Skip to Main Content

3-Dimensional Mirror Symmetry

Ben Webster
Philsang Yoo

Communicated by Notices Associate Editor Steven Sam

Article cover

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.⁠Footnote1 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 , 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, my⁠Footnote2 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.

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 ; and

2.

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

The singular affine variety 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 is the variety of nilpotent elements in a semisimple Lie algebra , and is the cotangent bundle of the flag variety of . You can reconstruct 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 of complex projective space. This can be written as

Projection to the second component is a resolution of , the space of matrices of rank . This cotangent bundle has a canonical symplectic form, which makes this resolution symplectic.

The cyclic group acts on , preserving its canonical symplectic form, by the matrices

The quotient has a unique symplectic resolution whose exceptional fiber is a union of copies of ’s that form a chain.

We have an isomorphism , but for , 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 on for which preserves the symplectic structure. One obvious invariant is the set of its fixed points of this torus.⁠Footnote3

3

is the diagonal matrices in ; the diagonal matrices in modulo the torsion.

The affine variety 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

We can make this stronger by noting that we match geometrically defined hyperplane arrangements on these spaces.⁠Footnote4

4

In , the vectors where the vanishing set of the corresponding vector field jumps in dimension; in , the Mori walls that cut out the ample cones of the different crepant resolutions of the same affine variety.

2.

Both torus actions have the same number of fixed points, which is ; this also shows that the sum of the Betti numbers of is .

3.

The stratifications on and have the same number of pieces, which is .⁠Footnote5

5

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

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 has a special category of representations that we call “category (see BLPW16, §3) and the categories of and 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:

, the cotangent bundle of the variety of complete flags in .

, the Hilbert scheme of points in .

After suitable modification⁠Footnote6 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 and . We can recover a bijection by only considering special orbits, of which there are the same number.

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 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 , 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 theory , there are two such choices, called the -twist and the -twist. Hence each 3d theory gives two symplectic singularities and called the Coulomb branch and Higgs branch of the theory.

The pairs of symplectic varieties and (similarly, , etc.) all turn out to be the Coulomb and Higgs branches of a single theory . 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: . Thus, we can also describe our dual pairs of symplectic varieties as the Higgs branches of dual theories .

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 -dimensional Riemannian manifold ;

2.

(fields) a fiber bundle over and the space of sections of over ;

3.

(action functional) a functional .

In very rough terms, should be viewed as the space of all possible states of a physical system, while the function 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 . A particularly important type is local operators at that depend only on the value of a field or its derivatives at .

Example 2.1 (Free scalar field theory).
1.

a (compact) Riemannian manifold ;

2.

so that ;

3.

given by , where is the Laplacian of the metric and is the volume form associated to .

In the case of , for any point , the functionals defined by and are local operators at .

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

1.

Let be a compact Lie group. When consists of connections on a principal -bundle over , such a field theory is called a gauge theory and is called the gauge group of the theory.

2.

Let be a manifold. When consists of maps from to , such a theory is called a -model and is called the target of the -model. In this case, .

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

However, in many cases, these integrals do not make sense because the space is often infinite-dimensional, and as a result, the Lebesgue measure cannot be defined.

More generally, given observables ’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

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 as a single observable.

2.2. TQFT

In the framework of Atiyah and Segal, a -dimensional topological quantum field theory (TQFT) is a symmetric monoidal functor from the category to the category of complex vector spaces. Objects of are closed oriented -manifolds , a morphism from to is a diffeomorphism class of a -dimensional bordism from to , and the monoidal structure is given by disjoint union with the empty set being the unit object.

Regarding a closed -manifold as a bordism from to yields a complex number . Physically, one should imagine that . On the other hand, the complex vector space attached to a closed -manifold is the Hilbert space of states on . The most important case is that of . In this case, the vector space will be the vector space of local operators in the TQFT. The principle that these spaces coincide is called “the state-operator correspondence.”

Suppose . Since any closed oriented 1-manifold is a disjoint union of copies of circles, it is enough to describe . Moreover, the map associated to a pair of pants yields a linear map and the one associated to a disk is a linear map :

Graphic without alt text

Topological arguments show that these maps and others from the reversed picture induce a commutative Frobenius algebra structure on .

Note that one can apply a similar idea to any -dimensional TQFT to show that obtains a commutative algebra structure for using the cobordism where we remove two disjoint -balls from the interior of a -ball. When we interpret as the space of local operators of the theory, this product has a physical meaning: it is precisely the operator product introduced above.⁠Footnote7

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, will not be finite-dimensional in the examples we consider.

Since this is a commutative -algebra, can be interpreted as the coordinate ring of an algebraic variety. In fact, the spectrum 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 , measurements at distant points cannot interfere so that . Thus, defines a ring map , 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 with the standard metric. In this case, the isometry group is called the Poincaré group and acts on by rotation and translation. We will only consider field theories on 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 is -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 carries an action of a Lie superalgebra called a super-Poincaré algebra whose even part is the Poincaré algebra and whose odd part consists of copies of spin representations of . A Lie bracket is given by the action of on , as well as a symmetric⁠Footnote8 pairing of -representations.

8

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

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 , supersymmetry algebra has odd part , where are the two spin representations of and . The pairing is induced by the isomorphism as -representations.

2.

The , supersymmetry algebra has odd part , where is the spin representation of and . The pairing is induced by the isomorphism as -representations.

Finally, in order to extract a TQFT from a SUSY theory, suppose that one has chosen a supercharge of a SUSY algebra such that . Since is odd, this means acts as zero in any representation of . Hence, one can consider or even itself as a graded complex, and take its -cohomology. Necessarily, this procedure results in a simpler theory, which one calls a twist or a twisting.

If an element is in the image of , then translation by will be trivial in the twisted theory. The most important case for us is if the image of fills in all of . 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 , and thus on and . Let (resp. ) be the supersymmetry algebra with (resp. ) and (resp. ) supersymmetry. By a standard argument (see, e.g., ESW22, §§11.2 & 12.1), we have:

In the case (resp. ) there is a topological twist if and only if (resp. ).

In the case where (resp. ), there are exactly 2 topological twists up to appropriate symmetry, which we denote by and .

2.4. Mirror symmetry

When is a Calabi–Yau manifold, there is a physics construction of a 2-dimensional SUSY -model with target . If we twist with respect to , the resulting TQFT is called the A/B-model . The A model depends on the symplectic topology of , and the B-model on the complex geometry of .

There is a remarkable duality, called mirror symmetry, on the set of such SUSY -models, which identifies and for another mirror dual Calabi–Yau manifold . Moreover, this duality is compatible with topological twists: the identification of and is compatible with an involution of the , SUSY algebra which exchanges and . Therefore, the TQFTs and should be equivalent. This idea has resulted in several marvelous predictions. The most famous is that the numbers of rational curves of degree on a quintic 3-fold, understood as the correlation functions of , should be equal to the correlation functions of , which can be more easily computed.

The remarkable success of mirror symmetry motivates the consideration of an analogous duality, called 3d mirror symmetry, for , SUSY field theories, which identifies two superficially different theories, say and . Just as before, there are still two interesting topological twists and in the super-Poincaré algebra, and an automorphism of which switches these. By the same logic, we have an equivalence of topologically twisted theories between and , which we write and , respectively, to emphasize the TQFT perspective.

We will focus on understanding the algebras . As discussed in Section 2.2, the algebraic varieties

are the moduli spaces of vacua of the respective theories. We will call these the Coulomb branch and Higgs branch of the theory . Of course, the identification of local operators in one theory with in the mirror theory is one of the most important features of mirror symmetry in this case as well:

Thus we call the varieties and 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 .

3. Higgs and Coulomb Branches

This section focuses on the Coulomb and Higgs branches in one particularly important case: the , SUSY -model into , gauged by the action of a subgroup . The fields corresponding to the map to are often called the “matter content” of the theory. It is often more convenient to forget the coordinates on and think of it as a general -module with a choice of norm and an action of . It will also simplify things for us to consider as a -vector space with complex structure and the induced action of the complexification of ; we can encode the action of the quaternions and in the holomorphic symplectic form . We will denote the corresponding theory by and denote the Higgs and Coulomb branches by .

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 for a -representation . In incredibly rough terms, this phase space comes from maps of into the cotangent bundle of the quotient satisfying certain properties. These are easiest to explain if we deform our to be the boundary of the cylinder

for some real number .

Graphic without alt text

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

(A)

The algebra is the algebra of locally constant functions on the space of maps of the cylinder to which are constant on the sides and holomorphic on the top and bottom.

(B)

The algebra is the algebra of holomorphic functions on the space of maps of the cylinder to 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 should be functions on constant maps . Here in addition to a point in , one should also consider a covector to this quotient (see BF19, §7.14).

We can define this more concretely using the moment map of on the symplectic vector space . If , then consists of all pairs of and covectors that vanish on the tangent space to the orbit through (and thus can be considered covectors on the quotient).

Definition 3.1.

The Higgs branch is defined as a holomorphic symplectic quotient, that is, one has , the complex polynomial functions on which are -invariant, and . The points of this space are in bijection with closed -orbits in .

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 is the cotangent bundle of a -representation of .

3.1.1. Abelian/hypertoric gauge theories

Assume that is abelian. Since it is connected and reductive, this means for some . For any -representation of , we can choose an isomorphism such that is a subgroup of the full group of diagonal matrices.

These ingredients are typically used in the construction of a toric variety: the GIT quotient , at any regular value of the moment map will give a quasi-projective toric variety for the action of the quotient .

The construction of the Higgs branch 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 . Probably the most familiar examples for readers are the following:

Example 3.2.

Let be the scalar matrices. We can consider the elements of as pairs of an column vector and a row vector , with the group acting by

The outer product is thus an matrix of rank , invariant under the action of . Thus, defines a map . The moment map is defined by the dot product, so if and only if is nilpotent. Thus, we have a map to the space of nilpotent matrices of rank .

The -orbit through is closed if and only if and are both nonzero or both zero; you can see from this that the map above is a bijection. Thus we find .

Example 3.3.

Let be the diagonal matrices of determinant 1. We can again think of as pairs . In this case, the moment map condition guarantees that for all , and the closed orbit condition that if for some