Coulomb branches have symplectic singularities

Abstract

We show that Coulomb branches for 3-dimensional ${\mathcal {N}}=4$ supersymmetric gauge theories have symplectic singularities. This confirms a conjecture of Braverman–Finkelberg–Nakajima.

1 Introduction

Let G be a complex reductive algebraic group and N a finite-dimensional representation of G. A mathematical definition of the Coulomb branch (of cotangent type) $\mathcal {M}_C(G,N)$ of a 3-dimensional ${\mathcal {N}}=4$ supersymmetric gauge theory associated to (G, N) was introduced in the seminal papers [5, 9]. They showed that Coulomb branches have a number of remarkable properties. Of relevance to us is the fact that they are irreducible normal Poisson varieties, where the Poisson structure is non-degenerate on the smooth locus. Therefore, it is natural to conjecture, as they do, that Coulomb branches have symplectic singularities in the sense of Beauville [1].

Using partial resolutions of singularities constructed from flavor symmetries, it was shown by Weekes [11] that most Coulomb branches arising from quiver gauge theories have symplectic singularities. In this note, we extend that result by showing that all Coulomb branches have symplectic singularities.

Theorem 1.1

$\mathcal {M}_C(G,N)$ has symplectic singularities.

This confirms the "optimistic conjecture” of Braverman–Finkelberg–Nakajima [5, 3(iv)]. As an immediate corollary, we note that:

Corollary 1.2

$\mathcal {M}_C(G,N)$ has finitely many symplectic leaves.

In the case of quiver gauge theories for finite type quivers, the symplectic leaves of $\mathcal {M}_C(G,N)$ have been explicitly described in [7]. See [11] for other consequences of the main theorem.

Our proof relies on an elementary observation about varieties with symplectic singularities. Namely, if there is a birational Poisson morphism $X \rightarrow Y$ between normal affine varieties and Y is known to have symplectic singularities then so too does X. We apply this observation twice—first in the case where G is a (connected) torus to allow us to reduce to the case where the Coulomb branch can be identified with a toric hyper-Kähler manifold and secondly to reduce from the case of a Coulomb branch for a general reductive group to one for a torus. In both cases, the birational Poisson morphism we require was already constructed by Braverman–Finkelberg–Nakajima [5].

2 The proof

2.1 An elementary observation

Throughout, variety will mean a integral, separated scheme of finite type over the complex numbers. We recall, following [1], that a variety X has symplectic singularities if it is a normal variety whose smooth locus admits a symplectic form $\omega$ such that for some (any) resolution of singularities $q :Z \rightarrow X$, $q^*\omega$ extends to a regular 2-form on Z.

The following elementary lemma is the key to the proof of the main theorem.

Lemma 2.1

Let X, Y be complex normal Poisson varieties. Assume that Y has symplectic singularities and the Poisson structure on the smooth locus of X is non-degenerate. If there exists a generically étale Poisson morphism $f :X \rightarrow Y$, then X has symplectic singularities.

Proof

The only thing to check is that the pull-back to some resolution of singularities of the symplectic form $\omega$ on the smooth locus of X is regular. Let $\omega _0$ denote the symplectic form on the smooth locus of Y.

We choose a resolution of singularities $p :W \rightarrow Y$. Let C denote the (unique) irreducible component of $W \times _Y X$ dominating both Y and X. By base change, $C \rightarrow X$ is a proper generically étale map. Taking a resolution of singularities $Z \rightarrow C$, we form the commutative diagram

Since all the maps f, g, p, q are generically étale, there exists a dense open subset U of Y such that the restrictions $p^{-1}(U) \rightarrow U, f^{-1}(U) \rightarrow U$ and $g^{-1}(p^{-1}(U)) \rightarrow p^{-1}(U), q^{-1}(f^{-1}(U)) \rightarrow f^{-1}(U)$ are étale. We check that $q^* \omega$ extends to a regular form on Z. This means that there exists some regular 2-form (necessarily unique) on Z whose restriction to some dense open subset, over which q is étale, agrees with $q^* \omega$. Since f is assumed Poisson, $f^*(\omega _0 |_U) = \omega |_{f^{-1}(U)}$. Therefore,

$$\begin{aligned} q^*(\omega |_{f^{-1}(U)}) = q^*(f^*(\omega _0 |_U)) =g^*(p^*(\omega _0 |_U)). \end{aligned}$$

Since Y is assumed to have symplectic singularities, there exists a regular 2-form $\eta$ on W whose restriction to $p^{-1}(U)$ agrees with $p^*(\omega _0 |_U)$. Thus, $q^*(\omega |_{f^{-1}(U)}) |_V$ equals $g^*(\eta ) |_{V}$, where $V = g^{-1}(p^{-1}(U)) \cap q^{-1}(f^{-1}(U))$ and $g^*(\eta )$ is a regular 2-form on Z. $\square$

Instead of using Y to deduce that X has symplectic singularities, one can ask if we can use X to deduce that Y has symplectic singularities. As shown in the result below, the answer is yes, provided the morphism is also assumed proper; see also [1, Proposition 2.4] or [3, Lemma 6.12]. The result is not required in this paper, but we provide a proof for completeness.

Proposition 2.2

Let X, Y be complex normal Poisson varieties. Assume that X has symplectic singularities and the Poisson structure on the smooth locus of Y is non-degenerate. If there exists a generically étale proper Poisson morphism $f :X \rightarrow Y$, then Y has symplectic singularities.

The outline of the proof is the same as that of Lemma 2.1. The difference is that we now have a meromorphic form $p^* \omega _0$ on W that we wish to show is regular. Diagram (2.1) implies that $g^*(p^* \omega _0) = q^*(f^* \omega )$ is regular on Z. We deduce from the key lemma below that $p^* \omega _0$ is regular.

Lemma 2.3

Let $g :Z \rightarrow W$ be a proper, generically étale morphism between smooth complex varieties. Then a meromorphic k-form $\omega$ on W is regular if and only if $g^* \omega$ is regular.

Proof

Our assumptions imply that g is surjective. First, we note that the locus where $\omega$ is not regular is a divisor on W; locally we can pick $w_1, \dots , w_n$ such that $d w_1, \dots , d w_n$ are a basis of $\Omega _W^1$. Then $\omega$ can be uniquely expressed as $\sum _{{\textbf{i}}} a_{{\textbf{i}}} d w_{{\textbf{i}}}$, and the non-regular locus of $\omega$ is the union of the non-regular loci of the meromorphic functions $a_{{\textbf{i}}}$.

Next, we claim that the locus (on W) where g is finite has complement of codimension at least two. Since g is assumed proper, Stein factorization says that we can factor $g = \phi \circ h$, where $h :Z \rightarrow T$ has connected fibers and $\phi :T \rightarrow W$ is finite. It suffices then to show that the locus of points t on T where $\dim h^{-1}(t) = 0$ has complement C of codimension at least two. But if c is a generic point of an irreducible component $C_0$ of C, then $\dim C_0 + \dim h^{-1}(c) < \dim Z$ since $h^{-1}(C_0)$ is a proper closed subset of Z. Since $\dim h^{-1}(c) \ge 1$, this implies that $\dim C_0 < \dim T - 1$.

Therefore, we may assume that g is a finite morphism. If $U \subset W$ is any open set such that g is étale on $g^{-1}(U)$ then it is clear that $\omega |_U$ is regular if and only if $g^* (\omega |_U)$ is regular. Thus, we just need to consider a generic point $w \in g(R_g) \subset W$, where $R_g$ is the ramification divisor of g. Let $D \subset g(R_g)$ be an irreducible component, and $E \subset R_g$ an irreducible component of $R_g$ mapping onto D. We may assume that $w_1 = 0$ is a local equation for D at w. Since Z and W are smooth, the local rings $\mathcal {O}_{W,D}$ and $\mathcal {O}_{Z,E}$ are (noetherian) regular local rings of dimension one; that is, they are discrete valuation rings. We have an embedding $g^* :\mathcal {O}_{W,D} \rightarrow \mathcal {O}_{Z,E}$, with $\mathcal {O}_{Z,E}$ finite over $\mathcal {O}_{W,D}$. The function $w_1$ is a uniformizer for $\mathcal {O}_{W,D}$, and choosing a uniformizer t for $\mathcal {O}_{Z,E}$, we have $g^*(w_1) = t^{\ell } u$ for some unit $u \in \mathcal {O}_{Z,E}^{\times }$. Here $\ell$ is the ramification index of D. The module $\Omega _{\mathcal {O}_{Z,E}}^1$ has basis $d t, d g^*(w_2), \dots , d g^*(w_n)$. Therefore, if $\omega _{{\textbf{i}}} = a_{{\textbf{i}}} d w_1 \wedge d w_{i_1} \wedge \cdots \wedge d w_{i_{k-1}}$ is a summand of $\omega$, for some $1< i_1< \cdots < i_{k-1} \le n$, then

$$\begin{aligned} g^* \omega _{{\textbf{i}}}{} & {} = \ell u g^*(a_{{\textbf{i}}}) t^{\ell -1} d t \wedge d g^*(w_{i_1}) \wedge \\{} & {} \qquad \cdots \wedge d g^*(w_{i_{k-1}}) + g^* (a_{{\textbf{i}}}) t^{\ell } d u \wedge d g^*(w_{i_1}) \wedge \cdots \wedge d g^*(w_{i_{k-1}}). \end{aligned}$$

If $a_{{\textbf{i}}} = w_1^{-r} s$ for some unit $s \in \mathcal {O}_{W,D}$ and $r \ge 1$, then $\ell u g^*(a_{{\textbf{i}}}) t^{\ell -1} = \ell u^{1-r} g^*(s) t^{-\ell (r-1) - 1}$ has a pole of order $\ell (r-1) + 1 \ge 1$. Since $d u \in \bigoplus _{i \ge 2} \mathcal {O}_{Z,E} \, d g^*(w_i)$, we deduce that $\omega _{{\textbf{i}}}$ is regular if and only if $g^* \omega _{{\textbf{i}}}$ is regular. $\square$

2.2 Toric hyper-Kähler manifolds

Consider a short exact sequence

$$\begin{aligned} 0 \rightarrow {\mathbb {Z}}^k {\mathop {\longrightarrow }\limits ^{B}} {\mathbb {Z}}^n {\mathop {\longrightarrow }\limits ^{A}} {\mathbb {Z}}^d \rightarrow 0. \end{aligned}$$

(2.2)

We assume that no row of B is zero. Write $T:= {\mathbb {C}}^{\times }$ for the one torus. The above sequence encodes an action of $T^d$ on ${\mathbb {C}}^n$ via

$$\begin{aligned} (t_1, \dots , t_d) \cdot x_i = t_1^{a_{i,1}} \cdots t_d^{a_{i,d}} x_i. \end{aligned}$$

Since (2.2) is exact, the stabilizer of any $x \in {\mathbb {C}}^n$, with $x_i \ne 0$ for all i, is trivial. In particular, the action is effective. The induced action on $T^* {\mathbb {C}}^n$ is Hamiltonian, and we write $\mu :T^* {\mathbb {C}}^n \rightarrow \mathfrak {t}_d^*$ for the associated moment map. Explicitly,

$$\begin{aligned} \mu (x,y) = \left( \sum _i a_{i,j} x_i y_i \right) _{j = 1}^d. \end{aligned}$$

If $\theta$ denotes a rational character of $T^d$ and $\zeta \in \mathfrak {t}_d^*$, then we can take Hamiltonian reduction

$$\begin{aligned} \mathcal {M}_H(\theta ,\zeta ):= \mu ^{-1}(\zeta )^{\theta } /\!\!/\!\, T^d. \end{aligned}$$

Here $\mu ^{-1}(\zeta )^{\theta }$ denotes the open subset of $\theta$-semistable points in $\mu ^{-1}(\zeta )$. The variety $\mathcal {M}_H(\theta ,\zeta )$ is a toric hyper-Kähler manifold (also called a hypertoric variety in the literature) and is a Higgs branch for the gauge theory $(T^d,{\mathbb {C}}^n)$. The fact that these varieties have symplectic singularities is well-known, but the proofs in the literature, [2, Proposition 4.11] or [8, Theorem 2.16], always assume that the matrix A is unimodular (so that the variety admits a symplectic resolution given by variation of GIT). Since we will need to consider matrices A that are not unimodular, we explain how to extend this result to general toric hyper-Kähler manifolds.

Lemma 2.4

Choose $\theta ,\theta '$ such that $\mu ^{-1}(\zeta )^{\theta '} \subset \mu ^{-1}(\zeta )^{\theta }$. Then there exists a projective birational Poisson morphism $\mathcal {M}_H(\theta ',\zeta ) \rightarrow \mathcal {M}_H(\theta ,\zeta )$.

Proof

Since we have assumed that no row of B is zero, $\mu ^{-1}(\zeta )$ is a reduced, irreducible complete intersection [2, Lemma 4.7]. Moreover, as shown in [2, Proposition 4.11] when A is unimodular and in [10] in general, the quotient $\mathcal {M}_H(\theta ,\zeta )$ is normal. Since the variety is constructed as a Hamiltonian reduction, the Poisson bracket on $\mathcal {O}_{T^* {\mathbb {C}}^n}$ descends to a Poisson bracket on $\mathcal {O}_{\mathcal {M}_H(\theta ,\zeta )}$.

The fact that there is a projective Poisson morphism $\pi :\mathcal {M}_H(\theta ',\zeta ) \rightarrow \mathcal {M}_H(\theta ,\zeta )$ is a direct consequence of Hamiltonian reduction; see for instance the proof of [3, Lemma 2.4]. We need to check that it is birational.

Let $\mu ^{-1}(\zeta )^{\theta \, \textrm{st}}$ denote the set of $\theta$-stable points in $\mu ^{-1}(\zeta )$ and $\mathcal {M}_H(\theta ,\zeta )^{\theta \, \textrm{st}}$ its image in $\mathcal {M}_H(\theta ,\zeta )$. The map $\pi$ is bijective over $\mathcal {M}_H(\theta ,\zeta )^{\theta \, \textrm{st}}$. Hence, we need to show that $\mathcal {M}_H(\theta ,\zeta )^{\theta \, \textrm{st}}$ (or equivalently, $\mu ^{-1}(\zeta )^{\theta \, \textrm{st}}$) is non-empty. Since $\mu ^{-1}(\zeta )^{0\, \textrm{st}}$ is contained in $\mu ^{-1}(\zeta )^{\theta \, \textrm{st}}$, it suffices to show that $\mu ^{-1}(\zeta )^{0\, \textrm{st}} \ne \emptyset$. In other words, there exists a closed orbit in $\mu ^{-1}(\zeta )$ with finite (in fact trivial) stabilizer. Let $U \subset T^* V$ consist of all points (x, y) with $x_i,y_i \ne 0$ for all $1 \le i \le n$. As noted previously, the stabilizer of any point in U is trivial. We claim that (a) every orbit in U is closed in $T^* V$, and (b) $\mu ^{-1}(\zeta ) \cap U \ne \emptyset$. Thus, (a) and (b) would imply $\emptyset \ne \mu ^{-1}(\zeta ) \cap U \subset \mu ^{-1}(\zeta )^{0\, \textrm{st}}$.

Let $(p,q) \in U$. If $p_i q_i =: \lambda _i \in {\mathbb {C}}^{\times }$, then the equation $x_i y_i = \lambda _i$ holds for all points in $\overline{T^d \cdot (p,q)}$. But this forces $x_i,y_i \ne 0$ for all points (x, y) in $\overline{T^d \cdot (p,q)}$. That is, $\overline{T^d \cdot (p,q)} \subset U$. Since all orbits in U are free, we have $\overline{T^d \cdot (p,q)} = T^d \cdot (p,q)$ proving (a).

For (b), the exactness of (2.2) implies that the rank of A is d. Therefore, permuting the $x_i$, we may assume that the first $d \times d$ block of A has non-zero determinant. Applying an automorphism to $T^d$ corresponds to multiplying A on the left by a unimodular $d \times d$ matrix U. Therefore, replacing A by UA, we may assume that A is in Hermite form. In particular, the moment map relations $\mu (x,y) = \zeta$ become

$$\begin{aligned} x_i y_i = a_{i,i}^{-1} \zeta _i - \sum _{j > i} a_{i,i}^{-1} a_{j,i} x_j y_j. \end{aligned}$$

(2.3)

Making further substitutions (and replacing $a_{i,i}^{-1} \zeta _i$ by some $\zeta _i'$), we may assume $a_{j,i} = 0$ for $j \le d$ in the relations (2.3). The fact that no row of B is zero translates into the fact that for each $1 \le i \le d$ there exists some $j > d$ with $a_{j,i} \ne 0$. This means that for generic $(x_{d+1},\dots ,x_n,y_{d+1},\dots ,y_n)$ with $x_j,y_j \ne 0$ the relations (2.3) can be satisfied, but only with $x_i y_i \ne 0$ for $1 \le i \le d$ too. Thus, $\mu ^{-1}(\zeta ) \cap U \ne \emptyset$. $\square$

Proposition 2.5

The toric hyper-Kähler manifold $\mathcal {M}_H(\theta ,\zeta )$ has symplectic singularities.

Proof

As noted in the proof of Lemma 2.4, the quotient $\mathcal {M}_H(\theta ,\zeta )$ is a normal Poisson variety. Moreover, since $T^d$ acts freely, with closed orbits, on the non-empty open set $\mu ^{-1}(\zeta )^{\theta \, \textrm{st}} \cap U$, the Poisson structure on $\mathcal {M}_H(\theta ,\zeta )$ is generically non-degenerate.

Choose a generic $\theta '$ such that $\mu ^{-1}(\zeta )^{\theta '} \subset \mu ^{-1}(\zeta )^{\theta }$. Then Lemma 2.4 says that there exists a projective birational Poisson morphism $\mathcal {M}_H(\theta ',\zeta ) \rightarrow \mathcal {M}_H(\theta ,\zeta )$. If $\mathcal {M}_H(\theta ',\zeta )$ admits symplectic singularities, then [3, Lemma 6.12] implies that $\mathcal {M}_H(\theta ,\zeta )$ will also admit symplectic singularities. Thus, we may assume that $\theta$ is generic.

To check that $\mathcal {M}_H(\theta ,\zeta )$ has symplectic singularities, it suffices to check étale locally. As explained in the proof of [6, Proposition 6.2], the fact that $\theta$ is generic means that the stabilizer under $T^d$ of each point in $\mu ^{-1}(\zeta )^{\theta }$ is finite. Therefore, the (étale) symplectic slice theorem, e.g., [3, Theorem 3.8],¹ says that étale locally $\mathcal {M}_H(\theta ,\zeta )$ is isomorphic (as a Poisson variety) to the quotient of a symplectic vector space by a finite (abelian) group acting symplectically. In particular, it has symplectic singularities by [1, Proposition 2.4] and hence the Poisson structure is non-degenerate on the whole of the smooth locus of $\mathcal {M}_H(\theta ,\zeta )$. $\square$

While this note was in preparation, the above statement also appeared as [4, Proposition 5.1].

2.3 Coulomb branches

Coulomb branches are normal varieties whose smooth locus admits a symplectic form [5]. Let $G^{\circ }$ be the connected component of the identity in G. Then, as noted in [5, Remarks 2.8(3)], $\mathcal {M}_C(G,N) \cong \mathcal {M}_C(G^{\circ },N)/ ( G / G^{\circ })$ as Poisson varieties. Hence, $\mathcal {M}_C(G,N)$ will have symplectic singularities by [1, Proposition 2.4] if we can show that $\mathcal {M}_C(G^{\circ },N)$ has symplectic singularities. Therefore, we may assume that G is connected. We first consider the abelian case.

Lemma 2.6

Assume $G = T^k$ is a torus. Then $\mathcal {M}_C(G,N)$ has symplectic singularities.

Proof

The action of $G = T^k$ on $N = {\mathbb {C}}^m$ is encoded in an integral $k \times m$ matrix $B_0$. Namely,

$$\begin{aligned} (t_1, \dots , t_k) \cdot x_i = t_1^{b_{1,i}} \cdots t_d^{b_{k,i}} x_i. \end{aligned}$$

If we decompose $N = N_0 \oplus N^{G}$, then [5, 3(vii)] says that $\mathcal {M}_C(G,N) = \mathcal {M}_C(G,N_0)$. Therefore, we may assume that $N = N_0$. In other words, no row of $B_0$ is zero; this will be important later.

The idea of course is to identify the Coulomb branch with a toric hyper-Kähler manifold and apply Proposition 2.5. However, this identification only holds if there is sufficient matter in the theory, specifically if the representation N is assumed to be a faithful T-module.

Let $N' = N \oplus {\mathbb {C}}^k = {\mathbb {C}}^n$, where $T^k$ acts on ${\mathbb {C}}^k$ in the natural way (so that the weights are encoded by the identity matrix) and $n = m + k$. By [5, 4(vi)], there is a birational Poisson morphism $\mathcal {M}_C(T^k,N) \rightarrow \mathcal {M}_C(T^k,N')$. Then Lemma 2.1 says that $\mathcal {M}_C(T^k,N)$ will have symplectic singularities if we can show that $\mathcal {M}_C(T^k,N')$ has symplectic singularities. The action of $T^k$ on $N'$ is encoded in the matrix $B = \left( \begin{array}{c} B_0 \\ \textrm{Id} \end{array} \right)$ and we may form a short exact sequence

$$\begin{aligned} 0 \rightarrow {\mathbb {Z}}^k {\mathop {\longrightarrow }\limits ^{B}} {\mathbb {Z}}^n {\mathop {\longrightarrow }\limits ^{A}} {\mathbb {Z}}^d \rightarrow 0, \end{aligned}$$

where $A = (\textrm{Id} | -\!B_0^T)$. We note that no row of B is zero. In this situation, it is noted in [5, 4(iv)] that $\mathcal {M}_C(T, N')$ is isomorphic to the affine toric hyper-Kähler manifold $\mathcal {M}_H((T^d)^{\vee },N')$. By Proposition 2.5, the latter has symplectic singularities. $\square$

Now we return to the general situation, where G is a connected reductive group. Let T be a maximal torus of G and W the associated Weyl group. It is shown in [5, Lemma 5.9, Lemma 5.10] that there exists a birational Poisson morphism $\mathcal {M}_C(G,N) \rightarrow \mathcal {M}_C(T, N |_T)/W$. Lemma 2.1 implies that $\mathcal {M}_C(T, N |_T)$ has symplectic singularities. It follows from [1, Proposition 2.4] that $\mathcal {M}_C(T, N |_T)/W$ also has symplectic singularities. Therefore, Theorem 1.1 is a consequence of Lemma 2.1.