3-Dimensional Mirror Symmetry

1. Introduction

1.1. The House of Symplectic Singularities

Some have compared research in mathematics to searching through a dark room for a light switch.⁠¹ In other circumstances, it can be like walking through the same house during the day—one can see all the furniture, but can still look through the drawers and cupboards for smaller nuggets of treasure. As enjoyable as such a treasure hunt is (and easier on the shins), discovering new rooms we haven’t seen before may lead to even greater rewards. In some fields, this is just a matter of walking down the hall; the hard part is simply knowing which door to open. But even more exciting is finding a secret passage between two rooms we already thought we knew.

Of course, if you are not playing a game of Clue, secret passages can be hard to find. You cannot just go tearing out walls and expecting them to be there. However, in the late 20th and early 21st centuries, mathematicians found one remarkable source of such secret passages: quantum field theory (QFT).

What are called “dualities” in QFT often provide connections between mathematical objects that were totally unexpected beforehand. For example, (2-dimensional) mirror symmetry has shown that algebraic and symplectic geometers were actually living in the same house, though the passage between them is still quite poorly lit and harder to traverse than we would like. Unfortunately, employing these dualities in mathematics is not just a matter of bringing in a physicist with their x-ray specs; it is more like receiving an incomplete and weather-worn set of blueprints, possibly written in an unknown language, that hint at the right place to look. Still, we get some very interesting hints.

¹

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated and you can see exactly where you were. —Andrew Wiles.

✖

For representation theorists, the most splendid and best explored of all mansions is the house of simple Lie algebras; while it is more than a century old, it still has many nooks and crannies with fascinating surprises. It also has a rather innocent-looking little pass-through between rooms, called Langlands duality. After all, it is just transposing the Cartan matrix; most of us cannot keep the Cartan matrix straight from its transpose without looking it up anyway. The Langlands program has revealed the incredible depths of this simple operation.

Many new wings have been found to this manor: Lie superalgebras, representations of algebraic groups in characteristic $p$, quiver representations, quantum groups, categorification, etc. Despite their diversity, they all rely on the same underlying framework of Dynkin diagrams. But in recent years, researchers have found a new extension more analogous to the discovery of many new series of Dynkin diagrams: the world of symplectic resolutions and symplectic singularities. According to an oft-repeated bon mot, usually attributed to Okounkov: “symplectic singularities are the Lie algebras of the 21st century.”

Interesting results about this particular annex started appearing around the turn of the 21st century, based on work of Kaledin, Bezrukavnikov, and others. Some time in 2007, my⁠² collaborators Tom Braden, Nick Proudfoot, Tony Licata, and I noticed hints of another secret passage, connecting pairs of rooms (i.e., symplectic resolutions) there. Many coincidences were needed for the different rooms to line up precisely, making space for a secret passage. However, we were not able to step into the passage itself. Nevertheless, we found one very intriguing example: the secret passages we were looking for would generalize Langlands duality to many new examples.

²

All pronouns in this section are from the perspective of BW.

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Of course, you can guess from the earlier discussion what happened. After I gave a talk at the Institute for Advanced Study in 2008, Sergei Gukov pointed out to me that physicists already knew that these secret passages should exist based on a known duality: 3-dimensional mirror symmetry. As explained above, this definitely did not resolve all of our questions; to this day, an explanation of several of the observations we had made remains elusive. More generally, this duality was poorly understood by physicists at the time (and many questions remain), but at least it provided an explanation of why such a passage should exist and a basis to search for it.

In the 15 years since that conversation, enormous progress has been made on the connections between mathematics and 3-dimensional QFT. The purpose of this article is to give a short explanation of this progress and some of the QFT behind it for mathematicians. It is, of necessity, painfully incomplete, but we hope that it will be a useful guide for mathematicians of all ages to learn more.

1.2. Plan of the paper

Let us now discuss our plan with a bit more precise language. A symplectic resolution is a pair consisting of

1.

a singular affine variety $X_0$; and

2.

a smooth variety $X$ with an algebraic symplectic form which resolves the singularities of $X_0$.

The singular affine variety $X_0$ is a special case of a symplectic singularity, which is a singular affine variety where the smooth locus is equipped with a symplectic form that is well-behaved at singularities.

The most famous example of a symplectic resolution is the Springer resolution, where $X_0$ is the variety of nilpotent elements in a semisimple Lie algebra $\mathfrak{g}$, and $X$ is the cotangent bundle of the flag variety of $\mathfrak{g}$. You can reconstruct $\mathfrak{g}$ from the geometry of this resolution. Thus, one perspective on the house of simple Lie algebras is that the Springer resolution is really the fundamental object in each room of a simple Lie algebra, with all other aspects of Lie theory determined by looking at the Springer resolution from various different angles.

Thus, simple Lie algebras lie at one end of a hallway, with many other doors that lead to other symplectic resolutions and singularities. This leads to the natural question of whether any given notion for Lie algebras generalizes to other symplectic resolutions if we treat them like the Springer resolution of a new Lie algebra that we have never encountered. For example, each symplectic singularity has a “universal enveloping algebra” which generalizes the universal enveloping algebra of a Lie algebra.

Two examples accessible to most mathematicians are:

•

The cotangent bundle $X_n^{(\text{A})}=T^*\mathbb{CP}^{n-1}$ of complex projective space. This can be written as$$\begin{multline*} T^*\mathbb{CP}^{n-1} =\{(\ell , \phi )\in \mathbb{CP}^{n-1}\times M_{n\times n}(\mathbb{C}) \\ \mid \phi (\mathbb{C}^n)\subset \ell , \phi (\ell )=\{0\}\}. \end{multline*}$$

Projection to the second component is a resolution of $M_{n\times n}^{\mathrm{rk}\leq 1}(\mathbb{C})$, the space of $n\times n$ matrices of rank $\leq 1$. This cotangent bundle has a canonical symplectic form, which makes this resolution symplectic.

•

The cyclic group $\mathbb{Z}/n\mathbb{Z}$ acts on $\mathbb{C}^2$, preserving its canonical symplectic form, by the matrices$$\begin{equation*} k\mapsto \begin{bmatrix} \exp (2\pi ik/n) & 0\\ 0& \exp (-2\pi ik/n)\end{bmatrix}. \end{equation*}$$

The quotient $\mathbb{C}^2/(\mathbb{Z}/n\mathbb{Z})$ has a unique symplectic resolution $X_n^{(\text{B})}$ whose exceptional fiber is a union of $n-1$ copies of $\mathbb{CP}^1$’s that form a chain.

We have an isomorphism $X_2^{(\text{A})}\cong X_2^{(\text{B})}$, but for $n>2$, these varieties have different dimensions. There are some intriguing commonalities when we look at certain combinatorial information coming out of these varieties. Central to this are two geometric objects:

•

The action of a maximal torus $T^{(*)}$ on $X_n^{(*)}$ for $*\in \{\text{A},\text{B}\}$ which preserves the symplectic structure. One obvious invariant is the set of its fixed points of this torus.⁠³

³

$T^{(\text{A})}$ is the diagonal matrices in $PGL_n(\mathbb{C})$; $T^{(\text{B})}$ the diagonal matrices in $SL_2(\mathbb{C})$ modulo the $n$ torsion.

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•

The affine variety $X_0$ has a unique minimal decomposition into finitely many smooth pieces with induced symplectic structures, generalizing the decomposition of nilpotent matrices into Jordan type.

There are some intriguing coincidences between this pair of varieties:

1.

We have isomorphisms$$\begin{equation*} \mathfrak{t}^{(\text{A})}\cong H^2(X_n^{(\text{B})})\qquad \mathfrak{t}^{(\text{B})}\cong H^2(X_n^{(\text{A})}). \end{equation*}$$

We can make this stronger by noting that we match geometrically defined hyperplane arrangements on these spaces.⁠⁴

⁴

In $\mathfrak{t}$, the vectors where the vanishing set of the corresponding vector field jumps in dimension; in $H^2(X_n^{(\text{B})})$, the Mori walls that cut out the ample cones of the different crepant resolutions of the same affine variety.

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2.

Both torus actions have the same number of fixed points, which is $n$; this also shows that the sum of the Betti numbers of $X_n^{(*)}$ is $n$.

3.

The stratifications on $X_0^{(\text{A})}$ and $X_0^{(\text{B})}$ have the same number of pieces, which is $2$.⁠⁵

⁵

The smooth locus is one stratum, and in both cases, the other one is a single point.

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It would be easy to dismiss these as not terribly significant, but they are numerical manifestations of a richer phenomenon. That is,

4.

the “universal enveloping algebra” of $X_n^{(*)}$ has a special category of representations that we call “category $\mathcal{O}$” (see BLPW16, §3) and the categories $\mathcal{O}$ of $X_n^{(\text{A})}$ and $X_n^{(\text{B})}$ are Koszul dual; the homomorphisms between projective modules in one category describe the extensions between simple modules in the other.

The other reason that we should not dismiss these “coincidences” is that the same statements 1.–4. apply to many pairs of symplectic singularities, which are discussed in BLPW16, §9. These include all finite and affine type A quiver varieties and smooth hypertoric varieties. Some examples are self-dual:

•

$Y_n^{(\text{A})}=Y_n^{(\text{B})}=T^*\operatorname {Fl}_n$, the cotangent bundle of the variety of complete flags in $\mathbb{C}^n$.

•

$Z_n^{(\text{A})}=Z_n^{(\text{B})}=\operatorname {Hilb}^{n}(\mathbb{C}^2)$, the Hilbert scheme of $n$ points in $\mathbb{C}^2$.

After suitable modification⁠⁶ of 3., it also includes the Springer resolutions of Langlands dual pairs of Lie algebras.

⁶

In this case, the strata are the adjoint orbits of nilpotent elements, and the number of these is different for types $B_n$ and $C_n$. We can recover a bijection by only considering special orbits, of which there are the same number.

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This mysterious duality on the set of symplectic singularities and their resolutions has obtained the name of “symplectic duality” for its connection of two apparently unrelated symplectic varieties.

Question 1.1.

Is there an underlying principle that explains statements 1.-4., that is, which explains the symplectic duality between these pairs of varieties?

As discussed above, work on QFT in dimension 3 suggests that the answer to this question is closer to “yes” than it is to “no.” Our aim in this article is to explain the basics of why this is so and what it tells us about mathematics.

We can break this down into two sub-questions:

Q1.

What are 3d $\mathcal{N}=4$ SUSY QFTs and their topological twists?

Q2.

What do they have to do with symplectic duality?

In Section 2, we will provide an answer to the questions, which we now briefly summarize.

First, every 3-dimensional topological quantum field theory (TQFT) gives us a Poisson algebra. In many cases, this ring is the coordinate ring of a symplectic singularity $X_0$, and all the examples discussed above can be constructed in this way. Given a QFT, a choice of a topological twist gives rise to a TQFT. In fact, for a 3d $\mathcal{N}=4$ theory $\mathcal{T}$, there are two such choices, called the $A$-twist and the $B$-twist. Hence each 3d theory gives two symplectic singularities $\mathfrak{M}_A(\mathcal{T})$ and $\mathfrak{M}_B(\mathcal{T})$ called the Coulomb branch and Higgs branch of the theory.

The pairs of symplectic varieties $X^{(\text{A})}$ and $X^{(\text{B})}$ (similarly, $Y,Z$, etc.) all turn out to be the Coulomb and Higgs branches of a single theory $\mathcal{T}$. Then statements 1.–4. can be understood in terms of the physical duality referenced in Section 1.1, called “3-dimensional mirror symmetry.”

This is a very large topic, and due to constraints on the length and number of references, we will concentrate on the relationship to symplectic resolutions of singularities, giving relatively short shrift to the long and rich literature in physics on the topic; the introduction of BDGH16 will lead the reader to the relevant references, starting from the original work of Intrilligator–Seiberg and Hanany–Witten, which laid the cornerstone of this theory.

Just as the 2-dimensional mirror symmetry known to mathematicians suggests that complex manifolds and symplectic manifolds (with extra structure) come in pairs whose relationship is hard to initially spot, 3-dimensional mirror symmetry rephrases our answer to Question 1.1: the Coulomb branch of one theory can also be thought of as the Higgs branch of its dual theory: $\mathfrak{M}_A(\mathcal{T})=\mathfrak{M}_B(\mathcal{T}^{\vee })$. Thus, we can also describe our dual pairs of symplectic varieties as the Higgs branches of dual theories $(\mathfrak{M}_B(\mathcal{T}), \mathfrak{M}_B(\mathcal{T}^{\vee }) )$.

This answer is not as complete as we would like, since we cannot construct 3-dimensional QFTs as rigorous mathematical objects. We can only work with mathematical rigor on certain aspects of some classes of theories, the most important of which are linear gauge theories. In these cases, we have mathematical definitions of the Higgs and Coulomb branches and thus can prove mathematical results about them.

In Section 3, we will review these constructions of the Higgs and Coulomb branches in the case of linear gauge theories. The former of these constructions has been known to mathematicians for many decades HKLR87, but the construction of Coulomb branches was a surprise even to physicists when it appeared in 2015 BFN18, and is key to the progress we have made since that time.

These varieties are the keystones of a rapidly developing research area that combines mathematics and physics. In particular, they point the way to understanding a mirror symmetry of 3-dimensional theories that is not only a counterpart to the mirror symmetry known to mathematicians (which is 2-dimensional mirror symmetry) but also provides an enrichment of the geometric Langlands program (which comes from a duality of 4-dimensional theories).

We will conclude the article in Section 4 with a brief discussion of interesting directions of current and future research to give the interested reader guidance on where to turn next.

2. Physical Origin

2.1. QFT

In this section, we will give a very short introduction to (Euclidean) QFT. Typically, a QFT has the following input data:

1.

(spacetime) a $d$-dimensional Riemannian manifold $(M,g)$;

2.

(fields) a fiber bundle $B$ over $M$ and the space $\mathcal{F}=\mathcal{F}(M)=\Gamma (M,B)$ of sections of $B$ over $M$;

3.

(action functional) a functional $S \colon \mathcal{F} \to \mathbb{R}$.

In very rough terms, $\mathcal{F}$ should be viewed as the space of all possible states of a physical system, while the function $S$ controls which states will likely be physically achieved.

In a classical physical system, we want to think about measuring quantities, such as the velocity or position of a particle. We can formalize this in the notion of an observable, which is, by definition, a functional $O \colon \mathcal{F}\to \mathbb{R}$. A particularly important type is local operators at $x$ that depend only on the value of a field or its derivatives at $x$.

Example 2.1 (Free scalar field theory).

1.

a (compact) Riemannian manifold $(M,g)$;

2.

$B=M\times \mathbb{R}$ so that $\mathcal{F}=C^\infty (M)$;

3.

$S \colon \mathcal{C}^\infty (M) \to \mathbb{R}$ given by $S(\phi )=\int _M \phi \Delta _g \phi \text{Vol}_g$, where $\Delta _g$ is the Laplacian of the metric $g$ and $\text{Vol}_g$ is the volume form associated to $g$.

In the case of $M=\mathbb{R}$, for any point $x\in \mathbb{R}$, the functionals $O_x, O_x^{(1)} \colon C^\infty (\mathbb{R}) \to \mathbb{R}$ defined by $O_x(\phi )= \phi (x)$ and $O_x^{(1)}(\phi ) = \phi '(x)$ are local operators at $x$.

Two other types of field theories play an important role for us:

1.

Let $G_c$ be a compact Lie group. When $\mathcal{F}$ consists of connections on a principal $G_c$-bundle over $M$, such a field theory is called a gauge theory and $G_c$ is called the gauge group of the theory.

2.

Let $X$ be a manifold. When $\mathcal{F}$ consists of maps from $M$ to $X$, such a theory is called a $\sigma$-model and $X$ is called the target of the $\sigma$-model. In this case, $B=M\times X$.

One insight of the quantum revolution in physics is that a physical system cannot be described by a single field, which would have a well-defined value for each observable. Instead, we can only find the expectation values of observables as integrals, where a measure depending on the action accounts for how probable states are. These integrals are often written notionally in the form

$$\begin{equation} \langle O\rangle \coloneq \int _{\phi \in \mathcal{F}(M)} e^{ - S(\phi )/ \hbar } O(\phi ) D \phi . \end{equation}$$

However, in many cases, these integrals do not make sense because the space $\mathcal{F}(M)$ is often infinite-dimensional, and as a result, the Lebesgue measure $D\phi$ cannot be defined.

More generally, given observables $O_i$’s which only depend on the values of the fields on open sets that do not overlap, we consider the integrals of the following form

$$\begin{equation*} \langle O_1,\cdots ,O_n\rangle \coloneq \int _{\phi \in \mathcal{F}(M)} e^{ - S(\phi )/ \hbar } O_1(\phi )\cdots O_n(\phi ) D \phi . \end{equation*}$$

These are called the correlation functions of the theory and the main objects of study in a QFT. One may also understand the integral as the correlation function of a single observable, as the notion of operator product allows one to express products of $O_1,\cdots ,O_n$ as a single observable.

2.2. TQFT

In the framework of Atiyah and Segal, a $d$-dimensional topological quantum field theory (TQFT) is a symmetric monoidal functor $Z$ from the category $(\operatorname {Bord}_d,\amalg ,\emptyset )$ to the category $(\operatorname {Vect}_{\mathbb{C}},\otimes ,\mathbb{C})$ of complex vector spaces. Objects of $\operatorname {Bord}_d$ are closed oriented $(d-1)$-manifolds $N$, a morphism from $N$ to $N'$ is a diffeomorphism class of a $d$-dimensional bordism $M$ from $N$ to $N'$, and the monoidal structure is given by disjoint union $\amalg$ with the empty set $\emptyset$ being the unit object.

Regarding a closed $d$-manifold $M$ as a bordism from $\emptyset$ to $\emptyset$ yields a complex number $Z(M)$. Physically, one should imagine that $Z(M)=\int _{\phi \in \mathcal{F}(M)} e^{ - S(\phi )/ \hbar } D \phi$. On the other hand, the complex vector space $Z(N)$ attached to a closed $(d-1)$-manifold $N$ is the Hilbert space of states on $N$. The most important case is that of $N=S^{d-1}$. In this case, the vector space $Z(S^{d-1})$ will be the vector space of local operators in the TQFT. The principle that these spaces coincide is called “the state-operator correspondence.”

Suppose $d=2$. Since any closed oriented 1-manifold is a disjoint union of copies of circles, it is enough to describe $Z(S^1)$. Moreover, the map associated to a pair of pants yields a linear map $m\colon Z(S^1)\otimes Z(S^1)\to Z(S^1)$ and the one associated to a disk is a linear map $u \colon Z(\emptyset )\to Z(S^1)$:

Topological arguments show that these maps and others from the reversed picture induce a commutative Frobenius algebra structure on $Z(S^1)$.

Note that one can apply a similar idea to any $d$-dimensional TQFT $Z$ to show that $Z(S^{d-1})$ obtains a commutative algebra structure for $d\geq 2$ using the cobordism where we remove two disjoint $d$-balls from the interior of a $d$-ball. When we interpret $Z(S^{d-1})$ as the space of local operators of the theory, this product has a physical meaning: it is precisely the operator product introduced above.⁠⁷

⁷

An important warning for the reader: we will considering topological twists of QFTs below, which do not always produce TQFTs in the framework above, since the maps defined by some cobordisms may not converge. For example, $Z(S^{d-1})$ will not be finite-dimensional in the examples we consider.

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Since this is a commutative $\mathbb{C}$-algebra, $Z(S^{d-1})$ can be interpreted as the coordinate ring of an algebraic variety. In fact, the spectrum $\mathfrak{M}=\operatorname {Spec}Z(S^{d-1})$ has a physical interpretation as well: it is the moduli space of vacua of the theory. This reflects the fact that at a vacuum state, which by definition is a linear map $\langle -\rangle \colon Z(S^{d-1})\to \mathbb{C}$, measurements at distant points cannot interfere so that $\langle O_1 O_2\rangle =\langle O_1\rangle \langle O_2\rangle$. Thus, $O \mapsto \langle O\rangle$ defines a ring map $Z(S^{d-1})\to \mathbb{C}$, and a point in the spectrum.

In many examples of applications of the idea of physics to mathematics, the perspective of TQFT provides a useful guiding principle. Before discussing how to use the idea, let us explain how one may obtain a TQFT starting from a QFT.

2.3. From QFT to TQFT

There are two well-known ways to construct a TQFT, that is, a theory which is independent of a metric of the spacetime manifold. One is to begin with a space of fields and action functional which do not depend on a metric. For example, Chern–Simons theory is one such theory. This approach is quite limited and leads to relatively few examples. Many more examples arise from applying a topological twist to a supersymmetric field theory (which depends on a metric). Let us briefly review the latter idea.

Consider $M=\mathbb{R}^d$ with the standard metric. In this case, the isometry group $\operatorname {ISO}(d,\mathbb{R})=\operatorname {SO}(d,\mathbb{R})\ltimes \mathbb{R}^d$ is called the Poincaré group and acts on $\mathbb{R}^d$ by rotation and translation. We will only consider field theories on $\mathbb{R}^d$ where the action functional is equivariant under the induced action on the space of fields.

We will also only consider theories where the space of sections $\mathcal{F}$ is $\mathbb{Z}/2\mathbb{Z}$-graded; this arises physically from the spin angular momentum of particles, and thus the natural classifications of particles into bosons (even) and fermions (odd). We call a field theory supersymmetric (SUSY) if it admits nontrivial “odd symmetries,” which one calls supercharges.

More precisely, this means that the space $\mathcal{F}$ carries an action of a Lie superalgebra $\mathfrak{a}$ called a super-Poincaré algebra whose even part is the Poincaré algebra $\mathfrak{a}_0=\mathfrak{so}(d)\ltimes \mathbb{R}^d$ and whose odd part $\mathfrak{a}_1=\Sigma$ consists of copies of spin representations of $\mathfrak{so}(d)$. A Lie bracket is given by the action of $\mathfrak{so}(d)$ on $\Sigma$, as well as a symmetric⁠⁸ pairing $\Gamma \colon \Sigma \otimes \Sigma \to \mathbb{R}^d$ of $\mathfrak{so}(d)$-representations.

⁸

In the world of super Lie algebras, a Lie bracket is symmetric if both inputs are odd!

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For simplicity, we work with a complexification of the supersymmetry algebra from now on; this is mostly harmless for the purpose of discussing twists.

Example 2.2.

1.

The $d=2$, $(\mathcal{N}_+,\mathcal{N}_-)$ supersymmetry algebra $\mathfrak{a}_{d=2}$ has odd part $S_+\otimes W_+\oplus S_-\otimes W_-$, where $S_\pm \cong \mathbb{C}$ are the two spin representations of $\mathfrak{so}(2)$ and $\dim W_\pm = \mathcal{N}_\pm$. The pairing $\Gamma \colon \Sigma \otimes \Sigma \to \mathbb{C}^2$ is induced by the isomorphism $S_+^{\otimes 2} \oplus S_-^{\otimes 2} \cong \mathbb{C}^2$ as $\mathfrak{so}(2)$-representations.

2.

The $d=3$, $\mathcal{N}=k$ supersymmetry algebra $\mathfrak{a}_{d=3}$ has odd part $S \otimes W$, where $S\cong \mathbb{C}^2$ is the spin representation of $\mathfrak{so}(3)$ and $\dim W=\mathcal{N}$. The pairing $\Gamma \colon \Sigma \otimes \Sigma \to \mathbb{C}^3$ is induced by the isomorphism $\operatorname {Sym}^2 S\cong \mathbb{C}^3$ as $\mathfrak{so}(3)$-representations.

Finally, in order to extract a TQFT from a SUSY theory, suppose that one has chosen a supercharge $Q$ of a SUSY algebra such that $[Q,Q]=0$. Since $Q$ is odd, this means $\frac{1}{2}[Q,Q]=Q^2$ acts as zero in any representation of $\mathfrak{a}$. Hence, one can consider $\mathcal{F}$ or even $\mathfrak{a}$ itself as a $\mathbb{Z}/2\mathbb{Z}$ graded complex, and take its $Q$-cohomology. Necessarily, this procedure results in a simpler theory, which one calls a twist or a twisting.

If an element $x\in \mathbb{R}^d\subset \mathfrak{a}$ is in the image of $[Q,-]$, then translation by $x$ will be trivial in the twisted theory. The most important case for us is if the image of $Q$ fills in all of $\mathbb{C}^d$. In this case, the dependence on position vanishes and the theory becomes topological; consequently, the twisted theory is called a topological twist of the original theory.

Whether a topological twist exists is purely dependent on the super-Poincaré algebra $\mathfrak{a}$, and thus on $d$ and $\mathcal{N}$. Let $\mathfrak{a}_{d=3}$ (resp. $\mathfrak{a}_{d=2}$) be the supersymmetry algebra with $d=3$ (resp. $d=2$) and $\mathcal{N}=n$ (resp. $\mathcal{N}=(n_+,n_-)$) supersymmetry. By a standard argument (see, e.g., ESW22, §§11.2 & 12.1), we have:

•

In the case $d=3$ (resp. $d=2$) there is a topological twist if and only if $n\geq 4$ (resp. $n_\pm \geq 2$).

•

In the case where $n= 4$ (resp. $n_\pm = 2$), there are exactly 2 topological twists up to appropriate symmetry, which we denote by $Q_{\operatorname {A}}$ and $Q_{\operatorname {B}}$.

2.4. Mirror symmetry

When $X$ is a Calabi–Yau manifold, there is a physics construction of a 2-dimensional $\mathcal{N}=(2,2)$ SUSY $\sigma$-model $\mathcal{T}^{d=2}(X)$ with target $X$. If we twist with respect to $Q_{\operatorname {A/B}}$, the resulting TQFT is called the A/B-model $\mathcal{T}^{d=2}_{\operatorname {A/B}}(X)$. The A model depends on the symplectic topology of $X$, and the B-model on the complex geometry of $X$.

There is a remarkable duality, called mirror symmetry, on the set of such SUSY $\sigma$-models, which identifies $\mathcal{T}^{d=2}(X)$ and $\mathcal{T}^{d=2}(X^\vee )$ for another mirror dual Calabi–Yau manifold $X^\vee$. Moreover, this duality is compatible with topological twists: the identification of $\mathcal{T}^{d=2}(X)$ and $\mathcal{T}^{d=2}(X^\vee )$ is compatible with an involution of the $d=2$, $\mathcal{N} =(2,2)$ SUSY algebra which exchanges $Q_{\operatorname {A}}$ and $Q_{\operatorname {B}}$. Therefore, the $d=2$ TQFTs $\mathcal{T}^{d=2}_{\operatorname {A}}(X)$ and $\mathcal{T}^{d=2}_{\operatorname {B}}(X^\vee )$ should be equivalent. This idea has resulted in several marvelous predictions. The most famous is that the numbers of rational curves of degree $d$ on a quintic 3-fold, understood as the correlation functions of $\mathcal{T}^{d=2}_{\operatorname {A}}(X)$, should be equal to the correlation functions of $\mathcal{T}^{d=2}_{\operatorname {B}}(X^\vee )$, which can be more easily computed.

The remarkable success of $d=2$ mirror symmetry motivates the consideration of an analogous duality, called 3d mirror symmetry, for $d=3$, $\mathcal{N}=4$ SUSY field theories, which identifies two superficially different theories, say $\mathcal{T}$ and $\mathcal{T}^\vee$. Just as before, there are still two interesting topological twists $Q_{\operatorname {A}}$ and $Q_{\operatorname {B}}$ in the super-Poincaré algebra, and an automorphism of $\mathfrak{a}$ which switches these. By the same logic, we have an equivalence of topologically twisted theories between $\mathcal{T}_{\operatorname {A}}$ and $\mathcal{T}_{\operatorname {B}}^{\vee }$, which we write $Z_{\operatorname {A}}^{\mathcal{T}}$ and $Z_{\operatorname {B}}^{\mathcal{T}^\vee }$, respectively, to emphasize the TQFT perspective.

We will focus on understanding the algebras $\mathcal{A}_{\operatorname {A/B}}(\mathcal{T})=Z_{\operatorname {A/B}}^{\mathcal{T}}(S^2)$. As discussed in Section 2.2, the algebraic varieties

$$\begin{equation*} \mathfrak{M}_{\operatorname {A}}(\mathcal{T})=\operatorname {Spec}Z_{\operatorname {A}}^{\mathcal{T}}(S^{2})\qquad \mathfrak{M}_{\operatorname {B}}(\mathcal{T})=\operatorname {Spec}Z_{\operatorname {B}}^{\mathcal{T}}(S^{2}) \end{equation*}$$

are the moduli spaces of vacua of the respective theories. We will call these the Coulomb branch $\mathfrak{M}_{\operatorname {A}}(\mathcal{T})$ and Higgs branch $\mathfrak{M}_{\operatorname {B}}(\mathcal{T})$ of the theory $\mathcal{T}$. Of course, the identification of local operators $\mathcal{A}_{\operatorname {A}}(\mathcal{T})$ in one theory with $\mathcal{A}_{\operatorname {B}}(\mathcal{T}^{\vee })$ in the mirror theory is one of the most important features of mirror symmetry in this case as well:

$$\begin{equation*} \mathfrak{M}_{\operatorname {A}}(\mathcal{T})\cong \mathfrak{M}_{\operatorname {B}}(\mathcal{T}^{\vee })\qquad \qquad \mathfrak{M}_{\operatorname {B}}(\mathcal{T})\cong \mathfrak{M}_{\operatorname {A}}(\mathcal{T}^{\vee }). \end{equation*}$$

Thus we call the varieties $\mathfrak{M}_{\operatorname {A}}(\mathcal{T})$ and $\mathfrak{M}_{\operatorname {B}}(\mathcal{T})$ mirror to each other, or symplectic duals in the terminology of BLPW16. These varieties have the virtue of being familiar types of mathematical objects, while still carrying much of the structure of the theory $\mathcal{T}$.

3. Higgs and Coulomb Branches

This section focuses on the Coulomb and Higgs branches in one particularly important case: the $d=3$, $\mathcal{N}=4$ SUSY $\sigma$-model into $\mathbb{H}^n$, gauged by the action of a subgroup $G_c\subset U(n,\mathbb{H})$. The fields corresponding to the map to $\mathbb{H}^n$ are often called the “matter content” of the theory. It is often more convenient to forget the coordinates on $\mathbb{H}^n$ and think of it as a general $\mathbb{H}$-module $X$ with a choice of norm and an action of $G_c$. It will also simplify things for us to consider $X$ as a $\mathbb{C}$-vector space with complex structure $I$ and the induced action of the complexification $G$ of $G_c$; we can encode the action of the quaternions $J$ and $K$ in the holomorphic symplectic form $\Omega (x,y)=\langle Jx,y\rangle +i\langle Kx,y\rangle$. We will denote the corresponding theory by $\mathcal{T}(X,G)$ and denote the Higgs and Coulomb branches by $\mathfrak{M}_{\operatorname {A/B}}(X,G)$.

Both of these varieties have concrete mathematical descriptions, which we will describe here as best we can in limited space. Both can be derived from manipulations in infinite-dimensional geometry, using the principle that the Hilbert space of a physical theory is obtained by geometric quantization of the phase space of the theory. This geometric quantization is easiest if $X=T^*N$ for a $G$-representation $N$. In incredibly rough terms, this phase space comes from maps of $S^2$ into the cotangent bundle of the quotient $N/G$ satisfying certain properties. These are easiest to explain if we deform our $S^2$ to be the boundary of the cylinder

$$\begin{equation*} \mathsf{Cyl}_\delta =\{(x,y,z) \mid x^2+y^2\leq \delta , |z|\leq \delta \} \end{equation*}$$

for some real number $\delta >0$.

We will frequently refer to the top, bottom, and sides of this cylinder, by which we mean the unit disks in the $z=\delta ,-\delta$ planes, and the portion of the boundary in between.

(A)

The algebra $\mathcal{A}_{\operatorname {A}}$ is the algebra of locally constant functions on the space of maps of the cylinder to $N/G$ which are constant on the sides and holomorphic on the top and bottom.

(B)

The algebra $\mathcal{A}_{\operatorname {B}}$ is the algebra of holomorphic functions on the space of maps of the cylinder to $N/G$ which are constant on the sides and locally constant on the top and bottom.

We have phrased this to emphasize the parallelism, that is, how the difference between the A- and B-twists is reflected by the placement of “locally constant” and “holomorphic.” In the sections below, we will unpack more carefully how we interpret the concepts in the formulations (A) and (B), since some generalization is necessary.

3.1. Higgs branches

First, we consider the B-twist. While second in alphabetical order, the associated Higgs branch is easier to precisely understand, and thus generally attracted more attention in the mathematical literature. According to the description (B), the algebra $\mathcal{A}_{\operatorname {B}}$ should be functions on constant maps $S^2\to N/G$. Here in addition to a point in $N/G$, one should also consider a covector to this quotient (see BF19, §7.14).

We can define this more concretely using the moment map $\mu \colon X\to \mathfrak{g}^*$ of $G$ on the symplectic vector space $(X,\Omega )$. If $X=T^*N$, then $\mu ^{-1}(0)$ consists of all pairs of $n\in N$ and covectors $\xi$ that vanish on the tangent space to the orbit through $n$ (and thus can be considered covectors on the quotient).

Definition 3.1.

The Higgs branch $\mathfrak{M}_{\operatorname {B}}(X,G)$ is defined as a holomorphic symplectic quotient, that is, one has $\mathcal{A}_{\operatorname {B}}(X,G)=Z_{\operatorname {B}}(S^2)=\mathbb{C}[\mu ^{-1}(0)]^G$, the complex polynomial functions on $\mu ^{-1}(0)$ which are $G$-invariant, and $\mathfrak{M}_{\operatorname {B}}=\operatorname {Spec}\mathcal{A}_{\operatorname {B}}$. The points of this space are in bijection with closed $G$-orbits in $\mu ^{-1}(0)$.

The resulting variety is typically singular symplectic. One can reasonably ask if this variety has a symplectic resolution. This does not happen in all cases, but in some cases it does.

There are two particular examples that we will focus on in this article: abelian and quiver gauge theories. In both cases, the target space $X=T^*N$ is the cotangent bundle of a $\mathbb{C}$-representation $N$ of $G$.

3.1.1. Abelian/hypertoric gauge theories

Assume that $G$ is abelian. Since it is connected and reductive, this means $G\cong (\mathbb{C}^{\times })^k$ for some $k$. For any $\mathbb{C}$-representation $N$ of $G$, we can choose an isomorphism $N\cong \mathbb{C}^n$ such that $G\hookrightarrow D$ is a subgroup of the full group $D$ of $n\times n$ diagonal matrices.

These ingredients are typically used in the construction of a toric variety: the GIT quotient $N\sslash _{\lambda }G$, at any regular value of the moment map will give a quasi-projective toric variety for the action of the quotient $F=D/G$.

The construction of the Higgs branch $\mathfrak{M}_{\operatorname {B}}(T^*N,G)$ of this theory is thus a quaternionic version of the construction of toric varieties. The resulting variety is called a hypertoric variety or toric hyperkähler variety. This variety has complex dimension $2(\dim N-\dim G)$. Probably the most familiar examples for readers are the following:

Example 3.2.

Let $G=\{\varphi I \mid \varphi \in \mathbb{C}^{\times }\}\subset D$ be the scalar $n\times n$ matrices. We can consider the elements of $T^*N$ as pairs of an $n\times 1$ column vector $\mathbf{a}$ and a $1\times n$ row vector $\mathbf{b}$, with the group $G\cong \mathbb{C}^{\times }$ acting by

$$\begin{equation*} \varphi \cdot (\mathbf{a},\mathbf{b})=(\varphi \mathbf{a},\varphi ^{-1} \mathbf{b}) . \end{equation*}$$

The outer product $\mathbf{ab}$ is thus an $n\times n$ matrix of rank $\leq 1$, invariant under the action of $G$. Thus, $(\mathbf{a},\mathbf{b})\mapsto \mathbf{ab}$ defines a map $\mathfrak{M}_{\operatorname {B}}(T^*\mathbb{C}^n,G)\to M_{n\times n}(\mathbb{C})$. The moment map $\mu (\mathbf{a},\mathbf{b})=\mathbf{ba}$ is defined by the dot product, so $\mu (\mathbf{a},\mathbf{b})=0$ if and only if $\mathbf{ab}$ is nilpotent. Thus, we have a map $\mathfrak{M}_{\operatorname {B}}(T^*\mathbb{C}^n,G)\to M_{n\times n}^{\mathrm{rk}\leq 1}(\mathbb{C})$ to the space of nilpotent matrices of rank $\leq 1$.

The $G$-orbit through $(\mathbf{a},\mathbf{b})$ is closed if and only if $\mathbf{a}$ and $\mathbf{b}$ are both nonzero or both zero; you can see from this that the map above is a bijection. Thus we find $\mathfrak{M}_{\operatorname {B}}( T^*\mathbb{C}^n,G) \cong M_{n\times n}^{\mathrm{rk}\leq 1}(\mathbb{C})$.

Example 3.3.

Let $G\subset D$ be the diagonal matrices of determinant 1. We can again think of $T^*N$ as pairs $(\mathbf{a},\mathbf{b})$. In this case, the moment map condition guarantees that $a_ib_i=a_jb_j$ for all $i,j$, and the closed orbit condition that if $a_i=0$ for some $i$, then $a_j=0$ for all other $j$, and similarly with $b_*$’s. By multiplying with a diagonal matrix, we can assume that $a_1=\cdots =a_n$ and $b_1=\cdots =b_n$ as well. This defines a surjective map $\mathbb{C}^2 \to \mathfrak{M}_{\operatorname {B}}$, sending $(x,y)$ to $\mathbf{a}=(x,\dots , x)$ and $\mathbf{b}=(y,\dots ,y)$. However, this is not injective: the diagonal matrices $e^{2\pi ik/n} I$ have determinant 1 and define an action of the cyclic group $\mathbb{Z}_n$ on the image of this map from $\mathbb{C}^2$. This matches the action of the matrix $\operatorname {diag} (e^{2\pi ik/n},e^{-2\pi ik/n})$ on $\mathbb{C}^2$. Thus, we find $\mathfrak{M}_{\operatorname {B}}(T^*\mathbb{C}^n,G)\cong \mathbb{C}^2/\mathbb{Z}_{n}$.

While the hypertoric varieties for other tori are less familiar and more complicated, they still have a very combinatorial flavor, and typically questions about them can be reduced to studying hyperplane arrangements, much as toric varieties can be studied using polytopes. Notably, they all possess symplectic resolutions, constructed with GIT quotients or equivalently hyperhamiltonian reduction at nonzero moment map values. For Example 3.2 above, this resolution is $T^*\mathbb{CP}^{n-1}$ and for Example 3.3, it is the unique crepant resolution obtained by iterated blowups at singular points.

3.1.2. Quiver gauge theories

The most famous examples of these reductions are Nakajima quiver varieties. That is, we fix a directed graph $\Gamma$, and a pair of vectors $\mathbf{v},\mathbf{w}$ whose components are indexed by the vertex set $I$. The group $G=\prod _{i\in I} \operatorname {GL}(v_i;\mathbb{C})$ has representations $\mathbb{C}^{v_i}$ for each $i\in I$. The representation we will consider is

$$\begin{equation*} N= \Big (\bigoplus _{i\to j}\operatorname {Hom}(\mathbb{C}^{v_i},\mathbb{C}^{v_j})\Big )\oplus \Big (\bigoplus _{i\in I} \operatorname {Hom}(\mathbb{C}^{v_i},\mathbb{C}^{w_i})\Big ). \end{equation*}$$

We want a left group action, so $(A,B)\in \operatorname {GL}(v_i;\mathbb{C})\times \operatorname {GL}(v_j;\mathbb{C})$ acts on $M\in \operatorname {Hom}(\mathbb{C}^{v_i},\mathbb{C}^{v_j})$ by $BMA^{-1}$. We call gauge theories $\mathcal{T}(T^*N,G)$ for this choice of $G$ and $N$ quiver gauge theories, and the Higgs branches $\mathfrak{M}_{\operatorname {B}}(T^*N,G)$ are Nakajima quiver varieties. These are geometric avatars of the $\mu$ weight space of a representation of highest weight $\lambda$ for the Kac–Moody algebra with Dynkin diagram $\Gamma$.

Physicists will typically draw two copies of each node, one in a circle filled with $v_i$, and one in a square filled with $w_i$, and draw in the edges of $\Gamma$ between the first copies, and then edges between the circle and square copies of vertex (not drawing vertices with 0’s).

Example 3.4.

If we have a quiver with a single vertex so that we have a single $v$ and $w$, then $N=\operatorname {Hom}(\mathbb{C}^v,\mathbb{C}^w)$, and $X=T^*N=\operatorname {Hom}(\mathbb{C}^v,\mathbb{C}^w)\oplus \operatorname {Hom}(\mathbb{C}^w,\mathbb{C}^v)$, that is a pair of matrices $(A,B)$ which are $w\times v$ and $v\times w$ respectively. The moment map in this case is $\mu (A,B)=BA$. If $v=1$, then this reduces to Example 3.2; more generally, the matrix $AB$ is unchanged by the action of the $\operatorname {GL}(v)$, and defines an isomorphism

$$\begin{equation*} \mathfrak{M}_{\operatorname {B}}(T^*N,G)\cong \{C\in M_{w\times w}(\mathbb{C}) \mid C^2=0,\ \operatorname {rk}(C)\leq v\}. \end{equation*}$$

Other important examples:

E1.

This quiver gives the space of $n\times n$ nilpotent matrices:

E2.

This quiver gives the symmetric power $\operatorname {Sym}^n\mathbb{C}^2=(\mathbb{C}^2)^n/S_n$.

There are many variations on these, but these will suffice as our main examples for the rest of this article. All of these examples have symplectic resolutions obtained by replacing the affine quotient with a GIT quotient: the space of rank $\leq v$ matrices is resolved by $T^*\operatorname {Gr}(w,v)$ if $v\leq w/2$, the nilpotent cone is resolved by the cotangent bundle of the flag variety (this is a special case of the Springer resolution), and the symmetric power $\operatorname {Sym}^n\mathbb{C}^2$ is resolved by the Hilbert scheme of $n$ points on $\mathbb{C}^2$.

3.2. Coulomb branches

Compared to Higgs branches, Coulomb branches are harder to describe. In older papers, one will generally see the statement that the “classical” Coulomb branch is $T^*{}^LT/W$ for ${}^LT$ Langlands dual to the maximal torus $T$ of $G$. However, this is not the true answer, as there are nontrivial “quantum corrections.” In certain special cases, the true Coulomb branch could be determined by other methods:

•

Work of Hanany–Witten and extensions identified the Coulomb branches of the quiver gauge theories E1. and E2. using string dualities.

•

It is implicit in Tel14 that the Coulomb branch of a pure gauge theory (meaning $N=0$) with gauge group $G$ is the (Lie algebra) regular centralizer variety for $G^{\vee }$, called the Bezrukavnikov–Finkelberg–Mirković space $\operatorname {BFM}(G^{\vee })$ there.

Work of Braverman, Finkelberg, and Nakajima BFN18 gives an explicit mathematical definition of the Coulomb branch based on the geometry of affine Grassmannians when $X=T^*N$. Let us try to roughly explain the source of this construction, which can look quite intimidating.

By the description (A), we should consider maps to $N/G$ that are holomorphic on the top and bottom disks of the cylinder and constant along the sides. We will shrink the parameter $\delta$ that defines the cylinder to be infinitesimally small and only consider the Taylor expansion at the origin in the top and bottom planes $z=\pm \delta$. If we let $t = x + iy$, then each holomorphic map $D\to N/G$ corresponds to a Taylor series $n(t)\in N[[t]]$, and two give the same map if they are in the same orbit of the $G$-valued Taylor series $G[[t]]$.

Finally, the map should be constant along the sides of the cylinder. Mapping to a quotient means that two things are equal if they are in the same orbit for a group-valued function on the circle. Since this is only on the sides of the cylinder, the group-valued function comparing the top and bottom might have a pole at the origin. Thus, if $n_\pm (t)$ is the Taylor expansion at $(0,0,\pm \delta )$, then we must have $g_{\pm } n_{-}(t)=n_+(t)$ for some $G$-valued Laurent series $g_{\pm }(t)\in G((t))$.

Definition 3.5.

The BFN space for $(G,N)$ is the quotient of the set $\{(n_+,n_-,g_{\pm }) \mid g_{\pm } n_{-}(t)=n_+(t)\}$ by the action of $G[[t]]\times G[[t]]$ given by

$$\begin{equation*} (h_+,h_-)\cdot (n_+,n_-,g_{pm})=(h_+n_+,h_-n_-,h_+g_{\pm }h_-^{-1}). \end{equation*}$$

By (A), we should consider the locally constant functions on this space. We have to be careful, and in fact, we need to consider the Borel–Moore homology (very carefully defined) of this quotient. That is:

Definition 3.6.

The algebra $\mathcal{A}_{\operatorname {A}}(T^*N,G)$ is the Borel–Moore homology of the BFN space.

While the action of $G[[t]]\times G[[t]]$ is not free, we can still interpret the homology of the quotient using equivariant topology; the interested reader should refer to BFN18 for a more precise discussion. We can still compute using usual methods from finite-dimensional topology and, in particular, identify the Coulomb branches in many cases.

The Coulomb branch comes equipped with a $\mathbb{C}^{\times }$ action, induced by the homological grading on Borel–Moore homology. Unfortunately, there can be operators of negative degree. We call such theories bad. Most other cases are called good.

One bad example which had already attracted the interest of mathematicians was the case of pure gauge theory where $N=0$. The Higgs branch is quite degenerate in this case, but $\mathfrak{M}_{\operatorname {A}}=\operatorname {BFM}(G)$ is the phase space of the rational Toda lattice, that is, the universal (Lie algebra) centralizer, restricted to the Kostant slice; see Tel14, §5.1 for a longer discussion of this variety. The other interesting cases we know fall into the two cases discussed before:

3.2.1. Abelian/hypertoric gauge theories

If $G$ is an abelian group and acts faithfully on $N$, then the Coulomb branch will coincide with the Higgs branch of another good abelian theory.

Recall that we can assume that $G\subset D$ is a subgroup of the group $D$ of diagonal matrices. The Langlands dual group ${}^LF$ of $F=D/G$ can be realized as the connected subgroup of $D$ whose Lie algebra is the perpendicular to $\mathfrak{g}\subset \mathfrak{d}=\mathbb{C}^n$; this is also isomorphic to $(\mathbb{C}^{\times })^k$, but the isomorphism $F\cong {}^LF$ is not canonical. We thus have two Langlands dual short exact sequences of tori:

$$\begin{align*} 1\to G\to & D \to F \to 1\\ 1\to {}^LG\leftarrow & D\leftarrow {}^L F \leftarrow 1. \end{align*}$$

Theorem 3.7.

We have isomorphisms $\mathfrak{M}_{\operatorname {A/B}}(T^*\mathbb{C}^n,G)\cong \mathfrak{M}_{\operatorname {B/A}}(T^*\mathbb{C}^n,{}^LF)$.

This isomorphism was widely expected earlier, but seems to have first been checked using the mathematical definition by Dimofte and Hilburn; see BDG17, §3.3 for a physical discussion of this isomorphism, and BFN18, §4(vii) for an elegant mathematical proof.

Let us give a sketch of it. First, the algebras $\mathcal{A}_{\operatorname {A/B}}$ carry a grading by the weight lattice of $F$/coweight lattice of ${}^LF$: (1) On $\mathcal{A}_{\operatorname {B}}(T^*\mathbb{C}^n,G)$, induced by the $D$-action on $T^*\mathbb{C}^n$; (2) On $\mathcal{A}_{\operatorname {A}}(T^*\mathbb{C}^n,{}^LF)$, induced by the bijection of components of the affine Grassmannian of ${}^LF$ to the coweight lattice. Second, in both algebras, the degree 0 elements form a copy of $\operatorname {Sym}(\mathfrak{f})$: (1) On $\mathfrak{M}_{\operatorname {B}}(T^*\mathbb{C}^n,G)$, the polynomial functions that factor through the moment map $\mathfrak{M}_{\operatorname {B}}(G,T^*\mathbb{C}^n)\to \mathfrak{f}^*$; (2) In $\mathcal{A}_{\operatorname {A}}(T^*\mathbb{C}^n,{}^LF)$, the Borel–Moore homology of a point modulo the action of ${}^LF[[t]]$, isomorphic to $\operatorname {Sym}(\mathfrak{f})\cong H^*_{{}^LF}(\operatorname {pt})$. The isomorphism of algebras is close to being determined by matching these two aspects of the Higgs and Coulomb branches.

3.2.2. Quiver gauge theories

The other class of theories we discussed before are the quiver gauge theories. The resulting Coulomb branches are thus the mirrors of the Nakajima quiver varieties. Those which are good in the sense discussed above correspond to pairs of a highest weight $\lambda$ for the Kac–Moody Lie algebra with Dynkin diagram given by the quiver, and a dominant weight $\mu$. These are characterized by:

1.

The highest weight vector has weight $w_i$ for the root $\mathfrak{sl}_2$ for the node $i$ (that is, $\alpha _i^{\vee }(\lambda )=w_i$).

2.

For simple roots $\alpha _i$, we have $\lambda -\mu =\sum v_i\alpha _i$.

When $\Gamma$ is an ADE Dynkin diagram, these Coulomb branches have an interpretation in terms of the affine Grassmannian of the finite-dimensional group $G_{\Gamma }$ whose Dynkin diagram is $\Gamma$. We can interpret $\lambda$ and $\mu$ as coweights of $G_{\Gamma }$, and thus consider the closure of the orbit $\overline{\mathsf{Gr}}^{\lambda } =\overline{G_{\mathcal{O}}t^{\lambda }G_{\mathcal{O}}/G_{\mathcal{O}}}\subset G_{\mathcal{K}}/G_{\mathcal{O}}$.

Proposition 3.8 (BFN19, Th. 3.10).

The Coulomb branch $\mathfrak{M}_{\operatorname {A}}(T^*N,G)$ for an ADE quiver gauge theory is isomorphic to the transverse slice to a generic point of $\overline{\mathsf{Gr}}^{\mu }$ inside $\overline{\mathsf{Gr}}^{\lambda }$.

Since these affine Grassmannian slices are not familiar to most readers, let us discuss a few examples:

Example 3.9.

The quivers E1., E2. both satisfy $\mathfrak{M}_{\operatorname {A}}\cong \mathfrak{M}_{\operatorname {B}}$! This is coincidental and usually doesn’t happen.

Example 3.10.

In the case of the quiver , where $\mathfrak{M}_{\operatorname {B}}\cong M_{n\times n}^{\mathrm{rk}\leq 1}(\mathbb{C})$, the Coulomb branch $\mathfrak{M}_{\operatorname {A}}$ is the affine variety $\mathbb{C}^2/\mathbb{Z}_{n}$, with the cyclic group $\mathbb{Z}_{n}$ acting by the matrices $\operatorname {diag} (e^{2\pi ik/n},e^{-2\pi ik/n})$.

Example 3.11.

In the quiver below, the Higgs and Coulomb branches are reversed from the previous example: $\mathfrak{M}_{\operatorname {B}}\cong \mathbb{C}^2/\mathbb{Z}_{n}$, $\mathfrak{M}_{\operatorname {A}}\cong M_{n\times n}^{\mathrm{rk}\leq 1}(\mathbb{C})$.

These are special cases of a much more general result. For good quiver theories where $\Gamma$ is an affine type A Dynkin diagram (that is, a single cycle), including the Jordan quiver (a single loop), the Coulomb branch is also a Higgs branch for a theory of an affine type A Dynkin diagram, but potentially of a different size, as proven in NT17. The combinatorics of this correspondence is a little complicated, but it matches the previously known combinatorics of rank-level duality. This suggests that the corresponding theories are mirror to each other.

4. Advanced Directions

Having given the definition of Higgs and Coulomb branches, the reader will naturally wonder what mathematics these lead to. There are a number of directions which are too deep to discuss in full detail, but which the interested reader might want to explore further:

4.1. Stable envelopes

Aganagić and Okounkov AO20, building on earlier work of Maulik–Okounkov, define classes called elliptic stable envelopes on each symplectic resolution with a Hamiltonian $\mathbb{C}^*$-action. There are many examples of these which arise as $\mathfrak{M}_{\operatorname {A/B}}$ for different $3$d, $\mathcal{N}=4$ gauge theories.

The equivariant stable envelopes are classes in equivariant elliptic cohomology which correspond to the thimbles flowing to the different $\mathbb{C}^*$-fixed points on the resolution (equivalently, the stable manifolds of the real moment map, thought of as a Morse function). These play an important role in the study of enumerative geometry and are expected to be one of the key mathematical manifestations of 3d mirror symmetry. The elliptic stable envelopes of mirror varieties are expected to be obtained from the specialization of a natural “Mother” class on the product $\mathfrak{M}_{\operatorname {A}}\times \mathfrak{M}_{\operatorname {B}};$ this is confirmed in the case of Example 3.4 by Rimányi–Smirnov–Varchenko–Zhou RSZV22. This identification switches two classes of parameters in the physical theory:

1.

“masses” which index resolutions of $\mathfrak{M}_{\operatorname {A}}$, and $\mathbb{C}^*$-actions on $\mathfrak{M}_{\operatorname {B}}$, and

2.

“Fayet–Iliopoulos (FI) parameters” which play the opposite role of indexing $\mathbb{C}^*$-actions on $\mathfrak{M}_{\operatorname {A}}$ and resolutions of $\mathfrak{M}_{\operatorname {B}}$.

4.2. Koszul duality of category $\mathcal{O}$’s

One of the mathematical phenomena which has attracted attention to 3-dimensional mirror symmetry is Koszul duality between categories $\mathcal{O}$. These are based on a deformation quantization of the algebras $\mathcal{A}_{\operatorname {A/B}}$ to noncommutative algebras. These deformations can be understood as incorporating the action of the rotation of $\mathbb{R}^3$ by $S^1$ around the $z$-axis; in physics terms, this is called an $\Omega$-background. The resulting algebra is noncommutative, since only the $z$-axis is invariant under the $S^1$-action, and two invariant points cannot switch places while staying on the $z$-axis.

1.

The algebra $\mathcal{A}_{\operatorname {A}}$ is deformed by considering the $S^1$-equivariant homology of the BFN space.

2.

The algebra $\mathcal{A}_{\operatorname {B}}$ is deformed by replacing $\mathbb{C}[X]=\operatorname {Sym} X^*$ by its Weyl algebra, which is defined by the relations $[x,y]=\hbar \Omega (x,y)$; we can replace the operations of taking the $G$-invariant functions on $\mu ^{-1}(0)$ with a noncommutative analogue of Hamiltonian reduction.

Category $\mathcal{O}$ is a category of special modules over these noncommutative algebras. This can be regarded as a categorification of the stable envelopes, in that instead of considering the homology classes of the thimbles flowing into fixed points, we consider sheaves of modules over a deformation quantization of $\mathfrak{M}_{\operatorname {A/B}}$ supported on these thimbles. See BLPW16, §3 for more details.

It was noticed by Soergel that the principal block of category $\mathcal{O}$ for a semisimple Lie algebra has an interesting self-duality property: It is equivalent to the category of ungraded modules over a graded algebra and inside the derived category $D^b(\mathcal{\tilde{O}})$ of graded modules over that algebra, there is a second “hidden” copy of the original category. For variations, such as singular blocks of category $\mathcal{O}$ or parabolic category $\mathcal{O}$, a similar phenomenon occurs, but it is a copy of another category that appears; for example, the singular and parabolic properties interchange. That is, the graded lifts of these categories are Koszul and their Koszul dual is another category (sometimes different, sometimes the same) of a similar flavor.

As discussed in the introduction, we can pretend that another symplectic singularity is the nilpotent cone of a new simple Lie algebra. The definition of category $\mathcal{O}$ for a general symplectic singularity with a $\mathbb{C}^*$-action was given by Braden, Licata, Proudfoot, and the first author in BLPW16. Computing numerous examples led these authors to the conjecture:

Conjecture 4.1.

The categories $\mathcal{O}$ of mirror dual symplectic singularities (i.e., the Higgs and Coulomb branch of a 3d $\mathcal{N}=4$ supersymmetric gauge theory) are Koszul dual.

A version of this conjecture (obviously, requiring more careful stating) is confirmed in Web. The physical interpretation of this Koszul duality is still uncertain, though one is proposed in BDGH16, §7.5.

4.3. Line operators

Just as local operators stand for observations one can make at a single point, there are line operators that describe observations one can make along a single line. Studying these is a natural way to extend our study of 3d mirror symmetry beyond the definition of the Higgs and Coulomb branches.

We can describe this category using the framework of $d$-dimensional extended TQFT, which assigns not just a Hilbert space to a $(d-1)$-manifold, but more generally a $k$-category to each manifold of codimension $k+1$. We can then generalize the description of the local operators as the space $Z(S^{d-1})$ by identifying the operators supported on a $k$-plane with the $k$-category $Z(S^{d-k-1})$. In particular, the category of line operators should be given by $Z(S^{d-2})$.

For the 3d $\mathcal{N}=4$ theories of interest to us, these can be understood after passing to the A- or B-twisted theory. Indeed, algebraic descriptions of these categories have been proposed by Hilburn and the second author. Here we only provide a rough description of $Z(S^1)$ with a similar flavor to (A) and (B) (see BF19 for a more precise statement of this proposal):

•

In the A-twist, we obtain locally constant sheaves (that is, D-modules) on holomorphic loops $N((t))/G((t))$ in the quotient $N/G$.

•

In the B-twist, we obtain holomorphic sheaves (that is, quasi-coherent sheaves) on a version of the locally constant loops (that is, the small loop space) in $N/G$.

This proposal is actually a good way to derive Definition 3.6: the trivial line is given by the pushforward D-module from $N[[t]]/G[[t]]$, and naively computing the endomorphisms of this pushforward as the Borel–Moore homology of the fiber product gives precisely Definition 3.6.

It is an intriguing but challenging problem to identify these categories in the already known dual pairs, which one may call the de Rham 3d homological mirror symmetry (see below for more context for the name). Recent work of Hilburn–Raskin HR22 confirms this in the case where $N=\mathbb{C}$, $G=\{1\}$.

4.4. Connections to 4d field theory and the Langlands program

Seminal work of Kapustin and Witten KW07 interprets a version of the geometric Langlands correspondence in terms of a physical duality between 4-dimensional field theories with $\mathcal{N}=4$ supersymmetry: the supersymmetric Yang–Mills theory in 4 dimensions for a pair of Langlands dual groups are related by S-duality. Elliott and the second author EY18 developed a mathematical framework to describe a variant of their proposal which yields the geometric Langlands correspondence upon (categorified) geometric quantization. Moreover, by applying this procedure to the A- and B-twists of a 3d $\mathcal{N}=4$ theory, one can obtain the aforementioned categories of line operators.

This connects to the 3-dimensional perspective discussed earlier in this paper, since 3d $\mathcal{N}=4$ theories appear as boundary conditions on 4d $\mathcal{N}=4$ super Yang–Mills theory. In particular, $S$-duality of these boundary conditions, as studied by Gaiotto and Witten GW09, is one of our most powerful tools for finding mirror theories. The theories associated to Nakajima quiver varieties for linear or cyclic (finite or affine type A) quivers arise this way, and this is the quickest route to understanding the duality of these theories discussed in Section 3.2.2.

The mathematical understanding of this perspective is still an emerging topic. Hilburn and the second author proposed a new relationship between the global/local geometric Langlands program and the statement of de Rham 3d homological mirror symmetry. In independent work, Ben-Zvi, Sakellaridis, and Venkatesh realized the physical perspective in the context of the relative Langlands program and have announced a number of interesting conjectures relating periods and special values of $L$-functions.

4.5. Betti 3d mirror symmetry

The Betti (singular) cohomology and de Rham cohomology of an algebraic variety are, of course, isomorphic, but they have different nonabelian generalizations. They manifest as the moduli spaces of local systems and of flat connections on a given variety, which are analytically isomorphic (via the Riemann–Hilbert correspondence) but algebraically different.

Most importantly for us, the complex structure on the de Rham moduli space depends on the complex structure of the underlying curve, whereas the Betti space does not. Analogously, Ben-Zvi and Nadler propose a “Betti” version of the geometric Langlands correspondence which gives an automorphic description of the quasi-coherent sheaves on the Betti moduli space, to complement the “de Rham” version of the geometric Langlands correspondence.

Our discussions in the earlier sections of this paper also belong to the de Rham world. Namely, in our description of moduli spaces of vacua, objects attached to $S^2$ depend on a complex structure on this curve; the appearance of structures which are holomorphic in one plane and constant on an orthogonal line is a sort of degenerate complex structure on $S^2$. On the other hand, for physicists, this perspective looks somewhat artificial, compared to treating all directions in $\mathbb{R}^3$ equally.

Indeed, the proposal of Kapustin and Witten KW07 is already phrased from a Betti perspective: It does not depend on the complex structure of the curve and needs to be modified to fit with the usual (de Rham) Langlands conjecture (as is done in EY18). Another key feature of Higgs and Coulomb branches, which is physically expected, but hard to see from a de Rham perspective, is the existence of a hyperkähler metric. These have been constructed for Higgs branches $\mathfrak{M}_{\operatorname {B}}(T^*N,G)$ using hyperhamiltonian quotients, but it is hard to imagine the construction of such a metric on the Coulomb branch in the framework of BFN18. Possibly the most intriguing aspect of the Betti perspective is that, as it does not depend on a complex structure, it is better suited to the approach of extended TQFT. Hence, this is the framework in which one can push the approach of homological mirror symmetry to the fullest.

In the case of 2-dimensional mirror symmetry, Kontsevich made the striking realization that we can capture the equivalence of the A-model of one theory and the B-model of another as an equivalence of two triangulated (dg/$A_{\infty }$) categories: from the Fukaya category of a symplectic manifold to the derived category of coherent sheaves on a complex variety. These are the categories of boundary conditions of the respective twisted theories, and hence the equivalence of theories can be reconstructed from the equivalence of categories. In terms of extended 2d TQFT, this is an equivalence of $Z(\operatorname {pt})$’s of the dual theories.

This provides an enticing model to follow in the 3d case. Ideally, we would assign a 2-category $Z(\operatorname {pt})$ of boundary conditions to the A- and B-twist of each theory $\mathcal{T}$ and conjecture the equivalence between those for dual theories, which we would call the Betti 3d homological mirror symmetry. This program was put forward by Teleman Tel14, based on a proposal of Kapustin–Rozansky–Saulina KRS09 for $\mathcal{T}(T^*N,G)$. Significant progress on the A-model 2-category has been made in the abelian case in recent work of Gammage–Hilburn–Mazel-Gee GHM22 and Doan–Rezchikov DR22 suggesting an ambitious program for a more general case.