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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} =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, electric-magnetic 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 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.

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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 [1,9,14,15] to be (1. 6) with k c ≡ 1 2 i n i |n i | (see section 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.
The corresponding charges of BPS vortices arise in a seemingly different way, from quantizing the vortex Fermion zero modes, 4  This formally gives a tower of 2 Nz degenerate states: treating the Ψ A (Ψ † A ) as raising (lowering) operators, the top and bottom vortex states in this tower are (1.9) Writing "|0 " q J as the naive (ignoring zero modes) groundstate for q J = 0, (1.10) 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. The N = 1 case is the classic N = 2 susy Abelian Higgs model, 5 and its vortices and zero modes have been studied in e.g. [18][19][20][21][22][23][24][25][26]. Its |q J | = 1 vortex has one complex Fermion zero mode, Ψ 1 and (1.7) leads to the BPS or anti-BPS doublet, |Ω ± q J . For N > 1, there is 3 The superconformal U(1)R * of the N = 2 SCFT at Qi = X± = 0, is a linear combination of those in (1.6), U(1)R * = U(1)R + j RjU(1)j, so ∆(Qi) = Ri, and ∆(X±) = R(X±) = 1 2 i |ni|(1 − Ri), with Ri determined by F-extremization [16] (or τRR minimization [14,17]). 4 Vs on S 2 × IR, where the σ = 0 in (1.2) lifts all of the monopole operator's Fermi zero modes [6]. This fits with the radial quantization map between energy on S 2 × IR and operator dimension. 5 Though the Chern-Simons term must be included, k ∈ Z + 1 2 , reflecting the parity anomaly.
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 |): (1.12) 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 section 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 JHEP10(2014)052 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 eff , and Chern-Simons coefficient k eff in (2.3) are shifted by integrating out massive matter, with ζ eff = ζ for m i = 0 and "Higgs" susy vacua have (Q i = 0, σ = 0), while "Coulomb" vacua have (Q i = 0, σ = 0) and k eff = ζ eff = 0. The asymptotic values of k eff for σ → ±∞ are So the σ → ±∞ asymptotic regions of the Coulomb branch only exist if (2.5) vanishes, i.e. if k = ∓k c , respectively. For non-zero k eff and ζ eff , there are also isolated "topological vacua", with Q i = 0 and σ = −ζ eff /k eff ; those vacua will not enter in our discussion.

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2.2 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 with ρ matter ≡ δLmatter δA 0 the matter contribution to the electric charge density (see the appendix for our sign conventions). The "k" in (2.6) is the classical value when we consider the theory at σ = 0, where the matter Fermions are massless and kept in the low-energy theory. On the other hand, for σ = 0, the matter Fermions are massive and can be integrated out, and then we should replace k in (2.6) with k eff , as in (2.4).
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] with k → k eff in (2.7) 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 = ±∞, we replace k → k eff (σ = ±∞) = k ± k c (2.5), and use q J → ±1 in (2.7) to obtain the U(1) gauge charges of X ± in (1.6). The σ → ±∞ Coulomb branch only exists if k eff = 0, which is the condition for X ± to be a gauge invariant, scalar operator: If k = ∓k c , then the X ± Coulomb branch exists. (2.8) The U(1) i and U(1) R global charges of X ± in (1.6) likewise follow immediately from the oneloop induced, mixed Chern-Simons terms between the gauge field and background gauge fields coupled to the global currents [1,9,14,15]. Integrating out the matter Fermion components of Q i in (1.6), of mass m i (σ) = n i σ, gives mixed CS terms k gauge,U(1) j eff = 1 2 n j sign(n j σ) and k 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 (2.10) A BPS particle, with m = Z, has Z > 0 :

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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 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:

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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 .
3.1 Review of the minimal matter example: a single matter field Q 1 of charge This is the basic N = 2 Abelian Higgs model, and its BPS vortices have been discussed e.g. in [18, 20-22, 32, 33]. We here review the discussion from [1]. By (2.2), here k ∈ Z + 1 2 , and the theory has Tr(−1) F = |k| + 1 2 vacua [1]; we here discuss the BPS vortices of the theory in the Higgs 8 vacuum of the theory with FI parameter ζ > 0, i.e.
, is not analytically known, nor is it needed: knowing its existence and number 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 (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 (2.7). This gives r = 1 2 and s = − 1 2 (k + 1 2 ) [1], as in (1.12) with k c = 1 2 i n i |n i | = 1 2 . The k = ∓ 1 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, z = z i=1...|q J | . For |q J | = 1, a charge n i matter field with Q vac i = 0 can have an order |n i | zero at the location z 1 of the BPS (resp. anti-BPS) vortex core For |k| > 1 2 , one could consider vortices in the other vacua, with Q1 = 0 and σ = 0, and domain walls between the vacua, as in [33], but we will not consider such configurations here.

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with f i ≡ f i (z, z) non-vanishing. Turning on Bosonic zero modes can resolve the zeros in (3.8) or, with multiple matter fields, eliminate the zeros, as in the examples of [28][29][30].
Using (3.7), the BPS equations (3.4) can be rewritten in terms of ordinary derivatives and U(1) gauge neutral ratios of fields, where we divide by any Q i with Q vac i = 0:

Vanishing theorem and its consequences
The non-zero solutions of (3. , D (n j ) )) Since the l.h.s. of (3.11) is non-negative, equations (3.4) have a Q j = 0 solution only if the second term on the r.h.s. of (3.11) has the correct sign. By i.e. the Higgs branch moduli, must be set to zero, e.g. the meson fields 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.

Bosonic zero modes of |q J | = 1 BPS vortices with multiple matter fields
Each matter field with sign(n i ) = sign(ζ) has |n i | complex Bosonic zero modes, one of which is the vortex core location, z 1 in (3.8). Since matter fields with sign(n i ) = −sign(ζ) are set to zero (3.12), they do not yield Bosonic zero modes. Consider (3.9), taking say Q 1 and Q j to have sign(n 1 ) = sign(n j ) = sign(ζ), and suppose that Q vac 1 = 0 and Q vac j = 0. 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 BPS |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 −cmγ |z| exponential falloff for large |z| (with m γ the Higgsed photon mass) into a diffuse, power-law 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 Q i ∼ ρ i /|z| for large |z| has finite energy, with ρ i non-normalizable, so unchanging in time. Likewise, Fermi zero modes are either normalizable, if Ψ A < O(1/|z|) for large |z|, or non-normalizable if Ψ A = O(1/|z|). As in [1], we quantize all the Fermion zero modes as in (1.7), including the non-normalizable ones. The tower of 2 Nz states discussed around (1.7) includes states in different Hilbert spaces, if related by a non-normalizable Fermi zero mode. The charges of the states, and in particular the states |Ω ± q J at the top and bottom of the tower, are affected by all the Fermi zero modes, with the product in (1.10) 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 :

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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 section 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 (3.14), with p = |n j | non-normalizable. Non-zero time derivatives of the normalizable c j,p and c j,p can contribute to the vortex's energy, momentum, and spin angular momentum.
We now consider the Fermi zero modes of the q J = 1 BPS vortex or q J = −1 anti-BPS vortex in the vacuum (4.1). Since the counting and quantum numbers of Fermi zero modes cannot depend on continuous variables, we can simplify things by setting all Bosonic zero modes to zero, in which case Here Q vortex 1 coincides with that of U(1) k with only the matter field Q 1 ; the Q i =1 matter fields do not affect the solution. Likewise, the Fermi zero mode equations (3.5) and (3.6) involving λ ± and ψ 1± decouple from those for the Q j>1 matter Fermions. The solution for the zero modes from λ ↑,↓ and ψ 1↑,↓ is the same as that of the minimal matter theory reviewed in section 3.1: for |q J | = 1 it gives one Fermion zero mode, Ψ 1 , and conjugate Ψ † 1 , corresponding to the non-trivial supercharges Q + and Q − in (2.12). Now consider the decoupled equations (3.6) for the Q j>1 matter Fermi zero modes: If k = 0, Gauss' law (2.6) implies that A 0 = ±σ is a complicated function. Fortunately, for any value of k, (4.3) and the simpler version (4.4) have the same number of zero mode solutions, with the same spins. Indeed, using (3.11) and (3.10), it follows that Consider the case of a q J = +1 BPS vortex; the anti-BPS case is analogous. For matter with n j > 0, (4.5) gives ψ j↓ = 0 and (4.3) reduces to (4.4). For n j < 0 matter, ψ j↓ is non-trivial and satisfies the same equation in (4.3) and (4.4). The difference between the ψ j↑ equations in (4.3) and (4.4) for n j < 0 is immaterial in terms of counting solutions: the solution for ψ j↑ in either equation is uniquely determined, as D (n i ) z has trivial kernel for n i < 0 (4.5). So we can always count Fermi zero modes via (4.4). Using
All the Fermi zero modes are quantized, as in (1.7) and (1.8), giving 2 |n i | states. The Ψ 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.
5 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 U(1) j global symmetries in (1.6) enhance to SU(N + ) × SU(N − ) × U(1) A , where the U(1) A charge is +1 for all Q i and Q i . We take N + > 0, and ζ > 0, and then (4.1) is the general JHEP10(2014)052 vacuum with BPS vortices; it spontaneously breaks SU(N + ) → SU(N + − 1) × U(1), so for N + > 1 the vacua contain the NG bosons ∼ = CP N + −1 . For N + N − = 0, the vacua also include non-compact directions, given by the mesons M i j = Q i Q j , with M i j of rank 1, but as in (3.12) BPS or anti-BPS vortices require Q i = 0, so M i j = 0. The Chern-Simons quantization condition (2.2) gives k + 1 2 ∆N ∈ Z, with ∆N ≡ N + − N − ; also, k c = 1 2 ∆N . The cases (N + , N − ) = (N, 0) were discussed in [1]. The minimal matter case, N = 1, was reviewed in section 3.1. The vortices of the N > 1 case is the N = 2 version of the "semi-local" 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. Now consider the Fermi zero modes of the q J = 1 BPS vortex. Again, the counting is independent of the ρ i in (5.1) (though ρ i = 0 does dramatically affect the shape of the solutions) so we set ρ i = 0 for simplicity. As discussed after (4. with j = 2 . . . N + , and j = 1 . . . N − . For each such j and j, (5.2) has one zero mode solution, with spin 1 2 sign(n i ). As we have argued, for counting solutions and spins, we can replace (5.2) with the simpler version (4.4), whose solutions are as in (4.5), (4.7) and (4.8): The spinors u j>1 and d j give N + + N − − 1 Fermi zero modes Ψ j>1 and Ψ j ; all are non-normalizable, since all lim |z|→∞ |ψ| ∼ 1/|z| in (5.3). The charges are as in (1.12):
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:  zero mode has spin + 1 2 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 eqs. (3.5), (3.6). 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: Parity is a symmetry for k = 0 and maps X+ ↔ X−. We can turn on a (P odd) real mass mQ for Q and Q and then there is only one Coulomb branch, X± if m(X±) = −mQ ± ζ = 0; mQ = 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(mQζ). Taking mQ → 0 requires both doublets in (5.7).

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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, |Ω ± q J =1 , 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.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. , 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 qJ , but that introduces sign differences with other literature, e.g. the definitions of X± in [1,9], so we will not do that here.