Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

Abstract

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the ()-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D theories decorated by BPS ’t Hooft-Wilson loops.

CALT-TH 2017-064

PUPT-2547

WIS/05/17-Dec-DPPA

Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

Mykola Dedushenko, Yale Fan, Silviu S. Pufu, and Ran Yacoby

 1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA 2Department of Physics, Princeton University, Princeton, NJ 08544, USA 3Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel

Abstract

1 Introduction

supersymmetry in three dimensions provides a rich middle ground between the availability of calculable supersymmetry-protected observables and nontrivial dynamics. As an example that will be relevant to us, gauge theories with matter hypermultiplets exhibit an infrared duality known as mirror symmetry [1], under which the Higgs and Coulomb branches of the vacuum moduli space of a given theory are mapped to the Coulomb and Higgs branches of the other. In particular, the half-BPS operators that acquire expectation values when the theory is taken to the Higgs/Coulomb branch, henceforth referred to as Higgs/Coulomb branch operators (HBOs/CBOs), are mapped to the CBOs/HBOs of the mirror dual theory. The duality is nontrivial for several reasons: while the Higgs branch is protected by a non-renormalization theorem and can simply be fixed classically from the UV Lagrangian [2], the Coulomb branch generically receives quantum corrections; the duality exchanges certain order operators and disorder operators; and non-abelian flavor symmetries visible in one theory may be accidental in the mirror dual. At the same time, supersymmetry allows for various calculations of protected observables that led to the discovery of the duality and to various tests thereof, such as the match between the infrared metrics of the Coulomb and Higgs branches [3], scaling dimensions of monopole operators [4], various curved-space partition functions [5, 6, 7], expectation values of loop operators [8, 9], and the Hilbert series [10].

Our goal in the present paper is to provide new insights into the mirror symmetry duality and, more generally, into 3D QFTs, by developing new techniques for calculating correlation functions of certain CBOs that include monopole operators. These techniques are related to the observation of [11, 12] that all superconformal field theories (SCFTs) contain two one-dimensional topological sectors, one associated with the Higgs branch and one associated with the Coulomb branch. These sectors are described abstractly as consisting of the cohomology classes with respect to a pair of nilpotent supercharges, and each cohomology class can be represented by a position-dependent linear combination of HBOs/CBOs that can be inserted anywhere along a line. For the Higgs branch case, it was shown in [13] that the 1D sector has a Lagrangian description that can be obtained by supersymmetric localization and that gives a simple way of computing all correlation functions of the 1D Higgs branch theory. The objective of this work is to provide an explicit description of the Coulomb branch topological sector. Having explicit descriptions of both the Higgs and Coulomb branch 1D sectors allows for more explicit tests of mirror symmetry, including a precise mapping between all half-BPS operators of the two theories.

For simplicity, in this work, we focus only on abelian gauge theories.111In fact, our results can easily be generalized to theories with both ordinary and twisted multiplets coupled through BF terms, first studied in [14]. Any abelian gauge theory has a known mirror dual, which is also abelian. The fundamental abelian mirror duality, proven in [4], states that the IR limit of SQED with one flavor coincides with a free (twisted) hypermultiplet. All other abelian mirror pairs can be formally deduced from the fundamental one by gauging global symmetries [15].

Compared to the Higgs branch 1D theory described in [13], the description of the Coulomb branch theory is more complicated because it involves monopole operators. Monopole operators in 3D gauge theories are local disorder operators, meaning that they cannot be expressed as polynomials in the classical fields. Instead, their insertion in the path integral is realized by assigning boundary conditions for the fields near the insertion point. Specifically, a monopole operator is defined by letting the gauge field approach the singular configuration of an abelian Dirac monopole at a point. Calculations involving monopole operators are notoriously difficult, even in perturbation theory. Following [16], the IR conformal dimensions of monopole operators have been estimated for various non-supersymmetric theories using the expansion [17, 18, 19, 20, 21, 22], the -expansion [23], and the conformal bootstrap [24]. In supersymmetric theories, one can also construct BPS monopole operators by assigning additional singular boundary conditions for some of the scalars in the vector multiplet. For such BPS monopoles, some nonperturbative results are known: for instance, in theories, their exact conformal dimension was determined in [4, 25, 26, 27].222The exact results mentioned above are valid for “good” or “ugly” theories, to use the terminology of [25]. We will only consider such theories in this paper. The correlation functions that we calculate in this paper provide additional nonperturbative results involving BPS monopole operators.

The Coulomb branch 1D theory whose description we will derive encodes information on the geometry of the quantum-corrected Coulomb branch. The Coulomb branch is constrained by supersymmetry to be a (singular) hyperkähler manifold which, with respect to a fixed complex structure, can be viewed as a complex symplectic manifold whose holomorphic symplectic structure endows its coordinate ring with Poisson brackets.333The description of the Coulomb branch as a complex symplectic manifold is not sufficient to reconstruct its hyperkähler metric. It would be interesting to understand whether, and how, information on this metric is encoded in the SCFT. The holomorphic coordinate ring of the Coulomb branch, which describes it as a complex variety, is believed to coincide with the ring of chiral CBOs. As explained in [11], the OPE of the 1D Coulomb branch theory provides a deformation quantization of the Poisson algebra associated with the chiral ring.

In brief, we obtain an explicit description of the Coulomb branch 1D theory as follows. First, we stereographically map the theory from to . While the 1D theory is defined on a straight line in , after the mapping to , it is defined on a great circle. Ideally, we would like to perform supersymmetric localization on with respect to a judiciously chosen supercharge such that the 3D theory localizes to a theory on the great circle (this is how the description of the 1D Higgs branch theory was obtained in [13]). Unfortunately, it is challenging to calculate functional determinants in the presence of an arbitrary number of disorder operators inserted along the great circle. To circumvent this problem, we develop another approach in which we cut the into two hemispheres glued along an that intersects the great circle at two points, and then calculate the wavefunction. Because we can add a localizing term on , it is sufficient to evaluate the wavefunction along a finite-dimensional locus in field space. For every insertion within the hemisphere, we derive a corresponding operator acting on the wavefunction. As we will explain, gluing two hemisphere wavefunctions allows us to compute arbitrary correlators of the 1D theory.

We hope that the methods presented in this paper can be generalized and applied also to non-abelian theories. In these theories, both the Coulomb branch geometry and mirror symmetry are less understood than in the abelian case. In particular, the mirror duals of non-abelian theories are not always known, and the Coulomb branch metric can no longer be simply computed due to nonperturbative effects that are absent in abelian theories. A general picture for the Coulomb branch geometry was recently proposed in [28], and it should be possible to verify it rigorously using correlators of CBOs (there have also been a number of papers on Coulomb branches of 3D theories in the mathematical literature [29, 30, 31, 32, 33]). Furthermore, correlators of CBOs and HBOs could shed light on non-abelian mirror symmetry, because this duality maps these two classes of operators to each other. We hope to report on progress in answering these interesting questions in the near future.

The remainder of this section contains a technical overview of our approach and a summary of our results. The rest of the paper is organized as follows. In Section 2, we introduce in detail the theories that we study and their 1D topological sectors. In Section 3, we perform supersymmetric localization on with monopole-antimonopole insertions at opposite points on the sphere. In Section 4, we perform supersymmetric localization on a hemisphere and on its boundary and explain how to glue two hemisphere wavefunctions. In Section 5, we explain how to compute correlators in the 1D theory with multiple operator insertions. In Section 6, we discuss, as applications of our results, a derivation of the chiral ring relations, and we provide several new tests of mirror symmetry. Several technical details are relegated to the appendices.

1.1 Technical Overview

Let us now describe the general logic behind our computation, which closely follows that of [13]. Consider an theory with gauge group and a hypermultiplet transforming in a (generally reducible) unitary representation of . The theory could also be deformed by real masses and FI parameters, which, for simplicity, we set to zero until further notice. The above information determines an preserving Lagrangian on and another Lagrangian on an with radius , both of which coincide when . Furthermore, the theories on and have the same IR limit, and we will consider examples in which it is a nontrivial SCFT.444The limit on is identical to the flat space IR SCFT. Instead, taking at fixed leads to an SCFT on whose correlators are equivalent to those of the IR SCFT on , by a conformal map from to . One subtlety in this procedure, first noted in [34], is that on , there can be mixing between operators of different conformal dimensions, though this mixing can always be resolved. From our point of view, the advantage of working on is that preserves certain supercharges and , which are only symmetries of the flat space theory at the IR fixed point. The attractive property of (or ) is that its cohomology contains local operators which have nontrivial correlation functions, and which form a subset of the full family of CBOs (or HBOs).555This cohomology is distinct from the chiral ring, as will be explained later. It follows that the correlators of these -closed (-closed) operators, which are known as twisted CBOs (HBOs), could possibly be computed using supersymmetric localization of the path integral on with respect to (). Indeed, the problem of localizing with respect to was fully solved in [13], thus making correlators of twisted HBOs calculable.

In this work, we are interested in correlators of twisted CBOs, which can be described abstractly as follows. First, each CBO is a Lorentz scalar transforming in a spin- irrep of an R-symmetry, such that in the IR SCFT, it is a superconformal primary of dimension .666Strictly speaking, the RG flow on only preserves a subgroup of the R-symmetry mentioned above. Nevertheless, it is useful (and possible) to group CBOs into irreps also along the flow, even if it only becomes a true symmetry in the IR. Each twisted CBO is given by a certain position-dependent linear combination of the R-symmetry components of a CBO, and is restricted to lie on the great circle fixed by the isometry generated by . Furthermore, at each point on this circle, the twisted CBOs are chiral with respect to a distinct subalgebra. More details will be given in Section 2. Restricting our 3D theories to the cohomology of , therefore, results in some 1D field theory on a circle whose local operators can be identified with cohomology classes of twisted CBOs, which, in turn, are in one-to-one correspondence with Coulomb branch chiral ring operators.

The above 1D theory provides a significant simplification of the original 3D problem of computing correlators of CBOs, due to the following properties. First, the IR two- and three-point functions of twisted CBOs in the 1D theory are sufficient to fix the corresponding correlators of CBOs in the full 3D SCFT, simply because a two- or three-point function of Lorentz scalar primary operators is fixed by conformal invariance up to an overall constant (see, e.g., Section 6.4 of [13]). Moreover, it turns out that the 1D theory is topological in the sense that its correlators are independent of the relative separation between insertions, but can depend on their order on the circle. We will refer to this theory as the Coulomb branch 1D topological quantum field theory (TQFT). The topological correlators could in principle be functions of dimensionless parameters along the flow. Because we set all the real masses and FI terms to zero, the only remaining dimensionless parameter is . However, the 1D theory is independent of (and therefore of ) because, as shown in [13], the Yang-Mills action is -exact. It follows that the correlators of twisted CBOs are RG-invariant and can be identified, all along the flow, with those of the IR SCFT. The same results also hold for twisted HBOs, whose associated 1D TQFT is obtained by passing to the cohomology of . The above properties of the 1D TQFTs turn them into a powerful framework to study correlators of half-BPS operators in theories.

The observation that some BPS operators in -dimensional theories with eight supercharges admit a lower-dimensional description was first made for SCFTs in [35]. Following that work, the 1D TQFTs associated with 3D SCFTs were studied in detail in [12, 11]. It was shown in [12, 11] that conformal bootstrap arguments can be used to fix the 1D TQFT in some simple examples, though doing this for general 3D SCFTs proved to be difficult. Finally, the fact that the 1D TQFTs can also be defined along RG flows on , as we just reviewed, was discovered in [13]. This fact allows for the use of supersymmetric localization to calculate correlators in the 1D TQFTs for 3D theories described in the UV by a Lagrangian. Moreover, it follows that the 1D theory is also defined along relevant deformations of the theory on by real masses and FI parameters. The correlators of twisted CBOs are in general sensitive to these deformations, providing nonperturbatively calculable examples of correlators along RG flows.777The topological invariance of the Coulomb (Higgs) branch 1D theory is lost upon turning on FI (real mass) parameters. However, the resulting position dependence of correlators turns out to be very simple.

We develop three complementary approaches to computing correlators of twisted CBOs. In Section 3, we use localization on in an -symmetric background created by a monopole-antimonopole pair to compute correlators involving two twisted monopole CBOs and an arbitrary number of non-defect twisted CBOs. In Sections 4 and 5, we explain how to vastly generalize these results by localizing on a hemisphere with half-BPS boundary conditions, which allows for insertions of twisted CBOs anywhere along a great semicircle. These insertions are conveniently described by certain operators acting on the wavefunction. Pairs of such wavefunctions can then be glued along their boundary to reproduce the partition function with an arbitrary number of twisted CBOs. In Section 5, we further show how to interpret our results as a dimensional reduction of the Schur index of 4D theories enriched by BPS ’t Hooft-Wilson loops.

1.2 Summary of Results

Let us now summarize our results and fix our notation. We consider theories with gauge group and hypermultiplets of gauge charges with . Viewing as an matrix, we demand that to avoid having subgroups of with no charged matter. The theory has flavor symmetry where acts on the hypermultiplet, while generally emerges in the IR and acts on the Coulomb branch. Only a maximal torus of is manifest in the UV as a ‘’topological symmetry” acting on monopole operators and generated by currents constructed from the field strength as .

Let the 1D theory live on a great circle parametrized by (see Figure 1). The -closed twisted CBOs are constructed from products of bare twisted monopole operators , labeled by their charge where is the monopole charge lattice determined by Dirac quantization, as well as twisted vector multiplet scalars corresponding to each factor of . As we will see in Section 2, is a position-dependent linear combination of the three real vector multiplet scalars, while can be described as a particular -invariant background for the vector multiplet fields, which inserts the appropriate Dirac monopole singularity. These singular backgrounds are described in detail in Appendix C.

In Section 5, we present a matrix model expression for a correlator with insertions of twisted CBOs , where . To describe this expression, it is useful to think of as a union of two hemispheres joined along their boundary, as depicted in Figure 1. The 1D TQFT circle intersects the boundary at its North and South poles labeled, respectively, by and in Figure 1. Under this decomposition, the path integral on can be thought of as an inner product (more accurately, a bilinear form) composing the wavefunctions of and . Moreover, in this language, the insertions of twisted CBOs can be represented as certain shift operators acting on the hemisphere wavefunctions.

Explicitly, consider the case in which the are all inserted along the semicircle inside the upper hemisphere () in the order . There is no loss of generality in inserting all operators in because the 1D TQFT is topological, so only the order of the insertions is important. Our analysis then implies that this correlator can be computed in terms of an ordinary -fold integral given by

 ⟨O(1)(φ1)⋯O(n)(φn)⟩S3 =1ZS3∑→B∈Γm∫Rr[d→σ]→BΨ−(→σ,→B)ˆO(1)N⋯ˆO(n)NΨ+(→σ,→B). (1.1)

Let us now unpack the notation in (1.1):

• The represent wavefunctions defined by the path integral on the hemispheres evaluated with certain half-BPS boundary conditions on . We will show in Section 4 that these boundary conditions are parametrized by constants and by the monopole charge . In particular, the vacuum wavefunctions , which have zero monopole charge, are given by

 Ψ±(→σ,→B)=δ→B,→0Nh∏I=11√2πΓ(12−i→qI⋅→σ). (1.2)

The variables arise from localization of scalars in the vector multiplet.

• In (1.1), each of the twisted CBOs is represented by a certain shift operator, denoted by , acting on the wavefunction . The label on the implies that it represents an insertion of through the North pole of , labeled by in Figure 1. The order in which the shift operators act on represents the order of insertions on the semicircle. There is a second set of shift operators representing insertions through the South pole (labeled by in Figure 1), such that the same correlator (1.1) is given by

 ⟨O(1)(φ1)⋯O(n)(φn)⟩S3 =1ZS3∑→B∈Γm∫Rr[d→σ]→BΨ−(→σ,→B)ˆO(n)S⋯ˆO(1)SΨ+(→σ,→B). (1.3)

The order in which the operators act on also represents the order of insertions on the semicircle, but in the opposite direction. The shift operators corresponding to the bare twisted monopoles and the vector multiplet scalars are written explicitly in (5.20), (5.21), and (5.13), respectively. It is important that the shift operators do not depend on the insertion point. This must be the case because the correlators are topological and depend only on the order of the insertions, which is reflected in the nontrivial commutation relations between the shift operators.

• The wavefunctions can be glued into a partition function on with the measure as in (1.1), where is given explicitly by

 [d→σ]→B =μ(→σ,→B)drσ, (1.4) μ(→σ,→B) =Nh∏I=1(−1)|→qI⋅→B|−→qI⋅→B2Γ(1+|→qI⋅→B|2+i→qI⋅→σ)Γ(1+|→qI⋅→B|2−i→qI⋅→σ). (1.5)

This measure is simply the partition function of chiral multiplets in a 2D theory, coupled to vector multiplets with magnetic charge [36]. We have normalized the correlators (1.1) by the partition function , such that .

• The above expressions can be generalized straightforwardly to include deformations by real masses and FI parameters. This will be described in Section 5.1.2.

The above description of correlators of twisted CBOs in terms of hemispheres and shift operators, while derived using localization in 3D, was inspired by computations of Schur indices with line defects in 4D theories [37, 38, 39].888In turn, the interpretation of loop operator insertions on as shift operators acting on half-indices in [37, 38, 39] was inspired by earlier works [40, 41, 42], where loop operator insertions on were also understood as shift operators acting on the wavefunction, as derived via localization in [43]. In fact, as we show in Section 5, these problems are closely related. The defect Schur index can be computed by a path integral on with ’t Hooft-Wilson loops wrapping the . To preserve supersymmetry, the defects should be inserted at points along a great circle in . As we will show, upon dimensional reduction of the 4D index along , the line defects become twisted CBOs in the 3D dimensionally reduced theory. The above expressions for correlators of twisted CBOs can all be derived from the 4D defect Schur index, providing a strong consistency check of our results.

2 Preliminaries

In this section, we set the stage for the problems that we study in the rest of the paper. We start by reviewing the construction of supersymmetric Lagrangians using vector multiplets and hypermultiplets on . We then describe a BPS sector of these theories that is captured by a 1D theory, focusing on the case of the Coulomb branch. Finally, we give a careful definition of BPS monopole operators, which are of primary interest in this paper, and explain some of their properties.

In this section, we try to be maximally general and define everything for non-abelian gauge theories. However, the actual localization computations in the rest of the paper will be performed only for abelian theories.

2.1 N=4 Theories on S3

The theories that we analyze in this paper are Lagrangian 3D gauge theories. We start by giving a short review of their structure and summarizing our conventions, referring the reader to [13] for more details.

2.1.1 Supersymmetry Algebra

supersymmetry on is based on the superalgebra or a central extension thereof. Its even subalgebra contains the isometries of , whose generators we denote by and , as well as the R-symmetry subalgebra generated by and . The odd generators are denoted by and .999Above, are spinor indices. They can be raised and lowered from the left with the anti-symmetric symbols , , where . The same raising/lowering convention will also be used for the fundamental indices of R-symmetries. See Appendix A for a full list of our conventions. The algebra obeyed by , , is

 [J(ℓ)i,J(ℓ)j] =iϵijkJ(ℓ)k, [J(ℓ)αβ,Q(ℓ±)γ] =12(εαγQ(ℓ±)β+εβγQ(ℓ±)α), (2.1) =±Q(ℓ±)α, {Q(ℓ+)α,Q(ℓ−)β} =−4ir(J(ℓ)αβ+12εαβRℓ), (2.2)

where we have set

 J(ℓ)αβ≡(−(J(ℓ)1+iJ(ℓ)2)J(ℓ)3J(ℓ)3J(ℓ)1−iJ(ℓ)2). (2.3)

The generators of obey the same relations with .

The generators and act by Lie derivatives and with respect to the left- and right-invariant vector fields and on . The generators and will often be important to us, and their corresponding vector fields are given by

 vℓ3=−i2(∂τ+∂φ),vr3=−i2(∂τ−∂φ). (2.4)

Above, we have used coordinates that exhibit as a fibration over a disk with the fiber shrinking at the boundary, which will be useful in the remainder of the paper (see Appendix A.1 for details). Explicitly, let us embed in as

 X21+X22+X23+X24=r2 (2.5)

and parametrize the by

 X1+iX2=rcosθeiτ,X3+iX4=rsinθeiφ, (2.6)

where and . In these coordinates, parametrizes the unit disk, and the fiber. We also sometimes use the notation

 Pτ=−(Jℓ3+Jr3),Pφ=−Jℓ3+Jr3 (2.7)

to denote the and rotation isometries of .

It is convenient to think of as a subalgebra of the 3D superconformal algebra , whose R-symmetry subalgebra is . This embedding is parametrized by the choice of the subalgebra of , which is specified by the Cartan elements

 hab∈su(2)H,¯¯¯h˙a˙b∈su(2)C, (2.8)

where () label the fundamental irrep of (). Here, and are traceless Hermitian matrices satisfying and . They determine a relation between the generators , of and the generators , of :

 12(Rℓ+Rr)=12habRba≡RH,12(Rℓ−Rr)=12¯¯¯h˙a˙b¯¯¯¯R˙b˙a≡RC. (2.9)

The superconformal symmetries of are parametrized by conformal Killing spinors satisfying the conformal Killing spinor equations on :

 ∇μξa˙a=γμξ′a˙a,∇μξ′a˙a=−14r2γμξa˙a, (2.10)

where are curved-space gamma matrices and is the radius of (the first equation implies the second via where ). Those that correspond to supersymmetries within the subalgebra satisfy the additional condition

 ξ′a˙a=i2rhabξb˙b¯¯¯h˙b˙a. (2.11)

To conform with previous works, we use the convention that

 hab=−σ2,¯¯¯h˙a˙b=−σ3. (2.12)

Different choices of are related by conjugation with and, as will be explained shortly, determine which components in the triplets of FI and mass parameters can be present on the sphere: and . In Appendix G, we describe how the algebra is obtained from the rigid limit of off-shell 3D conformal supergravity, following the philosophy of [44]. The latter point of view elucidates the origin of the matrices and as background values for scalar fields within a certain 3D Kaluza-Klein supergravity multiplet.

2.1.2 Lagrangians

The supersymmetry algebra just described acts in Lagrangian theories constructed from a vector multiplet and a hypermultiplet . The vector multiplet transforms in the adjoint representation of the gauge group and has components

 V=(Aμ,λαa˙a,Φ˙a˙b,Dab), (2.13)

consisting of the gauge field , gaugino , and scalars and , which transform in the trivial, , , and irreps of the R-symmetry, respectively. The hypermultiplet transforms in some unitary representation of and has components

 H=(qa,˜qa,ψα˙a,˜ψα˙a) (2.14)

where are scalars transforming as under the R-symmetry and as under , respectively, while are their fermionic superpartners and transform as under the R-symmetry. The SUSY transformations of and are collected in Appendix A.2.

The action for coupled to is

which actually preserves the full superconformal symmetry . The super Yang-Mills action preserves only the subalgebra and is given by

 SYM[V] =1g2YM∫d3x√gTr(FμνFμν−DμΦ˙c˙dDμΦ˙c˙d+iλa˙a⧸Dλa˙a−DcdDcd−iλa˙a[λa˙b,Φ˙a˙b] −14[Φ˙a˙b,Φ˙c˙d][Φ˙b˙a,Φ˙d˙c]−12rhab¯h˙a˙bλa˙aλb˙b+1r(habDba)(¯h˙a˙bΦ˙b˙a)−1r2Φ˙c˙dΦ˙c˙d). (2.16)

The theory (2.15) has flavor symmetry group , whose Cartan subalgebra we denote by . The factor acts on the hypermultiplets, while contains the topological symmetries that act on monopole operators.101010 may be enhanced to a non-abelian group in the IR. It is possible to couple the theory to a supersymmetric background twisted vector multiplet in , which on leads to a single FI parameter for every factor of the gauge group (as opposed to an triplet on ). The corresponding FI action is given by

 SFI[V] =idim(tC)∑I=1ζI∫d3x√g(hab(D(I))ba−1r¯h˙a˙b(Φ(I))˙b˙a), (2.17)

where and are the scalars in the vector multiplet gauging the factor of . Similarly, one can introduce real masses for the hypermultiplets by turning on background vector multiplets in . In order to preserve supersymmetry, all the components of are set to zero except for

 ˆm=−12¯h˙a˙b(Φb.g.)˙b˙a=r2hab(Db.% g.)ba. (2.18)

In particular, on , there is a single real mass parameter for every generator in (as opposed to an triplet on ). In the presence of nonzero real mass and FI parameters, the algebra is centrally extended by charges and for the respective factors of the superalgebra. The central charges are related to the mass/FI parameters by

 1r(Zℓ+Zr)=iˆm∈itH,1r(Zℓ−Zr)=iˆζ∈itC. (2.19)

A more detailed description of the superalgebras can be found in [13].

Finally, let us specify the contour of integration in the path integral. Because we work in Euclidean signature, the fermionic fields do not obey any reality conditions, while the bosonic fields satisfy

 q†a=˜qa,A†μ=Aμ,Φ†˙a˙b=−Φ˙a˙b,D†ab=−Dab, (2.20)

where the Hermitian conjugate is taken in the corresponding representation.

2.1.3 Abelian Gauge Theories

In the bulk of the paper, we will focus exclusively on abelian gauge theories. Specifically, we will consider a gauge theory coupled to hypermultiplets with gauge charges , where . The maximal tori of the global symmetry algebras in this case are given by and . The hypermultiplets transform under with weights , while monopole operators transform under the topological symmetry with charges . The monopole charge lattice is defined through Dirac quantization by the constraints where ranges over all matter charges allowed in the gauge theory.

2.2 Twisted Operators and the 1D Theory

Supersymmetric field theories with eight supercharges in various dimensions have subsectors of operators which can be described by lower dimensional theories. Our 3D theories are among those that have such sectors, which, moreover, turn out to furnish certain 1D theories. This fact was originally noticed for SCFTs in [35], further developed in [12, 11], and extended to non-conformal theories on in [13].

Following [13], we consider two pairs of supercharges within .111111The embedding of these supercharges inside is given in Appendix A.2. Those associated with the Higgs branch are

 QH1=Q(ℓ+)1+Q(r−)1,QH2=Q(ℓ−)2+Q(r+)2, (2.21)

and those associated with the Coulomb branch are

 QC1=Q(ℓ+)1+Q(r+)1,QC2=Q(ℓ−)2+Q(r−)2. (2.22)

Each of these four supercharges is nilpotent. There exists a 1D theory associated with cohomology classes of and another associated with those of . To see this, let us focus on the (equivariant) cohomology of or acting on local operators, for an arbitrary constant . Because of the relations

 (QHβ)2 =4iβr(Pτ+RC+irˆζ), (2.23) (QCβ)2 =4iβr(Pτ+RH+irˆm), (2.24)

local operators in the cohomology of or must be annihilated by the right-hand side of (2.23) or (2.24), respectively. This implies that local operators can only be inserted at the fixed points of the isometry, which form a great circle parametrized by at , where the -circle shrinks (see (2.6)).121212It also follows from (2.23) (or (2.24)) that the spins and R-charges of - (or -) closed operators should be related. However, this constraint turns out to be trivial because all these operators turn out to be Lorentz scalars transforming trivially under (or ). In flat space, is the rotation that fixes the line along which operators are inserted.

Another important property emphasized in [13] is that

 {QHβ,…} =Pφ+RH+irˆm, (2.25) {QCβ,…} =Pφ+RC+irˆζ, (2.26)

which leads to the definitions of twisted translations:

 ˆPHφ =Pφ+RH, (2.27) ˆPCφ =Pφ+RC. (2.28)

The twisted translations (or ) are - (or -) closed, and can therefore be used to translate cohomology classes along the great -circle. The cohomology classes of and therefore form two distinct 1D theories. Furthermore, when (or ), the twisted translation (or ) is exact under (or ). The twisted-translated cohomology classes then become independent of the position along the circle. In such a situation, the cohomology classes furnish a 1D TQFT, meaning that their OPE is independent of the separation between operators, but can depend on their ordering along the circle. This OPE therefore determines an associative but non-commutative product, which can be thought of as a star product on some variety.

The operators in the cohomology are most easily classified at the superconformal point, where the symmetry is enhanced to . In this case, one finds that for every fixed insertion point , the operators in the cohomology of and are in the Higgs and Coulomb branch chiral rings, respectively, with respect to some superconformal subalgebra of .131313In particular, the star product in the 1D TQFT then yields a deformation quantization of the chiral ring, which describes the Higgs or Coulomb branch of the moduli space of the theory as a complex variety; this point of view was advocated in [11]. Indeed, for SCFTs, we have the algebraic relations

 {QH1,QH†1} ={QH2,QH†2}=8(D−R11), (2.29) {QC1,QC†1} ={QC2,QC†2}=8(D−12(¯R˙1˙2+¯R˙2˙1)), (2.30)

where is the generator of dilatations. The relation (2.29), together with the state-operator map (which yields an inner product, hence a notion of adjoint in radial quantization) and the standard Hodge theory reasoning (which exhibits a unique harmonic representative of each cohomology class), implies that representatives of the cohomology of , when inserted at the origin, satisfy

 D=R11. (2.31)

Such operators belong to the Higgs branch chiral ring. They are the highest-weight components of HBOs , which are half-BPS superconformal primaries transforming in the spin- irrep of , and are Lorentz scalars of dimension . Similarly, (2.30) implies that the representatives of cohomology at the origin satisfy

 D=12(¯R˙1