Matching 3d N=2 Vortices and Monopole Operators

In earlier work with N. Seiberg, we explored connections between monopole operators, the Coulomb branch modulus, and vortices for 3d, N=2 supersymmetric, $U(1)_k$ Chern-Simons matter theories. We here extend the monopole / vortex matching analysis, to theories with general matter electric charges. We verify, for general matter content, that the spin and other quantum numbers of the chiral monopole operators match those of corresponding BPS vortex states, at the top and bottom of the tower associated with quantizing the vortices' Fermion zero modes. There are associated subtleties from non-normalizable Fermi zero modes, which contribute non-trivially to the BPS vortex spectrum and monopole operator matching; a proposed interpretation is further discussed here.


Introduction
Three-dimensional U (1) gauge theories exhibit IR-interesting phenomena and phases, with qualitative similarities to 4d non-Abelian gauge theories. For example, electricmagnetic dualities can be explored in this context, and the U (1) gauge group makes it easier to make the duality more precise, and potentially construct the duality-map between fields. This is particularly true for 3d theories with N ≥ 2 supersymmetry, where magnetically charged, BPS vortex solitons can be regarded as giving the dual quanta in terms of the electric variables, with corresponding chiral superfield monopole operators.
Building on [1], we here consider 3d, N = 2 supersymmetric, compact 1 U (1) k gauge theory (k is the Chern-Simons coefficient), with matter chiral superfields Q i , with general electric charges n i ∈ Z. A key aspect is that the theory has an exact 2 , conserved global U (1) J topological symmetry, with current j µ J = ǫ µρσ F ρσ /4π, and associated charge (1.1) The theory contains local operators, and particle states, with q J = 0, despite the fact that the photon and Q i have q J = 0. There are three distinct, related ways to get q J = 0: 1. Monopole operators: disorder the gauge field, with q J units of magnetic flux, around a point x µ 0 in spacetime [4,5,6]. It is a local, chiral N = 2 operator (the 3d reduction of 4d 't Hooft line operators). This short-distance definition of the operator is independent of IR data, e.g. the particular vacua, or the spacetime geometry. The chiral condition implies that the real scalar σ = Σ| of the N = 2 photon linear multiplet has [6,1] Upon taking ζ → 0, all Q vac i → 0, the BPS magnetic vortices become massless, and can potentially condense and give dual Higgs description of the Coulomb branch [9], in the sense of 3d mirror symmetry's exchange of the electric and magnetic Higgs and Coulomb branches [10]. See also [11,12] for vortices and partition functions.
Connections and distinctions between monopole operators, vortices, and the Coulomb branch, for the theories in flat space, were studied in [1,13], and will be further explored here. We determine, and match, the gauge and global charges of monopole operators and the vortices. For the monopole operators X ± , the charges are simply, and exactly, obtained by a one-loop calculation of induced Chern-Simons terms [9,14,15,1] to be with k c ≡ 1 2 i n i |n i | (see sect. 2). The operators X ± in (1.6) exist as gauge invariant operators 3 only if k = ∓k c ; this is the condition for the X ± Coulomb branch to exist. 3 The superconformal U (1) R * of the N = 2 SCFT at Q i = X ± = 0, is a linear combination of those in (1.6), U (1) R * = U (1) R + j R j U (1) j , so ∆(Q i ) = R i , and ∆(X ± ) = R(X ± ) = 1 2 i |n i |(1 − R i ), with R i determined by F-extremization [16](or τ RR minimization [17,14]).
The corresponding charges of BPS vortices arise in a seemingly different way, from quantizing the vortex Fermion zero modes 4 , Ψ A , with A = 1 . . . N z , i.e. from This formally gives a tower of 2 N z degenerate states: treating the Ψ A (Ψ † A ) as raising (lowering) operators, the top and bottom vortex states in this tower are Writing "|0 " q J as the naive (ignoring zero modes) groundstate for q J = 0, We identify the X ± quanta with the top and bottom vortex states: with |0 the q J = 0 vacuum. We verify that the vortex charges, computed from (1.10), are indeed compatible with (1.11) and the X ± charges in (1.6).
This matching was verified in [1] for theories with N matter fields Q i , with all n i = 1.
We here extend the analysis to theories with general matter charges n i . We find that, in the q J = ±1 vortex background (for ζ > 0), the Fermion component of Q i leads to |n i | zero modes, Ψ i,p=1...|n i | , with charges 6 and spin given by (again, k c ≡ 1 2 i n i |n i |): Quantizing the Ψ i,p gives a tower of 2 |n i | degenerate vortex states. The top and bottom states |Ω ± q J , as in (1.8), have quantum numbers that follow from (1.12) and (1.10); this gives the charges of |Ω ± q J =1 in (1.12). These |Ω ± q J =1 charges indeed agree with those of X + and X † − in (1.6), fitting with the proposed operator / state map in (1.11). As we will see, the |q J | = 1 Fermi zero modes in (1.12) have large z behavior (from (1.4)) |Ψ i,p | ∼ |z| p−1−|n i | , and the p = |n i | case is non-normalizable, for every matter field.
As in [1], we quantize all Fermi zero modes as in (1.7), including the non-normalizable ones, and interpret the non-normalizable Fermi zero modes as mapping between different Hilbert spaces. But some additional discussion is required here, particularly for theories with k = k c = 0. Then both X + and X − exist in the same theory, corresponding to the two Coulomb branches. Fitting with (1.11), both |Ω + q J =1 and |Ω − q J =1 in (1.12) have U (1) spin zero, and can condense to give the X + or X − branches. But |Ω + q J =1 and |Ω − q J =1 are are related via non-normalizable Fermi zero modes. The BPS quanta created by X + and X † − evidently must reside in different Hilbert spaces, which seems puzzling. Our (tentative) interpretation is that this reflects the fact that X + and X − label two disconnected branches of the moduli space of vacua, i.e. that X + X − ∼ 0 in the chiral ring. Quantum field theories typically do not have a Hilbert space of single-particle states, with a mapping between them via normalizable zero modes. To the extent that it can happen for BPS states relies on the x-independence of the chiral ring OPE. If a product of chiral operators is zero in the chiral ring, the associated BPS states can appear to reside in different Hilbert spaces. We discuss this further in sect. 5, e.g. for N f = 1 SQED, and its W = M X + X − dual. It would be good to have a more complete understanding.
The outline of the remaining sections is as follows. Section 2 briefly reviews some of the basic points, and sets up our notation and conventions; a few more details are in an appendix. Section 3 broadly discusses the BPS vortices, and their zero modes, for the general N = 2 susy, U (1) k charge n i matter theories. Section 4 discusses vortices and zero modes in general cases with a vev Q i ∝ δ i,1 , with Q 1 of charge n 1 = 1. Section 5 considers theories with N ± matter fields of charge n i = ±1, e.g. N = 2 SQED with N + = N − = N f flavors. Section 6 discusses cases where Q i = 0 for matter with charge n i = 1, where there can be an unbroken Z |n i | discrete gauge symmetry, i.e. an orbifold.
One could generalize to non-Abelian gauge theories; it will not be considered here.

2.
A few preliminaries (see also the appendix)

Lagrangian and effective Chern-Simons terms
The U (1) k gauge theory, with matter fields Q i of charges n i , has classical Lagrangian We will set the real masses m i = 0, and take W tree = 0. Dirac-quantization for monopole operators implies that the Chern-Simons coefficient k is quantized as The supersymmetric vacua have expectation values of the Coulomb modulus σ = Σ|, or the matter fields Q i = Q i |, subject to the conditions D = 0 and m j (σ)Q j = 0, where and m i (σ) ≡ m i + n i σ. The effective FI parameter ζ ef f , and Chern-Simons coefficient 3) are shifted by integrating out massive matter, with ζ ef f = ζ for m i = 0 and and k ef f = ζ ef f = 0. The asymptotic values of k ef f for σ → ±∞ are if k = ∓k c , respectively. For non-zero k ef f and ζ ef f , there are also isolated "topological vacua," with Q i = 0 and σ = −ζ ef f /k ef f ; those vacua will not enter in our discussion.

Chern-Simons contribution to Gauss' law, and charges and spin from q J
The Chern-Simons term affects Gauss' law (the A 0 EOM), as The Chern-Simons contribution in (2.6) implies that operators or states with q J = 0 acquire an associated electric charge, and a related contribution to their spin [31][32][33] if the Fermions are massive and integrated out. For vortices, if k = 0, the last term in (2.6) leads to A 0 = 0, which complicates the equations of motion.
The gauge and global charges of the X ± operators in (1.6) follow from (2.7), and its analogs for mixed gauge-flavor Chern-Simons coefficients. Since X ± extend to σ = ±∞, which is the condition for X ± to be a gauge invariant, scalar operator: If k = ∓k c , then the X ± Coulomb branch exists. i n i sign(n i σ). Taking σ → ±∞ for q J = ±1, the analog (2.7) for the global charges then gives the corresponding charges in (1.6).

BPS and anti-BPS particles
Particle states can be labelled by their U (1) spin , s, and it is convenient to convert the spinors to a rotational spin-diagonal basis (s = 1 for z = x 1 + ix 2 and ∂ z = 1 2 (∂ x 1 + i∂ x 2 )). For the supercharges, we define (fixing a minor notational issue vs [1]) so Q ± and Q ± have spin s = ± 1 2 . In terms of these, the N = 2 algebra is and the remaining two supercharges make a two-dimensional representation Likewise, an anti-BPS particle has m = −Z > 0, and is annihilated by Q + and Q − . Every BPS state has a CPT conjugate anti-BPS state, with opposite global charges and Z, but with the same U (1) spin spin s. The R-charges and spins of these states are [1]

BPS and anti-BPS vortices
The central term of the supersymmetry algebra (setting real masses m i = 0) is For Z > 0, the vortex can be BPS, annihilated by Q − and Q + (2.11). For Z < 0, the vortex is anti-BPS, annihilated by Q − and Q + . The condition that these supercharges annihilate the background implies the BPS equations for a static (all ∂ t → 0) vortex with with D given by (2.3). One must also impose Gauss' law (2.6). In our conventions, the chiral superfields, Q i , of a Z > 0 BPS vortex are anti 7 -holomorphic (resp holomorphic for a Z < 0 anti-BPS vortex). We will here be particularly interested in the zero modes.
The vortex's Fermi zero modes are the static ∂ t → 0 solutions of the Fermion equations of motion, from (2.1) with m i = 0, in the background of the static vortex's Bosonic fields: where ψ i↑,↓ and λ ↑,↓ have spin ± 1 2 , and U (1) R charge −1. As we discuss in section 4, the number of solutions of (3.5) and (3.6), and their quantum numbers, are as in (1.12): each matter field contributes |n i | Fermi zero modes, with spin correlated to the sign of n i .
, is not analytically known, nor is it needed: knowing its existence and number 7 This (unfortunately) is due to following [34]'s sign convention for A µ ; see the appendix.
of zero modes suffices. The vortex with U (1) J charge q J has |q J | complex Bosonic zero modes, and |q J | spin + 1 2 Fermionic zero modes. The q J = 1 vortex has one complex zero mode z 1 , the translational invariance zero mode of the BPS vortex core location, and one complex spin 1 2 Fermionic zero mode [20,21], Ψ 1 , a combination of the photino and the matter fermion that solves (3.5) and (3.6). The Bosonic field configuration is annihilated by Q − and Q + (2.11), while the other two supercharges give the Fermi zero mode, Ψ 1 ∼ Q + , and complex conjugate Ψ † 1 ∼ Q − , i.e. the photino and matter Fermi field configuration of Ψ 1 follows from acting with Q + on F vortex µν (z, z) and Q vortex 1 (z, z).
yields a BPS doublet (2.12); adding the q J = −1, anti-BPS, CPT conjugate states gives one copy of the spectrum (2.13). The U (1) R and U (1) spin quantum numbers there are found 2 theory is dual to a theory of a free chiral superfield, X ± [35]. The FI parameter ζ maps to a real mass m X in the dual. BPS vortices map to X-particle states.

Cases with multiple matter fields Q i : the (anti)-BPS equations for the Bosonic fields
By (3.4), the vortex gauge field configuration is completely determined by that of any non-zero matter field Q i : The condition that the gauge field (3.7) be smooth, with winding number q J (1.4), implies [36] that a charge n i = 1 matter field has Q i (z) with |q J | zeros, at the vortex core locations, with f i ≡ f i (z, z) non-vanishing. Turning on Bosonic zero modes can resolve the zeros in , D (n j ) )) Since the LHS of (3.11) is non-negative, equations (3.4) have a Q j = 0 solution only if the second term on the RHS of (3.11) has the correct sign. M i j = Q i Q j = 0 in a theory with vector-like matter. As discussed in [9], the fact that BPS vortices require M i j = 0 can have a simple dual perspective, e.g. for N f = 1 SQED it is clear from the W = M X + X − dual that the X ± quanta are only BPS for M = 0. See [38,39] for other, dynamical arguments leading to the same conclusion. The general solution of (3.9) for a q J = 1 BPS (or q J = −1 anti-BPS) vortex is then where the denominators are determined by the z → z 1 vanishing degree of Q 1 in (3.8), (which is the only singularity of the ratio) and the numerators by (anti) holomorphy and the condition that the ratio approaches the vacuum value, i.e. zero, for |z| → ∞: (3.14) The |n j | coefficients c j,p (or c j,p ) in (3.14) are the Bosonic zero modes for matter field Q j with Q vac j = 0 in a BPS (or anti-BPS) q J = 1 vortex. Matter field(s) Q i with Q vac i = 0 also yield |n i | Bosonic zero modes, one of which is the translational zero mode z 1 .

Normalizable vs non-normalizable zero modes
The Bosonic or Fermionic zero modes of the static vortex are replaced with dynamical variables on the vortex worldline theory, if the associated induced kinetic term is normalizable. Non-normalizable zero modes, on the other hand, are frozen parameters. For example, the translational zero mode of a |q J | = 1 vortex is quantized as z 1 → z 1 (t), which is normalizable, with finite induced kinetic term d 2 zL → 1 2 m BP S |ż 1 | 2 . Considering the c j,p or c j,p term in (3.13) for large |z| gives |Q j | ∼ |c j,p ||z| p−1−|n j | , so the induced coefficient of a |ċ j,p | 2 term involves ∼ d 2 z|z| 2(p−1−|n j |) , i.e. c j,p and c j,p are normalizable for 1 ≤ p < |n j | (requiring |n j | > 1) and log-IR-divergent non-normalizable for p = |n j |.
The non-normalizable ρ j ≡ c j,p=|n j | or ρ j ≡ c j,p=|n j | zero modes in (3.13) generalize the non-normalizable zero modes of "semi-local vortices" [27][28][29][30]. As found there, turning on ρ i = 0 dramatically changes the character of the vortex solution, removing the zero in (3.8) at the vortex core, and changing the flux F 12 in (3.3) from having the usual ∼ e −cm γ |z| exponential falloff for large |z| (with m γ the Higgsed photon mass) into a diffuse, powerlaw falloff. In our general n i case, each matter field with sign(n i ) = sign(ζ) and Q vac i = 0 yields one-such non-normalizable ρ i bosonic zero mode. If |n j | > 1, there are also |n j | − 1 additional normalizable, and hence dynamical, zero modes c j,p<|n j | or c j,p<|n j | .
The bosonic non-normalizable zero modes, ρ i , are interpreted, as in [1], as superselection parameters already of the q J = 0 vacuum, even before adding the vortex: taking including all normalizable and also non-normalizable Fermi zero modes.

Fermi zero modes of BPS vortices for somewhat general cases.
We will consider |q J | = 1 BPS and anti-BPS vortices, taking ζ/n 1 > 0, in the vacuum with σ = 0 and non-zero expectation value for only Q 1 : For the rest of this section, we assume that n 1 = 1, though we allow for general charges n j for the other Q j>1 matter fields in (4.1). We will discuss the n 1 = 1 case in sect. 6.
Each Q i matter field with n i > 0 has n i Bosonic zero modes, while Q i with n i < 0 have none. The Q 1 Bosonic zero mode is the normalizable, translational zero mode, z 1 .
For the matter fields Q j =1 , with n j > 0, the Bosonic zero modes are the c j,p or c j,p in   Since, for q J = 1, Q 1 has a degree one zero at z 1 , this gives (similar to (3.13)) n j > 0 (q J = 1) : with the n j coefficients, u j,p=1,...n j , Fermionic zero modes of spin p − 1 2 . Likewise, with the |n j | coefficients, d j,p , Fermionic zero modes of spin −(p − 1 2 ). As in the bosonic case, for either (4.7) or (4.8), the p = |n j | Fermi zero mode is non-normalizable. The spins of u j,p and d j,p follow from constructing the angular momentum generator, much as in [40], assigning spin +1 to z, and spin + 1 2 to ψ j,↑ in (4.7). By (1.5), Q n j 1 /(z − z 0 ) n j is θ independent for large |z|, so we assign spin + 1 2 to each term u j,p z p−1 in (4.7), and, likewise, spin − 1 2 to all d j,p z p−1 in (4.8). So u j,p has spin p− 1 2 and d j,p has spin −(p− 1 2 ). In sum, the q J = 1 vortex has the Ψ (q J =1) n j ,p in (1.12): |n j | Fermion zero modes, of spins sign(n j )(p − 1 2 ), for p = 1 . . . |n j |. The q J = −1 vortex is similar. The other quantum numbers likewise follow from those of ψ j,↑,↓ , and are as given in (1.12). We assign U (1) gauge charges in (1.12), even though U (1) gauge is spontaneously broken (screened) by (4.1).
The zero modes of a matter field Q i are in |n i | different N = (2, 0) chiral multiplets (i.e. a complex Boson and a complex Fermion) if sign(n i ) = sign(ζ), or |n i | N = (2, 0) chiral Fermi multiplets (i.e. a complex Fermion and an auxiliary field) if sign(n i ) = − sign(ζ).
All the Fermi zero modes are quantized, as in (1.7) and (1.8), giving 2 |n i | states. The zero mode should be regarded as Q + , i.e. neutral under U (1) gauge and the non-R-symmetry global symmetries; quantizing this zero mode yields BPS doublets (2.13). Including all zero modes yields 2 |n i |−1 BPS doublets.
Consider a theory with vector-like, charge-conjugation symmetric matter content, with pairs Q i and Q i , of charges ±n i . Then k c = 1 2 i n i |n i | = 0 in (2.5), and the k = 0 theory with ζ = 0 has asymptotic Coulomb branches X ± . The theory respects P and T if k = 0, and it respects C if ζ = 0. For every Fermi zero mode Ψ n j ,p , there is a Fermi zero mode Ψ −n j ,p of opposite spin, so the A Ψ A appearing in (1.10) has spin s = 0, and the top and bottom states |Ω ± q J =1 have s = − 1 2 k, so spin 0 for k = 0, This fits with (1.11): these states map to the quanta of X ± , |Ω + q J =±1 ∼ X ± |0 and |Ω − q J =±1 ∼ X † ∓ |0 , with X ± a gauge invariant operator for k = 0.

Examples: theories with N ± matter fields of charge n i = ±1
We denote the matter as Q i=1...N + , with n i = +1, and Q i=1...N − , with n i = −1. The charge is +1 for all Q i and Q i . We take N + > 0, and ζ > 0, and then The cases (N + , N − ) = (N, 0) were discussed in [1]. The minimal matter case, N = 1, was reviewed in sect. 3.1. The vortices of the N > 1 case is the N = 2 version of the "semilocal" vortices of [27][28][29], allowing also for Chern-Simons terms. Our present discussion in this section also includes cases with both N + N − = 0; we did not find much discussion of vortices in such theories in the literature, aside from some brief comments in [23,24].
For general (N + , N − ), a q J = 1 BPS vortex has N + complex bosonic zero modes. One is the normalizable, translational zero mode, z 1 , corresponding to the vortex core location.
The remaining N + − 1 bosonic zero modes are the non-normalizable ρ i parameters in The N − negatively charged matter fields Q i must identically vanish (3.12) in a BPS configuration, so they do not yield bosonic zero modes.
As discussed [1] and section 3.4, we quantize all N + + N − Fermi zero modes, including the non-normalizable ones. This leads to a tower of 2 N + +N − vortex states, with the top and bottom states |Ω ± q J =1 , with quantum numbers as in (5.4). The normalizable zero mode, Ψ 1 , is identified with Q − , so the states form 2 N + +N − −1 BPS doublets (2.12). These come from quantizing the non-normalizable Ψ j>1 and Ψ i Fermi zero modes: The omitted U (1) gauge charge is screened by Q vac If k = ∓k c ≡ ∓ 1 2 ∆N , the X ± Coulomb branch exists, and |Ω ± has spin 0, and is an SU (N + − 1) × SU (N − ) singlet, consistent with (1.11) and interpreting X ± as a condensate and is normalizable, and the Ψ 1 ≡ Ψ 2 zero mode has spin − 1 2 and is not normalizable. Quantizing Ψ 1 and Ψ 2 (1.7) gives two BPS doublets: The two BPS doublets in (5.8) and (5.9) reside in different Hilbert spaces, since they are connected via the non-normalizable Ψ 2 Fermi zero mode from ψ Q . For k = 0, both |Ω ± q J =1 have spin 0, and quantum numbers consistent with (1.11): |Ω + q J =1 ∼ X + |0 and |Ω − q J =1 ∼ X † − |0 . We interpret |Ω ± in different Hilbert spaces as corresponding to X + X − ∼ 0 in the chiral ring, and the disconnected X ± branches of the ζ = 0 theory 9 .
The W = M X + X − dual [9] must have the same structure: the map from the X + |0 to the X † − |0 BPS state must involve (in addition to the normalizable Q + zero mode), a ∼ 1/|z| non-normalizable ψ M = Q ψ Q zero mode. Again, we propose that this reflects that This tentative interpretation should be further clarified, perhaps in future work.
6. Cases with Q vac i = 0 for matter with n i = 1.
If a matter field Q 1 , with n 1 > 1, has an expectation value (4.1) (negative n 1 can be obtained via charge conjugation of the present discussion), Q vac 1 = 0 breaks U (1) gauge → Z n 1 , a discrete gauge symmetry, a.k.a. a Z n 1 orbifold. See [46], and references cited therein, for more about Z n 1 gauge theory. Before the Z n 1 orbifold projection, the Fermion zero modes are essentially the same as in section 4, with |n i | Fermion zero modes Ψ i=1,p=1...|n i | for each matter field Q i , and charges as in (1.12). This includes n 1 Fermi zero modes (one is the supercharge) coming from matter field Q 1 and the photino, from eqns. (3.5), (3.6). 9 Parity is a symmetry for k = 0 and maps X + ↔ X − . We can turn on a (P odd) real mass m Q for Q and Q and then there is only one Coulomb branch, X ± if m(X ± ) = −m Q ± ζ = 0; m Q = 0 also eliminates the non-normalizable ψ Q zero mode. There is then a BPS state matching either X + |0 , or X † − |0 , depending on sign(m Q ζ). Taking m Q → 0 requires both doublets in (5.7).
The Fermi zero modes are quantized as in (1.7), giving a tower of 2 i |n i | states, and one then projections to Z n 1 gauge invariant states. The top and bottom states |Ω ± q J =1 (1.8) survive the Z n 1 projection, with quantum numbers again matching with X + and X † − . As a special case, recall from [1] that if the charges all have a common integer factor, n i = n n i , with n and n i integer, the theory is simply a Z n orbifold of a rescaled theory: Note that q J ∈ Z, while q J ∈ nZ, and a has periodicity a ∼ a + 2π, while a ∼ a + 2π/n.
Consider e.g. the theory of a single matter field, Q 1 , with charge n 1 > 1, which is equivalent to a Z n 1 orbifold of the rescaled theory with matter of charge n 1 = 1. Since the q J = 1 vortex of the original theory maps (6.1) to a q J = n 1 vortex of the rescaled theory, it has n 1 complex Bosonic zero modes (the locations z 1 , . . . , z n 1 of the individual vortex cores in the rescaled theory), and n 1 Fermionic zero modes, Ψ 1 , . . . , Ψ n 1 , prior to the Z n 1 orbifolding.
Quantizing the Ψ A=1...n 1 as in (1.7), gives a tower of 2 n 1 states. The top and bottom states, , have charges as given by (1.10) and (1.12), here with k c = 1 2 n 2 1 . These states are Z n 1 invariant, and their charges match those of X + and X † − in (1.6). For k = ∓k c , the operator X = X ± is U (1) gauge neutral, with spin 0, and labels a half-Coulomb branch.
This theory is a Z n 1 orbifold of a free field theory [1], with X 1/n 1 the free field.
We can also consider BPS vortices in vacua with Q vac i = 0 for multiple fields, of different charges n i , with all sign(n i ) = sign(ζ) (3.10), i.e. a weighted projective space, with weights n i . The Fermi zero mode analysis for the general case is then complicated by the couplings among flavors in (3.6). In any case, the counting and charges of the Fermi zero modes cannot be affected by continuous moduli, so they must again be as as (1.12).
In conclusion, in all cases the BPS vortex states |Ω ± q J =1 have quantum numbers compatible with (1.11). For k = ∓k c , it is a spin 0 BPS state, which becomes massless for ζ → 0 and can condense to give a dual Higgs description of the X ± Coulomb branch.

Acknowledgments:
I would especially like to thank Nathan Seiberg for many illuminating discussions, key observations, and helpful suggestions. I would also like to thank Juan Maldacena, Ilarion Melnakov, Silviu Pufu, Sav Sethi, and David Tong, for useful discussions or correspondence.
I would like to thank the organizers and participants of the workshops String Geometry and Beyond at the Soltis Center, Costa Rica, and the KITP program New Methods in Nonperturbative Quantum Field Theory for the opportunities to discuss this work, and for many stimulating discussions. I would especially like to thank the KITP, Santa Barbara, for hospitality and support in the final stage of this work, in part funded by the National Science Foundation under Grant No. NSF PHY11-25915. This work was also supported by the US Department of Energy under UCSD's contract de-sc0009919, and the Dan Broida Chair.
Appendix A. Additional details, conventions, and notation In components, the lagrangian (2.1) is (A.1) We use [34] conventions 10 (reduced from 4d to 3d along the x µ=2 direction, see [47]), though this introduces an unfortunate, non-standard sign convention 11 for the gauge field. In a configuration where the fields asymptote to a zero of (A.2), the total energy of (A.1) (with m i = 0) can be written (using (3.11) and (2.6)), as (with F 12 ≡ F W &B

12
) with D as in (2.3). The BPS (resp. anti-BPS) configurations saturates the inequality for upper (resp. lower) sign choice and ζq J > 0 (resp. ζq J < 0). , which changes the names of BPS vs anti-BPS with respect to much of the vortex literature. This could be fixed by introducing a minus sign in the definition (1.1) of q J , but that introduces sign differences with other literature, e.g. the definitions of X ± in [9,1], so we will not do that here.