Deschamps and Mottez (hereafter DM) argue that the Gauss-Matuyama terrestrial magnetic field reversal may have left a vanishing main dipole moment to the field for a time of order 10,000 years.

They say this may have allowed an enhanced cosmic ray flux, boosting the effect we proposed in Melott and Thomas (2019) [1].

We point out that the bulk of the cosmic ray flux from a nearby supernova should be too energetic, up to a million times more energetic than the limits of deflection by the terrestrial magnetic field.

In fact, only those highly energetic ones will directly reach the troposphere, relevant for cloud-to-ground lightning.


DM argue that the Gauss-Matuyama magnetic field reversal at the Pliocene-Pleistocene boundary would have greatly lowered the overall ability of the Earth’s magnetic field for of order 104 years, allowing many more cosmic rays to reach the Earth’s atmosphere and surface.

It is argued that this would enhance the effect we proposed earlier.

This argument is valid for most of the cosmic rays we received today, which are below a billion (109) eV (1 GeV) in energy.

These cosmic rays mostly affect only the upper atmosphere directly.


However, as shown in Figures 2 and 3 of our paper, the input from nearby supernova have energies up to a million times greater.

Earth’s magnetic field is only effective for particles with rigidity below about 20 GV (Smart and Shea 2009) [3]; for protons this roughly 20 GeV, well below the energies to which our modeled supernova cosmic rays extend.

This is the basis of their ability to penetrate to the troposphere and ionize it, and the basis of our hypothesis on the great increase of cloud-to-ground lightning. But since such cosmic rays are almost not at all deflected by the Earth’s magnetic field, they would therefore not be measurably enhanced by a magnetic field reversal.

Ionization of the stratosphere and ozone depletion is possible by the cosmic rays that are deflected by the terrestrial magnetic field, and might be slightly enhanced by the reversal.

But this was not the topic of our paper.


The moduli space of a supersymmetric system is defined as the set of classically mass- less, or light, degrees of freedom.

The beauty of this concept lies in the fact that much of the low-energy behaviour of the system may be encoded as geometrical features on the moduli space.

Whether the subject be string compactifications, the dynamics of gauge theories, or the interactions of solitons, the moduli space approximation provides an effective, and tractable, approach to extract the infra-red quantum properties of the system.


One particularly useful geometric feature of the moduli space is the metric, describing the kinetic interactions of the system.

Our interest in this paper will be focused on the moduli space of solitons, specifically vortices.

In this case, the relevance of the metric was first revealed by Manton who showed that geodesics on the moduli space track the classical scattering of solitons [1].


It is common lore that for dynamics exhibiting 8 or more supercharges, the metric on the moduli space is exactly calculable.

For theories with 4 supercharges or less, the metric can, in general, only be computed in asymptotic regimes.

In the context of solitons, both Yang-Mills instantons and monopoles preserve up to 8 supercharges and indeed exact, albeit somewhat implicit, expressions for the metrics are known using the techniques of [2, 3] 1 .

In contrast, vortices preserve a maximum of only 4 supercharges, and knowledge of the metric is currently restricted to the situation where the solitons are well-separated [5, 6].


Nevertheless, it is possible to make progress in supersymmetric quantum field theo- ries even when the moduli space metric is not known.

This is because, as first empha- sised by Witten [7], many of the simplest quantities of interest depend only on topo- logical characteristics of the moduli space.

For example, the supersymmetric bound states of solitons are related to various cohomology classes of the moduli space [7, 8] 2 .

Similarly non-perturbative contributions to BPS correlation functions, which involve integrals over the moduli space of instantons, often reduce to topological invariants [9, 10].

Thus, for many purposes it suffices to know only crude topological information about the moduli space.


The purpose of this paper is to describe the moduli space of vortices in U(N) Yang- Mills-Higgs theories where the gauge group is broken completely by N fundamental scalar fields.

The theory has a mass gap and exhibits vortices, labeled by the winding number k of the magnetic field,


Here we summarise our main results.

We start in Section 2 with a study of the moduli space of charge k > 0 vortices which we shall denote as V k,N .

Our first result concerns the real dimension of the moduli space which, using index theory techniques, we show to be


In Section 3 we present a brane construction of the vortices, from which we extract a description of V k,N as a U(k) symplectic quotient of C k(N +k) .

This quotient construc- tion is most easily described as the Higgs branch of a U(k) gauge theory with four supercharges, coupled to a single adjoint chiral multiplet and N fundamental chiral multiplets.


The moduli space V k,N naturally inherits a metric from the Kähler quotient construc- tion.

This does not agree with the Manton metric describing the classical scattering of solitons.

Given our discussion above, this is neither unexpected nor an obstacle to utilising our construction for further calculations.

As we shall see, the inherited metric is a deformation of the Manton metric, preserving the Kähler property, the isometries and the asymptotic form.


The parametric scaling of the dimension (1.2) is reminiscent of the moduli space of k instantons in a U(N) gauge theory, which we shall denote as I k,N .

Recall that the real dimension of the instanton moduli space is dim(I k,N ) = 4kN Moreover, those familiar with instanton moduli spaces will have recognised the quotient construction of V k,N as a truncated version of the ADHM quotient [2].

In Section 4, we make this relationship more explicit and show that the moduli space of vortices V k,N is a complex middle-dimensional submanifold (or, since I k,N is hyperKähler, equivalently a special Lagrangian submanifold) of the resolved instanton moduli space I k,N .

We further show that V k,N may be realised as the fixed point set of a holomorphic U(1) action on I k,N , descending from the rotations of instantons in a plane.


In Section 5, we generalise this construction to vortices in U(N) gauge theories with N f = N + M flavours.

For abelian gauge theories, such objects have been well studied and are known as semi-local vortices.

In the strong coupling limit these vortices become lump solutions on the CP M Higgs branch of the theory.

For the non-abelian theory, these vortices are related to lumps in the G(N, N f ) Grassmannian sigma-model of N planes in C N f .

We denote these moduli spaces of vortices asV k,(N,M ) (note that V k,N ∼ =V k,(N,0) ).

The dimension of the moduli space is,


We again give a brane construction as well as a quotient construction of the mod- uli space and explain how it can be described as the fixed point set of a (different) holomorphic action on the moduli space of instantons I k,N f .


In Section 6, we consider the Yang-Mills-Higgs theory defined on the spatial non- commutative plane with [x 1 , x 2 ] = −iϑ. We describe how the moduli space of vortices changes as ϑ is varied.

We show that the moduli spaces may become singular, cease to exist, or undergo interesting topology changing transitions for different values of ϑ. We end in Section 7 with conclusions and a discussion.


Our starting point is the maximally supersymmetric theory admitting vortex solutions which, for concreteness, we choose to live in d = 2 + 1 dimensions with N = 4 super- symmetry 3 .

Our theory includes a U(N) vector multiplet, consisting of a gauge field A µ , a triplet of adjoint scalar fields φ r , r = 1, 2, 3 and their fermionic partners.

To these we couple N fundamental hypermultiplets, each of which contains two complex scalars q andq, and their partner fermions.

As well as the U(N) G gauge symmetry, the Lagrangian also enjoys a SU(N) F flavour symmetry.

Under these two groups, the q field transforms as (N,N), whileq transforms as (N, N).

In the following we take both q andq to represent N × N matrices, q = q a i ,q =q i a a, i = 1, . . . , N where the a index furnishes a representation under U(N) G while the i index refers to SU(N) F .

In this notation, the bosonic part of the Lagrangian reads 4 ,


The final term in the Lagrangian is a D-term and includes a Fayet-Iliopoulos parameter ζ, which we take to be strictly positive ζ > 0.

The presence of this parameter induces symmetry breaking with the unique vacuum, up to Weyl permutations, given by


The ground state of the theory is a gapped, colour-flavour locking phase with the symmetry breaking pattern


The breaking of the overall U(1) G gauge symmetry ensures the existence of vortex solutions in the theory.

These vortices obey a Bogomoln'yi bound which is the natural generalisation of the usual abelian vortex bound [11] and may be simply determined by the standard trick of completing the square in the Hamiltonian.

It will turn out that the most general vortex solutions involve only the fields q and B = F 12 , and we choose to set the remaining fields to zero at this stage.

Restricting to time independent configurations, the Hamiltonian reads,


where k ∈ Z is the winding number of the configuration defined in (1.1).

Choosing k > 0, the bound is saturated by configurations satisfying the first order Bogomoln'yi equations which, for once, we write with all indices explicit to emphasise their matrix nature B a b = e 2 (q a i q i† b − ζδ a b ) D z q a i = ∂ z q a i − i(A z ) a b q b i = 0 (2.4) where we have introduced the complex coordinate on the spatial plane z = x 1 + ix 2 .

4 Our conventions: we choose a Hermitian connection with F µν = ∂ µ A ν − ∂ ν A µ − i[A µ , A ν ] and D µ q = ∂ µ q−iA µ q. All gauge and flavour indices are implicit and assumed summed, with the exception of r, s = 1, 2, 3 which is explicit and summed.


The main purpose of this paper will be to study the moduli space of solutions to these equations.

We denote the moduli space of charge k vortices in the U(N) Yang-Mills- Higgs theory as V k,N .

We start with a study of the linearised equations to determine the dimension of V k,N .

The reader uninterested in the details of the index theorem may skip to the following subsection where basic properties of V k,N are discussed, taking with them the following punchline:


In this section, we prove the result (2.5) by studying the fluctuations (Ȧ,q) around a given solution.

Our method follows closely the work of E. Weinberg [12] who analysed the moduli space in the abelian case N = 1.

The linearised Bogomoln'yi matrix equations are D a zȦz − D ā zȦ z = ie 2 2 (qq † + qq † ) D f zq = iȦ z q (2.6) and are to be augmented with a gauge fixing condition, for which we choose Gauss' law, D aȦz + D ā zȦ z = − ie 2 2 (qq † − qq † ) (2.7) which can be combined with the first of the equations in (2.6) to give


The observant reader will have noticed the appearance of superscripts on the covariant derivatives, which are there to remind us of the U(N) G representation of the field on which they act:


Before proceeding, notice that it is possible to rescale the gauge field A → A/e and coordinate z → ez to remove e 2 from the equations.

The number of zero modes is therefore independent of e 2 and we use this freedom to set e 2 = 2 which simplifies the linearised Bogomoln'yi equations somewhat so they can be written as, ∆η ≡ iD ā z −q r† q r iD f z Ȧ ż q = 0 where the superscript in q r denotes the fact that the matrix q acts as right multiplica- tion.

We now define the index of ∆ as J = lim M 2 →0 J (M 2 ) ≡ lim M 2 →0 Tr M 2 ∆ † ∆ + M 2 − Tr M 2 ∆∆ † + M 2 which counts the number of complex zero modes of ∆ minus the number of zero modes of ∆ † .

Let us firstly show that ∆ † is strictly positive definite, and therefore admits no zero modes by examining the norm squared of a putative zero mode


where the vanishing of the cross-terms occurs when evaluated on a solution to (2.4).

With all terms on the right-hand side positive definite, the last two terms ensure that X = Y = 0.

Thus ∆ † admits no zero modes and J counts the number of zero modes of ∆. We now turn to the task of evaluating J .

For theories in which the fields have suitable fall-off at spatial infinity (faster than 1/r in our case - see the second reference in [12]), the quantity J (M 2 ) is independent of M 2 and the index J may be computed more simply in the opposite limit M 2 → ∞. It is a simple matter to derive an explicit expression for the two composite operators,


where the various operators are defined as,


where the . . . vanish in the M 2 → ∞ limit.

Taking the trace over the adjoint action of B a causes this term to vanish, and we are left only with the left action of B on the space of N × N matrices q. We thus have,


which counts the complex dimension of V k,N to give the promised result.


Let us now discuss a few basic facts about the vortex moduli space.

On general grounds, the space decomposes as, V k,N = C ×Ṽ k,N where C parameterises the center of mass of the vortex configuration, while informa- tion about the relative and internal vortex motion is contained within the 2(kN − 1)- dimensional centered vortex moduli spaceṼ k,N . Supersymmetry, and the BPS-nature of the vortices, ensures that the moduli space admits a natural Kähler metric defined by the overlap of the zero modes, L = Tr d 2 x 2 e 2Ȧ zȦz +qq † = g abż ażb (2.9) where z a are complex coordinates on V k,N .

This is the Manton metric, descending from the kinetic terms of the Lagrangian (2.3) and is such that geodesics of g describe the classical scattering of vortices [1].


For the case of the abelian-Higgs model, N = 1, many properties of the vortices and the metric have been studied.

Taubes showed long ago that, as expected, the collective coordinates of V k,N correspond to the positions of k unit charge vortices moving on the plane and may be identified with the zeros of the Higgs field q [13].

The metric on V k,1 can be shown to be geodesically complete and although the exact form of the metric remains unknown for k ≥ 2, several interesting properties were uncovered by Samols [5].

Asymptotically, the metric approaches the flat metric on C k /S k where S k is the permutation group of k elements, reflecting the fact that the vortices are indistinguishable particles.

The interactions of the vortices resolve the orbifold singularities of C k /S k as the cores overlap.

The leading order corrections to the flat metric, which are exponentially suppressed in the separation between vortices, were recently calculated by Manton and Speight [6].


The moduli space of vortices in the non-abelian Yang-Mills-Higgs model does not appear to have been studied in the literature.

Here we make a few elementary remarks.

The dimension dim(V k,N ) = 2kN suggests that the charge k vortex again decomposes into k unit charge vortices, each of which is alloted a position on the plane together with (N − 1) complex internal degrees of freedom describing the orientation of the vortex in the SU(N) diag group.

Indeed the action SU(N) diag on the fields descends to a natural action onṼ k,N , resulting in a holomorphic SU(N) isometry of the metric g. For k ≥ 2, there is a further isometry ofṼ k,N resulting from spatial rotations of the vortices.


Let us examine the moduli space of a single vortex in further detail.

Given a specific solution (B ⋆ , q ⋆ ) to the abelian vortex equations, one can always construct a solution to the non-abelian equations (2.4) by simply embedding (B ⋆ , q ⋆ ) in the upper-left corner of the N × N matrices B and q. In the case of a single vortex k = 1, acting on this configuration with the SU(N) diag symmetry sweeps out the full moduli space of solutions.

Since the vortex embedded in the upper-left corner breaks SU(N) diag → SU(N − 1) × U(1), the vortex moduli space is V 1,N ∼ = SU(N)/(SU(N − 1) × U(1)) ∼ = CP N −1 endowed with the round Fubini-Study metric.

The only information that we still need to determine is the overall scale of the moduli space.

This will be important later in matching to the instanton moduli space.

Since CP N −1 is a homogeneous space, we can fix the scale by calculating the overlap of any two suitable zero modes arising from the SU(N) diag action.

For Ω(z,z) ∈ su(N), the zero modes associated to an SU(N) diag rotation are given by,Ȧ = D a Ω ,q = i(Ωq − qΩ 0 ) (2.10) where Ω → Ω 0 as |z| → ∞. The transformation of q arises because the left action is by the U(N) G gauge symmetry, while the right action is by the SU(N) F flavour symmetry.

The z dependence of Ω is required in order to satisfy the gauge fixing condition (2.7) which becomes


For the initial configuration embedded in the upper-left corner of B and q, these equa- tions are solved by the (N − 1) rotations,


and it is a simple matter to compute the overlap (2.9) of the zero modes (2.10) to determine the overall radius of the moduli space to be Radius 2 Ṽ 1,N ∼ 1 e 2 (2.11) Finally, let us make a brief comment on the spectrum of vortices in the quantum theory.

In the d = 2 + 1 theory with N = 4 supersymmetry, ground states of the vortices in a given sector are associated to harmonic forms onṼ k,N .

For the case of a single vortex, there are therefore χ(CP N −1 ) = N such states, implying that the vortex transforms in the fundamental representation of SU(N) diag .


We now turn to the brane realisation of these vortices.

We keep the same basic structure as Section 3, simply adding M further semi-infinite D3-branes to provide the extra flavours.

We choose to add these to the right-hand NS5-brane, so the final set-up is x 3,4,5 x x 6 9 N D3−Branes NS5−Branes M D3−Branes k D1−Branes shown in Figure 3.


Once again, it is a simple matter to read off the theory on the k D1-branes [17].

It consists of a U(k) field theory, still coupled to the chiral multiplets Z and ψ as in Section 3, but now augmented with M further chiral multipletsψ which transform in thek representation of the gauge group.

We shall write,


These fields also transform under their own U(M) E flavour symmetry, so the full global symmetry group of the theory is therefore G = SU(2) R × S(U(N) D × U(M) E ) × U(1) F where the overall U(1) of the U(N) D ×U(M) E flavour symmetry lies in the U(k) gauge group.


As in Section 3, we are interested in the Higgs branch of the D1-brane theory, which we denote asM k,(N,M ) .

This Higgs branch is expected to be isomorphic to the vortex moduli space,V k,(N,M ) ∼ =M k,(N,M ) .


Let us examine the Higgs branch in more detail.

It is given by a U(k) quotient of C k(N +M +k) , parameterised by Z, ψ andψ.

The D-term moment map is D m n = N i=1 ψ m i ψ i† n − M w=1ψ m† wψ w n − [Z, Z † ] m n − rδ m n = 0 (5.27) where we have, for once, abandoned the summation convention in order to highlight the ranges of the various indices.

The D-term imposes k 2 real constraints which are augmented by the restriction to U(k) invariant coordinates.

Thus the real dimension of the Higgs branch is dim(M k,(N,M ) ) = 2k(N + M) in agreement with the vortex moduli space.


The symmetry group G imprints itself as a holomorphic S(U(N) × U(M)) × U(1) isometry of the Higgs branch and thus, as before, the symmetries of the metric on the Higgs branch defined by the Kähler quotient construction are the same as those of the Manton metric.

However, as in Section 3, here the agreement stops.

In particular, the metric onM k,(N,M ) defined by the Kähler quotient construction is finite, and sees nothing of the non-normalisable modes of the vortex.

Given our remarks in the introduction, one would expect that for many supersymmetric problems in soliton and instanton physics, one can replace the Manton metric onV k (N,M ) with the Kähler quotient metric onM k,(N,M ) and in this manner avoid the subtleties of the non-normalisable directions.


The FI parameter of (5.27) is once again related to the gauge coupling, r = 2π e 2 so that the limit in which the semi-local vortices reduce to sigma model lumps is simply r → 0.

Let examine how the Higgs branch changes with r.


For the vortex theory described in Section 3, the Higgs branch only exists for r > 0.

When r = 0, the D-term (3.17) requires ψ = 0, and the metric on the Higgs branch becomes the flat, singular metric on C k /S k .

This reflects the fact that the vortices of Section 2 become point-like objects when e 2 → ∞.


Note that for semi-local vortices the Higgs branch defined by (5.27) even makes sense for r < 0.

One may want to interpret this as a "continuation past infinite coupling" (e 2 → ∞ and then some).

In the following Section we shall give a different interpretation for the regime r < 0.


Let us examine the claim (3.18) in more detail.

Firstly, note that the center of mass po- sition of the D1-branes, given by Z = z1, decouples from the other fields, guaranteeing that the Higgs branch decomposes as M k,N ∼ = C ×M k,N in agreement with the vortex moduli space.

To make further comparisons, let us consider specific examples, starting with the description of a single vortex k = 1 in the U(N) theory.

In this case the vortex dynamics is abelian so Z decouples and the D-term constraint reduces to |q| 2 = r where q is an N-vector.

We are left with the well known gauged linear sigma-model construction of CP N −1 , and we have,


The size, or Kähler class, of the Higgs branch is determined by the FI parameter r = 2π/e 2 in agreement with the vortex moduli space (2.11).


The second example we consider is that of k vortices in the abelian-Higgs model with N = 1.

This vortex quantum mechanics was previously studied in [19] as a matrix model for identical particles moving on the plane.

Prior to that, the D-term constraints (3.17) were solved in a somewhat different context by Polychronakos [20], who showed that a given solution is uniquely determined by a set of eigenvalues for Z, up to Weyl permutations.

Thus


In these two, simple cases, we have therefore confirmed that the Higgs branch and vortex moduli spaces are indeed isomorphic.

We now turn to the question of the metric.

The Higgs branch M k,N inherits a natural Kähler metric from the Kähler quotient construction described above.

The presence of the flavour symmetry SU(N) D guarantees that this metric exhibits an SU(N) holomorphic isometry.

For k ≥ 2, M k,N also enjoys a U(1) holomorphic isometry, arising from U(1) F , corresponding to rotating the branes in the x 1 + ix 2 plane.

Thus the quotient metric on M k,N and the Manton metric on V k,N share the same isometries.

Indeed, from the brane picture it is clear that the SU(N) diag and SU(N) D symmetry groups of the D3-brane and D1-brane theories, share the same origin.


Do further properties of the metrics coincide?

In the case of k = 1, the metric onM 1,N is the round Fubini-Study metric on CP N −1 , in agreement with the Manton metric onṼ 1,N .

However, in this case the agreement is a consequence of the symmetries of the problem.

In general, the metrics are not the same.

To see this, let us return to the case of the abelian-Higgs model with N = 1.

Importantly, the asymptotic metric on M k,1 is the flat metric on C k /S k , in agreement with the Manton metric.

This is crucial to ensure that the Higgs branch describes the moduli space of indistinguishable particles since mere topological information does not suffice (topologically C k /S k ∼ = C k as any polynomial will confirm).

However, in the case of the Kähler quotient, the leading order corrections to the flat metric are power-law.

This is to be contrasted with the exponential corrections of the Manton metric.

To be concrete, consider the case k = 2, N = 1.

The metrics on bothṼ 2,1 andM 2,1 take the form, ds 2 = f 2 (σ)(dσ 2 + σ 2 dθ 2 ) (3.19) where σ is the separation between vortices, or D1-branes, and θ ∈ [0, π) so that the moduli space looks like a cone.

For the Higgs branch, the explicit Kähler quotient construction was performed in [19] and the conformal factor is given by,


The calculation of the leading order scattering of vortices was performed in [6], and the equivalent metric onṼ 2,1 was computed to be, f 2 V (σ) ≈ 1 − λ 2 4π σ e −2σ + . . .

( 3.21) where λ is a coefficient which parameterises the asymptotic return to vacuum of the Higgs field in the solution to (2.4).

This coefficient is not known analytically but it was shown in [14] that T-duality between the A N singularity and fully localised NS5-branes requires a worldsheet instanton effect and holds only if λ = 8 1/4 ≈ 1.682.

This is in agreement with the numerical result λ ≈ 1.683 of [15].

To summarise, we see that, while the metrics on V k,1 and M k,1 are asymptotically, and qualitatively, similar they differ in the details.


The moduli spaceV k,(N,M ) of k vortices in U(N) theories with N f = (N + M) flavours is again a complex submanifold of the moduli space of instantons.

This time we must look at I k,N f describing k instantons in U(N f ) Yang-Mills theory.

The ADHM theory for these instantons was described in Section 4 where, obviously, we must replace N with N f .

So, for example, the global symmetry group of the ADHM theory is,


As in Section 4, we define the submanifold describing vortices by weakly gauging a symmetry or, equivalently, by the fixed point set of a S 1 action on I k,N f .

The new ingredient here is that the U(1) action includes a component from the SU(N f ) F flavour symmetry.

In fact, it will prove to be simpler to phrase the discussion in terms of U(1) G × SU(N f ) F = U(N f ), where U(1) G ⊂ U(k) is the overall U(1) gauge symmetry.

To this end, consider the Cartan subalgebra of U(N f ), N f i=1 U(1) (i) F Write the associated charges as Q (i) F , where i = 1, . . . , N f .

In this notation, the charge under the overall U(1) gauge symmetry is,


The theory describing semi-local vortices can be obtained from the ADHM theory by a weak gauging which gives mass to all fields carrying non-vanishing charge, Q = Q A + Q R − Q L − N i=1 Q (i) F + M i=N +1 Q (i) F and the vortex moduli spaceV k,(N,M ) is isomorphic to the fixed point set of the associ- ated U(1) action on the instanton moduli space I k,N +M .

In this case, the U(1) action arises from a simultaneous rotation of the instantons in the x 3 − x 4 plane, together with a gauge rotation in U(N f ).


While the above discussion has been in terms of the ADHM gauge theory, the deforma- tion also has a simple description directly in terms of the instanton moduli space I k,N .

The U(1) symmetry of the gauge theory descends to anŜ 1 action on I k,N , endowing the metric on I k,N with a Killing vectork.

This Killing vector is holomorphic, preserv- ing the preferred complex structure while rotating the remaining two.

The mass terms introduced above by weakly gauging U(1) induce to a potential V on I k,N proportional to the length 2 of the Killing vector, V ∼k 2 Such potentials have been widely used in soliton physics recently (see for example [24],[25]), although usually in the context of supersymmetry-preserving tri-holomorphic Killing vectors.

We therefore have a description of the vortex moduli space V k,N directly in terms of the instanton moduli space


The zeroes of the Killing vectork are precisely the fixed points of theŜ 1 action.

The meaning of this action can be determined from the assignment of chargesQ in (4.24).

Recall that U(1) R × U(1) L ⊂ SU(2) R × U(1) L is the subgroup of the rotations SO(4) ∼ = SU(2) R × SU(2) L of R 4 that are left unbroken by the non-commutative deformation (4.22).

We find therefore that the actionQ corresponds to rotating the instantons in the x 3 − x 4 plane, and the vortices are related to instantons which are invariant under this U(1) action.


Let us now turn to some examples: the moduli spaceĨ 1,N of a single instanton in U(N) non-commutative Yang-Mills is given by the cotangent bundle T ⋆ (CP N −1 ) en- dowed with the Calabi metric [26].

The potentialk 2 vanishes on the zero section of the bundle CP N −1 , reducing to the moduli space of a single vortexṼ 1,N .


Another example: the moduli spaceĨ 2,1 of two instantons in U(1) gauge theory is the Eguchi-Hanson met- ric on T ⋆ (S 2 ).

The explicit hyperKähler quotient construction was performed in [27].

Note that this case is special sinceĨ 2,1 ∼ =Ĩ 1,2 , which is not true for k > 2.


However, the tri-holomorphic SU(2) isometry of T ⋆ (S 2 ) has a different origin in these two cases.

In the notation of (4.23), the isometry is SU(2) F forĨ 1,2 , while it is SU(2) L forĨ 2,1 .

Since, from (4.24), the potential on the instanton moduli space involves SU(2) L , but not SU(2) F , the vortex moduli spacesṼ 2,1 andṼ 1,2 are given by different holomorphic submanifolds of T ⋆ (S 2 ).

It is a simple exercise to show that the vacua of the potential onĨ 2,1 is the two dimensional cone endowed with the metric (3.20).


From the perspective of the D4-brane, the above deformation of the instanton theory involves locking the U(1) L/R symmetries tangent to the D4-brane, with the U(1) A symmetry normal to the D4-branes.

This is reminiscent of the twisting of the tangent and normal bundles of branes when wrapped on cycles [28].

In this section, we give evidence suggesting that the two are indeed related.


To see this connection, let us first return to the brane set-up of Section 3 as depicted in Figure 1.

We perform a T-duality in the x 9 direction, and describe the resulting IIA string theory set-up.

Under T-duality, the two NS5-branes are replaced by the back- ground geometry C 2 /Z 2 .

( The duality between NS5-branes and ALF spaces was first conjectured by Hull and Townsend [29].

A proof from the worldsheet sigma-model, including the breaking of translation symmetry associated to the localization of the NS5-brane, was given in [14]).

The separation of the NS5-branes in the x 6 direction resolves the orbifold singularity, resulting in the background spacetime T ⋆ (S 2 ) 7 .

Topo- logically, this space can be thought of as an S 1 fibration, parameterised by x 9 , over R 3 , parameterised by r = (x 6 , x 7 , x 8 ).

In Gibbons-Hawking coordinates, the metric takes the form,


The 3-vector r 0 resolves the orbifold singularity and, for the T-dual of Figure 1, is given by r 0 ∼ (1/e 2 , 0, 0).

The S 1 fiber degenerates at the two points r = ±r 0 , resulting in the Christmas cracker topology shown in Figure 2.

The zero section S 2 , which can be clearly seen in this picture, contains a paper hat and a 20 year old joke.


What becomes of the D-branes after T-duality? The D3-branes of Figure 1 become D4-branes with worldvolume spanning x 1 , x 2 , x 6 and x 9 .

They wrap the compact S 2 , and one half of the cracker as depicted by shading in the Figure.


The vortices are a little more mysterious.

Had the D1-branes in Figure 1 been infinite in the x 9 direction, they would become D0-branes in the IIA description.

Since the D1-branes actually stretch only a fraction of the distance, we expect that they become fractional D0-branes.

However, such objects are usually understood in terms of a D2-D2 pair wrapping a vanishing S 2 , through which an NS-NS B-field threads in order to provide the D0- brane charge.

Yet in our case the S 2 has finite size, and such an interpretation breaks down, as can easily be seen by computing the mass of such a D2-D2 pair.

It would be interesting to get a better understanding of these fractional D0-branes in this picture, and complete the relationship to the wrapped D0-D4 system.


In this Section, we discuss a generalisation of the vortices to U(N) Yang-Mills with N f = (N + M) flavours.

The Lagrangian takes the same form as previously (2.3) except the fundamental scalars are now N × (N + M) dimensional matrices


Rather than the unique, isolated vacuum of Section 2, the theory now has a Higgs branch of vacua.

However, ifq develops an expectation value, then there are no BPS vortex solutions.

This may be easily seen from an analysis of the Bogomoln'yi equa- tions, and follows from the mathematical fact truth that a line bundle of negative degree admits no holomorphic sections (see, for example, [30] for the translation).

We therefore restrict attention to the reduced Higgs branch of vacua, denoted N N,M ob- tained by insistingq = 0.

For example, for abelian theories with N = 1, the Higgs branch of vacua is the cotangent bundle T ⋆ (CP M ), while the reduced Higgs branch describing the vacua which admit BPS vortex solutions is simply N 1,M = CP M .

In general, the reduced Higgs branch is the Grassmannian of N planes in C N +M ,


This is a symmetric space, and we may choose to work in any of the vacua without loss of generality.

We pick,


In this vacuum the SU(N + M) F flavour symmetry of the theory is broken in the pattern,


The theory admits BPS vortices with the Bogomoln'yi equations taking the same form as previously (2.4) with q now interpreted as a matrix of the appropriate size.

We denote the moduli space of vortices in this model asV k,(N,M ) .

Note that, in the notation of the previous sections, we have V k,N ∼ =V k,(N,0) .

It is a simple matter to generalise the index theorem of Section 2 to the present case.

We omit the details, stating only the result J (M 2 → ∞) = k(N + M).

We therefore have


Note that since we have taken the limit M 2 → ∞, this computation ignores a surface term contribution which comes from fields dropping off as 1/r [12].

Indeed, as we shall review below, it is known that the counting (5.26) includes zero modes which are not L 2 normalisable and which one would not, therefore, expect to be included in J = J (M 2 = 0).

Nevertheless, these modes corresponds to collective coordinates of the semi-local vortex and we wish to keep them in our discussion, so the result (5.26) is the relevant one.


In the abelian case N = 1, the vortex equations with multiple Higgs fields have been well studied in the literature, where they go by the name of semi-local vortices.

For a review of their properties and their relationship to electroweak strings, see [31].

The result dim(V k,(1,M ) ) = 2k(1 + M) was previously determined from a direct analysis of the Bogomoln'yi equations in [32] and subsequently from a brane picture in [33].


An interesting feature of the semi-local vortices is that they may remain non-singular in the limit e 2 → ∞. This is in contrast to the vortices considered in Section 2 whose size scales as (e 2 ζ) −1/2 and thus become point-like objects in this limit.

In fact, the semi- local vortices reduce to another familiar topological object as e 2 → ∞: they become sigma-model lumps (a.k.a sigma model instantons, or textures) on the reduced target space N N,M .

While the vortices are supported by Π 1 (U(N)), the lumps are supported by Π 2 (N N,M ).

For example, the semi-local vortices of the abelian N = 1 model become lumps on CP M .

A nice description of how the metamorphosis from vortex to lump occurs may be found in [34].

Thus, in the limit e 2 → ∞, the moduli space of semi-local vorticesV k,(N,M ) becomes the moduli space of Grassmannian G(N, N + M) lumps.


It is well known that sigma-model lumps share several properties with Yang-Mills instantons.

In particular, they may have arbitrary size and, in the e 2 → ∞ limit, k of the collective coordinates of (5.26) may be thought of as the scales of the k lumps.

Since the lumps may have any size, they can also shrink to a singular solution.

Thus, just like the unresolved instanton moduli spaces, the moduli space of lumps contains singularities.

These singularities are removed by introducing a gauge field with a finite coupling e 2 and returning to the full vortex equations (2.4).

In this way, the inverse gauge coupling 1/e 2 plays a role in the vortex dynamics reminiscent of the non-commutivity parameter θ in Yang-Mills instantons.

This similarity was previously noted in [30, 35], and we shall make the analogy more explicit in the following section.


The Manton metric onV k,(N,M ) may be once again defined in terms of the overlap of zero modes.

The resulting metric is Kähler and inherits a S(U(N) × U(M)) holomor- phic isometry from the surviving symmetry group (5.25), together with a further U(1) isometry from the rotational symmetry.

However, the Manton metric onV k,(N,M ) suffers from a sickness since some of the zero modes are (logarithmically) non-normalisable.

This well known problem for lumps in the CP M sigma-model [36] is not ameliorated by a finite gauge coupling e 2 as shown in [37].

Classically this non-normalisability ensures that certain collective coordinates (for example, the scaling size of the lump) have infi- nite moment of inertia and are thus constants of the dynamics.

The non-normalisability of modes leads to subtleties when treating these objects quantum mechanically.


In this Section we examine our Yang-Mills-Higgs theories defined the spatial non- commutative plane [x 1 , x 2 ] = −iϑ (6.28) and ask how this deformation affects the moduli space of vortices.

Various aspects of non-commutative vortices in the abelian Higgs model have been considered in [40, 41, 42, 43].


Before recalling the results of these papers, let us start by regaling ourselves with the beautiful tale of non-commutative instantons.

We have already covered this is Section 4, but a good story is always worth retelling.

It was shown by Nekrasov and Schwarz [22] that a non-commutative deformation of R 4 as given, for example, in (4.22) induces a FI parameter r = 4θ in the ADHM instanton gauge theory.

This FI term resolves the singularities of the instanton moduli space in the manner described previously by Nakajima [23].

One may expect that a similar phenomenon occurs in our vortex theory.


However, we have seen that the job of resolving the singularities on the vortex moduli space is already adequately performed by the gauge coupling since the FI parameter is r = 2π/e 2 .

So what role could the non-commutative deformation (6.28) play? To avoid undue suspense, we shall first reveal the answer, followed by a derivation, and then an analysis of the consequences.


We shall show that the effect of a non-commutative background is to change the FI parameter of the vortex theory to r = 2π 1 e 2 + ϑζ To see this, we return once more to the brane picture of Section 3.

We want now to deform the D3-brane dynamics so that at low energies it is described by the Yang- Mills-Higgs theory (2.3) defined on the non-commutative plane (6.28).

The string theory background that achieves this feat is well known [38]: a background NS NS B field is added, with components B 12 = 0.

We wish to understand the effect of this B field on the dynamics of the D1-strings.


In fact, a very similar situation was analysed by Hashimoto and Hashimoto in [39].

These authors considered the situation of a D-string suspended between two D3-branes in a background B field, a set-up which describes a monopole in non-commutative Yang-Mills.

The basic physics is very simple to describe.

The background B field may be absorbed into the D3-brane as a magnetic flux F 12 .

The end of the D1-brane acts as a magnetic source in the D3-brane, and therefore experiences a force due to F 12 .

To see how force acts, recall that our D3-brane lies in the 0126 directions and note that F 12 = ⋆ F 06 .

The magnetic end of the D-string therefore feels the same force as an electric charge in a background electric field F 06 .

In other words, the string end moves in the x 6 direction.

This displacement continues until the force due to the B field is canceled by the excess tension of the D1-brane.

In our case, one end of the D1-brane is attached to the NS5-brane, and cannot move in the x 6 direction.

The final configuration is therefore given by the tilted D-strings, as shown in Figure 4.


An analysis of the supersymmetry generators was performed in [39] which, translated to the present set-up, reveals that these tilted D1-branes continue to preserve four supercharges.

The effect of this tilt on the theory on the D1-branes is to change the FI parameter r, which is now given (up to a normalisation of 1/g s - see (3.15)) by the distance between the end of the D1-brane and the left-hand NS5-brane.

The displacement δ of the D1-brane from its original position was calculated in [39], and is given by


With the parameters e 2 and ζ still defined in terms of the distances ∆x 6 and ∆x 9 between the NS5-branes as in (3.12), we find the promised result,


We now turn to studying some simple consequences of this equation.

Consider first the vortices of Section 2, in which we have a U(N) gauge group with N hypermultiplets.

The most striking feature is that the vortex moduli space only exists for r ≥ 0 or, alternatively, for ϑ above the critical value ϑ c ϑ > ϑ c = −1/e 2 ζ Moreover, at the critical value ϑ = ϑ c , the moduli space becomes singular.

For example, in the abelian case N = 1, we have V k,1 ∼ = C k /S k , endowed with the flat, singular metric at ϑ = ϑ c .

In fact neither of these results are new: both were previously derived in the abelian N = 1 theories by Bak, Lee and Park by an explicit study of the solutions to the non-commutative Bogomoln'yi equations [41] (see also the phase diagram in [43]).

It is pleasing to see these properties reproduced from our D1-brane theory.

From equation (6.29) it is also clear that vortex moduli space V k,N is non-singular in the e 2 → ∞ limit provided ϑ > 0.

This point was previously made in [43] and the Manton metric on V k,1 was explicitly calculated in this limit.

Once again, the Manton metric differs from the metric induced on the Higgs branch by the Kähler quotient construction.

For example, in the case of two vortices, the non-commutative metric in the e 2 → ∞ limit takes the form of a cone (3.19), now with the conformal factor given by


which coincides with neither (3.20) nor (3.21).


We now turn to the consequences of (6.29) for semi-local vortices.

Firstly, note that if we set e 2 → ∞ so that we are studying the moduli space of sigma-model lumps, then the non-commutivity parameter ϑ resolves the singularities just as in the case of Yang-Mills instantons.

Similar observations were made from a field theory perspective in [44].

However, for semi-local vortices the moduli space defined by the D-term (5.27) continues to make sense for r < 0 or, alternatively, for ϑ < ϑ c .

Moreover, the topology of the moduli space differs for r > 0 and r < 0.

Thus, as we decrease the non-commutivity parameter ϑ past its critical value of −1/e 2 ζ, the moduli space of non-commutative vortices undergoes a topology changing transition.


A familiar example of this occurs for a single k = 1 semi-local vortex in a U(2) gauge theory with 4 hypermultiplets.

In this case the moduli space is related to the Calabi-Yau 3-fold known to string theorists simply as The Conifold.

For r = 0, it is the singular space defined by the complex equation xy − wz = 0.

The FI parameter resolves the singularity and we have,V


As we decrease the non-commutivity parameter ϑ to pass from r > 0 to r < 0, the moduli space of vortices undergoes a flop transition.

For other values of N and M, the topology change is more dramatic and the moduli space of vortices may have different Betti numbers for r > 0 and r < 0.


Finally, note that for ϑ < ϑ c the equation (6.29) implies a duality between different non-commutative gauge theories in the sense that they share the same moduli spaces of vortices.

Specifically, consider the U(N) theory with N f = (N + M) flavours.

The moduli space of k vorticesV k,(N,M ) is defined by the moment map (5.27) with FI parameter r given by (6.29).

For ϑ < −1/e 2 ζ we have r < 0.

From the D-term (5.27) we see that the moduli space of vortices in this theory coincides with the moduli space of vortices in a U(M) gauge theory, also with N f = (N + M) flavours. ( We must also perform a parity transformation in going from the U(N) theory to the U(M) theory, so that Z → Z † ).

Denote the gauge coupling of the U(M) theory as e ′2 , and the non-commutivity parameter as ϑ ′ .

Then for the moduli spaces of vortices to coincide, we simply require


This duality is, like many, reminiscent of the (N c = N) → (N f − N c = M) duality of Seiberg.


Let us begin this ending with a summary of our results.

We have studied vortices in U(N) Yang-Mills theories with N f ≥ N flavours.

These theories have a FI parameter ζ which ensures the gauge group is completely broken in the vacuum.

The gauge coupling parameter is e 2 and the theories may be defined on the non-commutative plane with [x 1 , x 2 ] = −iϑ.


We have shown that the moduli space of charge k vortices in this theory is described by the Higgs branch of a U(k) gauge theory with four supercharges, coupled to N chiral multiplets ψ transforming in the k representation of the gauge group, N f − N chiral multipletsψ transforming in thek representation, and a single chiral multiplet Z transforming in the adjoint representation.

This Higgs branch is defined by a U(k) symplectic quotient of C k(k+N f ) with moment map, [Z † , Z] + ψψ † −ψ †ψ = r (7.30) where the level of the moment map r is a FI parameter defined by


We further showed that the vortex moduli space may be constructed as a complex submanifold (or, for N f = N, a special Lagrangian submanifold) of the moduli space of k instantons in non-commutative U(N f ) Yang-Mills theory.


Relationships between the instanton and vortex equations have been noted in the past.

In particular, a reduction of instantons in SU(2) Yang-Mills on R 2 × S 2 gives rise to the U(1) vortex equations [45].

While this relationship appears to share several characteristics of our correspondence, it differs in many important details.

It would be interesting to elucidate the connections between these two approaches.


As we discussed in detail, the Manton metric on the moduli space of vortices does not coincide with the metric induced on the Higgs branch (7.30) by the Kähler quotient construction.

Indeed, all our experience with gauge theories with four supercharges suggests that it is too ambitious an enterprise to determine the metric on the moduli space of vortices.

Other, more topological, questions can be asked with greater success and the construction (7.30) provides the answers.

As stressed in the introduction, these topological questions include certain quantum correlation functions in supersymmetric gauge theories.


Given the relationship described in Section 4 between the moduli space of vortices and the moduli space of Yang-Mills instantons, one may expect quantitative agreement between topological correlation functions of two and four dimensional gauge theories.

These would receive non-perturbative instanton corrections in four dimensions, and vortex corrections in two dimensions.

Indeed, it is well known that N = 1 SU(M + 1) super-Yang-Mills in four dimensions shares many features with the N = (2, 2) CP M sigma-model in two dimensions including, most pertinently, its low-energy effective action [46, 47].

A second, more quantitative, example was given in [48] where N = 2 theories in four dimensions were shown to have a spectrum of monopoles that coincides with the spectrum of kinks in certain N = (2, 2) theories in two dimensions.

Both monopole and kink masses receive identical corrections from non-perturbative effects, namely Yang-Mills instantons and vortices respectively.

It seems likely that the semi- classical reason for these agreements can be traced to the relationship between the vortex and instanton moduli spaces described here.


We show that the construction of vortex solitons of the noncommutative Abelian-Higgs model can be extended to a critically coupled gauged linear sigma model with Fayet-Illiopolous D-terms.

Like its commutative counterpart, this fuzzy linear sigma model has a rich spectrum of BPS solutions.

We offer an explicit construction of the degree k static semilocal vortex and study in some detail the infinite coupling limit in which it descends to a degree k CPN instanton.

This relation between the fuzzy vortex and noncommutative lump is used to suggest an interpretation of the noncommutative sigma model soliton as tilted D-strings stretched between an NS5-brane and a stack of D3-branes in type IIB superstring theory.


1. Introduction


A little more than a decade ago, the study of electroweak strings in a modified Abelian-Higgs theory initiated in [35] revealed a curious new vortex solution.

As the story goes, vortices are indeed enigmatic objects [34] and the semilocal vortices found in [35] are no exception.


Firstly, standard lore holds that a non-simply-connected vacuum manifold is a necessary condition for the existence of stable, finite energy cosmic string solutions.

If this is anything to go by, the very existence of these semilocal vortices should be called into question since the vacuum manifold of the modified Abelian-Higgs theory is S3.

Yet exist they do.

Consequently, a more consistent condition was offered in [17].


Semilocal vortices (actually, this holds for other defects as well) form in theories exhibiting spontaneous symmetry breaking and whose vacuum manifold is fibred by the action of the gauge group in some non-trivial way.

In this same work it was realised also that the low momentum dynamics of these vortices bear a striking resemblance to the 2-dimensional lump solutions of the CPN nonlinear sigma model.


Since then, this similarity between the modified Abelian-Higgs theory (a.k.a gauged linear sigma model) and the CPN (or, more generally, Grassmannian) sigma model has found itself the subject of much attention [10, 30, 40].

Nevertheless, much of what is known about the semilocal vortex is only asymptotic.


Even its descent to the lump in the infinite coupling limit is only exact at spatial infinity and suffers Skyrme term corrections at smaller radial distances.

This is the allure and frustration of vortices; as simple as their defining equations seem, they are also remarkably unyielding.


Until a short time ago, the only avenue toward tractable vortex equations was a curvature deformation of the background space in which the vortices live [32, 38].

These are, of course, not without their own puzzles.

The recent renaissance in noncommutative geometry (due in no small part to the seminal work of [31]) offers new recourse.

Fuzzy deformations of the background space have, in only a few years, not only yielded a wealth of new solitonic solutions but also several new insights into old solutions to a host of field theories (see [4, 12, 33] for excellent reviews).

The noncommutative Abelian-Higgs model, for example, exhibits exact vortex solutions [1, 2, 23, 37] whose moduli space metric can be computed explicitly in the large noncommutativity limit [34].


In this work we extend this idea to the (2+1)-dimensional, critically coupled, gauged linear sigma model with an N + 1 component Higgs field.

The BPS spectrum of the resulting fuzzy theory is studied and, like its commutative counterpart, shown to have quite rich structure.

In particular, we use an extension of the computational technique of [23] to explicitly construct a family of exact semilocal vortices.

As expected, our family contains the Abelian-Higgs vortices of [1, 23, 37] as well as the fluxons of [14] as special cases.


As suggested by the title, the metamorphosis of the semilocal vortex is of central importance in this paper.

By turning up the gauge coupling, we demonstrate conclusively, at the level of the solutions, the descent of the semilocal vortex into the instanton solution of the fuzzy CPN model of the same degree.

Interestingly, unlike the commutative case, this “transmogrification” of the vortex is exact at a certain point in the parameter space of the theory.


Finally, we turn our attention towards the physical interpretation of the k-lump solution of the noncommutative CPN model of [22].

Without much additional work, the brane configuration in type II-B string theory that realises the fuzzy lump may be read off from the construction of [10] as tilted D-strings suspended between an NS5-brane and D3-brane.

We conclude, as is conventional, with the conclusions.


2. The Gauged Linear Sigma Model


2.1 Definitions


Among the many extensions to the Abelian-Higgs model, one of the most natural is the gauged linear sigma model with Fayet-Illiopolous D-terms [30, 40].

This is certainly true if the aim is the construction of a model that supports solitonic excitations saturating BPS-like bounds.

With its CN+1-valued scalar fields and U(1)N+1 gauge symmetry, the linear sigma model is a natural springboard for our discussion of the relation between noncommutative semilocal vortices, fuzzy sigma model lumps and the braney systems they are associated with.

To this end then it will prove useful to briefly review some of the ideas and notation used to extract the vortex excitations from the solution spectrum of the semilocal model.

Following [30] we write the linear sigma model action as


SSL = ∫ d3x (DµΦ)(DµΦ)† + ∑_{a=1}^{N+1} (1/4e2_a)(F^a_µν)^2 + ∑_{a=1}^{N} (e2_a/2)(R_a − Φτ_aΦ†)^2


The dynamical degrees of freedom in this model are encoded in a CN+1-valued spacetime scalar Φ = (φ1, ..., φN+1) and the N + 1 U(1)-valued connection forms A^a = A^a_µdx^µ with associated curvature 2-forms F^a = dA^a.

The τ^a are the N + 1 generators of U(1)N+1.

The gauge covariant derivative we will take as


D := d − i ∑_{a=1}^{N+1} τ^aA^a


There are two sets of parameters in the theory; the N + 1 coupling constants e_a of dimension (mass)1/2 and N + 1 Fayet-Illiopolous (FI) parameters R_a - effectively the vacuum expectation values of the components of Φ. Without loss of generality (and because we can always re-scale the fields to absorb them anyway) we set the latter to unity.

The coupling constants we retain because they control the energy scales of the model.


In the temporal gauge, the static energy corresponding to the action (2.1) is


E = ∫ d2x [(D_iΦ)(D_iΦ)† + ∑_{a=1}^{N+1} (1/4e2_a)(F^a_ij)^2 + (e2_a/2)(Φτ_aΦ† − 1)^2]


For instance, in the case N = 1, following [30] the energy functional becomes (in exhaustive detail)


E = ∫ d2x [(D_1Φ)(D_1Φ)† + (D_2Φ)(D_2Φ)† + (1/4e2_1)(F_ij)^2 + (1/4e2_2)(G_ij)^2 + (e2_1/2)(Φτ_1Φ† − 1)^2 + (e2_2/2)(Φτ_2Φ† − 1)^2]


where the GL(2, R)-valued connections A = τ_1A_1 and B = τ_2A_2 are associated to the curvature forms F = dA and G = dB respectively.

For our purposes, it will suffice to turn off B and e2 and take τ1 = 1/2 giving


E = ∫ d2x [(D_1Φ)(D_1Φ)† + (D_2Φ)(D_2Φ)† + (1/2e2_1)(F_ij)^2 + (e2_1/4)(ΦΦ† − 1)^2]


Even under such restricted circumstances, the resulting linear sigma model is still remarkably rich [30, 40], exhibiting a wealth of solitonic structure and enjoying intimate relations with nonlinear sigma models on toric varieties.


In what follows, it will prove convenient to rewrite the energy in terms of the complex coordinates z := (x1+ix2)/√2 and ¯z := (x1−ix2)/√2.

This particular normalization means that


∂z := ∂/∂z = (1/√2)(∂1 − i∂2)


∂¯z := ∂/∂¯z = (1/√2)(∂1 + i∂2)


This in turn induces a complexification of the gauge covariant derivative so that


Dz := (1/√2)(D1 − iD2)


D¯z := (1/√2)(D1 + iD2)


when Az := (A1 − iA2)/√2 and A¯z := (A1 + iA2)/√2.

These are of course now GL(2, C)-valued objects.

With these definitions,


E = ∫ d2z [(DzΦ)(DzΦ)† + (D¯zΦ)(D¯zΦ)† + (1/2e2)(F12)^2 + (e2/2)(ΦΦ† − 1)^2]


2.2 Solitons on the Plane


To see the emergence of the semilocal vortex in the spectrum of the gauged linear sigma model, the usual method of “completing the square” in the energy functional may be followed.

After some straightforward manipulations, (2.8) may be put into the form


E = ∫ d2z [2(D¯zΦ)(D¯zΦ)† + (1/2e2)F12 + e2(ΦΦ† − 1)^2] + ∫ d2z T + ∫ d2z F12 = 2πk


where T = ∂¯z(ΦDzΦ†) − ∂z(ΦD¯zΦ†).

As such, the second to last term is a total derivative whose integral vanishes.

Consequently, a nonvanishing lower bound of 2πk is established on finite energy field configurations.

As usual, the bound E ≥ 2πk saturates when the first order system


{D¯zΦ = 0, F12 = e2(ΦΦ† − 1), ∫ d2z F12 = 2πk}


is satisfied.

The first of these is, of course, really two equations, one for each component of the C2-valued field Φ. The equations in (2.10) form a closed system whose solutions are precisely the semilocal vortices of [16, 30, 35].


Although such solitonic solutions are vortex-like in many respects, a little analysis soon reveals that their asymptotic behavior is very different from the exponential falloff of Abelian-Higgs vortices [16, 17].

In fact the fields of the semilocal model exhibit a distinctive power-law behavior at spatial infinity, a symptom of the fact that the width of the flux tube is an arbitrary parameter of the theory.

This should be contrasted with Abelian-Higgs vortices where the width is controlled by the Compton wavelength of the gauge boson.


In this sense, these vortex solutions are rather reminiscent of CPN instantons.

This is no mere coincidence.

In fact, the correspondence can be made precise in the large coupling limit in which the semilocal vortices of the U(1)N+1-gauged linear sigma model descend to the instanton solutions of a CPN nonlinear sigma model [30, 40].

While this is quite clear at the levels of the action and equations of motion, its realization at the level of the solutions is marginally obscured by the fact that only the asymptotic forms of the vortex solutions are known to exist.


This is not unlike the situation with the conventional Nielsen-Olesen vortex.

However this particular hurdle was recently surmounted in [1, 2, 23, 37] where a noncommutative deformation of the two-dimensional configuration space of the Abelian-Higgs model allows for the construction of exact vortex solutions.

The fact that the noncommutative version of the theory seems so much richer than its commutative counterpart is by now not surprising [7, 8].

It would seem then, that a noncommutative deformation of the base space of the two-dimensional gauged linear sigma model might offer, if nothing else, an interesting avenue to explore the construction of exact semilocal vortices.


3. And then everything became fuzzy...


Conventionally a noncommutative deformation of the two-dimensional configuration space is imposed by a Moyal-deformation of the algebra of functions over R2 and implemented by replacing ordinary pointwise multiplication of functions by Moyal *-multiplication.

Consequently, coordinates on the noncommutative plane R2_ϑ satisfy the Heisenberg algebra


[xi, xj] = xi * xj − xj * xi = iθij


where θij = ϑǫij is a nondegenerate, antisymmetric matrix of constants and ϑ is a positive deformation parameter of dimension (mass)2.

In terms of the complex coordinates on R2_ϑ, the commutator [z, ¯z] = ϑ is easily seen to be isomorphic to the algebra of annihilation and creation operators for the simple harmonic oscillator.

Thus a function on the noncommutative space may be associated, via a Weyl transform [12], to an operator acting on an auxiliary one-particle Hilbert space H built from harmonic oscillator eigenstates.

After a mild redefinition of z = √ϑ a and ¯z = √ϑ a†, the action of the coordinate operators on the basis states is given by


a |n⟩ = √n |n−1⟩, a† |n⟩ = √n+1 |n+1⟩


with the vacuum |0⟩ defined by a |0⟩ = 0.

Further, under the Weyl map


∂z → −(1/√ϑ) a†, ∂¯z → (1/√ϑ) a


∫ d2z f(z, ¯z) → 2πϑ Tr_H Of(z, ¯z)


3.1 The Noncommutative Semilocal Model


With this prescription at hand, the noncommutative semilocal energy functional (2.8) can be written as


E_ϑ = 2πϑ Tr_H [(Dz Φ)(Dz Φ)† + (D¯z Φ)(D¯z Φ)† + (1/2e2) F12 + (e2/2) (Φ Φ† − 1)^2]


where, now Dz Φ = (1/√ϑ)(a† + C) Φ, D¯z Φ = (1/√ϑ)(a + C†) Φ and the gauge field Az = (i/√ϑ)(a† + C).

As in the commutative case, this can be massaged into a Bogomol’nyi form which is saturated when the BPS equations


(D¯z Φ) = 0


F12 = ϑ e2 (Φ Φ† − 1)


Tr_H (1 + [C†, C]) = k


are satisfied.

As in the commutative case, this is a system of three first order equations, subject to the flux constraint.

Solutions of this system will be the noncommutative generalizations of the semilocal vortex of [17].

In the spirit of [2, 23], we begin with an ansatz for the Higgs doublet and the gauge field.

To this end the most general vortex-like solution of the BPS equations which maintain the cylindrical symmetry is of the form


Φ = |I⟩ ⊗ φ1 + |II⟩ ⊗ φ2


where


|I⟩ = (1, 0), |II⟩ = (0, 1) and


φi = ∑_{m=0}^{∞} f^{(i)}_m |m⟩⟨m + q^{(i)}|


C = ∑_{m=0}^{∞} g_m |m⟩⟨m + 1|


Without loss of generality all coefficients are taken to be real.

The construction of exact vortex solutions to the semilocal model now hinges on determining the various coefficients in the above ansatze that satisfy the appropriate boundary conditions.

In terms of the coefficients f_n ≡ f^{(1)}_n and h_n ≡ f^{(2)}_n, the first of eqs.(3.6) become


m + q + 1 + g_m f_{m+1} = 0


h_m √m + 1 + g_m h_{m+1} = 0


with the choice q^{(1)} = q and q^{(2)} = 0.

An explanation for this will follow in due course.

For the moment though, notice that eqs.(3.10) mean that the coefficients of each of the components of the Higgs doublet are not independent.

Indeed


h_{m+1} = √(m + 1)/(m + q + 1) f_{m+1}/f_m h_m.


This is a simple recurrence relation which is easily solved for h_m to give


h_m = √(m! q!)/(m + q)!

κ f_m


with κ = h_0/f_0 determining the relationship between the initial conditions of each coefficient sequence.

With the convenient definitions of Q_n ≡ f^2_n and P_n ≡ h^2_n, this may be combined with the second of the BPS equations to give


Q_1 = (q + 1)Q_0/(1 + γ − γ(1 + κ^2)Q_0)


Q_{m+1} = Q_m + (m + q)Q_{m−1} − (m + q + 1)Q^2_m/(1 + m! q!/(m + q)!

κ^2) γ Q_m


m > 0


In principle then, the noncommutative vortex solution of the critically coupled linear sigma model may be determined by solving the recurrence relation (3.13) and consequently (3.12) subject to the “boundary conditions” f_n → 1, h_n → 0 as n → ∞. Well, almost.

The attentive reader would of course have noticed that there is still the matter of the arbitrary integer q. Fortunately, there is also the third of the BPS equations, the flux constraint.

Using (3.10) and the ansatz for the gauge field it may be shown

that


Tr_H (1 + [C†, C]) = Tr_H (1 + g^2_{m−1} − g^2_m) = lim_{M→∞} (M + 1)/(M + q + 1) − Q_M/Q_{M+1}


In the last step, the cutoff of [37] was employed to regulate the trace.

In the large M limit, the convergence of the coefficient sequence means that the ratio of successive Q’s approaches unity.

Consequently, the flux constraint equation implies that q = k. Indeed, a quick comparison with the analogous commutative result confirms that this is the only physically meaningful conclusion; the index of φ1 is equal to the topological number of the vortex.

Interestingly enough, choosing q^{(2)} = 0 does not affect this conclusion.

Again, this is not altogether unexpected since q^{(2)} is just the angular momentum quantum number of the vortex [16].

Convergence of the coefficient sequence for φ2 bounds the angular momentum quantum number to the range 0 ≤ q^{(2)} < k. However, since none of the arguments presented here depends essentially on q^{(2)} we can, without any loss of generality, set q^{(2)} = 0.

It is also worth noting that when q = 0, h_m = κ f_m and both boundary conditions can only be simultaneously satisfied if h_m ≡ 0 which reduces to the k = 0 vortex of the noncommutative Abelian-Higgs model [1, 2, 23, 34, 37].

Instead of solving eqs.(3.13) in full generality, it is perhaps more illuminating to focus on a few examples.


3.2 Examples


To begin with we consider the case h_m = 0 for all m. In this case the Higgs doublet Φ = (φ1, 0) satisfies exactly the equations of motion of the noncommutative Abelian-Higgs model and it is quite easy to check that the solutions of (3.13) reduce to the degree k vortices found in [23] for which


Q_1 = (q + 1)Q_0/(1 + γ(1 − Q_0))


Q_{m+1} = Q_m + (m + q)Q_{m−1} − (m + q + 1)Q^2_m/(1 + γ) γ Q_m


This set of equations has been studied extensively and numerically shown to exhibit regular vortex solutions with +k units of magnetic flux for a large γ range.

In particular, for small ϑ (and consequently γ) the regular commutative Neilsen-Olesen vortex solutions of [3, 28] are obtained.

In addition, an obvious solution to (3.15) that satisfies the boundary conditions of the semilocal model is Q_m ≡ 1.

As noted in [23], these are exactly the fluxon solutions of [14].


Moving on now to the more interesting case of non-vanishing h_m, it will suffice to restrict our attention to k = 1 for which the BPS recurrence relations become


Q_{m+1} = Q_m + (m + 1)Q_{m−1} − (m + 2)Q^2_m/(1 + γ) γ Q_m


P_m = (1 + κ^2)/(m + 1) Q_m − (m + 1)Q_m


The vortex solutions of the noncommutative semilocal model are constructed by solving eqs.(3.16) subject to the convergence constraint (P_m, Q_m) → (0, 1) as n → ∞. From the first of these it is clear that when the Q_m sequence converges and κ is of order unity, P_m ∼ 1/m for large m. Again, this remains true for any fixed value of the angular momentum quantum number.

We solve the above system numerically using a double precision, split-step shooting algorithm.

At first glance, the shooting-parameter space looks to be two-dimensional (corresponding to the different values of the pair (P_0, Q_0)) but a prescient choice of κ^2 = 1/ϑ fixes one of these parameters in terms of the other and reduces the dimension to one.

With the initial value Q_0 as the shooting parameter, we solve (3.16) for various values of γ and tabulate our results below.


ϑ e2 γ Q0
0.2 1 0.2 0.099732894
0.2 4 0.8 0.140471163
0.2 16 3.2 0.158732886334
0.2 36 7.2 0.16297094403243935
0.5 1 0.5 0.215729007
0.5 4 2 0.2895665841653
0.5 16 8 0.32043540606185
0.5 36 18 0.32737242959721649


Each of these initial values for the Q_m results in a coefficient sequence that converges (with varying degrees of accuracy) to one.

Once determined, the P_m and Q_m may then be used to compute other characteristic quantities associated with the semilocal vortex.

For example, the magnetic field of the semilocal vortex may easily be computed as


B = (γ/ϑ) ∑_{n=0}^{∞} Q_n |n⟩⟨n + 1 + ϑ^{-1}/(n + 1)|


Substituting this, together with the covariant derivative


Dz Φ = (1/√ϑ) ∑_{m=0}^{∞} (f_m √m + 1 + g_m−1 f_{m−1}) |m⟩⟨m| ⊗ |I⟩ + (h_{m+1} √m + 1 + g_m f_m) |m + 1⟩⟨m| ⊗ |II⟩


into eq.(3.5) allows for the energy density of the vortex to be computed quite straightforwardly as


E = (1/ϑ) ∑_{m=0}^{∞} [(1 + (m + 1)/(m + 1 + ϑ^{-1}) Q_m − Q_{m−1})^2 + γ (P_m − P_{m−1})^2] |m⟩⟨m|


It may be verified numerically that up to the first few hundred terms the above expression for the energy density sums to 1/(2πϑ) to within a few percent as is expected for the 1-vortex solution.

To make contact with the primary aim of this paper, it will be convenient to visualize the profile of the vortex, especially as γ is turned up.

However, both eqs.(3.17) and (3.19) are Fock space representations.

Fortunately, these can be turned into (noncommutative) coordinate space representations relatively easily with the inverse Weyl map under which the projector |n⟩⟨n| 7→ (−1)^n exp(−2r^2/ϑ) L_n(2r^2/ϑ) where L_n is the n’th Laguerre polynomial.

In fig.1 we plot the magnetic field as a function of r for various values of the dimensionless parameter γ.

Fig.2. contains a series of snapshots of the energy profile of the vortex as gamma increases from 0.2 to 28.8.


4. The large coupling limit


Having presented a general algorithm for the construction of degree k semilocal vortex solutions of the gauged noncommutative linear sigma model and explicitly constructed the 1-vortex solution we proceed now to study one of the more interesting limits of the semilocal model: its large coupling limit.

At the level of the action (3.5), the e2 → ∞ limit decouples the gauge field dynamics and any finite energy static solution has


E = 2πϑ Tr_H [(Dz Φ)(Dz Φ)† + (D¯z Φ)(D¯z Φ)†]


subject to the constraint Φ Φ† = 1.

In this limit, the gauge field is relegated to an auxiliary field, completely determined by Φ. Recalling that Φ is an (n + 1)-component complex vector leads to the conclusion that this is, of course, nothing but the noncommutative version of the CPN sigma model.

At the level of the action this observation is certainly not new; in the commutative case, this relation has been commented on by several authors in many different contexts [16, 17, 30, 40].

However, it remains to be seen whether this correspondence persists at the level of the solutions.

If it does we will have produced an explicit descent from the vortices of the fuzzy linear sigma model to the instantons of the noncommutative CPN model.

In the interests of self-containment, we review now the derivation of the lump solutions of the sigma model.


With eq.(4.1) as a starting point, a reparameterization of the (N + 1)-component Higgs field as Φ = (1/W†) W and subsequent definition of the Hermitian projector P ≡ W (W† W)−1 W† allows for the static energy (or two-dimensional action) to be written as


E = 2π Tr_H tr([P, a†][a, P])


In this form, the CPN energy is remarkably similar to the kinetic term of the static energy of a (2 + 1)-dimensional noncommutative scalar field (see eq.(2.2) of ref.[8]) with the crucial difference of the additional matrix trace in eq.(4.2).

Indeed it was shown in [20, 21, 25] that the quantity Tr_H tr([a, a†] P) contributes a nonvanishing boundary term to the energy and some care needs to be exercised in the derivation of the noncommutative Bogomol’nyi bound.

With this in mind, the energy may correctly be written as


E = 2π Tr_H tr(2F+(P)†F+(P)) + 2πQ+ ≥ 2πQ+


with the topological charge Q+ ≡ Tr_H tr(P [a, a†]) and F+(P) ≡ [a, P].

A similar expression holds for the anti-BPS states.

Focusing on the BPS states though, saturation of the bound on the energy is obtained when F+(P) = 0.

As first shown in [22], solutions are not difficult to find; any Hermitian projector constructed from an (n + 1)-vector W whose components are holomorphic polynomials in z will satisfy the above BPS equation.

These are precisely the noncommutative extension of the instanton solutions of the conventional CPN sigma model.

For example, the static, lump solutions of the noncommutative CP1 model are given by


W1 = (z/a1, b1)


W2 = (z^2/a2, 2b2 z + c2)


where the soliton parameters a1, ..., d2 ∈ C are chosen to coincide with the standard way of writing the solutions in the commutative theory [36].

These are the complex moduli of the CP1 instanton.

To facilitate comparison with the vortices, these may be written in the harmonic oscillator basis so that, for example, the 1-lump solution becomes


W1 = ∑_{n=0}^{∞} √(ϑ(n + 1))/(ϑ(n + 1) + 1) |n + 1⟩⟨n| ⊗ |I⟩ + ∑_{n=0}^{∞} 1/(ϑ(n + 1) + 1) |n⟩⟨n| ⊗ |II⟩


Returning to the degree k semilocal vortex of the last section, notice that eq.(3.13) may be recast as


1 + (n! k!/(n + k)!) κ^2 Q_n − 1 + 1/γ (n + k + 1) Q_n Q_{n+1} − (n + k) Q_{n−1} Q_n − 1 = 0


In the infinite coupling limit e2 → ∞ (or equivalently γ → ∞), the above recurrence relation may be be solved exactly to give


Q_n = 1 + (n! k!/(n + k)!) κ^2


In particular, for k = 1 we find


Q_n = (n + 1)/(n + 1 + κ^2)


P_n = κ^2/(n + 1 + κ^2)


Finally, matching coefficients to all orders in eqs.(4.5) and (4.8) means that the descent from noncommutative vortex to fuzzy lump only occurs when κ^2 = 1/ϑ. Indeed, this is exactly the choice we made in our numerical computations to reduce the dimension of the shooting-parameter space.

As a check, we expect that for a fixed value of ϑ, Q_0 → ϑ/(ϑ + 1) as γ → ∞. A quick glance at the table of our numerical results verifies that this is indeed the case for ϑ = 0.2 and 0.5.

Moreover, hindsight reveals that the set of energy densities in figure 2. is in fact a series of snapshots of the k = 1 vortex of the noncommutative semilocal model morphing into a fuzzy CP1 1-lump.

The case k = 2 is no less straightforward.

With its center of mass localised at the origin, the CP1 2-lump in eq.(4.4) can be written as


W2 = ∑_{n=0}^{∞} √(ϑ^2(n + 1)(n + 2))/(ϑ^2(n + 1)(n + 2) + 1) |n⟩⟨n + 2| ⊗ |I⟩ + ∑_{n=0}^{∞} 1/(ϑ^2(n + 1)(n + 2) + 1) |n⟩⟨n| ⊗ |II⟩


when b2, the frozen out modulus [5] is set to vanish.

A comparison with the general expression for the infinite coupling coefficients (4.7) reveals a matching at all levels only if κ^2 = 1/2ϑ. Generalisation to larger k follows in much the same way so no further attention is paid to it here.


At this juncture, a few comments are in order.

The Bogomol’nyi equations of the commutative gauged linear sigma model admit a one parameter family of vortex solutions [17].

This single complex parameter w is to the commutative theory what the ratio of initial coefficients κ is to our noncommutative model with w = 0 corresponding to the conventional Neilsen-Olesen string.


One of the distinguishing characteristics of the w ≠ 0 semilocal vortices is the power law behavior exhibited by the scalar and gauge fields as they relax to their respective vacuum values.

Consequently, the magnetic field B ∼ 2/ξ^4 and the width of the flux tube trapped in the vortex core is an arbitrary parameter instead of the Compton wavelength of the vector boson as in the Neilsen-Olesen vortex.


In the noncommutative model we once again find a one parameter family of vortices only now the parameter, κ, is not at all arbitrary.

Indeed, we find that there exists a point in the κ parameter space dependent on the degree of the vortex and the deformation parameter ϑ at which the semilocal vortex exactly descends to the corresponding noncommutative CPN lump.

Correspondingly, the width of the magnetic flux tube associated with the semilocal vortex is set by the scale of noncommutativity.


This observed exact metamorphosis of the vortex into the lump should be compared to the results of section 3. of [17].

There an expansion of the 1-instanton solution of the commutative CPN model in powers of |z/w| was used to establish that the vortex-instanton matching was exact at spatial infinity with differences emerging at O(|z/w|^4) in this expansion.


5. Brane Realisations


Quite apart from their intrinsic field theoretic value [35, 16, 17], the vortices of gauged linear sigma models also have a remarkably rich stringy structure.

Beginning with the ground-breaking work of [11] in which the (2 + 1)-dimensional, N = 4 U(N) Yang-Mills-Higgs theory was recognised as the worldvolume theory on a stack of N D3-branes, an intricate tapestry of ideas can be woven, leading inexorably to a realisation of the noncommutative semilocal vortex as a D-brane configuration in type IIB string theory [10].

In this section, we review some of these ideas and cast them into a form that better facilitates comparison with our results.


As in [10] the description of the system begins with a (2 + 1)-dimensional, N = 4, U(N) Yang-Mills-Higgs theory.

The field content of the theory consists of a U(N) vector multiplet made up of a gauge field A_µ and a triplet of adjoint scalars φ_r together with their fermionic super partners.

Coupled to these are N fundamental hypermultiplets each of which contain a doublet of complex scalars q and q† and their super partners.

The Lagrangian for the theory is endowed with a global SU(N + M) flavour symmetry as well as a local U(N) gauge symmetry.

Consequently, under these two groups and with N_f ≡ N + M denoting the number of flavours, q and q† transform as (N, N_f) and (N, N_f) respectively; the fundamental scalars are represented by (N + M) × N matrices.

The dynamical content of the bosonic sector of the theory is contained in the Lagrangian


L = Tr [1/4e^2 F^2 + 1/2e^2 (Dφ_r)^2 + (Dq)^2 + (Dq†)^2 + (q† q + q q†)φ_r φ_r + e^2/2 (q† q − ζ 1_N)^2 + e^2 q q†]


where the Fayet-Illiopolous (FI) parameter, ζ, in the final D-term in (5.1) is chosen to be positive.

This theory exhibits a Higgs branch of vacua which possess BPS vortices only if q and φ_r both vanish.

This constraint defines a so-called reduced Higgs branch, NN,M = Gr(N, N + M), the Grassmannian manifold of N-dimensional hyperplanes in C^{N+M}. A particular vacuum choice is made by picking


q_vac = (√ζ δ^a_i, 0)


In our abelian case, for example, N = 1, the reduced Higgs branch N1,M = Gr(1, 1 + M) ≃ CPM and q_vac = (√ζ, 0).

Relabeling q as Φ, setting the FI parameter ζ = 1 and restricting to time-independent solutions trivially establishes the equivalence of the action in this branch with the static energy (2.5).

As discussed earlier, the spectrum of solutions of this theory is rich with BPS vortices.

The brane realisation of these vortices is built up from the U(N) Yang-Mills-Higgs described in (5.1).

It consists of N D3-branes suspended between two parallel NS5-branes and a further N + M D3’s attached to the right hand NS5-brane to add flavour (see figure 3).


In the Higgs branch, one of the NS5-branes is separated from the others.

This separation is proportional to the FI parameter ζ.

The degree k BPS vortices manifest as k D-strings stretched between the D3-branes and the separated NS5-brane - an identification made on the basis of the fact that the stretched D1-branes are the only BPS states of the brane configuration with the correct mass.

More than just a pretty picture, the geometry of the D-brane configuration in figure 3 encodes vital information about the FI parameter, ζ as well as the gauge coupling e as


1/e^2 = ∆x^6/(2πg_s)


ζ = ∆x^9/(4π^2 g_s l_s^2)


where l_s and g_s are the string length and coupling respectively and ∆x^6 and ∆x^9 are the separation distances between the NS5-branes in the 6 and 9 directions respectively.

It is now clear that the sigma model limit (e^2 → ∞) of the vortex occurs precisely when the separation of the NS5-branes in the 6 direction vanishes.

The configuration that realises the k-lump solution of the (commutative) CPM nonlinear sigma model then is as above only with N = 1.


In string theory the transition from commutative to noncommutative worldvolume theories is achieved by turning on an NS-NS B-field in the appropriate direction [31].

In the present context, the transition from the semilocal action (2.5) to its noncommutative counterpart (3.5) translates into turning on a constant NS-NS B-field B_{12} = ϑ dx^1 ∧ dx^2 in the (1, 2) directions in a background of two NS5-branes with a D3-brane stretched between them and a further M + 1 D3’s attached to the right hand NS5-brane.


What of the vortices?

The effect of the B-field on the D3-brane and the D3-brane is quite remarkable.

The basic physics is analogous to the situation of a D-string suspended between two D3-branes studied in [15] and was first described for the vortex case in [10].

The NS-NS 2-form manifests on the D3-brane worldvolume as a constant magnetic field F_{12} while the D-string endpoint appears as a magnetic source.

Since on the 4-dimensional worldvolume of the D3-brane F_{12} = ⋆ F_{06}, the magnetic endpoint of the D1-brane feels the same force as an electric charge in a constant electric field in the 6 direction.

However, as other end of the D-string remains married to the NS5-brane, the D-string responds to this force by tilting as in figure 4.

The effect of the tilting was investigated in [15] by studying the D-string Born-Infeld action at weak string coupling


S = (1/2πl_s^2) ∫_{0}^{∆x^9} dx^9 (1/g_s) √(1 + (dx^6/dx^9)^2) + A_{06} (dx^6/dx^9)


where the RR 2-form A_{06} that couples to the D-string worldvolume is induced by B_{06}. The result of that investigation translated into the language of the vortex theory [10] is that the displacement of the D1-brane endpoint is given by δ = (ϑ ∆x^9)/(2πl_s^2).

With this and some straightforward algebra, the distance between the D-string endpoint and the left NS5-brane can be computed.

With the choice of ζ = 1 for the FI parameter, the result is


r = 2πg_s (1/e^2 + ϑ)


This distance is, in fact, the FI parameter of the theory living on the D1-branes (see [10] for a lucid discussion of this aspect).

Having fixed ∆x^9 with the choice ζ = 1 the magnitude of r is completely determined by the size of the gauge coupling as determined by the NS5-brane separation in the 6 direction and the noncommutativity.

Since the latter is also fixed, the transition from vortex to lump can be studied by changing the distance between the NS5-branes.

As ∆x^6 is decreased to zero, the separation between the D-string endpoint and the left NS5-brane decreases to r* = 2πg_s ϑ. It is this configuration of the k tilted D-strings stretched between the (formerly right hand) NS5-brane and the D3-brane (on which a stack of M semi-infinite D3-branes end) that realises the degree k instanton of the CPM sigma model.

This concludes our treatment of D-brane realisation of the noncommutative CPM lump.


More than just an academic exercise, this identification of the semilocal vortex and CPM instanton has proven invaluable in the understanding of the low energy dynamics of both the vortex and instanton as encoded in the geometry of their respective moduli spaces [34].

We refer the interested reader to [10] for a nice discussion of the structure of the moduli spaces and content ourselves with merely summarising some of their most pertinent results.

The moduli space of degree k semilocal vortices ˆVk,(1,M) is a 2k(1 + M)-dimensional space with a natural Kähler metric defined by the overlap of zero modes.

However, this metric is afflicted with some non-normalisable zero modes that, classically, correspond to the moduli with infinite moments of inertia and that make the quantum mechanical treatment of these solitonic objects quite subtle.

Fortunately these subtleties may be circumvented with a little help from the branes.

A study of the theory on the D1-brane predicts that the Higgs branch, ˆMk,(1,M), constructed by a U(k) Kähler quotient of Ck(1+M+k) is isomorphic to the moduli space ˆVk,(1,M).

While the metric on ˆMk,(1,M) retains all the symmetries of the Kähler metric on the vortex moduli space, it is finite and suffers from none of the non-normalisablity problems of the latter.

Consequently, the study of the quantum theory of semilocal vortices may be simplified somewhat by replacing the natural metric on the vortex moduli space with the metric on the Higgs branch of the D-brane theory inherited from the Kähler quotient construction of [10].


Conclusions and Discussion


The primary concern of this work has been the construction and study of a noncommutative extension of (2 + 1)-dimensional critically coupled, gauged linear sigma model.

Like its commutative counterpart this theory possesses a rich spectrum of BPS solutions.

By extending the systematic construction of [23] we have explicitly constructed a family of vortex solutions to the BPS equations (3.6) for arbitrary positive values of the noncommutativity parameter ϑ. As expected, these fuzzy vortices reduce to the exact Neilsen-Olesen strings of the noncommutative Abelian-Higgs model [1, 2, 23, 37] on the co-dimension one surface κ = 0 of the parameter space.

Despite retaining many of the properties of their commutative cousins [17, 35], the introduction of a new length scale set by the noncommutativity parameter ϑ induces several remarkable differences.

Among these we find that the width of the magnetic flux tube trapped in the vortex core no longer exhibits the characteristic arbitrariness of the commutative semilocal vortex.

In the noncommutative model, this width is set by the scale of the noncommutativity.


The detailed investigation of the large coupling (e^2 → ∞) regime of the ϑ-deformed gauged linear sigma model carried out in section 4.

confirms, both numerically and analytically, the commutative intuition of the vortex morphing into a lump of the (fuzzy) CPM sigma model.

Additionally, while the agreement between vortex and lump in the ϑ = 0 case is precise only asymptotically [17], we find an exact matching at all levels of the harmonic oscillator expansion at finite ϑ. Indeed, insisting that this agreement holds selects a preferred set of values for κ, dependent on the scale of noncommutativity and the degree of the vortex.

This effectively reduces the dimension of the parameter space by one.

While we have explicitly constructed solutions for the 1-vortex and 2-vortex cases, the construction of higher degree solutions follows in much the same way and we do not expect any further surprises.


Finally, we reviewed the elegant constructions of [10] that lead to a realisation of the noncommutative CPM k-lump as k tilted D-strings stretched between an isolated NS5-brane and the D3-brane (on which a stack of M semi-infinite D3-branes end) and a semi-infinite D3 whose one endpoint ends on a second NS5 (see figure 4).

This identification is built on the foundation of a study of the N = 4 U(N) Yang-Mills-Higgs D3-brane worldvolume theory hinges on the metamorphosis of vortices into lumps.

Of course, to be sure that this configuration really does correspond to the lump solution requires more work than just a comparison of the masses of both configurations; the spectrum of fluctuations around each object needs to be computed and compared.


This is a more difficult endeavor which, together with a more thorough investigation of the spectrum of BPS objects of the noncommutative gauged linear sigma model is left to future work [26].

Curiously, this realisation of fuzzy CPM lumps is not unique, at least for M = 1.

Drawing on the tree level equivalence between N = 2 open string theory and self-dual Yang-Mills theory in (2 + 2) dimensions [29], it was argued in [20, 21] that the effective field theory induced on the worldvolume of N D2-branes by a Kähler B-field background is a noncommutative U(N) sigma model.

Using a modified “method of dressing” soliton solutions of the latter were constructed and their various scattering properties investigated.


In this context, the k-lump solution of the CP1 sigma model may be interpreted as k D0-branes in the worldvolume of a stack of D2-branes [21, 18].

Again, while this assertion needs to be tested beyond the level of a mass comparison, the possibility of a duality between N = 2 open string theory and the type II-B superstring is, to say the least, intriguing and certainly deserves more attention.


Acknowledgments


J.M. would like to thank the KITP (Santa Barbara) for the warm hospitality and a most stimulating environment during the early stages of the work.

Our gratitude is extended to Robert de Mello Koch, George Ellis, Clifford Johnson, David Tong and Amanda Weltman for valuable discussions at various stages of this work and to Bill Watersson who inspired the title. And finally, a word from our sponsors: J.M. is supported by the Sainsbury-Lindbury Trust and a research associateship of the University of Cape Town.

A.M acknowledges the NRF and the University of Cape Town for financial support.


We study the low-energy quantum dynamics of vortex strings in the Higgs phase of = 2 supersymmetric QCD.

The exact BPS spectrum of the stretched string is shown to coincide with the BPS spectrum of the four-dimensional parent gauge theory.

Perturbative string excitations correspond to bound W-bosons and quarks while the monopoles appear as kinks on the vortex string.

This provides a physical explanation for an observation by N. Dorey relating the quantum spectra of theories in two and four dimensions.

Two-dimensional sigma-models have long acted as a playground in which to test aspects of four-dimensional gauge dynamics.

The two systems share many qualitative features including asymptotic freedom, a dynamically generated mass gap, anomalies, and instantons.

Some years ago, N. Dorey proved a more quantitative correspondence between supersymmetric theories in two and four dimensions [1].

He showed that the BPS spectrum of = (2; 2) CPN ¡1 sigma-model coincides with the BPS spectrum of four-dimensional = 2 SU(N ) supersymmetric QCD.

The correspondence is exact, holding at the quantum level in both weak and strong coupling regimes.

Generalisations to other two-dimensional sigma-models were later found [2].

However, despite some insight from brane constructions [3, 2], the underlying reason for the agreement remained mysterious.


The purpose of this paper is to provide a ﬂeld theoretic explanation for the correspondence.

The key to our story lies in the recent progress in understanding the dynamics of various soliton conﬂgurations in the Higgs phase of = 2 SQCD.

Of particular relevance are the non-abelian vortices [4], which are string-like objects in four dimensions, and monopoles [5] which, due to the Meissner eﬁect, are conﬂned in the Higgs phase and come attached to two semi-inﬂnite vortex strings.1 We shall show that the two-dimensional theory considered by Dorey in [1] is precisely the theory describing the vortex string.

As we explain below, the BPS excitations of the string have an interpretation as four dimensional states: the perturbative string excitations correspond to W boson-string bound states, while the solitonic kinks of the string correspond to the conﬂned monopoles in four dimensions [5].

The results presented here ﬂt into the growing body of work devoted to understanding the dynamics of solitons in the Higgs phase of = 2 theories.


In recent years we have found that these theories admit a remarkably rich structure of classical BPS solitons.

As well as the strings and conﬂned monopoles mentioned above, there is an intricate system of domain walls [9, 10], domain wall junctions [11] and, perhaps most remarkably, D-branes [12, 13] in which the vortex string terminates on a domain wall where its end is electrically charged under a gauge ﬂeld.

While ﬂeld theoretic D-branes have been known to exist for some time in strong coupling regimes [14] the objects described in [12, 13] are amenable to semi-classical analysis.


The correspondence discovered in [1] holds in both strong coupling and weak coupling regimes of the two theories.

In the latter regime, the central charge of the theory may be expanded in an inﬂnite series of instanton contributions.

Since the BPS spectra coincide, this expansion agrees term by term, suggesting a quantitative correspondence between two dimensional instantons (which are vortices) and four-dimensional Yang-Mills instantons.

Indeed, in [4], an ADHM-like construction of the vortex moduli space was presented and it was shown that the moduli space of vortices is a particular submanifold of the moduli space of instantons.

It would be interesting to prove explicitly that the integrals over the relevant moduli spaces coincide. From the interpretation of the correspondence presented here, this agreement suggests another solitonic connection: a vortex in a vortex string looks like a Yang-Mills instanton in four dimensions.

In section 3, we present the Bogomoln’yi equations describing such a solution.


The conclusion of this paper | that the quantum dynamics of solitons, speciﬂcally vortex strings, may be used to extract information about the strong coupling dynamics of the underlying four dimensional gauge theory | is reminiscent of the stringy games played in ten dimensions.

For example in the the old-new Matrix theory, D-brane solitons contain much information about the bulk dynamics.

It would be interesting to see if this analogy can be pushed further.


The plan of the paper is as follows.

In section 2, we study = 2 supersymmetric U(Nc) gauge theory with Nf = Nc (cid:176)avours.

We review the exact central charge on the Coulomb branch which can be determined from the Seiberg-Witten solution.

We then follow the states as you slide onto the Higgs branch, breaking the gauge group completely.

We shall show that the monopoles remain BPS, but are now conﬂned.

At the same time, a new BPS object appears: the vortex string.

In section 3 we describe the low-energy dynamics of the vortex string and show that it is governed by the two-dimensional theory studied in [1].

We review the computation of the BPS spectrum and conﬂrm that it does indeed coincide with that of the four-dimensional parent theory.

In particular, we shall see that elementary excitations of the string are associated to W-bosons, while kinks in the string are monopoles in four dimensions.

In section 4, we repeat this story for Nf > Nc (cid:176)avours, and the associated \semi-local" vortices, giving a rationale for the generalisation discovered in [2].

This includes the case of the conformal vortex string.


Note added: after ﬂnishing this work we were informed that similar conclusions have been reached by M. Shifman and A. Yung [15].

We would like to thank M. Shifman for communicating their results to us prior to publication.


One of the longstanding problems that continues to plague researchers interested in "perception without aware- ness" is the difficulty in ensuring that the observer is in- deed unaware of the information s/he is purported to have processed perceptually.

The study of the nature and extent of perceptual processing in the contralesional field in in- dividuals with hemispatial neglect is especially appealing, then, because it offers a methodological alternative to cir- cumvent the need to ensure that "unattended" information is indeed "unattended." Given that these patients do not orient to the contralesional side and that they do not voli- tionally report the information on the unattended side, the assumption is that they are simply not aware of this infor- mation.

One can then ask how much perceptual processing takes place under these conditions of inattention.


Since the early studies of Volpe, Ledoux, and Gaz- zaniga (1979) and Marshall and Halligan (1988), many neuropsychological investigations have examined the fate of the unattended contralateral information, and MM and their collaborators have contributed substantially to this literature (McGlinchey-Berroth, 1997; McGlinchey- Berroth, Milberg, Verfaellie, Alexander, & Kilduff, 1993; McGlinchey-Berroth et al., 1996).

Our study is conducted within this neuropsychological tradition and focuses on whether contralesional perceptual information (appearing on the left side in patients with a right-hemisphere lesion) can be grouped or organized perceptually in the absence of attention.

We also explore whether the perceptual fate of this unattended information varies with the severity of the hemispatial neglect.


To address this issue, we adopt a procedure that has been extensively and successfully used in investigations that ask similar questions with normal observers.

In brief, in our study, 8 patients with hemispatial neglect (diag- nosed using a standardized, normed bedside battery, and confirmed and quantified using a covert spatial attentional paradigm) performed a matrix change detection task on a black and white checkerboard placed in their intact ipsile- sional field.

The matrix could remain the same across two successive displays, or a single square could be shifted.

Concurrent with this demanding perceptual task, task- irrelevant information, consisting of grayscale circles, was


Attention, Perception, & Psychophysics 2010mental task (change detection/task-irrelevant grouping), because we measured the patients' attentional status on a different task using the covert spatial cuing paradigm.

They bolster their claim by pointing to studies that show poor correlation among neglect tasks.

Although this psy- chometric variability is worthy of notice, this particular point is not directly relevant here.

At the outset, we note that MM refer to the fact that we assess the patients' at- tentional status on the covert attentional cuing paradigm and do not acknowledge that all patients in our study were initially tested on the well-known Behavioural In- attention Test as well (Wilson, Cockburn, & Halligan, 1987).

The BIT is probably the most popular test for di- agnosing neglect and is widely used for this purpose.

It comprises nine conventional tests of neglect, it is well normed with a correlation .92 ( p .0001) between the subtests, and the BIT neglect score is highly correlated with the responses of clinicians to a short questionnaire, .67 ( p .001).

One study has confirmed that seven of the nine BIT behavioral subtests differentiated signifi- cantly between subjects with visual neglect and those without neglect and, furthermore, that most of the sub- tests correlated significantly with the Activities of Daily Living checklist (Hartman-Maeir & Katz, 1995).

These findings support the construct and predictive validity of most of the BIT behavioral subtests as functional mea- sures of unilateral neglect.


Of note too, the covert attentional paradigm task (ad- opted from Posner, Snyder, & Davidson, 1980) was not used to determine simply whether the participants did or did not have neglect; that determination was made on the basis of the BIT scores.

The covert attentional task was used: (1) to confirm the presence of attentional neglect with a sensitive psychophysical test, thereby increasing our confidence in the neglect diagnosis; and (2) to de- termine the severity of attentional neglect for each in- dividual, a point that is not acknowledged by MM.

We also confirmed that the BIT neglect scores and an index of attentional dysfunction (derived from the covert cuing paradigm) of the patients were highly correlated.

We note, too, that previous studies have documented a strong association between clinical measures of neglect and performance on the cued attentional paradigm that we have used (Friedrich, Egly, Rafal, & Beck, 1998).

There is no question, then, that the diagnosis of neglect in the patients was correct and, moreover, that the quantifica- tion of the extent of the neglect, as derived from the BIT and cuing task, was accurate and reliable.


Our approach of diagnosing and quantifying the severity of the neglect impairment across multiple tasks goes well beyond what is typically done in the existing literature.

With few ex- ceptions, the characterization of neglect in existing neu- ropsychological studies of perception without awareness is done on the basis of lesion site, clinical observation, and/or some bedside testing (horizontal line bisection, line cancellation, figure drawing, single word reading).

We are confident that we have thoroughly characterized the nature and severity of the attentional impairments in presented in the contralesional field.