A. Okounkov said, "symplectic resolutions are Lie algebras of the 21st century." Is there a conjecture on the classification of symplectic resolutions? Do Braverman-Finkelberg-Nakajima Coulomb branches give most known examples of symplectic singularities (and do BFN Coulomb branches have explicit descriptions)? Where can one find a list of all known examples of symplectic resolutions? What are the consequences of the classification of symplectic resolutions in representation theory etc.? Is classification of symplectic resolutions a very hard problem (or, if it is intractable, is there a nice class of symplectic resolutions analogous to semisimple Lie algebras that can be classified)? What are some directions in this problem that can be approachable (cf. results of Bellamy-Schedler)? Also, is there an object "Lie group of the 21st century" which fits into an analogy [Lie group of the 21st century] : [symplectic resolution (Lie algebra of the 21st century)] = Lie group : Lie algebra (I suppose quantizations of symplectic resolutions loosely correspond to universal enveloping algebras in this analogy)?
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4$\begingroup$ but there is no reasonable classification of Lie algebras, only of semisimple ones. $\endgroup$– user145520Commented Apr 21, 2020 at 11:46
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4$\begingroup$ @vrz so maybe there is a reasonable class of symplectic resolutions that can be classified? $\endgroup$– Yellow PigCommented Apr 21, 2020 at 11:55
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2$\begingroup$ maybe, I was nitpicking really. $\endgroup$– user145520Commented Apr 21, 2020 at 11:56
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1$\begingroup$ Do you have a reference for the Okounkov quote? $\endgroup$– LSpiceCommented Apr 28, 2020 at 22:45
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2$\begingroup$ @LSpice This is a talk of Okounkov with this title: youtu.be/3ezeRLjQgI0 $\endgroup$– Yellow PigCommented Apr 29, 2020 at 3:50
2 Answers
I don't have enough reputation to comment so I will post it as an answer. Some classification of symplectic resolutions was done by Namikawa (Poisson deformations and birational geometry). As observed in Kubrak and Travkin - Resolutions with conical slices and descent for the Brauer group classes of certain central reductions of differential operators in characteristic $p$, given a singular variety $Y$ over a field of char $0$ and provided there exists at least one symplectic resolution $\pi: X \rightarrow Y,$ the vector space $V_{\mathbb{R}}=\operatorname{Pic}(X) \otimes_{\mathbb{Z}} \mathbb{R}$ can be partitioned into a union of rational cones, and there is an action of a finite group $W$ on $V_{\mathbb{R}}$ that maps cones to cones. The set of symplectic resolutions $\pi: X \rightarrow Y$ is then identified with the set of cones modulo the action of $W$.
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$\begingroup$ Thanks a lot! I guess I was wondering about the possibility of a more explicit classification (like the classification of semisimple Lie algebras) $\endgroup$ Commented Apr 29, 2020 at 3:59
Here is an answer by Gwyn Bellamy, which he let me post here:
1) Is there a conjecture on the classification of symplectic resolutions? No, not that I am aware of. I think this is the wrong question anyway. Rather, one should first try to classify all conic symplectic singularities. There is an amazing result of Namikawa that says that if you bound the degrees of your algebra of functions on the singularity then there are only countably many isomorphism classes. So it is not inconceivable that a classification is possible. I believe that Namkiawa is trying to develop such a classification program. See in particular the papers of his PhD student T. Nagaoka. I think if we had such a classification then it would be relatively straightforward to decide when they admit symplectic resolutions.
2) Do Braverman-Finkelberg-Nakajima Coulomb branches give most known examples of symplectic singularities? Maybe. First, it is not known how many of these are actually conic (to fit into (1)). If we consider first the Higgs branch rather than the Coulomb branch then I think it is a reasonable question to ask if most conic symplectic singularities can be realised as Hamiltonian reductions of a symplectic vector space with respect to a (possibly disconnected) reductive group. One gets all nilpotent orbit closures of classical type this way for instance (I don’t know if this is still true for more general Slodowy slices). Now if this is the case and we believe symplectic duality then one should also realise most conic symplectic singularities as coulomb branches. I think there’s a slight issue here though. The definition as given by BFN does not work so well for disconnected groups. For instance if we take the gauge group to finite then the coulomb branch is just a point. Another way to see that one probably can’t get many quotient singularities (V/G for G \subset Sp(V) finite) is that the coulomb branch is always rational (has same field of fractions as affine space). I don’t think V/G is always rational even for type E Kleinian singularities, so can’t be realised via BFN construction. Maybe there is a way to modify their construction.
3) Do BFN Coulomb branches have explicit descriptions? No (though I am not an expert) outside the quivers gauge theories of finite type (or affine type A) there is no geometric or moduli description.
4) The case of quotient singularities is the one I am most familiar with (work with Travis). Here the classification of symplectic resolutions is almost complete, except for a finite number of exceptional groups. I believe that a PhD student of U. Thiel is looking at these. We also know precisely when quiver varieties admit symplectic resolutions, and I believe there is a classification due to Fu/Namikawa for (normalizations of) nilpotent orbit closures.
5) Also, is there an object "Lie group of the 21st century" which fits into an analogy [Lie group of the 21st century] : [symplectic resolution (Lie algebra of the 21st century)] = Lie group : Lie algebra? Yes, I would say this picture is very well understood. See the Asterique article by Braden-Licata-Proudfoot-Webster and subsequent work by Losev.
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1$\begingroup$ On 2), Coulomb branch is conical with respect to a certain standard choice of $\mathbb C^\times$-action if and only if the gauge theory is good or ugly. See e.g, Section 3 of arxiv.org/pdf/1510.03908.pdf. There is a belief that instanton moduli spaces for exceptional groups (e.g., minimal nilpotent orbits for exceptional groups) cannot be realized as a hamiltonian reduction of a symplectic vector space. $\endgroup$ Commented May 12, 2020 at 2:07
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1$\begingroup$ (cont'd.) In Lemma 6.9 in arxiv.org/pdf/1601.03586.pdf, it was shown that only type A,D simple singularities appear in the two dimensional case. Therefore type E simple singularities do not appear as a Coulomb branch. Since minimal nilpotent orbits and simple singularities are believed to be symplectic dual, this is compatible with the belief above. $\endgroup$ Commented May 12, 2020 at 2:09
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1$\begingroup$ On 3), I do not think that this is a well-defined question unless one specify what `explicit descriptions' means. Besides mathematically proved statements, Hanany and his collaborators identify many Coulomb branches with nilpotent orbits and Slodowy slices. $\endgroup$ Commented May 12, 2020 at 2:16