# Self-Dual Chern-Simons Theories^{1}^{1}1Lectures presented at the International Symposium “Field Theory and Mathematical Physics”,
Mt. Sorak, Korea (June-July 1994); hep-th/9410065

###### Abstract

In these lectures I review classical aspects of the self-dual Chern-Simons systems which describe charged scalar fields in dimensions coupled to a gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These self-dual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol’nyi bound which is saturated by solutions to a set of first-order self-duality equations. In the nonrelativistic case the self-dual potential is quartic, the system possesses a dynamical conformal symmetry, and the self-dual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic self-duality equations are integrable and all finite charge solutions may be found. In the relativistic case the self-dual potential is sixth order and the self-dual Lagrangian may be embedded in a model with an extended supersymmetry. The self-dual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the self-dual nature of the potential.

## 1 Introduction : Self-Dual Theories

“Self-duality” is a powerful notion in classical mechanics and classical field theory, in quantum mechanics and quantum field theory. It refers to theories in which the interactions have particular forms and special strengths such that the second order equations of motion (in general, a set of coupled nonlinear partial differential equations) reduce to first order equations which are simpler to analyze. The “self-dual point”, at which the interactions and coupling strengths take their special self-dual values, corresponds to the minimization of some functional, often the energy or the action. This gives self-dual theories crucial physical significance. For example, the self-dual Yang-Mills equations have minimum action solutions known as instantons, the Bogomol’nyi equations of self-dual Yang-Mills-Higgs theory have minimum energy solutions known as monopoles, and the Abelian Higgs model has minimum energy self-dual solutions known as vortices. In these lectures, I discuss a new class of self-dual theories, self-dual Chern-Simons theories, which involve charged scalar fields minimally coupled to gauge fields whose ‘dynamics’ is provided by a Chern-Simons term in dimensions. The physical context in which such self-dual models arise is that of anyonic quantum field theory. An interesting novel feature of these self-dual Chern-Simons theories is that they permit a realization with either relativistic or nonrelativistic dynamics for the scalar fields. In the nonrelativistic case, the self-dual point corresponds to a quartic scalar potential, with overall strength determined by the Chern-Simons coupling strength. The nonrelativistic self-dual Chern-Simons equations may be solved completely for all finite charge solutions, and the solutions exhibit many interesting relations to two dimensional (Euclidean) integrable models. In the relativistic case, while the general exact solutions are not explicitly known, the solutions correspond to topological and nontopological solitons and vortices, many characteristics of which can be deduced from algebraic and asymptotic data. These self-dual Chern-Simons theories also have the property that, at the self-dual point, they may be embedded into a model with an extended supersymmetry, a general feature of self-dual theories.

Before introducing the self-dual Chern-Simons theories, I briefly review some other important self-dual theories, in part as a means of illustrating the general idea of self-duality, but also because various specific properties of these theories appear in our analysis of the self-dual Chern-Simons systems. More details concerning some of these models can be found in the lectures of Professor C. Lee on “Instantons, Monopoles and Vortices” from this symposium.

Perhaps the most familiar, and in a certain sense the most fundamental, self-dual theory is that of four dimensional self-dual Yang-Mills theory. The Yang-Mills action is

(1) |

where is the gauge field curvature. The Euler-Lagrange equations form a complicated set of coupled nonlinear partial differential equations:

(2) |

where is the covariant derivative. However, in four dimensional Euclidean space the Yang-Mills action (1) is minimized by solutions of the self-dual (or anti-self-dual) Yang-Mills equations:

(3) |

where is the dual field strength. Note that the self-dual equations (3) are first order equations (in contrast to the second order equations of motion (2)), and their “instanton” solutions are known in detail [1]. We shall see that the nonrelativistic self-dual Chern-Simons equations have an interesting connection with these self-dual Yang-Mills equations.

Another important class of self-dual equations are the “Bogomol’nyi equations”

(4) |

which arise in the theory of magnetic monopoles in dimensional space-time. These equations arise from a minimization of the static energy functional of a Yang-Mills-Higgs system in a special parametric limit known as the BPS limit [2, 3]. It is interesting to note that these Bogomol’nyi equations can be obtained from the (anti-) self-dual Yang-Mills equations (3) by a ‘dimensional reduction’ in which all fields are taken to be independent of , and is identified with :

(5) | |||||

(6) | |||||

(7) |

We shall see that the nonrelativistic self-dual Chern-Simons equations may also be obtained from the self-dual Yang-Mills equations by a dimensional reduction. Furthermore, the relativistic self-dual Chern-Simons equations involve a special algebraic embedding problem (that of embedding into the gauge algebra) which also plays a crucial role in the analysis of the Bogomol’nyi equations (4).

The abelian Higgs model in dimensions is a model of a complex scalar field interacting with a gauge field with conventional Maxwell dynamics. For a special quartic potential, with a particular overall strength, the static energy functional is minimized by solutions to the following set of self-duality equations:

(8) | |||||

(9) |

These self-duality equations have vortex solutions [4, 3, 5] which are important in the phenomenological Landau-Ginzburg theory of superconductors. The self-duality equations we find in the self-dual Chern-Simons systems also arise from minimizing the energy functional in a dimensional theory, and the resulting Chern-Simons self-duality equations have a similar form to the abelian Higgs model self-duality equations (9).

Yang [6] proposed an approach to the four dimensional self-dual Yang-Mills equations (3) in which they can be viewed as the consistency conditions for a set of first order differential operators. This idea is fundamental to the notion of “integrability” of systems of differential equations, a subject with many connections to self-dual theories [7, 8]. If the self-dual Yang-Mills equations (3) are rewritten in terms of the null coordinates and , they become

(10) | |||||

(11) | |||||

(12) |

These express the consistency conditions for the first order equations

(13) | |||||

(14) |

where is known as a “spectral parameter”. The first two equations in (12) can be solved locally to give

(15) | |||

(16) |

where and are gauge group elements. Then, defining , the third of the self-duality equations in (12) becomes

(17) |

If we now make a dimensional reduction in which the fields are chosen to be independent of and , this equation becomes the two dimensional equation

(18) |

which is known as the chiral model equation. The chiral model equation will play a very important role in our analysis of the nonrelativistic self-dual Chern-Simons equations. Also note that if and is further restricted to satisfy the condition , then (18) is the equation of motion for the model [1, 9].

The final class of models which we shall recall in this introduction are known as Toda theories. The original Toda system [10] described the displacements of a line of masses joined by springs with an exponential spring tension. The equations of motion for the Toda lattice are

(19) |

where the matrix is the tridiagonal discrete approximation to the second derivative, and can be chosen for periodic or open boundary conditions. This system is classically integrable in the limit of an infinite number of masses, in the sense that it possesses an infinite number of conserved quantities in involution. The Toda lattice system also has a deep algebraic structure due to the fact that the matrix in (19) is the Cartan matrix of the Lie algebra (or its affine extension). Indeed, this relationship allows one to extend the original Toda system to a Toda lattice based on other Lie algebras [11, 12, 13].

The Toda system generalizes still further, to an integrable set of partial differential equations

(20) |

which is not only integrable, but also solvable, in the sense that the solution may be written in terms of arbitrary functions, where is the rank of the classical Lie algebra whose Cartan matrix appears in (20) [11, 12]. For the classical Toda system reduces to the nonlinear Liouville equation

(21) |

which was solved by Liouville [14]. Both the Liouville and Toda equations, together with their solutions, appear prominently in the analysis of the nonrelativistic self-dual Chern-Simons models. Moreover, the Toda equations also arise from the Bogomol’nyi equations (4) when one looks for spherically symmetric monopole solutions [15]. This reduction involves an algebraic embedding problem very similar to one that appears in the treatment of the relativistic self-dual Chern-Simons models.

The self-dual Chern-Simons theories discussed in these lectures describe charged scalar fields in dimensional space-time, minimally coupled to a gauge field whose dynamics is given by a Chern-Simons Lagrangian rather than the conventional Maxwell (or Yang-Mills) Lagrangian. The possibility of describing gauge theories with a Chern-Simons term rather than with a Yang-Mills term is particular to odd-dimensional space-time, and dimensions is special in the sense that the derivative part of the Chern-Simons Lagrangian is quadratic in the gauge fields. To conclude this introduction, I briefly review some of the important properties [16, 17, 18] of the Chern-Simons Lagrange density:

(22) |

The gauge field takes values in a finite dimensional representation of the gauge Lie algebra . The totally antisymmetric -symbol is normalized with . The Euler-Lagrange equations of motion derived from this Lagrange density are simply

(23) |

which follows directly from the fact that

(24) |

The equations of motion (23) are gauge covariant under the gauge transformation

(25) |

and so the Lagrange density (22) defines a sensible gauge theory even though (22) is not invariant under the gauge transformation (25). Indeed, under a gauge transformation transforms as

(26) |

For an abelian Chern-Simons theory, the final term in (26) vanishes and the change in is a total space-time derivative. Hence the action is gauge invariant. However, for a nonabelian Chern-Simons theory the final term in (26) is proportional to the winding number of the group element , and the action changes by a constant. To ensure that remains invariant, the Chern-Simons Lagrange density (22) must be multiplied by a dimensionless coupling parameter which assumes quantized values [16, 17]

(27) |

The Chern-Simons term describes a topological gauge field theory [17] in the sense that there is no explicit dependence on the space-time metric. This follows because the Lagrange density (22) can be written directly as a 3-form . This fact implies that if the Chern-Simons Lagrange density is coupled to other fields, then it will not contribute to the energy momentum tensor. This may also be understood by noting that is first order in space-time derivatives

(28) |

The time derivative part of contributes to the canonical structure of the theory, the part contributes to the Gauss law constraint, and there is no contribution to the Hamiltonian. It is very important that is first order in space-time derivatives, because in the self-dual Chern-Simons theories discussed in these lectures the self-duality equations (which should be first order) involve the Chern-Simons equations of motion directly.

## 2 Nonrelativistic SDCS Theories

2.1 Nonrelativistic Self-Dual Chern-Simons Equations

The nonrelativistic self-dual Chern-Simons system is a model in dimensional space-time describing charged scalar fields with nonrelativistic dynamics, minimally coupled to gauge fields with Chern-Simons dynamics [19, 20, 21]. The Lagrange density for such a system is:

(29) |

where is the Chern-Simons Lagrange density (22). I have chosen to work with adjoint coupling of the scalar and gauge fields (for other couplings of matter and gauge fields see [18]), with the covariant derivative in (LABEL:nrlag) being . The scalar fields and the gauge fields take values in the same representation of the gauge Lie algebra . In these lectures, will usually be taken to be , but much of the formal structure generalizes straightforwardly to other gauge algebras. The parameter appearing in (LABEL:nrlag) is the dimensionless Chern-Simons coupling constant, while denotes the scalar field mass. Notice that the scalar field potential appearing in (LABEL:nrlag) has a particular quartic form, with an overall scale depending on both and . This form of the potential is fixed by the condition of self-duality, as shown below.

The Euler-Lagrange equations of motion that follow from the nonrelativistic self-dual Chern-Simons Lagrange density (LABEL:nrlag) are:

(31) |

(32) |

where is the gauge curvature, and is the covariantly conserved () nonrelativistic matter current

(33) |

In addition there is an abelian current

(34) | |||||

(35) |

which is ordinarily conserved (). The matter field equation of motion (31) is referred to as the gauged planar nonlinear Schrödinger equation [19]. The study of the nonlinear Schrödinger equation in -dimensional space-time is partly motivated by the significance of the -dimensional nonlinear Schrödinger equation. Here we consider a gauged nonlinear Schrödinger equation in which we have not only the nonlinear potential term for the matter fields, but also we have a coupling of the matter fields to the gauge fields. The gauge equation of motion (32) relates the matter and gauge fields via a Chern-Simons coupling. Notice that even though the Chern-Simons Lagrange density is not strictly invariant under a gauge transformation, the equations of motion (31,32) are gauge covariant.

The Hamiltonian density corresponding to the Lagrange density (LABEL:nrlag) is

(36) |

where we recall that the Chern-Simons term does not contribute to the energy density since it is first order in space-time derivatives. The energy density (36) is supplemented by the Gauss law constraint

(37) |

which is the component of the gauge equations of motion (32). To obtain a Bogomol’nyi - style lower bound for the energy density we employ the following useful identity:

(38) |

where .

Using this identity in (36), together with the Gauss law constraint (37) which relates the “magnetic field” to the nonrelativistic matter charge density , we see that the energy density can be written as

(39) |

This energy density is therefore minimized by solutions of the nonrelativistic self-dual Chern-Simons equations :

(40) |

(41) |

Notice that these self-duality equations are indeed first-order in derivatives of the fields, in contrast to the gauged nonlinear Schrödinger equation (31) which is second order.

Since the self-dual solutions minimize the Hamiltonian density, they provide static solutions to the Euler-Lagrange equations of motion (31,32). Alternatively, one can see this directly from inspection of the static equations of motion. Note that if , then the currents take the simple form

(42) | |||||

The gauge equation of motion (32) then implies that . Together with the identity

(43) | |||||

this reduces the matter equation of motion (31) to

(44) |

the RHS of which which vanishes for self-dual solutions.

In fact, owing to a remarkable dynamical symmetry of the nonrelativistic self-dual Chern-Simons model (LABEL:nrlag), it is possible to show that the self-dual solutions (40,41) saturate all static solutions of the equations of motion [21, 22]. For the Abelian models, this fact has recently been formulated in terms of a Kaluza-Klein reduction of a relativistic symmetry [23].

An important property of the nonrelativistic self-dual Chern-Simons equations (40,41) is that they can be obtained by dimensional reduction from the four dimensional self-dual Yang-Mills equations for a nonAbelian gauge theory. The signature SDYM equations are

(45) |

Taking all fields to be independent of and , these reduce to

(46) |

which are just the nonrelativistic self-dual Chern-Simons equations (40,41) with the identification . These dimensionally reduced self-dual Yang-Mills equations have been studied in the mathematical literature [24, 25].

2.2 Algebraic Ansatze and Toda Theories

Before classifying the general solutions to the nonrelativistic self-dual Chern-Simons equations, it is instructive to consider certain special cases in which simplifying algebraic Ansätze for the fields reduce (40,41) to familiar integrable nonlinear equations. Note that since we are considering static fields, the self-duality equations have the appearance of equations of motion in two dimensional Euclidean space.

First, choose the fields to have the following Lie algebra decomposition

(47) |

Here, refers to the Cartan subalgebra generators and to the simple root step operator generators of the gauge Lie algebra, normalized according to a Chevalley basis (for ease of presentation we consider only simply-laced algebras here) [26] :

(48) | |||||

(49) | |||||

(51) | |||||

(52) | |||||

(53) | |||||

(54) |

The indices and run over , where is the rank of the gauge algebra . The matrix is the Cartan matrix of , which expresses the inner products of the simple roots :

(55) |

For , the classical Cartan matrix is the symmetric tridiagonal matrix (familiar from the theory of numerical analysis):

(56) |

With the ansatze (47) for the fields, the first of the nonrelativistic self-dual Chern-Simons equations, , reduces to the set of equations

(57) |

When combined with its adjoint, and with the other nonrelativistic self-dual Chern-Simons equation, we find the classical Toda equations

(58) |

where . For , and the Cartan matrix is just the single number 2, so the Toda equations (58) reduce to the Liouville equation

(59) |

which Liouville showed to be integrable and indeed ”solvable” [14] - in the sense that the general real solution can be expressed in terms of a single holomorphic function :

(60) |

Kostant [11], and Leznov and Saveliev [12] have shown that the classical Toda equations (58) are similarly integrable (and indeed solvable), with the general real solutions for being expressible in terms of arbitrary holomorphic functions, where is the rank of the algebra. For it is possible to adapt the Kostant-Leznov-Saveliev solutions to a simple form reminiscent of the Liouville solution (60):

(61) |

where is the rectangular matrix , with being an -component column vector containing arbitrary holomorphic functions , , , :

(62) |

An alternative, extended, ansatz for the fields involves the matter field choice

(63) |

where is the step operator corresponding to minus the maximal root. With the gauge field still as in (47), the nonrelativistic self-dual Chern-Simons equations then combine to give the affine Toda equations

(64) |

where is the affine Cartan matrix. These affine Toda equations are also known to be integrable [11, 12, 13], although it is not possible to write simple convergent expressions such as (61) for the solutions.

There is another useful way to understand these various algebraic reductions of the nonrelativistic self-dual Chern-Simons equations. In two dimensional space we can express the gauge field as

(65) | |||||

(66) |

where is an element of the complexification of the gauge group [6, 27]. can be decomposed as

(67) |

where is hermitean and is unitary. Note that only with does (66) correspond to a pure gauge. Gauge transformations on correspond to different choices of the unitary factor . In general, the field strength corresponding to (66) is

(68) |

With the gauge field represented as in (66), the solution to the self-duality equation is trivially:

(69) |

for any . Inserting this solution in the other self-duality equation yields the gauge invariant equation for :

(70) |

Thus far, no special choices have been made and equation (70) is still completely general. Now, if we choose to write as

(71) |

where is restricted to the Cartan subalgebra, then (70) simplifies to

(72) |

These equations follow as equations of motion from the two-dimensional Euclidean Lagrange density

(73) |

If is now chosen to be the constant field then this Lagrangian (73) becomes that of the classical Toda theory, while if is chosen to be the constant field then it becomes that of the affine Toda theory. With these choices for the self-duality equation (72) reduces to the classical or affine Toda system, respectively.

2.3 Chiral Model, Unitons and General Solutions

Having considered some special cases in which the nonrelativistic self-dual Chern-Simons equations reduce to well-known integrable equations in two-dimensional Euclidean space, we now consider the question of finding the most general solutions. The key to the possibility of finding all solutions lies in the fact that there exists a special gauge transformation which converts the two equations (40,41) into a single equation

(74) |

where is the gauge transformed matter field . The existence of such a gauge transformation follows from the following zero-curvature formulation of the self-dual Chern-Simons equations [21, 28]. Define

(75) |

Then the nonrelativistic self-dual Chern-Simons equations (40,41) together imply that the gauge curvature associated with vanishes:

(76) | |||||

(77) |

Therefore, at least locally, one can write as a pure gauge

(78) |

for some in the gauge group. Gauge transforming the nonrelativistic self-dual Chern-Simons equations (40,41) with this group element leads to the single equation (74).

Equation (74) can be converted into the chiral model equation

(79) |

upon defining

(80) |

for some in the gauge group (the fact that it is possible to write in this manner is a consequence of (74)). Given any solution of the chiral model equation (79), or alternatively any solution of (74), we automatically obtain a solution of the original nonrelativistic self-dual Chern-Simons equations:

(81) |

The chiral model equations are also referred to as the “harmonic map equations” because if we regard as a connection, then it satisfies both

(82) | |||||

(83) |

and so has zero divergence and zero curl.

The global condition which permits the classification of solutions to the chiral model equation (79) is the condition of finiteness of the chiral model “action functional” (also referred to in the literature as the “energy functional”)

(84) |

This finiteness condition has direct physical relevance in the related nonrelativistic Chern-Simons system because

(85) | |||||

(86) | |||||

(87) |

where is the conserved gauge invariant matter charge in (35). Thus, the finite action solutions of the chiral model equations correspond precisely to the finite charge solutions of the nonrelativistic self-dual Chern-Simons equations.

In addition to being physically significant, this finiteness condition is mathematically crucial because it permits the chiral model solutions on to be classified by conformal compactification to the sphere [30, 31]. Indeed, Uhlenbeck has classified all finite action chiral model solutions for in terms of “uniton” factors (which will be discussed below).

Before discussing the general classification of finite charge solutions, we introduce the simplest such solutions, the “single unitons”, upon which the general solutions are constructed. A “single uniton” solution, , of the chiral model equation (79) has the form

(88) |

where is a “holomorphic projector” satisfying:

(89) | |||||

(90) | |||||

(91) |

These single uniton solutions are fundamental to the chiral model system because as a consequence of the conditions (91) we find that

(92) |

From this it immediately follows that satisfies the chiral model equation (79). These single uniton solutions are also solutions of the model since satisfies the additional condition, , as a result of being a projector. In terms of the field defined in (80), the single uniton solutions take the simple form

(93) |

It is straightforward to check that, as a consequence of the conditions (91) satisfied by , satisfies the equation (74), and therefore gives a solution to the nonrelativistic self-dual Chern-Simons equations as in (81).

The general holomorhic projector satisfying the conditions (91) can be expressed as

(94) |

where is any (rectangular) matrix such that [30]. It is easy to see that such an is a hermitean projector. The third condition (91) is equivalent to , which follows immediately from the fact that

(95) |

The next step towards the construction of general solutions involves the process of “composing” uniton solutions, as follows. Suppose is a single uniton solution with satisfying the conditions (91) for a holomorphic projector. Further, let be such that and . Then is a solution of the chiral model equation (79) provided the following first-order algebro-differential conditions are met:

(96) | |||

(97) |

Given these conditions,

(98) |

and so, once again, the chiral model equation (79) is immediately satisfied.

This procedure of composing uniton-type solutions can be continued, but since the matrices involved are projectors, there is a limit to how many independent projections can be made. For , at most such terms can be combined in this manner, as expressed in Uhlenbeck’s theorem:

THEOREM (K. Uhlenbeck [30]; see also J. C. Wood [32]): Every finite action solution h of the SU(N) chiral model equation (79) may be uniquely factorized as a product of “uniton” factors

(99) |

where:

a) is constant;

b) each is a Hermitean projector ( and );

c) defining , the following linear relations must hold:

(100) |

d) .

The sign in (99) has been inserted to allow for the fact that Uhlenbeck and Wood considered the gauge group rather than .

An important implication of this theorem is that for all finite action solutions of the chiral model have the “single uniton” form

(102) |

with being a holomorphic projector satisfying the conditions (91). These single uniton solutions are essentially the model solutions [34, 9]. For with one must consider composite unitons in addition to the single unitons. It becomes increasingly difficult to give a simple characterization of all possible projectors satisfying the subsidiary linear conditions specified in Uhlenbeck’s construction. However, Wood has presented a systematic parametrization of these higher unitons, for any , in terms of a sequence of projectors into Grassmannian factors. A detailed analysis of the and cases is also given in [35].

At this point, it is important to ask how these multi-uniton solutions to the chiral model equations relate to the special explicit Toda-type solutions discussed previously in (58-61). While the algebraic Ansätze (47,63) each leads to a non-Abelian charge density which is diagonal, the chiral model solutions (81) have charge density which need not be diagonal. However, is always hermitean, and so it can be diagonalized by a gauge transformation. It is still a nontrivial algebraic task to implement this diagonalization explicitly, but this can be done for the solutions, revealing an interesting new link between the chiral model and the Toda system [28].

It is instructive to illustrate this procedure with the case first. Here, Uhlenbeck’s theorem implies that the only finite charge solution has the form , where is a holomorphic projector as in (94). For we can only project onto a line in , so we take

(103) |

This then leads to the projection matrix

(104) |

and the corresponding solution can be expressed in terms of the single holomorphic function :

(105) |

The corresponding matter density is

(106) |

which may be diagonalized by the unitary matrix

(107) |

to yield the diagonalized charge density

(108) | |||||

(109) |

In this diagonalized form we recognize Liouville’s solution (60) to the classical Toda equation. It is worth emphasizing that for the nonrelativistic self-dual Chern-Simons equations (40,41) can be converted, by suitable algebraic ansatze as discussed in the previous section, into the classical Toda (i.e. Liouville) equation or the affine Toda (i.e. sinh-Gordon) equation. However, the above analysis shows that only the classical Toda case (i.e. Liouville) corresponds to finite charge.

A similar construction is possible for the case [28, 29]. Specifically, let

(110) |

be a product where each is a holomorphic projector onto the -dimensional subspace spanned by the columns of the rectangular matrix in (61,62):

(111) |

Then is a finite action solution of the chiral model equation (79) and the corresponding solution of the nonrelativistic self-dual Chern-Simons equations is

(112) |

The charge density may be diagonalized by an gauge transformation yielding a diagonal form

(113) |

where the are the Cartan subalgebra generators of in the Chevalley basis. This diagonal form of the charge density corresponds precisely to the Toda solution (61).

Another useful result which follows from the relationship between the nonrelativistic self-dual Chern-Simons equations (40,41) and the chiral model equation (79) is that the chiral model energy (84) is quantized in integral multiples of [33]. This implies that the abelian Chern-Simons charge is quantized in integral multiples of . A related quantization condition has been noted in [18], where the non-Abelian charges are quantized in integral multiples of for the Toda-type solutions (61). In this case the abelian charge is the sum of the individual nonAbelian charges : .

## 3 Relativistic SDCS Theories

3.1 Relativistic Self-Dual Chern-Simons Equations

In this section we discuss the relativistic generalization of the nonrelativistic self-dual Chern-Simons theories. The existence of vortex solutions in -dimensional relativistic gauge-Higgs models including Chern-Simons terms has been known for some time [36]. The importance of self-duality was first noticed in the context of abelian theroies [37, 38], where vortices in the relativistic Chern-Simons-Higgs model were shown to be related to a self-duality condition reminiscent of Bogomol’nyi’s analysis [3] of vortices in the abelian Higgs model. With a particular sixth order scalar potential there is a lower bound on the energy functional which is saturated by topological solitons and nontopological vortices [39]. An extension of these abelian models is possible, to nonabelian relativistic self-dual Chern-Simons theories with a global symmetry [40], once again with a special sixth order potential. However, while the self-dual structure of the model generalizes in a relatively straighforward manner, the analysis of the nonabelian relativistic self-dual Chern-Simons equations themselves is significantly more complicated, and correspondingly more interesting. The richness of the nonabelian theory is compounded by the many available choices: of gauge group, of representation, of matter coupling, etc… [40, 41, 42, 43]. Matter fields in the defining representation have been studied in [41], while the most interesting case once again seems to be the case of adjoint coupling [40, 42, 44, 45]. The self-dual structure of these relativistic self-dual Chern-Simons systems is related at a fundamental level to extended supersymmetry in dimensions [46, 47, 48], in the sense that the self-dual Lagrangian is the bosonic portion of a Lagrangian with an extended supersymmetry. This is in accordance with a general relationship between self-duality and extended supersymmetry [49].

Consider the Lagrange density

(114) |

where is the Chern-Simons Lagrange density in (22), and the scalar field potential is

(115) |

The space-time metric is taken to be and, as before, refers to the trace in a finite dimensional representation of the compact simple Lie algebra to which the gauge fields and the charged matter fields and belong. The parameter appearing in the potential (115) will play the role of a mass parameter (see (183)). Under a gauge transformation both the potential and the scalar field kinetic term remain invariant. However, the Chern-Simons Lagrange density is not invariant and the dimensionless coupling coefficient must be quantized in order for the corresponding quantum theory to be invariant under large gauge transformations [16]. The particular sixth-order form of the scalar field potential (115), together with its overall strength depending on the Chern-Simons coupling parameter , are fixed by the condition of self-duality, as shown below.

The Euler-Lagrange equations of motion obtained from the Lagrange density (114) are:

(116) |

(117) |

In the matter equation of motion (116), is defined by the change in the potential under a variation of :

(118) |

In the gauge equation of motion (117), is the relativistic nonabelian current

(119) |

which is covariantly conserved : . This system also has an abelian current, ,

(120) |

which is ordinarily conserved : .

The energy density corresponding to the Lagrange density (114) is

(121) |

supplemented by the Gauss law constraint

(122) |

which is the zeroth component of the gauge field equations of motion (117). Notice that, as is familiar for Chern-Simons theories, the Chern-Simons term in the Lagrange density (114) does not contribute to the energy, while it does affect the canonical structure and the constraints [16, 17, 18].

To find self-dual solutions which minimize the energy, we re-express the energy density in a modified form, using an adaptation of the Bogomol’nyi method for vortices in the abelian Higgs model [3]. Using the identity (38) together with the Gauss law constraint (122), we can write

(125) | |||||

where we recall that . The second term on the RHS of (LABEL:bogol2) may be cancelled in the energy density (121) by a term from if we write