June 19, 2021 hep-th/0306150

MIT-CTP-3388

Vortices, Instantons and Branes

Amihay Hanany and David Tong

Center for Theoretical Physics,
Massachusetts Institute of Technology,

Cambridge, MA 02139, U.S.A.

The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge vortices in Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of instantons in non-commutative Yang-Mills theories. This submanifold is the fixed point set of a action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the and Grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities between quantum field theories in two and four dimensions.

## 1 Introduction and Preview

The moduli space of a supersymmetric system is defined as the set of classically massless, 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 [2].

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 [2].

####
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
[3, 4]^{1}^{1}1For particularly simple cases, more explicit descriptions
also exist. See, for example [5]..
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 [6, 7].

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
[3, 4]^{1}^{1}1For particularly simple cases, more explicit descriptions
also exist. See, for example [5]..
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 [6, 7].

####
Nevertheless, it is possible to make progress in supersymmetric quantum
field theories
even when the moduli space metric is not known. This is because, as first
emphasised by Witten [8],
many of the simplest quantities of interest depend only on topological
characteristics of the moduli space. For example, the supersymmetric
bound states of solitons are related to various cohomology classes of
the moduli space
[8, 9]^{2}^{2}2Questions of normalisability
mean that asymptotic behaviour of the metric is also required..
Similarly non-perturbative contributions to
BPS correlation functions, which involve integrals over the moduli space
of instantons, often reduce to topological invariants
[10, 11]. Thus, for many purposes it suffices to know
only crude topological information about the moduli space.

Nevertheless, it is possible to make progress in supersymmetric quantum
field theories
even when the moduli space metric is not known. This is because, as first
emphasised by Witten [8],
many of the simplest quantities of interest depend only on topological
characteristics of the moduli space. For example, the supersymmetric
bound states of solitons are related to various cohomology classes of
the moduli space
[8, 9]^{2}^{2}2Questions of normalisability
mean that asymptotic behaviour of the metric is also required..
Similarly non-perturbative contributions to
BPS correlation functions, which involve integrals over the moduli space
of instantons, often reduce to topological invariants
[10, 11]. 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
Yang-Mills-Higgs theories where the gauge group is broken
completely by fundamental scalar fields.
The theory has a mass gap and exhibits vortices, labeled
by the winding number of the magnetic field,

(1.1)

Here we summarise our main results.
We start in Section 2 with a study of the moduli space of charge vortices
which we shall denote as . Our first result concerns the
real dimension of the moduli space which, using index theory techniques, we
show to be

(1.2)

In Section 3 we present a brane construction of the vortices, from which we
extract a description of as a symplectic
quotient of .
This quotient construction is most easily described as the Higgs branch
of a gauge theory with four supercharges, coupled to a single
adjoint chiral multiplet and fundamental chiral multiplets.

The purpose of this paper is to describe the moduli space of vortices in Yang-Mills-Higgs theories where the gauge group is broken completely by fundamental scalar fields. The theory has a mass gap and exhibits vortices, labeled by the winding number of the magnetic field,

(1.1) |

Here we summarise our main results. We start in Section 2 with a study of the moduli space of charge vortices which we shall denote as . Our first result concerns the real dimension of the moduli space which, using index theory techniques, we show to be

(1.2) |

In Section 3 we present a brane construction of the vortices, from which we extract a description of as a symplectic quotient of . This quotient construction is most easily described as the Higgs branch of a gauge theory with four supercharges, coupled to a single adjoint chiral multiplet and fundamental chiral multiplets.

####
The moduli space naturally inherits a metric from the Kähler
quotient construction. 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 moduli space naturally inherits a metric from the Kähler quotient construction. 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 instantons in a gauge theory, which we shall
denote as . Recall that the real dimension of the instanton moduli
space is

Moreover, those familiar with instanton moduli spaces will have recognised
the quotient construction of as a truncated version of the
ADHM quotient [3]. In Section 4, we make this
relationship more explicit and show that the moduli space of vortices
is a complex middle-dimensional submanifold (or, since is hyperKähler,
equivalently a special Lagrangian submanifold) of the resolved instanton moduli space
. We further show that may be realised as the fixed point set of a
holomorphic action on , descending from the rotations of
instantons in a plane.

The parametric scaling of the dimension (1.2) is reminiscent of the moduli space of instantons in a gauge theory, which we shall denote as . Recall that the real dimension of the instanton moduli space is

Moreover, those familiar with instanton moduli spaces will have recognised the quotient construction of as a truncated version of the ADHM quotient [3]. In Section 4, we make this relationship more explicit and show that the moduli space of vortices is a complex middle-dimensional submanifold (or, since is hyperKähler, equivalently a special Lagrangian submanifold) of the resolved instanton moduli space . We further show that may be realised as the fixed point set of a holomorphic action on , descending from the rotations of instantons in a plane.

####
In Section 5, we generalise this construction to vortices in gauge
theories with 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 Higgs branch of the theory. For the
non-abelian theory, these vortices are related to lumps in the
Grassmannian sigma-model of planes in .
We denote these moduli spaces of vortices as
(note that ).
The dimension of the moduli space is,

We again give a brane construction as well as a quotient construction of the moduli space and
explain how it can
be described as the fixed point set of a (different) holomorphic action on
the moduli space of instantons .

In Section 5, we generalise this construction to vortices in gauge theories with 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 Higgs branch of the theory. For the non-abelian theory, these vortices are related to lumps in the Grassmannian sigma-model of planes in . We denote these moduli spaces of vortices as (note that ). The dimension of the moduli space is,

We again give a brane construction as well as a quotient construction of the moduli space and explain how it can be described as the fixed point set of a (different) holomorphic action on the moduli space of instantons .

####
In Section 6, we consider the Yang-Mills-Higgs theory defined on the spatial non-commutative
plane with .
We describe how the moduli space of vortices changes as is varied.
We show that the moduli spaces may become singular, cease to exist, or
undergo interesting topology changing transitions for different values of .
We end in Section 7 with conclusions and a discussion.

In Section 6, we consider the Yang-Mills-Higgs theory defined on the spatial non-commutative plane with . We describe how the moduli space of vortices changes as is varied. We show that the moduli spaces may become singular, cease to exist, or undergo interesting topology changing transitions for different values of . We end in Section 7 with conclusions and a discussion.

## 2 Vortices

Our starting point is the maximally supersymmetric theory admitting
vortex solutions which, for concreteness, we choose to live in
dimensions with supersymmetry^{3}^{3}3Supersymmetric
theories which admit vortices exist in any
dimension between 1+1 and 5+1. The discussion of quantum effects,
particular to each case, is very interesting but will be left for future work..
Our theory includes a vector multiplet, consisting of a gauge field ,
a triplet of adjoint scalar fields , and their fermionic partners.
To these we couple fundamental
hypermultiplets, each of which contains two complex scalars
and , and their partner fermions.
As well as the gauge symmetry, the Lagrangian also
enjoys a flavour symmetry. Under these two groups, the field
transforms as , while transforms as
. In the following we take both and
to represent matrices,

where the index furnishes a representation under while the
index refers to . In this notation,
the bosonic part of the Lagrangian reads^{4}^{4}4Our conventions: we choose a
Hermitian connection with and . All gauge and
flavour indices are implicit and assumed summed, with the exception of
which is explicit and summed.,

(2.3) | |||||

The final term in the Lagrangian is a D-term and includes a Fayet-Iliopoulos parameter , which we take to be strictly positive . 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 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 [12] 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 and , and we choose to set the remaining fields to zero at this stage. Restricting to time independent configurations, the Hamiltonian reads,

where is the winding number of the configuration defined in (1.1). Choosing , 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

(2.4) |

where we have introduced the complex coordinate on the spatial plane .

####
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 vortices in the Yang-Mills-Higgs theory as .
We start with a study of the linearised equations to determine the
dimension of . The reader uninterested in the details of the
index theorem may skip to the following subsection where basic properties of
are discussed, taking with them the following punchline:

(2.5)

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 vortices in the Yang-Mills-Higgs theory as . We start with a study of the linearised equations to determine the dimension of . The reader uninterested in the details of the index theorem may skip to the following subsection where basic properties of are discussed, taking with them the following punchline:

(2.5) |

### An Index Theorem

In this section, we prove the result (2.5) by studying the fluctuations around a given solution. Our method follows closely the work of E. Weinberg [13] who analysed the moduli space in the abelian case . The linearised Bogomoln’yi matrix equations are

(2.6) |

and are to be augmented with a gauge fixing condition, for which we choose Gauss’ law,

(2.7) |

which can be combined with the first of the equations in (2.6) to give

(2.8) |

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

Before proceeding, notice that it is possible to rescale the gauge field and coordinate to remove from the equations. The number of zero modes is therefore independent of and we use this freedom to set which simplifies the linearised Bogomoln’yi equations somewhat so they can be written as,

where the superscript in denotes the fact that the matrix acts as right multiplication. We now define the index of as

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 . Thus admits no zero modes and counts the number of zero modes of . We now turn to the task of evaluating . For theories in which the fields have suitable fall-off at spatial infinity (faster than in our case – see the second reference in [13]), the quantity is independent of and the index may be computed more simply in the opposite limit . It is a simple matter to derive an explicit expression for the two composite operators,

where the various operators are defined as,

Expanding in terms of , we have

where the vanish in the limit. Taking the trace over the adjoint action of causes this term to vanish, and we are left only with the left action of on the space of matrices . We thus have,

which counts the complex dimension of to give the promised result.

### The Structure of the Vortex Moduli Space

Let us now discuss a few basic facts about the vortex moduli space. On general grounds, the space decomposes as,

where parameterises the center of mass of the vortex configuration, while information about the relative and internal vortex motion is contained within the -dimensional centered vortex moduli space . 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,

(2.9) |

where are complex coordinates on . This is the Manton metric, descending from the kinetic terms of the Lagrangian (2.3) and is such that geodesics of describe the classical scattering of vortices [2].

####
For the case of the abelian-Higgs model, , many
properties of the vortices and the metric have been studied.
Taubes showed long ago that, as expected, the collective coordinates
of correspond to the positions of unit charge vortices
moving on the plane and may be identified with the zeros of the Higgs
field [14]. The metric on can be
shown to be geodesically complete and although the exact form of the metric
remains unknown for , several interesting properties were uncovered
by Samols [6]. Asymptotically, the metric approaches the
flat metric on where is the permutation group of
elements, reflecting the fact that the vortices are indistinguishable
particles. The interactions of the vortices
resolve the orbifold singularities of 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
[7].

For the case of the abelian-Higgs model, , many properties of the vortices and the metric have been studied. Taubes showed long ago that, as expected, the collective coordinates of correspond to the positions of unit charge vortices moving on the plane and may be identified with the zeros of the Higgs field [14]. The metric on can be shown to be geodesically complete and although the exact form of the metric remains unknown for , several interesting properties were uncovered by Samols [6]. Asymptotically, the metric approaches the flat metric on where is the permutation group of elements, reflecting the fact that the vortices are indistinguishable particles. The interactions of the vortices resolve the orbifold singularities of 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 [7].

####
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 suggests that the
charge vortex again decomposes into unit charge vortices, each of
which is
alloted a position on the plane together with complex
internal degrees of freedom describing the orientation of the vortex in
the group. Indeed the
action on the fields descends to a natural action
on , resulting in a holomorphic isometry of the metric .
For , there is a further isometry of resulting from
spatial rotations of the vortices.

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 suggests that the charge vortex again decomposes into unit charge vortices, each of which is alloted a position on the plane together with complex internal degrees of freedom describing the orientation of the vortex in the group. Indeed the action on the fields descends to a natural action on , resulting in a holomorphic isometry of the metric . For , there is a further isometry of resulting from spatial rotations of the vortices.

####
Let us examine the moduli space of a single vortex in further detail.
Given a specific solution to the abelian vortex equations,
one can always construct a solution to the non-abelian equations (2.4) by
simply embedding in the upper-left corner
of the matrices and . In the case of a single vortex
, acting on this configuration with the symmetry
sweeps out the full moduli space of solutions. Since the vortex embedded
in the upper-left corner breaks ,
the vortex moduli space is

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 is a homogeneous space, we can fix the scale by
calculating the overlap of any two suitable zero modes arising from
the action. For , the zero modes
associated to an rotation are given by,

(2.10)

where as . The
transformation of arises because the left action is by the
gauge symmetry, while the right action is by the
flavour symmetry. The dependence of 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 and
, these equations are solved by the 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

(2.11)

Finally, let us make a brief comment on the spectrum of vortices in the quantum
theory. In the theory with supersymmetry, ground states of
the vortices in a given sector are associated to harmonic forms on . For
the case of a single vortex, there are therefore
such states, implying that the vortex transforms in the fundamental
representation of .

Let us examine the moduli space of a single vortex in further detail. Given a specific solution to the abelian vortex equations, one can always construct a solution to the non-abelian equations (2.4) by simply embedding in the upper-left corner of the matrices and . In the case of a single vortex , acting on this configuration with the symmetry sweeps out the full moduli space of solutions. Since the vortex embedded in the upper-left corner breaks , the vortex moduli space is

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 is a homogeneous space, we can fix the scale by calculating the overlap of any two suitable zero modes arising from the action. For , the zero modes associated to an rotation are given by,

(2.10) |

where as . The transformation of arises because the left action is by the gauge symmetry, while the right action is by the flavour symmetry. The dependence of 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 and , these equations are solved by the 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

(2.11) |

Finally, let us make a brief comment on the spectrum of vortices in the quantum theory. In the theory with supersymmetry, ground states of the vortices in a given sector are associated to harmonic forms on . For the case of a single vortex, there are therefore such states, implying that the vortex transforms in the fundamental representation of .

## 3 Branes

In this section, we discuss a brane realisation of the vortices in type IIB string theory. We start with the , Yang-Mills-Higgs theory described in the Lagrangian (2.3). The brane realisation of this is well known [17] and consists of D3 branes, suspended between two parallel NS5-branes. A further semi-infinite branes connect to the right-hand NS5-brane to provide the hypermultiplets.

####
In Figure 1 we draw this brane configuration, firstly on the Coulomb branch
with , and secondly on the Higgs branch in which one NS5-brane is
separated from the other branes, inducing a non-zero FI parameter .
In the second picture, we also include the BPS vortices which appear as
D1-branes stretched between the D3-branes and the isolated NS5-brane. To
see that these D1-branes are indeed identified with vortices, note that they
are the only BPS states of the brane configuration with the correct mass.
The spatial worldvolume directions of the branes follow official convention:

Both the FI parameter , and the gauge coupling, are encoded in the
separation between the two NS5-branes. We have

(3.12)

where
and are the string length and coupling respectively.
To take the gauge theory decoupling limit, we want to
send , while insisting that the field
theory excitations are much smaller than other stringy and
Kaluza-Klein modes. The two mass scales of the field theory are
the mass of the photon and the
mass of the vortex . An interesting curiosity about
vortices is that while their mass is , their size is .
In order to decouple the gauge theory from the string dynamics, we
require

while the ratio
remains fixed. The
decoupling limit can therefore
be achieved by setting and
and , taking .

In Figure 1 we draw this brane configuration, firstly on the Coulomb branch with , and secondly on the Higgs branch in which one NS5-brane is separated from the other branes, inducing a non-zero FI parameter . In the second picture, we also include the BPS vortices which appear as D1-branes stretched between the D3-branes and the isolated NS5-brane. To see that these D1-branes are indeed identified with vortices, note that they are the only BPS states of the brane configuration with the correct mass. The spatial worldvolume directions of the branes follow official convention:

Both the FI parameter , and the gauge coupling, are encoded in the separation between the two NS5-branes. We have

(3.12) |

where and are the string length and coupling respectively. To take the gauge theory decoupling limit, we want to send , while insisting that the field theory excitations are much smaller than other stringy and Kaluza-Klein modes. The two mass scales of the field theory are the mass of the photon and the mass of the vortex . An interesting curiosity about vortices is that while their mass is , their size is . In order to decouple the gauge theory from the string dynamics, we require

while the ratio remains fixed. The decoupling limit can therefore be achieved by setting and and , taking .

####
Let us now turn to the vortices. It is a simple matter
to read off the theory living on the worldvolume of the D1-branes
(similar configurations were
considered previously in the T-dual picture [18, 19]). The dynamics of the
D1-branes is controlled by an supersymmetric, gauged quantum
mechanics. The relevant representations of the supersymmetry algebra are
simply the dimensional reduction of the familiar vector and chiral multiplets
in dimensions. The vortex theory involves
a vector multiplet, consisting of a gauge field together with three
adjoint scalar fields , parameterising the motion of
the D1-branes in the directions.
These are coupled to an adjoint chiral multiplet whose complex scalar we
denote . The eigenvalues of parameterise the position of the
D1-branes in the plane. A further fundamental
chiral multiplets, with complex scalars , arise from the D1-D3
strings. The global symmetry group of the theory is

(3.13)

where is an R-symmetry rotating the scalars in the vector
multiplet^{5}^{5}5For vortex solutions whose worldvolume is
-dimensional, this R-symmetry group is .,
is a flavour symmetry rotating the phase of and is a
flavour symmetry acting on in the anti-fundamental
representation. The fields may be represented as
matrices, with
the gauge group acting by left multiplication, and the
flavour symmetry acting by right multiplication. We use the notation

All of these fields come with fermionic superpartners which we suppress.
The bosonic Lagrangian is given by

(3.14)

Once again, the gauge coupling and FI parameter of this theory
are determined by the separation of the NS5-branes, although with
reciprocal relations to the D3-brane theory (3.12)

(3.15)

We see that taking the decoupling limit of the D3-brane theory implies
the strong coupling limit of the vortex theory .
However, the FI parameter remains finite and in fact is identified
with the gauge coupling

(3.16)

For , there is no Coulomb branch, so that taking
the strong coupling limit decouples the
vector multiplet fields and restricts attention to the
Higgs branch of the theory. We shall denote this Higgs branch as
. It is given by a Kähler quotient of
, parameterised by and . The
associated moment map is simply the D-term from (3.14)

(3.17)

This imposes real constraints on , while modding
out by the gauge group reduces the dimension of the Higgs branch
by another . Thus the real dimension of the Higgs branch is

which we recognise as the dimension of the vortex moduli space (2.5).
Indeed, the main result of this paper is the brane-predicted isomorphism

(3.18)

Let us now turn to the vortices. It is a simple matter to read off the theory living on the worldvolume of the D1-branes (similar configurations were considered previously in the T-dual picture [18, 19]). The dynamics of the D1-branes is controlled by an supersymmetric, gauged quantum mechanics. The relevant representations of the supersymmetry algebra are simply the dimensional reduction of the familiar vector and chiral multiplets in dimensions. The vortex theory involves a vector multiplet, consisting of a gauge field together with three adjoint scalar fields , parameterising the motion of the D1-branes in the directions. These are coupled to an adjoint chiral multiplet whose complex scalar we denote . The eigenvalues of parameterise the position of the D1-branes in the plane. A further fundamental chiral multiplets, with complex scalars , arise from the D1-D3 strings. The global symmetry group of the theory is

(3.13) |

where is an R-symmetry rotating the scalars in the vector
multiplet^{5}^{5}5For vortex solutions whose worldvolume is
-dimensional, this R-symmetry group is .,
is a flavour symmetry rotating the phase of and is a
flavour symmetry acting on in the anti-fundamental
representation. The fields may be represented as
matrices, with
the gauge group acting by left multiplication, and the
flavour symmetry acting by right multiplication. We use the notation

All of these fields come with fermionic superpartners which we suppress. The bosonic Lagrangian is given by

(3.14) | |||||

Once again, the gauge coupling and FI parameter of this theory are determined by the separation of the NS5-branes, although with reciprocal relations to the D3-brane theory (3.12)

(3.15) |

We see that taking the decoupling limit of the D3-brane theory implies the strong coupling limit of the vortex theory . However, the FI parameter remains finite and in fact is identified with the gauge coupling

(3.16) |

For , there is no Coulomb branch, so that taking the strong coupling limit decouples the vector multiplet fields and restricts attention to the Higgs branch of the theory. We shall denote this Higgs branch as . It is given by a Kähler quotient of , parameterised by and . The associated moment map is simply the D-term from (3.14)

(3.17) |

This imposes real constraints on , while modding out by the gauge group reduces the dimension of the Higgs branch by another . Thus the real dimension of the Higgs branch is

which we recognise as the dimension of the vortex moduli space (2.5). Indeed, the main result of this paper is the brane-predicted isomorphism

(3.18) |

### Some Examples and the Metric

Let us examine the claim (3.18) in more detail. Firstly, note that the center of mass position of the D1-branes, given by , decouples from the other fields, guaranteeing that the Higgs branch decomposes as

in agreement with the vortex moduli space. To make further comparisons, let us consider specific examples, starting with the description of a single vortex in the theory. In this case the vortex dynamics is abelian so decouples and the D-term constraint reduces to where is an -vector. We are left with the well known gauged linear sigma-model construction of , and we have,

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

####
The second example we consider is that of vortices in the abelian-Higgs
model with . This vortex quantum mechanics was previously studied in
[20] 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 [21], who
showed that a given solution is uniquely determined by a set of
eigenvalues for , 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
inherits a natural Kähler metric from the Kähler
quotient construction described above. The presence of the flavour symmetry
guarantees that this metric exhibits an holomorphic isometry.
For , also enjoys a holomorphic
isometry, arising from , corresponding
to rotating the branes in the plane. Thus the quotient metric
on and the Manton metric on share the same isometries.
Indeed, from the brane picture it is clear that the
and symmetry groups of the D3-brane and D1-brane theories,
share the same origin.

The second example we consider is that of vortices in the abelian-Higgs model with . This vortex quantum mechanics was previously studied in [20] 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 [21], who showed that a given solution is uniquely determined by a set of eigenvalues for , 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 inherits a natural Kähler metric from the Kähler quotient construction described above. The presence of the flavour symmetry guarantees that this metric exhibits an holomorphic isometry. For , also enjoys a holomorphic isometry, arising from , corresponding to rotating the branes in the plane. Thus the quotient metric on and the Manton metric on share the same isometries. Indeed, from the brane picture it is clear that the and 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 , the
metric on is the round Fubini-Study metric on
, in agreement with the Manton
metric on . 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 . Importantly, the asymptotic metric on
is the flat metric on , 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
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
, . The metrics on both and
take the form,

(3.19)

where is the separation between vortices, or D1-branes, and
so that the moduli space looks like a cone. For the Higgs
branch, the explicit Kähler quotient construction was performed in
[20] and the conformal factor is given by,

(3.20)

The calculation of the leading order scattering of vortices was performed
in [7], and the equivalent metric on
was computed to be,

(3.21)

where 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
[15] that T-duality between the singularity and
fully localised NS5-branes requires a worldsheet instanton effect
and holds only if .
This is in agreement with the numerical result
of [16]. To summarise, we see that, while the metrics on and
are asymptotically, and
qualitatively, similar they differ in the details.

Do further properties of the metrics coincide? In the case of , the metric on is the round Fubini-Study metric on , in agreement with the Manton metric on . 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 . Importantly, the asymptotic metric on is the flat metric on , 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 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 , . The metrics on both and take the form,

(3.19) |

where is the separation between vortices, or D1-branes, and so that the moduli space looks like a cone. For the Higgs branch, the explicit Kähler quotient construction was performed in [20] and the conformal factor is given by,

(3.20) |

The calculation of the leading order scattering of vortices was performed in [7], and the equivalent metric on was computed to be,

(3.21) |

where 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 [15] that T-duality between the singularity and fully localised NS5-branes requires a worldsheet instanton effect and holds only if . This is in agreement with the numerical result of [16]. To summarise, we see that, while the metrics on and are asymptotically, and qualitatively, similar they differ in the details.

## 4 Instantons

The Kähler quotient construction of the vortex moduli space is reminiscent of the hyperKähler quotient of the moduli space of instantons in Yang-Mills theory. We denote this latter space as . In this Section, we make the connection between and more explicit. We begin with a review of the ADHM gauge theory describing instantons on non-commutative , with the specific anti-self-dual, commutation relations

(4.22) |

with all other commutators vanishing. Recall that the ADHM construction of , as proposed in [3], can be elegantly described in terms of an auxiliary gauge theory with 8 supercharges [22]. The matter content of this theory includes an adjoint valued hypermultiplet and fundamental hypermultiplets. The instanton moduli space is described as a hyperKähler quotient as the Higgs branch of this gauge theory, parameterised by the hypermultiplet scalar fields. Denote the two complex scalars in the adjoint multiplet as and , and the complex scalars in the remaining hypermultiplets as and . While transforms in the representation of the gauge group, transforms as , and we represent both of these fields as a (respectively ) matrix,

Theories with 8 supercharges have a triplet of D-terms which, in 4 supercharge language, can be decomposed into a D-term and F-term. These constraints, which provide the triplet of moment maps in the hyperKähler quotient construction, read

The FI parameter appears only in the D-term, a fact related to the specific choice of non-commutative background (4.22) as shown by Nekrasov and Schwarz [23]. The relationship is simply

The role of is to resolve the singularities of in the manner proscribed by Nakajima [24]. In doing so, it picks out a preferred complex structure on .

####
Note that we have used the same
notation in the ADHM gauge theory as we did in the
the previous section, and we will shortly explain the deformation which
takes us from ADHM to the vortex theory. Before doing so, it will do us well
to dwell a little on the symmetries of the ADHM theory. To compare with
the previous section we choose to define the ADHM theory in
dimensions, describing particles in dimensional Yang-Mills or,
alternatively, D0-branes moving in the background of D4-branes.
The global symmetry group of the ADHM theory is

(4.23)

The symmetry rotates the scalars in the vector multiplet^{6}^{6}6For
instantons with a -dimensional worldvolume, this R-symmetry group is
.. The is what remains of the
spatial rotation group of with the anti-self-dual
non-commutative deformation (4.22), and the descends from
the gauge symmetry on the D4-branes.
The adjoint doublet
transforms as under ,
while transforms
as and transforms
as , where the subscripts denote the
charge under the R-symmetry.

Note that we have used the same notation in the ADHM gauge theory as we did in the the previous section, and we will shortly explain the deformation which takes us from ADHM to the vortex theory. Before doing so, it will do us well to dwell a little on the symmetries of the ADHM theory. To compare with the previous section we choose to define the ADHM theory in dimensions, describing particles in dimensional Yang-Mills or, alternatively, D0-branes moving in the background of D4-branes. The global symmetry group of the ADHM theory is

(4.23) |

The symmetry rotates the scalars in the vector multiplet^{6}^{6}6For
instantons with a -dimensional worldvolume, this R-symmetry group is
.. The is what remains of the
spatial rotation group of with the anti-self-dual
non-commutative deformation (4.22), and the descends from
the gauge symmetry on the D4-branes.
The adjoint doublet
transforms as under ,
while transforms
as and transforms
as , where the subscripts denote the
charge under the R-symmetry.

####
We are now in a position to describe the deformation which takes us
to the vortex theory by adding masses to all the unnecessary fields.
We accomplish this
by weakly gauging a particular symmetry. This involves
gauging a symmetry in a manner consistent with supersymmetry. The scalars
in this new vector multiplet are then endowed with vacuum expectations
values (vevs) and the new vector multiplet is subsequently decoupled. The
only remnant of the whole process is the vevs, which give masses to any
field charged under the symmetry. If is
taken to be a flavour symmetry, then this process preserves the full 8 supercharges
of the ADHM theory. In contrast, if is a generic
R-symmetry, this process breaks all supersymmetry. However, there are
specific combinations of R-symmetries which one may gauge which preserve
a fraction of the supersymmetry and it is this combination that we shall
employ. Let be such that it rotates two of the
vector multiplet scalars, leaving the remaining three untouched;
let have the Pauli matrix generator ; and
let be the overall gauge rotation.
Then we choose the combination of symmetries that act on fields with
charge , such that

(4.24)

The fields and , together with two of the five vector
multiplet scalars, have . These all receive
masses. The fields and , and the three remaining scalars of the
vector multiplet all have and survive unscathed.
We are left with the vortex theory of Section 2, with the relationship
between the FI parameter and parameters giving,

We are now in a position to describe the deformation which takes us to the vortex theory by adding masses to all the unnecessary fields. We accomplish this by weakly gauging a particular symmetry. This involves gauging a symmetry in a manner consistent with supersymmetry. The scalars in this new vector multiplet are then endowed with vacuum expectations values (vevs) and the new vector multiplet is subsequently decoupled. The only remnant of the whole process is the vevs, which give masses to any field charged under the symmetry. If is taken to be a flavour symmetry, then this process preserves the full 8 supercharges of the ADHM theory. In contrast, if is a generic R-symmetry, this process breaks all supersymmetry. However, there are specific combinations of R-symmetries which one may gauge which preserve a fraction of the supersymmetry and it is this combination that we shall employ. Let be such that it rotates two of the vector multiplet scalars, leaving the remaining three untouched; let have the Pauli matrix generator ; and let be the overall gauge rotation. Then we choose the combination of symmetries that act on fields with charge , such that

(4.24) |

The fields and , together with two of the five vector multiplet scalars, have . These all receive masses. The fields and , and the three remaining scalars of the vector multiplet all have and survive unscathed. We are left with the vortex theory of Section 2, with the relationship between the FI parameter and parameters giving,

### The Moduli Spaces

While the above discussion has been in terms of the ADHM gauge theory, the deformation also has a simple description directly in terms of the instanton moduli space . The symmetry of the gauge theory descends to an action on , endowing the metric on with a Killing vector . This Killing vector is holomorphic, preserving the preferred complex structure while rotating the remaining two. The mass terms introduced above by weakly gauging induce to a potential on proportional to the length of the Killing vector,

Such potentials have been widely used in soliton physics recently (see for example [25],[26]), although usually in the context of supersymmetry-preserving tri-holomorphic Killing vectors. We therefore have a description of the vortex moduli space directly in terms of the instanton moduli space

The zeroes of the Killing vector are precisely the fixed points of the action. The meaning of this action can be determined from the assignment of charges in (4.24). Recall that is the subgroup of the rotations of that are left unbroken by the non-commutative deformation (4.22). We find therefore that the action corresponds to rotating the instantons in the plane, and the vortices are related to instantons which are invariant under this action.

####
Let us now turn to some examples: the moduli space
of a single instanton in non-commutative Yang-Mills is given
by the cotangent bundle
endowed with the Calabi metric [27]. The potential
vanishes on the zero section of the bundle , reducing
to the moduli space of a single vortex .
Another example: the moduli space of
two instantons in gauge theory is the Eguchi-Hanson metric on
. The explicit hyperKähler quotient construction
was performed in [28]. Note that this case is special
since , which is
not true for . However, the tri-holomorphic isometry
of has a different origin in these two cases. In the
notation of (4.23), the isometry is for ,
while it is for . Since, from (4.24),
the potential on the instanton moduli space involves , but not ,
the vortex moduli spaces and
are given by different holomorphic submanifolds of .
It is a simple exercise to show that the vacua of the potential on
is the two dimensional cone endowed with the
metric (3.20).

Let us now turn to some examples: the moduli space of a single instanton in non-commutative Yang-Mills is given by the cotangent bundle endowed with the Calabi metric [27]. The potential vanishes on the zero section of the bundle , reducing to the moduli space of a single vortex . Another example: the moduli space of two instantons in gauge theory is the Eguchi-Hanson metric on . The explicit hyperKähler quotient construction was performed in [28]. Note that this case is special since , which is not true for . However, the tri-holomorphic isometry of has a different origin in these two cases. In the notation of (4.23), the isometry is for , while it is for . Since, from (4.24), the potential on the instanton moduli space involves , but not , the vortex moduli spaces and are given by different holomorphic submanifolds of . It is a simple exercise to show that the vacua of the potential on is the two dimensional cone endowed with the metric (3.20).

### A Wrapped Brane Realisation

From the perspective of the D4-brane, the above deformation of the instanton theory involves locking the symmetries tangent to the D4-brane, with the symmetry normal to the D4-branes. This is reminiscent of the twisting of the tangent and normal bundles of branes when wrapped on cycles [29]. 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 direction, and describe the resulting IIA string theory
set-up. Under T-duality, the two NS5-branes are replaced
by the background geometry . (The duality between NS5-branes
and spaces was
first conjectured by Hull and Townsend [30]. A proof from the
worldsheet sigma-model, including the breaking of translation symmetry
associated to the localization of the NS5-brane, was given in [15]).
The separation of the NS5-branes in the direction resolves
the orbifold singularity, resulting in the background spacetime
^{7}^{7}7Note that this ubiquitous space has already
appeared twice as the instanton moduli spaces
and . Here it appears in an
unrelated context as the background spacetime in string theory..
Topologically, this space can be thought of as an
fibration, parameterised by , over , parameterised by
. In Gibbons-Hawking coordinates, the metric
takes the form,

where and

The 3-vector resolves the orbifold singularity and, for
the T-dual of Figure 1, is given by .
The fiber degenerates at the two points
, resulting in the Christmas cracker topology
shown in Figure 2. The zero section , 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 and .
They wrap the
compact , 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 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- pair wrapping a vanishing ,
through which an NS-NS B-field threads in order to provide the
D0-brane charge. Yet in our case the has finite size,
and such an interpretation breaks down, as can easily be seen by
computing the mass of such a 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.

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 direction, and describe the resulting IIA string theory
set-up. Under T-duality, the two NS5-branes are replaced
by the background geometry . (The duality between NS5-branes
and spaces was
first conjectured by Hull and Townsend [30]. A proof from the
worldsheet sigma-model, including the breaking of translation symmetry
associated to the localization of the NS5-brane, was given in [15]).
The separation of the NS5-branes in the direction resolves
the orbifold singularity, resulting in the background spacetime
^{7}^{7}7Note that this ubiquitous space has already
appeared twice as the instanton moduli spaces
and . Here it appears in an
unrelated context as the background spacetime in string theory..
Topologically, this space can be thought of as an
fibration, parameterised by , over , parameterised by
. In Gibbons-Hawking coordinates, the metric
takes the form,

where and

The 3-vector resolves the orbifold singularity and, for the T-dual of Figure 1, is given by . The fiber degenerates at the two points , resulting in the Christmas cracker topology shown in Figure 2. The zero section , 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 and . They wrap the compact , 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 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- pair wrapping a vanishing , through which an NS-NS B-field threads in order to provide the D0-brane charge. Yet in our case the has finite size, and such an interpretation breaks down, as can easily be seen by computing the mass of such a 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.

## 5 Semi-Local Vortices and Sigma-Model Lumps

In this Section, we discuss a generalisation of the vortices to Yang-Mills with flavours. The Lagrangian takes the same form as previously (2.3) except the fundamental scalars are now dimensional matrices

Rather than the unique, isolated vacuum of Section 2, the theory now has a Higgs branch of vacua. However, if develops an expectation value, then there are no BPS vortex solutions. This may be easily seen from an analysis of the Bogomoln’yi equations, and follows from the mathematical fact truth that a line bundle of negative degree admits no holomorphic sections (see, for example, [31] for the translation). We therefore restrict attention to the reduced Higgs branch of vacua, denoted obtained by insisting . For example, for abelian theories with , the Higgs branch of vacua is the cotangent bundle , while the reduced Higgs branch describing the vacua which admit BPS vortex solutions is simply . In general, the reduced Higgs branch is the Grassmannian of planes in ,

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 flavour symmetry of the theory is broken in the pattern,

(5.25) |

The theory admits BPS vortices with the Bogomoln’yi equations taking the same form as previously (2.4) with now interpreted as a matrix of the appropriate size. We denote the moduli space of vortices in this model as . Note that, in the notation of the previous sections, we have . It is a simple matter to generalise the index theorem of Sect