In https://arxiv.org/abs/2001.06911 the authors also refer to Nakajima's paper "Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras" for a discussion of this question but I could not find such a discussion there in a direct form, at least there is no mentioning of the words 'integrable system' there.

http://arxiv.org/abs/2001.06911 referred my paper. But I meant more formal analogy between quiver varieties and Hitchin moduli spaces, such as hyper-Kaehler structure, S^1 action scaling the symplectic form, etc. I also pointed out that an analog of the Hitchin integrable system is the affinization' morphism. It is often resolution of singularities, and generic fibers are points. Hence it is not an integrable system.

A comet-shaped quiver is skelton' of punctured Riemann surface, and its quiver variety is additive analog of the Hitchin moduli space. This analogy goes back (at least) to https://arxiv.org/abs/math/0103101. Therefore it is natural to look for an integrable system in this case. (I do not mean that http://arxiv.org/abs/2001.06911 was known before.)

Another example is a quiver variety of affine type A. It is known to be isomorphic to Coulomb branch of a quiver gauge theory of affine type A, different from the original in general. (https://arxiv.org/abs/1606.02002) By a general property of the Coulomb branch, it is equipped with an integrable system. https://arxiv.org/abs/1601.03586.

• Thank you so much for your clarification! Do you think there are more special cases (aside from comet-shaped quivers and quivers of affine type A) where quiver varieties are likely to have integrable systems on them? May 7, 2021 at 17:02
• I never ask the question to myself, hence I do not know whether likely or unlikely. In the above two examples, integrable systems are given by first identifying quiver varieties with other moduli spaces. Such an identification does not happen often. May 8, 2021 at 23:33
• Thank you so much for your comment! May 9, 2021 at 5:42

Yes of course, the supernova graphs generalise the star-shaped graphs. They are like the stars, but have "more going on in the middle"; for example their core can be any complete multipartite graph. See "Simply-laced isomonodromy systems, Publ. Math. IHES 116 (2012) no. 1, 1–68" especially section 4 on isospectral integrable systems, and the remark there that our big symplectic vector space generalises the "generalised Moser space" of Adams-Harnad-Hurtubise-Previato. This big symplectic vector space is the space of representations of a doubled quiver whose underlying graph is complete k-partite. E.g. we get the triangle, the square, the tetrahedron and the octahedron as simple examples. The stars appear since a complete bipartite graph with just one node in one of the two parts, is star-shaped. Then there is a general story about gluing "legs" on to the core (complete multipartite) graph.

A typical picture might be as in the linked image: 1, or as in the backdrop/poster here:

https://webusers.imj-prg.fr/~philip.boalch/Mars2017/index.html

Beyond this simply-laced setting, a more general "quiver modularity theorem" (**) was conjectured in appendix C of my 2008 paper "Irregular connections and Kac-Moody root systems" and proved by Hiroe-Yamakawa in their 2014 Adv. Math. paper.

For me the key conceptual breakthrough, that took about 10 years to realise, was the following: in the general linear case the Birkhoff orbits O_B (in my 2001 Adv. Math paper) should be viewed as the large symplectic vector space in the Nakajima construction.

In more detail: I understood the orbits O_B since 1999 (my thesis), and knew from a result of Vergne that they had global Darboux coordinates, but it wasn't until the 2008 paper that I saw the relation to quiver varieties; in fact the first observation was really Exercise 3 in the 2007 paper I wrote for the John McKay volume ("Quivers and difference Painleve equations"), and then I saw that the bipartite case essentially came from the "generalised Moser spaces" of Adams et al (and that this already contained all the stars): this bipartite observation was at the end of my 2007 talk in Princeton, archived recently here:

http://web.archive.org/web/20211224113418/https://webusers.imj-prg.fr/~philip.boalch/files/boalch_2007_IAS%20talk,%20Geometry%20of%20irregular%20connections%20on%20curves.pdf

One reason this took so long was because, for a while, I only wanted to use methods that worked for all reductive groups, not just for GL_n.

There are several things worth mentioning that follow from this link between 2d gauge theory and quiver varieties, such as the global Weyl group, the Lax project (the quivers are like Dynkin diagrams for some [wild] nonabelian Hodge spaces), a new theory of multiplicative quiver varieties, graphical Deligne-Simpson problems,...

Finally, if you know the Atiyah-Gowers discussion about the two cultures of mathematics, solitons versus graph theory, it may be worth pointing out that one thus can get somewhere in integrable systems by "sitting in an armchair and trying to understand graphs better"...

(**) i.e. identifying Nakajima quiver varieties with symplectic spaces of matrices of rational one-forms, that we think of as moduli spaces M^* of meromorphic connections on the trivial bundle on the Riemann sphere (or equivalently as meromorphic Higgs bundles).

• Thank you very much for your very interesting answer! Sorry, is the existence of an integrable system structure on quiver varieties for supernova graphs a conjecture or is it already established by someone? Jan 13 at 16:27