Vortices and Vermas

Vortices and Vermas

Mathew Bullimore Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UKPerimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, CanadaDepartment of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, USADepartment of Mathematics, University of Oregon, Eugene, OR 97403, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USADepartment of Physics, POSTECH, Pohang 790-784, Korea    Tudor Dimofte Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UKPerimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, CanadaDepartment of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, USADepartment of Mathematics, University of Oregon, Eugene, OR 97403, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USADepartment of Physics, POSTECH, Pohang 790-784, Korea    Davide Gaiotto Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UKPerimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, CanadaDepartment of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, USADepartment of Mathematics, University of Oregon, Eugene, OR 97403, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USADepartment of Physics, POSTECH, Pohang 790-784, Korea    Justin Hilburn Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UKPerimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, CanadaDepartment of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, USADepartment of Mathematics, University of Oregon, Eugene, OR 97403, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USADepartment of Physics, POSTECH, Pohang 790-784, Korea    Hee-Cheol Kim mathew.bullimore@maths.ox.ac.uk tudor@math.ucdavis.edu dgaiotto@perimeterinstitute.ca jhilburn@math.upenn.edu heecheol1@gmail.com Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UKPerimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, CanadaDepartment of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, USADepartment of Mathematics, University of Oregon, Eugene, OR 97403, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USADepartment of Physics, POSTECH, Pohang 790-784, Korea
Abstract

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d gauge theories in the presence of an -background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices. The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the Coulomb-branch algebra itself, related to that of Braverman-Finkelberg-Nakajima. By introducing additional boundary conditions, we find a construction of vortex partition functions of 2d theories as overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras, generalizing work of Braverman-Feigin-Finkelberg-Rybnikov on a finite version of the AGT correspondence. In the case of 3d linear quiver gauge theories, we use brane constructions to exhibit vortex moduli spaces as handsaw quiver varieties, and realize monopole operators as interfaces between handsaw-quiver quantum mechanics, generalizing work of Nakajima.

July 27, 2019

1 Introduction

1.1 Summary

In this paper, we study various setups involving a three-dimensional gauge theory with supersymmetry placed in an -background (Figure 1). Such a theory is labelled by a compact gauge group and a quaternionic representation describing the hypermultiplet content. We will require that the theory has only isolated massive vacua when generic mass and FI parameters are turned on, and place the system in such a vacuum at infinity in the plane of the -background.

Figure 1: A 3d theory in the -background, with a fixed vacuum at spatial infinity.

The vacuum and -background effectively compactify this system to one-dimensional supersymmetric quantum mechanics at the origin of , with a Hilbert space of supersymmetric ground states. By analyzing solutions of the BPS equations that are independent of the coordinate along , we find the following description of the Hilbert space:

  • The half-BPS particles of the three-dimensional gauge theory that preserve the same supersymmetry as the -background are vortices localized at the origin of . They are characterized by a vortex number : the flux of the abelian part of the gauge field through . The Hilbert space

    (1)

    is the direct sum of the equivariant cohomology of vortex moduli spaces with respect to the symmetries preserved by the vacuum .

We also provide a mathematical description of the vortex moduli space as the moduli space of based holomorphic maps from into the Higgs branch of the theory where the vortex number corresponds to the degree of the map. More precisely, it is the moduli space of such maps into a Higgs-branch “stack.” We expect that this description is more general and holds even when the theory does not have vortex solutions in the standard sense, for example when is a pure gauge theory.

The theory has monopole operators labelled by cocharacters of the gauge group , which create or destroy vortices. Together with vectormultiplet scalar fields, the monopole operators generate a Coulomb-branch chiral ring, which is the coordinate ring of the Coulomb branch in a given complex structure. The Coulomb-branch chiral ring is quantized to a noncommutative algebra in the presence of the background. A systematic construction of this ring and its quantization was the topic of BDG-Coulomb (); Nak-Coulomb (); BFN-II (). One motivation for the present paper is to provide a new construction of from its action on vortices.

We will compute the action of monopole operators on by analyzing solutions of three-dimensional BPS equations in . Schematically, a monopole operator labelled by a cocharacter takes a vortex with charge to one with charge . The following statement is one of the main results of this paper:

The action of monopole operators on endows it with the structure of a Verma module for the quantized Coulomb branch algebra .

Intuitively, this corresponds to the statement that the entire Hilbert space is generated from the vacuum state by acting with monopole operators of positive charge. We will demonstrate it explicitly for various theories with unitary gauge groups, and prove it given some assumptions on the structure of the Coulomb branch.

Figure 2: A 3d theory in the -background, with a boundary condition.

We can now enrich the setup of Figure 1 by adding a boundary condition at some point in the direction and filling , as in Figure 2. We will consider boundary conditions that preserve 2d supersymmetry on the boundary and are therefore compatible with the background. Such a boundary condition defines a state in the supersymmetric quantum mechanics:

(2)

The state is characterized by the additional relations obeyed by operators in when acting on it. In physical terms, the state is characterized by the behavior of monopole operators brought to the boundary.

We will also consider the setup shown in Figure 3, with boundary conditions and at either end of an interval . At low energies, this system has an effective description as a 2d theory in the -background. The partition function of this system admits two equivalent descriptions: directly as the partition function of the two-dimensional theory, or as an inner product

(3)

in the Hilbert space of the three-dimensional theory .

Figure 3: A 3d theory in the -background, sandwiched between half-BPS boundary conditions.

It is particularly interesting to consider boundary conditions and that preserve the gauge symmetry . Such ‘Neumann’ boundary conditions were studied extensively in BDGH (). They depend on a choice of -invariant Lagrangian splitting of the hypermultiplet representation, and on complex boundary FI parameters . The states created by these boundary conditions have an explicit description as an equivariant cohomology class in the vortex moduli space, or rather a sum of classes for all vortex numbers. They turn out to be coherent states in the supersymmetric quantum mechanics, satisfying an equation of the form

(4)

Mathematically, these conditions identify as a generalized “Whittaker vector” for the Coulomb-branch algebra .

If we now consider an interval with Neumann boundary conditions and at either end, we will find at low energies a 2d gauge theory with gauge group , chiral matter content transforming in the representation , and FI parameter . Its partition function acquires two equivalent descriptions:

  • The partition function is a standard vortex partition function Shadchin-2d (); DGH () of the two-dimensional theory . This is an equivariant integral

    (5)

    of a fundamental class over the moduli space of vortices in the two-dimensional gauge theory .

  • The partition function is an inner product

    (6)

    of states defined by the boundary conditions in the Hilbert space of the three-dimensional theory on .

The equivalence of these two descriptions means that

The vortex partition function of a 2d theory arising from a 3d theory on an interval with Neumann boundary conditions and is equal to an overlap of generalized Whittaker vectors for the quantized Coulomb branch algebra in .

As we explain in Section 1.2.3 below, this can be seen as a “finite” version of the AGT correspondence. Indeed, in very particular examples the Coulomb-branch algebra is known to be a finite W-algebra, motivating the name. One consequence of writing the partition function as an inner product of vectors that satisfy the Whittaker-like condition (4) is that the vortex partition function itself must satisfy differential equations in the parameters that quantize the twisted chiral ring of .

Throughout the paper, we find it useful to describe the physics of half-BPS vortex particles via an supersymmetric quantum mechanics on their worldlines. For each vacuum and vortex number , there is an supersymmetric quantum mechanics whose Higgs branch is the moduli space of vortices . Its space of supersymmetric ground states coincides with subspace of the 3d Hilbert space (1) of vortex number ,

(7)

Both the complex mass parameters of and the -background deformation parameter are twisted masses in the supersymmetric quantum mechanics; they are the equivariant parameters for the symmetry preserved by the vacuum .

The quantum mechanics can be given a simple description as a 1d gauge theory (with finite-dimensional gauge group) when itself is a type-A quiver gauge theory. Then can be engineered on a system of intersecting D3 and NS5 branes HananyWitten () and a vortex of charge corresponds to adding finite-length D1 branes to this geometry in appropriate positions HananyHori (); HananyTong-branes (). From the brane construction one reads off as a quiver quantum mechanics whose moduli space is precisely .

The monopole operators of change vortex number and so should correspond to interfaces between different supersymmetric quantum mechanics. Very schematically, a monopole operator is represented as an interface between the quantum mechanics and . It defines a correspondence between the moduli spaces; roughly speaking, this is a map

(8)

from a monopole moduli space to the product of vortex moduli spaces. Upon taking cohomology, this induces a map of Hilbert spaces (7). We will construct such correspondences for general theories , and explain how they reproduce the Coulomb-branch algebra.

When is an -type quiver gauge theory, we find an explicit description of these interfaces by coupling the supersymmetric quantum mechanics (as a 1d gauge theory) to matrix-model degrees of freedom at the interface. This provides a physical setup for a construction of Nakajima Nakajima-handsaw () (see Section 1.2.4 below) and extends it to more general -type quivers.

1.2 Relation to other work

There are numerous connections between this paper and previous work and ideas. We briefly mention a few prominent ones.

1.2.1 Vortices, J-functions, and differential equations

BPS vortices have a very long history in both mathematics and physics. They were initially discovered in abelian Higgs models, i.e.  gauge theories with scalar matter Abrikosov (); NielsenOlesen (). Vortex moduli spaces were later studied by mathematicians, e.g. Taubes-LG (); JaffeTaubes (), who established an equivalence between vortices and holomorphic maps. See Tong-TASI () for a review with further references.

Vortices in 2d theories played a central role in early work on mirror symmetry MorrisonPlesser (); Vafa-MS (); Givental-MS (); HoriVafa (). In mathematics, vortex partition functions such as (5) (and its K-theory lift) arose in Gromov-Witten theory, and are sometimes known as equivariant J-functions, cf. GiventalKim (); Givental-homgeom; GiventalLee (); CoatesGivental () and references therein.

From these early works it became clear that vortex partition functions should be solutions to certain differential equations — interpreted either as Picard-Fuchs equations or more intrinsically as “quantizations” of twisted-chiral rings of 2d theories (in the spirit of CV-tt* ()). Such differential equations have shown up over and over again in various guises, from (e.g.) topological string theory IntegrableHierarchies () to the AGT correspondence with surface operators AGGTV () and the 3d-3d correspondence DGG (). We re-derive them here using the construction of vortex partition functions as overlaps of Whittaker vectors.

1.2.2 -background

The -background was originally introduced for four-dimensional gauge theories with supersymmetry in Nek-SW (), building on the previous work LNS-SW (); Moore:1998et (); Moore:1997dj (). The idea that an -background is related to quantization of moduli spaces goes back to the work of Nekrasov-Shatashvili NShatashvili () and related works such as DG-RMQ (); AGGTV (); DGOT (); GMNIII ().

As explained in BDG-Coulomb (); BDGH (), a 3d gauge theory admits two distinct families of -backgrounds that provide a quantization of either the Higgs branch and Coulomb branch in a given complex structure. These -backgrounds may be viewed as deformations of the two distinct families of Rozansky-Witten twists introduced in RW (); BT-twists (). The former class was studied recently in Yagi-quantization () in the context of a sigma model onto the Higgs branch. In this paper, we study the latter: the -background that quantizes the Coulomb branch. This is a dimensional reduction of the usual four-dimensional -background in the Nekrasov-Shatashvili limit where the deformation is confined to a single plane NShatashvili ().

In Section 3, we demonstrate that the Hilbert space is given by the equivariant cohomology of a moduli spaces of vortices. This observation is not unexpected: a 2d theory with at least supersymmetry localizes to BPS vortex configurations in the presence of an -background Shadchin-2d (). It is therefore natural to find a Hilbert space populated by BPS vortex particles in three dimensions.

1.2.3 Finite AGT correspondence

One of our main results is that the Hilbert space of a 3d theory in the -background is a Verma module for the quantized Coulomb branch algebra, and that 2d vortex partition functions arise as overlaps of Whittaker-like vectors in . Special cases of these statements were discovered in mathematics by Braverman and Braverman-Feigin-Finkelberg-Rybnikov Brav-W (); BFFR-W (). Much earlier, Kostant Kostant-Whittaker () introduced overlaps of Whittaker vectors to construct eigenfunctions of the Toda integrable system, which happen to be examples of 2d vortex partition functions.

The physical setup for these references is the 3d theory and its generalization , introduced by Gaiotto and Witten as an S-duality interface in 4d gauge theory GW-Sduality (). The Higgs branch of is the cotangent bundle of a partial flag variety for , and its quantized Coulomb-branch algebra is a finite W-algebra for the Langlands dual algebra.111Finite W-algebras originated in Tjin-W (); dBT-W () and thereafter explored extensively in mathematics, as summarized in the review Losev-W (). Notice that holomorphic maps to are all supported on the base . By studying the Hilbert space of in an -background, one therefore expects to find (roughly) that the equivariant cohomology

(9)

is a Verma module for . Moreover, one expects that the vortex partition function for a 2d sigma-model with target is an overlap of Whittaker vectors for . These are precisely the claims made by Brav-W (); BFFR-W () (after modifying (9) by partially compactifying the space of based maps and passing to intersection cohomology to to account for the fact that this compactification is not necessarily smooth).

When is of type , the theory is a linear-quiver gauge theory. Moreover, in the presence of generic mass and FI deformations, it has isolated massive vacua. It is thus amenable to the gauge-theory methods of the current paper, and we will discuss it in many examples. We also generalize to theories whose Higgs branches are intersections of nilpotent orbits and Slodowy slices in .

One of the main goals of Brav-W (); BFFR-W () was to develop and prove a ‘finite’ analogue of the AGT conjecture. To relate to AGT, recall that the AGT conjecture AGT () states that instanton partition functions of 4d theories of class are conformal blocks for a W-algebra. In mathematics (see for instance BFN-instW (); SchiffmannVasserot (); MaulikOkounkov ()), this conjecture has been viewed as a consequence of two more fundamental statements: 1) that a W-algebra acts on the equivariant cohomology of instanton moduli spaces; and 2) that the instanton partition function is an inner product of Whittaker vectors for the W-algebra. Together, (1) and (2) imply that the instanton partition function satisfies conformal Ward identities that ensure it is W-algebra conformal block. By analogy, the statement that 2d vortex partition functions arise as inner products of Whittaker vectors (4) for finite W-algebras can be viewed as a finite version AGT.

We expect it should be possible to understand the full AGT conjecture using a higher-dimensional analogue of the setup in this paper, as outlined in Tachikawa-instanton () (cf. Gaiotto-states (); MMM-AGT (); Taki-AGT (). Specifically, one would like to consider a 5d theory in an background , with instanton operators generating a W-algebra or generalization thereof. Compatifying on an interval with half-BPS Neumann boundary conditions would lead to a 4d theory, whose instanton partition function naturally becomes interpreted as an overlap of Whittaker vectors. This geometry could be further enriched with codimension-two defects along or , leading to similar statements about ‘ramified’ instanton partition functions and affine Lie algebras Alday:2010vg (); KT-chainsaw (). This would be very interesting to explore.

1.2.4 Handsaw quivers and interfaces

In Section 6, we employ a description of BPS vortex-particles using supersymmetric quantum mechanics. For type-A quiver gauge theories whose Higgs branches are cotangent bundles of partial flag varieties, the supersymmetric quantum mechanics describing vortex particles are precisely the “handsaw” quivers that appeared in work of Nakajima Nakajima-handsaw (). The infrared images of the interfaces that represent the action of monopole operators were defined in Nakajima-handsaw () as correspondences between pairs of vortex moduli spaces, as in (8). Here we develop gauge-theory definitions of these interfaces and extend the discussion to more general type-A quiver gauge theories . The interfaces are closely analogous to those found in GaiottoKim () for five-dimensional gauge theories.

1.2.5 Symplectic duality

There are many relations known between geometric structures assigned to Higgs and Coulomb branches of 3d gauge theories, often referred to collectively as “symplectic duality” BPW-I (); BLPW-II (). This includes an equivalence of categories of modules associated to the Higgs and Coulomb branches, whose physical origin was studied in BDGH (). The relation proposed in this paper might also be included in the symplectic duality canon. It is somewhat different in character from the equivalence of categories discussed in BDGH (), most notably in its asymmetric treatment of the Higgs and Coulomb branches. The -background that quantizes the Higgs branch (related to the one studied here by mirror symmetry) should lead to a relation between quasi-maps to the Coulomb branch and Verma modules for Higgs-branch algebras.

1.3 Outline of the paper

We begin in Section 2 by reviewing the basic structure of 3d theories, their BPS operators and excitations, and the -background. In Section 3 we describe the Hilbert space , and give it a mathematical definition in terms of holomorphic maps to a Higgs-branch stack. In Section 4 we then derive the action of monopole operators (more generally, the Coulomb-branch algebra) on . We construct this action mathematically in terms of correspondences, leading to a new “definition” of the Coulomb-branch algebra complementary to that of Braverman-Finkelberg-Nakajima. In Section 5 we introduce half-BPS boundary conditions and 2d vortex partition functions as overlaps of Whittaker vectors. Finally, in Section 6 we use D-branes to derive quiver-quantum-mechanics descriptions of the 1d theories on the worldlines of vortices, and describe the matrix-model interfaces corresponding to monopole operators. In Section 7 we demonstrate our various constructions in the case of a simple 3d abelian quiver gauge theory, whose Higgs branch is a resolved singularity and whose Coulomb-branch algebra is a central quotient of .

2 Basic setup

We begin with a review of 3d theories, their symmetries, and their moduli spaces, setting up some basic notation. We then describe various half-BPS excitations and operators in 3d theories. Notably, half-BPS monopoles, vortices, and boundary conditions can be aligned so as to preserve two common supercharges. The BPS equations for this pair of supercharges will feature throughout the paper. In Section 2.4 we rewrite the 3d theory on as a 1d quantum mechanics along (with infinite-dimensional gauge group and target space). In terms of the quantum mechanics, vortices are simply identified as supersymmetric ground states, and monopoles as half-BPS operators or interfaces. The quantum-mechanics perspective also gives us an easy way to describe the -background, as an ordinary twisted-mass deformation.

2.1 3d theories

We consider a 3d supersymmetric gauge theory with compact gauge group and hypermultiplets transforming in the representation where is a unitary representation of .

Recall that this theory has an R-symmetry , where the two factors rotate vectormultiplet and hypermultiplet scalars, respectively. (Alternatively, these are metric isometries that rotate the ’s of complex structures on the Coulomb and Higgs branches.) The theory also has flavor symmetry , acting via tri-Hamiltonian isometries of the Coulomb and Higgs branches. Explicitly, is the Pontryagin dual

(10)

In the infrared, may be enhanced to a nonabelian group. This Higgs-branch symmetry is the group of unitary symmetries of acting independently of ; it fits into the exact sequence

(11)

Momentarily, we will fix a choice of complex structures on the Coulomb and Higgs branches, left invariant by a subgroup of the R-symmetry. All choices are equivalent. In the fixed complex structures, we denote the holomorphic hypermultiplet scalars as , with charges ; the vectormultiplet scalars split into a holomorphic field of charge , and a real that enters the construction of holomorphic monopole operators.

The Higgs branch can be described either as a hyperkähler quotient or an algebraic symplectic quotient

(12)

where are the real and complex moment maps for the action of on the representation . The moment maps are given by

(13)

where are the generators of . The Coulomb branch was constructed in full generality in BDG-Coulomb (); Nak-Coulomb (); BFN-II (). It takes the form of a holomorphic Lagrangian fibration

(14)

where the base is parameterized by -invariant polynomials in , and generic fibers are ‘dual complex tori’ . The fibers are parameterized by expectation values of monopole operators, which we will return to later.

The theory admits canonical mass and FI deformations that preserve 3d supersymmetry. Masses are constant, background expectation values of vectormultiplet scalars associated to the flavor symmetry, and thus take values in the Cartan subalgebra of ,

(15)

By combining the masses with the dynamical vectormultiplet scalars, we can lift them to elements in the Cartan of the full symmetry of the hypermultiplets, schematically denoted and . One can think of as generating an infinitesimal complexified action on the Higgs branch, and as generating a corresponding action on the hypermultiplets. We shall mostly be interested in complex masses, which deform the ring of holomorphic functions on the Coulomb branch (the Coulomb-branch chiral ring).

Similarly, FI parameters are constant, background values of twisted vectormultiplet scalars associated to the Coulomb-branch symmetry,

(16)

These transform as a triplet of rather than the usual . We shall mostly be interested in real FI parameters , which resolve the Higgs branch,

(17)

Algebraically, we also have

(18)

where the stable locus is a certain open subset of determined by the choice of .

We make a major simplifying assumption: that for generic and the theory is fully massive, with a finite set of isolated massive vacua. Geometrically, this means that the Higgs branch is fully resolved and the action on the Higgs branch has isolated fixed points; or equivalently that the Coulomb branch is fully deformed to a smooth space on which the action has isolated fixed points. In either description, the fixed points correspond to the massive vacua .

2.2 The half-BPS zoo

We are interested in the interactions of half-BPS monopole operators, vortices, and boundary conditions in a 3d theory. Each of these objects preserves a different half-dimensional subalgebra of the 3d algebra, which we summarize in Table 1.

Here and throughout the paper the Euclidean spacetime coordinates are denoted , or

(19)

The 3d supercharges are denoted , where is an Lorentz index, and are , R-symmetry indices. There is a distinguished that preserves the complex -plane in spacetime, rotating with charge one. Similarly, there are distinguished subgroups of the R-symmetry that preserve a fixed choice of complex structures on the Higgs and Coulomb branches. We index the supercharges so that they transform with definite charge under , namely

(20)

where denote charges , . The superalgebra then takes the form

(21)

where are the standard Pauli matrices, and the central charges act as infinitesimal gauge or flavor transformations with parameters

(22)

We can partially align the half-BPS subalgebras preserved by various objects by requiring that the subalgebras all have a common R-symmetry. This fixes the algebras to the form in Table 1. Although we are mainly interested in Coulomb-branch chiral ring operators, vortices, and boundary conditions, it is instructive to include a few other half-BPS objects as well.

Table 1: Supercharges preserved by various half-BPS boundary conditions, line operators, and local operators in 3d theory.

Some brief comments are in order:

  • There exist half-BPS boundary conditions preserving any 2d subalgebra with . The 2d b.c. shown here are rather special in that this subalgebra is preserved under 3d mirror symmetry, which swaps dotted and undotted R-symmetry indices on the ’s. Such b.c. were studied in BDGH ().

  • The half-BPS particles come in two varieties, related by mirror symmetry. In a gauge theory they can be identified as ordinary “electric” particles and vortices. Each preserve a particular 1d subalgebra. Similarly, a 3d theory has two types of half-BPS line operators (Wilson lines and vortex lines), discussed in GomisAssel (), which preserve the same 1d subalgebras as the BPS particles.

  • There are two half-BPS chiral rings. They only contain bosonic operators, whose expectation values are holomorphic functions on either the Higgs or Coulomb branches. Two supercharges ( and ) are preserved by both types of operators; these are the supercharges that would define the chiral ring of a 3d theory, which has no distinction between Higgs and Coulomb branches.

Most importantly for us, there is a pair of supercharges and preserved by all three of the objects we want to study: boundary conditions, vortices, and Coulomb-branch chiral ring operators. We will denote these as

(23)

in the remainder of the paper. Their sum is the twisted Rozansky-Witten supercharge . They do not quite commute with each other, but rather have

(24)

In other words, their commutator in a gauge theory is a combined gauge and flavor rotation, with parameters , . This is good enough for many purposes. In particular, if we consider a path integral with operator insertions and boundary conditions all of which preserve and (and thus are invariant under ), the path integral will localize to field configurations that are invariant under both and .222The localization can be understood as a two-step procedure. First, one localizes with respect to the twisted RW supercharge . Its fixed locus is invariant under , and thus has an action of . Then one can localize with respect to .

2.3 The quarter-BPS equations

The field configurations in a 3d gauge theory preserved by both and from (23) satisfy an interesting set of equations. They can easily be derived by considering the action of and on the various fields of the 3d theory; however, a more conceptual derivation follows from the quantum-mechanics perspective of Section 2.4.

To describe the equations, we introduce the complexified covariant derivatives333Throughout the paper we will assume the gauge field (and the scalar ) to be Hermitian, so and .

(25)

The equations state that the chiral scalars in a hypermultiplet are holomorphic in the -plane and constant in “time” with respect to the modified derivative

(26)

In addition, the derivative is constant in time, and real and complex moment-map constraints are imposed as

(27)

Finally, the vectormultiplet scalars obey

(28)

and

(29)

As usual, we write or to mean the action of a combined gauge and flavor transformation on in the appropriate representation — say for or for . Most of the equations in (28)–(29) can be understood as requiring that the anticommutator vanish when acting on any field. The final equation in (28) requires that the complex scalar lie in a Cartan subalgebra .

These equations have several specializations, corresponding to the fact that and are simultaneously preserved by 3d SUSY vacua, vortices, and Coulomb-branch operators (Table 1).

2.3.1 Supersymmetric vacua

The classical supersymmetric vacua of the 3d gauge theory correspond to solutions of the BPS equations that are independent of and , and have vanishing gauge field:

(30)

We are interested in situations where and vanish, but and are generic. We require that for generic and these equations have a finite number of isolated solutions , i.e. that the theory is fully massive. As mentioned at the end of Section 2.1, these solutions can be identified as fixed points on the Higgs branch of a complexified flavor symmetry generated by .

2.3.2 Vortices

Next, let us consider time-independent solutions of the BPS equations. In temporal gauge , equations (27) and (26) imply that

(31)

These are generalized vortex equations, which describe half-BPS vortex excitations of the 3d gauge theory. They generally only have solutions when . Quantizing the moduli space of solutions to these equations is the main goal of Section 3.

The generalized vortex equations should be supplemented by the additional constraints and

(32)

from (26) and (29). When , we can simply set to satisfy these constraints. In this case, the time-independent BPS equations are fully equivalent to (31). As is turned on, the additional constraints (32) have the effect of restricting the moduli space of solutions of (31) to fixed points of a combined gauge and flavor rotation. This will lead to the use of equivariant cohomology when quantizing the moduli space of vortices.

2.3.3 Monopoles

Finally, if we turn off the FI parameters and set the hypermultiplets to zero, , equations (27) become the monopole equations

(33)

Together with from (28), these describe half-BPS monopole solutions of the 3d gauge theory.

We recall that near the center of a monopole the field has a profile

(34)

where is Euclidean distance from the center and is an element of the cocharacter lattice of that specifies the magnetic charge. (Charges related by an element of the Weyl group are equivalent.) In the quantum theory, one defines a corresponding monopole operator by requiring that fields have a singularity of the form (34) near a given point. The Coulomb-branch chiral ring is then generated by such monopole operator and by guage-invariant polynomials in .

Altogether, the full set of BPS equations can be understood intuitively as describing vortices in the -plane that propagate in time , and that can be created or destroyed at the location of monopole operators. Of the four supercharges preserved by BPS vortices, only two are preserved by monopoles.

2.4 3d as 1d quantum mechanics

When describing the interactions of vortices, monopoles, and boundary conditions, an extremely useful perspective is to view the 3d theory as a one-dimensional quantum mechanics, whose supersymmetry algebra involves the same four supercharges preserved by vortices in Table 1.444This sort of gauged quantum mechanics played a prominent role in Witten-path (). Many of the basic results there are directly applicable here. Then vortices can be understood as supersymmetric ground states in the quantum mechanics. Similarly, boundary conditions that fill the -plane become half-BPS b.c. in the quantum mechanics (preserving and ); and monopoles become half-BPS operators (also preserving and ).

We can give a rather explicit description of the quantum mechanics — though the details will not be relevant for most of this paper. We use the language of 2d superfields and superspace, reduced to one dimension. The quantum mechanics is a gauge theory, whose gauge group

(35)

is the group of gauge transformations in the -plane. Its fields are valued in functions (or sections of various bundles) on the -plane.

The fields in a 3d hypermultiplet become chiral fields in the quantum mechanics, as does the -component of the gauge connection . A more gauge-covariant way of saying that is that the covariant derivative should be treated as a chiral field. There is a natural superpotential

(36)

that contains the kinetic terms for and . The 1d vectormultiplet contains all the 3d scalars as well as the gauge field ; they fit in the the vector superfield

(37)

The field is a twisted chiral, the leading component of the twisted-chiral superfield

(38)

The natural Kähler potential then takes the form

(39)

where schematically denotes the exponentiated action of on , and similarly for .

To include masses , we may introduce a background vectormultiplet for the Higgs-branch flavor symmetry. The complex masses become background values of twisted-chiral fields. Similarly, a real FI parameter enters in a standard twisted superpotential .

The vortex equations of Section 2.3.2 are easily derived as equations for supersymmetric vacua in this quantum mechanics. Namely, arise as F-term equations for the superpotential (36), the real equation is the D-term, and the supplemental constraints (etc.) arise as twisted-mass terms from the Kähler potential.

The quarter-BPS equations for and in Section 2.3 can be derived from the quantum mechanics in a similar way. In particular, equations (26)–(27) are a combination of F-terms and Morse flow

(40)

with respect to a Morse function

(41)

Here denotes the moment map for gauge group (35) of the quantum mechanics, which contains a contribution from the chiral field and its conjugate.

Such Morse flows may be more familiar in quantum mechanics, where instantons that preserve a single supercharge appear as Morse flow for a single real function Witten-Morse (). In the present case, our quantum mechanics has many subalgebras embedded inside. Each subalgebra is labelled by a phase , and contains the two supercharges and , which obey for any . The instantons that preserve take the form of Morse flow with respect to

(42)

The instantons that preserve both and individually must be Morse flows for (42) for all , and therefore obey (40).

2.5 -background

We would also like introduce an -deformation associated to the vector field

(43)

that rotates the -plane, with a complex parameter . There are many equivalent ways to understand this deformation. A standard approach (analogous to the -background in 4d theory Nek-SW (), see Section 1.2.2) is to work in the twisted-Rozansky-Witten topological twist, and to deform the supercharge and the Lagrangian in such a way that . Alternatively, one may view the -background as a twisted-mass deformation of the 1d quantum mechanics of Section 2.4. This latter approach, which we describe here, makes several important properties manifest.

The four supercharges of the quantum mechanics (the “vortex” row of Table 1) are all left invariant by a simultaneous rotation in the -plane and a R-symmetry rotation. Let us call this diagonal subgroup

(44)

It is an ordinary flavor symmetry of the 1d quantum mechanics, and thus we can introduce a background vectormultiplet for it, with a nonzero complex-scalar field (analogous to in (38)). Thus becomes a twisted-mass deformation in the quantum mechanics.

Formulated this way, it is clear that the -background preserves all four supercharges of the quantum mechanics.555This conclusion was also reached from a different viewpoint in (ClossetCremonesi, , Sec. 5), which constructed the -background by coupling to supergravity. Moreover, it is easy to see how it will deform the quarter-BPS equations of Section 2.3: any appearance of should be replaced by

(45)

representing a simultaneous transformation with parameters . (Here ‘’ is the generator of .)

Notably, this means that the nondynamical constraints (29) in the quarter-BPS equations, or (32) in the vortex equations, are deformed to

(46)

We have used here the fact that to replace with . Since and transform in conjugate representations of and , the parameters and (viewed as actual complex numbers) will typically appear with opposite signs in these two equations. On the other hand, and both have R-charge under , leading to the extra term in each equation.

3 Hilbert space

In this section, we analyze in some detail the Hilbert space of a 4d theory in the -background, with a fixed massive vacuum at spatial infinity.

From the perspective of supersymmetric quantum mechanics (Section 2.4), is a space of supersymmetric ground states. By standard arguments Witten-Morse (), we expect that should be realized as the cohomology

(47)

of a classical moduli space . The moduli space is the space of time-independent solutions to the BPS equations of Section 2.3.2. As discussed there, it is a particular generalization of a vortex moduli space. We will describe some general features of in Section 3.1, and related it to a space of holomorphic maps to the Higgs-branch stack in Section 3.2.

In the presence of complex masses and the -background, should be replaced by an equivariant cohomology group

(48)

where is an appropriate group of symmetries acting on . We will only consider theories where the action of has isolated fixed points. Then, by virtue of the localization theorem in equivariant cohomology AtiyahBott (), acquires a distinguished basis labelled by the fixed points. We will describe this abstractly in Section 3.3. Then, in Sections 3.43.5, we will analyze and very explicitly for families of abelian and non-abelian theories, including SQED, SQCD, and triangular-quiver gauge theories.

Here and throughout the rest of the paper we set , to allow nontrivial vortex configurations. We leave generic, so that the Higgs branch is fully resolved. We also usually set for simplicity, as this parameter does not affect the BPS sector that we are considering.

3.1 General structure

We begin by studying time-independent solutions to the BPS equations in the absence of -background and with mass parameters set to zero . We can then set , and reduce the BPS equations to the generalized vortex equations given in (31).

Suppose is a vacuum that survives mass deformations, and becomes fully massive in the presence of generic . This can be thought of as a point on the resolved Higgs branch where the gauge symmetry is fully broken, but a maximal torus of the flavor symmetry and the R-symmetry are preserved. Let denote the -orbit of in the space of hypermultiplet scalars.

We are interested in the moduli space of solutions to the time-independent BPS equations that tend to the vacuum at spatial infinity,

(49)

where is the infinite-dimensional group of gauge transformations on that are constant at infinity. The last condition ensures that gauge transformations preserve the orbit at infinity. We call this the moduli space of ‘generalized vortices’.

If we compactify the -plane to , a point in this moduli space may be equivalently described by the following data:

  1. A -bundle on , trivialized near .

  2. Holomorphic sections of an associated bundle in the representation tending to at infinity and satisfying and .

The moduli space will split into components labelled by a ‘vortex number’ . This labels topological type of the -bundle on ,

(50)

This number can also be defined as the winding number of a gauge transformation on the equator of that relates trivializations of the bundle on the northern and southern hemispheres. This makes it clear that . We will mainly be interested in cases where is a free abelian group, namely, with and products thereof. It is only in such cases that solutions of (49) are ‘vortices’ in the traditional sense. Nevertheless, we expect our construction to valid more generally and continue to use the term ‘vortex number’ for .

The moduli space of solutions splits into disconnected components

(51)

where labelled by the vortex number . In Section 3.2, we will see that not all vortex numbers are realized: whether or not the component is empty depends on the precise choice of vacuum .

The components of the moduli space are Kähler manifolds with rather large abelian symmetry groups

(52)

Here is the maximal torus of the Higgs-branch flavor symmetry preserved by the vacuum ; and is the combination (44) of Higgs-branch R-symmetry and rotation in the -plane that acts as a flavor symmetry of quantum-mechanics. We will work equivariantly with respect to and when turning on and , respectively.

The assumption that is an isolated fixed point of on a smooth Higgs branch ensures that has isolated fixed points under the combined symmetry . We will describe them in Section 3.3. Similarly, the fact that symmetry is fully broken at the vacuum ensures that is fully broken in a neighborhood of each fixed point on , and therefore that a neighborhood of each fixed point is smooth. More generally, we expect that the action in (49) is free and that the whole space is smooth, but we will not prove this here.

3.2 Algebraic description

We expect the moduli space of generalized vortices to have a complex-algebraic description as well. This is obtained by dropping the real moment-map equation and dividing by complex gauge transformations,

(53)

Usually, a stability condition must be imposed in the algebraic quotient. However, any solution that tends to a massive vacuum at infinity is automatically stable, so no further conditions are necessary in (53). This construction makes manifest that the moduli space is Kähler. The equivalence of descriptions (49) and (53) is a version of the Hitchin-Kobayashi correspondence for the generalized vortex equations, which we will not attempt to prove here. (Algebraically, (53) could be taken as a definition of .)

From the algebraic point of view, a point in is specified by

  1. A choice of -bundle on , trivialized near .

  2. Holomorphic sections of an associated bundle in the representation , satisfying and sitting inside the orbit at infinity.

Once we allow for complex gauge transformations, we may pass to a ‘holomorphic frame’ where . The holomorphic sections can then be described concretely as polynomial matrices in the affine coordinate . We must still quotient by holomorphic gauge transformations that preserve the choice of gauge. These are polynomial valued group elements . The resulting description of the moduli space is familiar in the physics literature, for example in the work of Morrison and Plesser MorrisonPlesser () and in the ‘moduli matrix’ construction of Eto-matrix (); Eto-moduli ().

Mathematically, we have described what are based maps from into the stack . Recall from (18) that that the actual Higgs branch involves a stability condition that depends on the real FI parameter . The stability condition prevents certain combinations of the hypermultiplet fields from vanishing. Provided is a faithful representation of , maps from into the stack only differ from ordinary holomorphic maps into the Higgs branch in that they may violate the stability condition at various points . Since must map to the vacuum , which is a point on the actual Higgs branch, holomorphicity ensures that the points where stability is violated are isolated and finite.

Thus we can simply say that

(54)

In this picture the decomposition 3.1 comes from looking at the fibers of the map