- Email: [email protected]
Bogomol'nyi solitons and Hermitian symmetric spaces
Vol. 43 (1999)
REPORTS
ON MATHEMATICAL
No. 112
PHYSICS
BOGOMOL’NYI SOLITONS AND HERMITIAN SYMMETRIC SPACES PHILLIAL
OH
Department of Physics, Sung Kyun Kwan University, Suwon 440-746, Republic of Korea and Center for Theoretical Physics, MIT., Cambridge, MA 02139-4307, USA (e-mail: [email protected]) (Received
March
23, 1998 -
Revised
August
20, 1998)
We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chem-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol’nyi bounded from below by a topological charge. The Bogomol’nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N - 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.
1.
Introduction
A coadjoint orbit method allowing to formulate a nonlinear sigma model defined on the target space of a homogeneous space G/H was proposed recently [l]. It was first applied to a nonrelativistic spin system whose Poisson bracket between dynamical variables defined on the coadjoint orbit satisfies axioms of the classical 4 algebra. The Euler-Lagrange equation of motion yields the generalized continuous Heisenberg ferromagnet [2, 31. When the target space of a coadjoint orbit is given by a Hermitian symmetric space (HSS), which is a symmetric space equipped with a complex structure [4], the generalized ferromagnet system becomes completely integrable in l+l dimension [l]. This method was exploited later to produce a class of integrable extensions of relativistic nonlinear sigma models (NLSM) in l+l dimension [5]. It was also discovered that incorporation of the Chern-Simons (CS) gauge field into 2+1 dimension on the same target space produces a class of self-dual field theories which admit Bogomol’nyi self-dual equations saturating the energy functional [6]. A detailed numerical investigation in the case of compact SU(2) [7] showed a rich structure of self-dual solitons in the system. In this paper, we apply the coadjoint orbit method to construct relativistic NLSM on the target space of coadjoint orbits coupled with the CS gauge field and we study self-dual solitons. When the target space is HSS, the Hamiltonian is bounded [2711
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from below by a topological charge, and the resulting self-dual CS solitons satisfy a vortex-type equation, thus producing a class of new self-dual theories on HSS. This construction provides a unified framework for treating an older O(3)-gauged model on S2 and the CP(N - 1) models [8] which are the well-known examples of the coadjoint orbit G/H with S2 = SO(3)/SO(2) M SU(2)/U(l) and CP(N - 1) = SU(N)/SU(N - 1) x U(1). We also study the self-dual solitons in a noncompact HSS with the SU(1, 1) group in which the target space is given by the upper sheeted hyperboloid and we find various topological and nontopological solitons. For completeness, we first give a brief summary of NLSM on the target space of a coadjoint orbit. Consider a group G, Lie algebra 6 and its dual G* : X E S; u E 6*. The adjoint action of G on the Lie algebra is defined by Ad(g)X = gXg-‘,
g E G.
(1.1)
Denoting the inner product between G and 6’ by (u, X), the coadjoint action of the group on G* is defined in the way which makes the inner product invariant: (Ad*(g).lL,X) = (u, Ad(g-l)X). The coadjoint orbit is given by the orbit of coadjoint point u E G*, then the orbit is generated by 0, = (51~ = Ad*(g)u,
(1.2) action of the group G: Fix a
g E G}.
(1.3)
It can be shown that 0, RZG/H, where H is the stabilizer of the point U. Let us assume that the inner product is given by the trace: (u, X) = Tr(Xu). Then Q and Q* are isomorphic and the coadjoint orbit can be parameterized by Q = gKg_’ = QAtBq~~;
tA,K E 6
(A = l,...,dimG),
(1.4)
where nAB is the G-invariant metric given by Tr(tAtB) = --inAB with tA’s being the generators of G. The action for the NLSM on the target space of the coadjoint orbit can be constructed as S(g) = ETr
J
d3~8,QPQ;
(1.5)
E = +l for the compact case, -1 for the noncompact case [9]. Let us first choose the element K to be the central element of the Cartan subalgebra of Q whose centralizer in E is H. Then, for the HSS, we have J = Ad(K) acting on the coset is a linear map satisfying the complex structure condition J2 = -1, which gives a useful identity
PI: [Q,[Q,+Qll = -apQ. This paper is organized as follows: In Section 2, starting from a CS gauged (1.5) on an arbitrary HSS, we derive self-dual equations and Bogomol’nyi We give explicit expressions in the CP(N - 1) case. In Section 3, we a noncompact minimal SU( 1,1) model and we discuss the rotationally solutions in detail. In Section 4, we present the conclusion.
(1.6) action of potential. deal with symmetric
BOGOMOL’NYI
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The compact model Let us consider the following CS gauged action of (1.5): ~ep”pTr(d~A,A~ + $A,A,A,)],
so = ld32[-~E(o~QAo~QB17aB)-WG(QA)-
(2.1)
J
where the covariant derivative is defined by
DpQ = a,Q + [A,, &I, We assume that the potential
A, = A,AtBTIAB.
(2.2)
is given by
(2.3)
W'G(Q~)= iIABQAQB, where IAB is a symmetric tensor and its meaning will be determined ality condition. The equations of motion are given by
D,[QP‘Q]
+ [o,Q] CpJPF 2
.yp
= 0
(0 =
by the self-du-
IABQAtB),
(2.4)
= [Q,D"Q].
We first treat the compact case with ~)AB = GAB. To study self-dual solitons, we bring the energy functional into Bogomol’nyi’s form EG = =
J
d2x [i ((Do&~)~ + (DiQA)2)
s
d2x
[$(Dc,&~)~ + $(DiQA
+W(QA)
where the topological
f
+ W(QA)]
f q[Q,
DjQ]A)2
(2.5)
c3jQlA - 2~ij&(Q~AjA)].
(2.6)
;cijFi;QA] f ‘hrT~,
charge TG is given by
TG = & / d2x[cijQA[&Q,
In deriving (2.5) we used the gauged version of (1.6), where 8, has been replaced by the covariant derivative D, [6]. Thus, the Hamiltonian is bounded from below by the topological charge TG when the potential Wo is chosen such that IV, f $&QA
= 0.
(2.7)
Here, Fi$ is determined in terms of QA by the Gauss law which is the time-component of (2.4). The minimum energy arises when the self-duality equation is satisfied:
DiQ = wij[Q,
DjQl.
(2.8)
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A consistency condition with the static equations of motion (2.4) forces Fij = 0,
A0 = +Q,
(2.9)
which in turn puts the potential WG = 0 and I AB = 0. Note that the gauge field can be chosen as a pure gauge in this case and the contents of the Bogomol’nyi solitons are precisely the two-dimensional instantons which were completely classified on each HSS [lo]. More interesting cases in which the system offers other solitons arise when we gauge the subgroup H. We consider the gauging, of the maximal torus subgroup of G: d32 [-$ (D,Q*D~Q*)
sH =
- We
(2.10)
+ @~,A;A;].
J
Here, the index a = 1, . . . , rankG denotes the maximal abelian subgroup. Again, the meaning of the potential WH will be determined from the self-duality condition. Using the Gauss law given by (2.11)
EtijFG = -[Q, DO&]“, we find that the energy functional satisfies EH = f
+ +(DiQ* f cij[Q, DjQ]A)2
1
f 47rTH, (2.12)
TH = with the Bogomol’nyi potential WH chosen as WH
=
&&&IA)27
QH = Q;P
= (Q” - Va)ta.
(2.13)
Note that P’S are free parameters associated with breaking of the vacuum symmetry [ll]. When the self-duality equations DiQ* =
&Q* = +$Q,QHI* r~ij[Q,DjQ]*~
(2.14)
are satisfied, we see that the energy is saturated by the topological charge EH = 4XlT~l.
(2.15)
The first-order equation (2.14) in the static case fixes A$ as A; = *;Q;,
(2.16)
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BOGOMOL’NYI SOLITONS AND HERMITIAN SYMMETRIC SPACES
which automatically solves the Euler-Lagrange equations of motion of the action (2.10) with the potential given by (2.13). Let us examine (2.11) and (2.14) more closely in the C’P(N - 1) case. We use Q in the form [l], Q=i++
where the column vector (a = 1,2,...,N1):
(2.17)
9 can be expressed
by the Fubini-Study
coordinates
$I,
1
*=& :
@N--l
(2.18)
,
“’ I
with ]?_/I]~ = 1$,112+ . . . + [$N-I~~. Using the (complex) notation 2 = z + iy, Z = 2 - iy, AZ = $(A1 - iA2), Ai = ;(A1 + iA2), and D, = i(Dl - iDa), D,- = i(Dl + iD2), we . obtain an alternative expression of the self-duality equation
DzQ = +[Q, DzQl.
(2.19)
With the above parametrization of Q, the self-duality sign becomes a set of N - 1 equations [6], D”=&+; I
A;+-A,+ ;
2
. ..+
/&A;-‘+
equation
(2.19) for the plus
/VA;),
(2.20)
DQ& = 0.
(2.21)
Similarly, for the minus sign, we have . ..+
f&A;-‘+
D;&
/-A:>,
= 0.
(2.22) (2.23)
From now on we concentrate on the plus sign. With 6, = w, exp(i&) we find that (2.11), (2.14) and (2.21) produce the following new vortex-type equation V2 logw, + @$& where P
=
(2.24)
is given by +dabcQc
-Q”Qb
. 1
(2.25)
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We used the normalization {XA,XB} = (4/N)SABI + 2dABCXC. Also the Bogomol’nyi potential (2.13) can be expressed by WH = &(V
- Q”)(Vb - Qb)
Let us give an example in the case of CP(l). potential becomes
1
r3
LNdab+ dabcQc -
Q"Qb]
.
With w1 = w, $i = #J, V1 = V, the above
&(v- Q3j2(1 - (Q3j2),
WH =
(2.26)
(2.27)
which is exactly the same as the potential in the O(3) model [ll]. Next, we find that (2.24) becomes [V - S]
[l-(G)‘].
(2.28)
A detailed numerical study of the above equation showed that it has various kinds of rotationally symmetric soliton solutions connected with symmetric and broken phases, and that they are anyons carrying fractional angular momentum [ll]. Similar results are expected in the more complicated higher CP(N) case, but a detailed study will be addressed elsewhere. 3.
Noncompact SU(1, 1) soliton
In this section we consider a noncompact HSS with E = -1. We restrict ourselves to the SU(1, 1) group with 77,~~= (-, -, +). The target space is given by the two-sheeted hyperboloid H = SU(1, 1)/U(l). Using the expression for the group element g of (1.4) given by
(3.1) which satisfies gMgt = M with M = diag(1, -l),
(
we have (with K = ig3/2)
1 + ]7#
Q = 2(1&)
.
2?&
-2lj* -(l
+ ]?/I]“) .
(3.2)
)
We restrict ourselves to ]$I < 1, which corresponds to the upper sheet of M = SU(l,l)/U(l). A couple of remarks concerning the ungauged case are in order at this point. First, some soliton solutions associated with a noncompact NLSM were discussed in connection with the Ernst equation [12], which is not self-dual. Second, using the above expression, one can check that there actually exist self-dual soliton solutions which are analytic or anti-analytic as in the compact case [13], but the energy and topo-
BOGOMOL’NYI
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277
logical charge diverge at the boundary [$I = 1. A coupling with a CS gauge field greatly improves the situation because the gauge field effectively provides a potential barrier to the boundary (see (3.7)) and prevents the system from diverging. Again, with the parameterization ?i, = w exp($), we find the Bogomol’nyi potential (2.13) and the self-dual equation (2.24) are produced as follows:
%=&
[v-(+=J12
V210gW+Eij8i8.jf$= Let us look for rotationally coordinates (T, 8) by w
=
$
[ V-
g$]
[(+g2-1].
(3.4)
symmetric solutions with an Ansatz given in the cylindrical
tar&f(T)
$ = &,
Ai
2 ’
Then, the Gauss law and self-dual equation a’(r) = (r/k2)(-V rf’(r)
(3.3)
[(s)z-l])
=
yu(r).
(3.5)
become (’ = d/dr)
+ cash f(r))(l
- cosh2 f(r)),
= (u(r) - n) sinh f(r).
(3.6)
The combined equation of motion in (3.4) becomes now an analogue of the one -dimensional Newton equation for T > 0, if we regard T as the “time” and U(T) as the “position” of the hypothetical particle with unit mass under a log tanh q time-dependent friction, (l/r)u’, and an effective potential I&: VeR(u) = &
coth2 ‘1~+ 5 cothu.
(3.7)
The exerted force also includes an impact term at T = 0 due to &&iajd = :a(~) in (3.4). Inspection of the effective potential suggests that solitons are basically of two types; the nontopological vortices with n # 0 (negative integer) and the non-topological solitons with n = 0. In the former case, the “particle” starts from u = -00, reaches a turning point where it stops, changes the direction, and finally rolls down to u = --oo. In the latter case, the “particle” starting at some finite position, either rolls down to u = --oo directly, or moves to a turning point, changes the direction, and rolls down to u = --o;). Let us look at the solutions more closely. Near T = 0, the condition for Ai to be nonsingular forces a(O) = 0. First, when n # 0, we must have f(0) = 0. When n = 0, cxE f(0) can be arbitrary. The behaviour of the solution near T = co can be also read off from the conditions f’(m) = a’(m) = 0; ,D= f(m) = 0 for (1,s < 0) near T = 00, arbitrary y = u(o0) and V. Putting f(r) = f,@, a(r) = y+u,rs we find 1 = y - n, s = 2y + 2 - 2n for V # 1. Since I, s < 0, we have the consistency
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conditiony 1, y must be equal to n (# 0). The solution which shows oscillatory behaviour before it comes to rest does not exit. Near T = co, we assume an exponential approach f(~-) = ,0 + f,r’e--ar, a(r) = y + a,rSeebr (a, b > 0). Then the substitution leads to a contradictory output, 1 = s and I = s + 1. The power law approach with a, b = 0 and 1, s < 0 also leads to a contradiction, In view of the Bogomol’nyi potential (3.3) this excludes any solitons in the broken vacuum with V = coshf(oo), and all the solitons are in the symmetric phases. Let us focus on the vicinity of T = 0. (a) V 5 1, in which the effective potential (3.7) is a monotonically decreasing function. (a-i) n # 0; trying power solutions of the form f(r) = fa+, a(r) = aarQ (p, q > 0), we find p = -n, q = 2 - 2n for V < 1. Hence n must be a negative integer. When V = 1, we have p = -n, q = 2 - 4n. (a-ii) n = 0, a # 0; let us try f(r) = cr + fo+‘, a(r) = aa@ (p,q > 0). We find p = q = 2. We note that both a0 and f. turn out to be negative, so that the solution rolls down to T = cc. The solution climbing at first up and then rolling down the hill does not exist. When (b) V > 1, the effective potential (3.7) develops a pool with a local minimum at fm = cash-’ V. (b-i) n # 0; the behaviour is similar to (a-i) except of the fact that u(r) passes the minimum at r, twice in the process of climbing up, passing the turning point, and rolling down the hill to its original position. (b-ii) n = 0, Q # 0; there are two cases. When cx < coshh’ V, the solution, if it exists, will behave similarly to (b-i) except that it starts at some fi-
0
-0.5 -1 -1.5
u(r) -2 -2.5 -3
‘. ---__
Fig.
----_
-3.5 -
l
I
I
I
0
1
2
3
4
1. a(r)
as a function
---_____________--
of T for n = 1 with various dashed
line, and (-1,1.5)
I
I
I
5
6
7
---8
(n, V): (0,O)for the solid for the dotted
line
---9
line,
(-1,O)
10 for the
BOGOMOL’NYI
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SOLITONS AND HERMITIAN
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I
2
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4
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279
I
I
I
I
6
7
8
9
.:
:: : : : : ., : : : ., : : : ‘, : ‘,
0.7 0.6 0.5 f(r)
0.4 0.3 0.2 0.1 0 0
Fig. 2. f(r)
1
10
r as a function of T for K. = 1 with various (71,V): (0,O)for the solid line, (-1,0) dashed line, and (-l,l.S)
for the
for the dotted line
nite point (Y. However, it cannot exist for the following reason; the initial “velocity” of the particle is given by u’(0) cx f’(0) = 0 (f’(r) 0: T from (a-ii)). Hence the particle does not carry enough kinetic energy to return to its starting point in this dissipative system with a conservative potential. Note that when n # 0, even though the initial velocity is in general equal to 0 except for n = -1, the solutions are possible because of the impact term at r = 0. In the opposite case cx > cash-’ V, it is similar to (a-ii) and only rolling down the hill is permitted. A detailed numerical study given in Figs. 1 and 2 indeed confirms the existence of these solitons. Note that there do not exist any topological lump solutions because nn(M) = 0. And the topological vortices do not exist because there is no bump in the effective potential where the particle can stop at the top. In the solutions, the magnetic flux is given by Cp = 27ry, and the energy is saturated by the topological charge; E = 4nlTI = 2x17(1 - V)l. The system also carries a nonvanishing angular momentum. Let us define J =
s
d2xEijZiDOQADjQBvAB.
(3.8)
By a simple calculation using the Gauss law (2.11) (with the plus sign in the righthand side due to the E factor), and the self-dual equations (2.14) and (3.6), we find J = m((+y-r1)~1~~). Thus the solitons in general carry a fractional angular momentum, representing anyons. For the nontopological solitons it is simply J = rny2.
280 4.
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Conclusions
We have shown that the coadjoint orbit approach for the relativistic NLSM coupled with a CS gauge field leads to a class of new self-dual field theories on the target space of HSS which contain the previous O(3) and CP(N - 1) models, and a new noncompact SU(1, 1) model. We have also found an explicit expression of the Bogomol’nyi potential when the maximal torus subgroup is gauged, and we have shown that the noncompact NLSM admits self-dual soliton solutions which are saturated by the Bogomol’nyi bound, and gave a complete description of the rotationally symmetric solutions. There remains, several further issues to be discussed. First, note that the identity (1.6) and its gauged version on HSS is essential for the existence of self-duality. In this respect, it would be an intriguing problem to extend the above formalism to other non-HSS coadjoint orbits, and also to higher noncompact groups. Quantization of the model is another problem to be addressed. Second, it would be interesting to see whether there exists a well-defined procedure in which the nonrelativistic NLSM of the generalized CS Heisenberg ferromagnet system defined on the coadjoint orbits [6] could emerge as a nonrelativistic limit of the present relativistic NLSM. This would require finding a connection between the symplectic structure of HSS [6] for the non-relativistic NLSM and the phase-space structure of relativistic NLSM.
Acknowledgements
I thank D. Chae, Y. Kim, K. Kimm, K. Lee, Q-H. Park, and C. Rim for useful discussions, Sung-Soo Kim for his invaluable help, and Prof. A. Strasburger for his hospitality at XVI Workshop on Geometric Methods in Physics. This work is supported in part by the Korea Science and Engineering Foundation through the project number (950702-04-Ol-3), and by the Ministry of Education through the Research Institute for Basic Science (BSRI/97-1419). REFERENCES [ll P. Oh and Q-H. Park: Phys. Len. B 383 (1996), 333. 121 M. Lakshmanan: Phys. Len. A 61 (1977), 53; L. A. Takhtajan: Phys. Let?. A 64 (1977), 235; J. Tjon and J. Wright: Phys. Rev. B 15 (1977), 3470. 131 L. D. Faddeev and L. A. Takhtajan: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin 1987. [41 A. P. Fordy and P. P. Kulish: Commum. Math. Phys. 89 (1983), 427. 151 P. Oh: /. Phys A: Math. Gen. 31 (1998), L325. [61 P. Oh and Q-H. Park: Phys. Lett. B 400 (1997), 157; (E) 416 (1998), 452. [71 Y. Kim, P. Oh and C. Rim: Mod. Phys. Lett. A 12 (1997), 3169. 181 G. Nardelli: Phys. Rev. Left. 73 (1994), 2524; B. J. Schroers: Phys. Left. B 356 (1995), 291; J. Gladikowski, B. M. A. G. Piette and B. J. Schroers: Phys. Rev. D 53 (1996), 844; K. Kimm, K. Lee and T. Lee: Phys. Rev. D 53 (1996), 4436; K. Arthur, D. H. Tchrakian and Y. Yang: Phys. Rev. D 54 (1996), 5245; P. K. Ghosh and S. K, Ghosh: Phys. Left. E 366 (1996), 199; Y. M. Cho and K. Kimm: Phys. Rev. D 52 (1995), 7325; K. Kimm, K. Lee and T. Lee: Phys. Lett. II 380 (1996), 303; P. K. Ghosh: Phys. Lett. B 384 (1996), 185.
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[9] We will only consider the simplest noncompact case SU(l,l). Here E = -1 is necessary to render the energy to be nonnegative in the representation in which the element K is anti-Hermitian. Higher orbits of noncompact group need a more careful treatment. [lo] A. M. Perelomov: Whys. Rep. 146 (1987), 135. [ll] K. Kimm, K. Lee and T. Lee: Whys. Len. B 380 (1996), 303. [12] S. lhkeno: Progr. Theor. Whys. 66 (1981), 1250; .I. Gruszczak: J. Phys. A: Math. Gen. 14 (1981), 3247. [13] A. A. Belavin and A. M. Polyakov: JETP Lett. 22 (1975), 245.
REPORTS
ON MATHEMATICAL
No. 112
PHYSICS
BOGOMOL’NYI SOLITONS AND HERMITIAN SYMMETRIC SPACES PHILLIAL
OH
Department of Physics, Sung Kyun Kwan University, Suwon 440-746, Republic of Korea and Center for Theoretical Physics, MIT., Cambridge, MA 02139-4307, USA (e-mail: [email protected]) (Received
March
23, 1998 -
Revised
August
20, 1998)
We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chem-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol’nyi bounded from below by a topological charge. The Bogomol’nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N - 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.
1.
Introduction
A coadjoint orbit method allowing to formulate a nonlinear sigma model defined on the target space of a homogeneous space G/H was proposed recently [l]. It was first applied to a nonrelativistic spin system whose Poisson bracket between dynamical variables defined on the coadjoint orbit satisfies axioms of the classical 4 algebra. The Euler-Lagrange equation of motion yields the generalized continuous Heisenberg ferromagnet [2, 31. When the target space of a coadjoint orbit is given by a Hermitian symmetric space (HSS), which is a symmetric space equipped with a complex structure [4], the generalized ferromagnet system becomes completely integrable in l+l dimension [l]. This method was exploited later to produce a class of integrable extensions of relativistic nonlinear sigma models (NLSM) in l+l dimension [5]. It was also discovered that incorporation of the Chern-Simons (CS) gauge field into 2+1 dimension on the same target space produces a class of self-dual field theories which admit Bogomol’nyi self-dual equations saturating the energy functional [6]. A detailed numerical investigation in the case of compact SU(2) [7] showed a rich structure of self-dual solitons in the system. In this paper, we apply the coadjoint orbit method to construct relativistic NLSM on the target space of coadjoint orbits coupled with the CS gauge field and we study self-dual solitons. When the target space is HSS, the Hamiltonian is bounded [2711
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from below by a topological charge, and the resulting self-dual CS solitons satisfy a vortex-type equation, thus producing a class of new self-dual theories on HSS. This construction provides a unified framework for treating an older O(3)-gauged model on S2 and the CP(N - 1) models [8] which are the well-known examples of the coadjoint orbit G/H with S2 = SO(3)/SO(2) M SU(2)/U(l) and CP(N - 1) = SU(N)/SU(N - 1) x U(1). We also study the self-dual solitons in a noncompact HSS with the SU(1, 1) group in which the target space is given by the upper sheeted hyperboloid and we find various topological and nontopological solitons. For completeness, we first give a brief summary of NLSM on the target space of a coadjoint orbit. Consider a group G, Lie algebra 6 and its dual G* : X E S; u E 6*. The adjoint action of G on the Lie algebra is defined by Ad(g)X = gXg-‘,
g E G.
(1.1)
Denoting the inner product between G and 6’ by (u, X), the coadjoint action of the group on G* is defined in the way which makes the inner product invariant: (Ad*(g).lL,X) = (u, Ad(g-l)X). The coadjoint orbit is given by the orbit of coadjoint point u E G*, then the orbit is generated by 0, = (51~ = Ad*(g)u,
(1.2) action of the group G: Fix a
g E G}.
(1.3)
It can be shown that 0, RZG/H, where H is the stabilizer of the point U. Let us assume that the inner product is given by the trace: (u, X) = Tr(Xu). Then Q and Q* are isomorphic and the coadjoint orbit can be parameterized by Q = gKg_’ = QAtBq~~;
tA,K E 6
(A = l,...,dimG),
(1.4)
where nAB is the G-invariant metric given by Tr(tAtB) = --inAB with tA’s being the generators of G. The action for the NLSM on the target space of the coadjoint orbit can be constructed as S(g) = ETr
J
d3~8,QPQ;
(1.5)
E = +l for the compact case, -1 for the noncompact case [9]. Let us first choose the element K to be the central element of the Cartan subalgebra of Q whose centralizer in E is H. Then, for the HSS, we have J = Ad(K) acting on the coset is a linear map satisfying the complex structure condition J2 = -1, which gives a useful identity
PI: [Q,[Q,+Qll = -apQ. This paper is organized as follows: In Section 2, starting from a CS gauged (1.5) on an arbitrary HSS, we derive self-dual equations and Bogomol’nyi We give explicit expressions in the CP(N - 1) case. In Section 3, we a noncompact minimal SU( 1,1) model and we discuss the rotationally solutions in detail. In Section 4, we present the conclusion.
(1.6) action of potential. deal with symmetric
BOGOMOL’NYI
2.
SOLITONS AND HERMITIAN
SYMMETRIC
SPACES
273
The compact model Let us consider the following CS gauged action of (1.5): ~ep”pTr(d~A,A~ + $A,A,A,)],
so = ld32[-~E(o~QAo~QB17aB)-WG(QA)-
(2.1)
J
where the covariant derivative is defined by
DpQ = a,Q + [A,, &I, We assume that the potential
A, = A,AtBTIAB.
(2.2)
is given by
(2.3)
W'G(Q~)= iIABQAQB, where IAB is a symmetric tensor and its meaning will be determined ality condition. The equations of motion are given by
D,[QP‘Q]
+ [o,Q] CpJPF 2
.yp
= 0
(0 =
by the self-du-
IABQAtB),
(2.4)
= [Q,D"Q].
We first treat the compact case with ~)AB = GAB. To study self-dual solitons, we bring the energy functional into Bogomol’nyi’s form EG = =
J
d2x [i ((Do&~)~ + (DiQA)2)
s
d2x
[$(Dc,&~)~ + $(DiQA
+W(QA)
where the topological
f
+ W(QA)]
f q[Q,
DjQ]A)2
(2.5)
c3jQlA - 2~ij&(Q~AjA)].
(2.6)
;cijFi;QA] f ‘hrT~,
charge TG is given by
TG = & / d2x[cijQA[&Q,
In deriving (2.5) we used the gauged version of (1.6), where 8, has been replaced by the covariant derivative D, [6]. Thus, the Hamiltonian is bounded from below by the topological charge TG when the potential Wo is chosen such that IV, f $&QA
= 0.
(2.7)
Here, Fi$ is determined in terms of QA by the Gauss law which is the time-component of (2.4). The minimum energy arises when the self-duality equation is satisfied:
DiQ = wij[Q,
DjQl.
(2.8)
274
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A consistency condition with the static equations of motion (2.4) forces Fij = 0,
A0 = +Q,
(2.9)
which in turn puts the potential WG = 0 and I AB = 0. Note that the gauge field can be chosen as a pure gauge in this case and the contents of the Bogomol’nyi solitons are precisely the two-dimensional instantons which were completely classified on each HSS [lo]. More interesting cases in which the system offers other solitons arise when we gauge the subgroup H. We consider the gauging, of the maximal torus subgroup of G: d32 [-$ (D,Q*D~Q*)
sH =
- We
(2.10)
+ @~,A;A;].
J
Here, the index a = 1, . . . , rankG denotes the maximal abelian subgroup. Again, the meaning of the potential WH will be determined from the self-duality condition. Using the Gauss law given by (2.11)
EtijFG = -[Q, DO&]“, we find that the energy functional satisfies EH = f
+ +(DiQ* f cij[Q, DjQ]A)2
1
f 47rTH, (2.12)
TH = with the Bogomol’nyi potential WH chosen as WH
=
&&&IA)27
QH = Q;P
= (Q” - Va)ta.
(2.13)
Note that P’S are free parameters associated with breaking of the vacuum symmetry [ll]. When the self-duality equations DiQ* =
&Q* = +$Q,QHI* r~ij[Q,DjQ]*~
(2.14)
are satisfied, we see that the energy is saturated by the topological charge EH = 4XlT~l.
(2.15)
The first-order equation (2.14) in the static case fixes A$ as A; = *;Q;,
(2.16)
275
BOGOMOL’NYI SOLITONS AND HERMITIAN SYMMETRIC SPACES
which automatically solves the Euler-Lagrange equations of motion of the action (2.10) with the potential given by (2.13). Let us examine (2.11) and (2.14) more closely in the C’P(N - 1) case. We use Q in the form [l], Q=i++
where the column vector (a = 1,2,...,N1):
(2.17)
9 can be expressed
by the Fubini-Study
coordinates
$I,
1
*=& :
@N--l
(2.18)
,
“’ I
with ]?_/I]~ = 1$,112+ . . . + [$N-I~~. Using the (complex) notation 2 = z + iy, Z = 2 - iy, AZ = $(A1 - iA2), Ai = ;(A1 + iA2), and D, = i(Dl - iDa), D,- = i(Dl + iD2), we . obtain an alternative expression of the self-duality equation
DzQ = +[Q, DzQl.
(2.19)
With the above parametrization of Q, the self-duality sign becomes a set of N - 1 equations [6], D”=&+; I
A;+-A,+ ;
2
. ..+
/&A;-‘+
equation
(2.19) for the plus
/VA;),
(2.20)
DQ& = 0.
(2.21)
Similarly, for the minus sign, we have . ..+
f&A;-‘+
D;&
/-A:>,
= 0.
(2.22) (2.23)
From now on we concentrate on the plus sign. With 6, = w, exp(i&) we find that (2.11), (2.14) and (2.21) produce the following new vortex-type equation V2 logw, + @$& where P
=
(2.24)
is given by +dabcQc
-Q”Qb
. 1
(2.25)
276
P. OH
We used the normalization {XA,XB} = (4/N)SABI + 2dABCXC. Also the Bogomol’nyi potential (2.13) can be expressed by WH = &(V
- Q”)(Vb - Qb)
Let us give an example in the case of CP(l). potential becomes
1
r3
LNdab+ dabcQc -
Q"Qb]
.
With w1 = w, $i = #J, V1 = V, the above
&(v- Q3j2(1 - (Q3j2),
WH =
(2.26)
(2.27)
which is exactly the same as the potential in the O(3) model [ll]. Next, we find that (2.24) becomes [V - S]
[l-(G)‘].
(2.28)
A detailed numerical study of the above equation showed that it has various kinds of rotationally symmetric soliton solutions connected with symmetric and broken phases, and that they are anyons carrying fractional angular momentum [ll]. Similar results are expected in the more complicated higher CP(N) case, but a detailed study will be addressed elsewhere. 3.
Noncompact SU(1, 1) soliton
In this section we consider a noncompact HSS with E = -1. We restrict ourselves to the SU(1, 1) group with 77,~~= (-, -, +). The target space is given by the two-sheeted hyperboloid H = SU(1, 1)/U(l). Using the expression for the group element g of (1.4) given by
(3.1) which satisfies gMgt = M with M = diag(1, -l),
(
we have (with K = ig3/2)
1 + ]7#
Q = 2(1&)
.
2?&
-2lj* -(l
+ ]?/I]“) .
(3.2)
)
We restrict ourselves to ]$I < 1, which corresponds to the upper sheet of M = SU(l,l)/U(l). A couple of remarks concerning the ungauged case are in order at this point. First, some soliton solutions associated with a noncompact NLSM were discussed in connection with the Ernst equation [12], which is not self-dual. Second, using the above expression, one can check that there actually exist self-dual soliton solutions which are analytic or anti-analytic as in the compact case [13], but the energy and topo-
BOGOMOL’NYI
SOLITONS AND HERMITIAN
SYMMETRIC
SPACES
277
logical charge diverge at the boundary [$I = 1. A coupling with a CS gauge field greatly improves the situation because the gauge field effectively provides a potential barrier to the boundary (see (3.7)) and prevents the system from diverging. Again, with the parameterization ?i, = w exp($), we find the Bogomol’nyi potential (2.13) and the self-dual equation (2.24) are produced as follows:
%=&
[v-(+=J12
V210gW+Eij8i8.jf$= Let us look for rotationally coordinates (T, 8) by w
=
$
[ V-
g$]
[(+g2-1].
(3.4)
symmetric solutions with an Ansatz given in the cylindrical
tar&f(T)
$ = &,
Ai
2 ’
Then, the Gauss law and self-dual equation a’(r) = (r/k2)(-V rf’(r)
(3.3)
[(s)z-l])
=
yu(r).
(3.5)
become (’ = d/dr)
+ cash f(r))(l
- cosh2 f(r)),
= (u(r) - n) sinh f(r).
(3.6)
The combined equation of motion in (3.4) becomes now an analogue of the one -dimensional Newton equation for T > 0, if we regard T as the “time” and U(T) as the “position” of the hypothetical particle with unit mass under a log tanh q time-dependent friction, (l/r)u’, and an effective potential I&: VeR(u) = &
coth2 ‘1~+ 5 cothu.
(3.7)
The exerted force also includes an impact term at T = 0 due to &&iajd = :a(~) in (3.4). Inspection of the effective potential suggests that solitons are basically of two types; the nontopological vortices with n # 0 (negative integer) and the non-topological solitons with n = 0. In the former case, the “particle” starts from u = -00, reaches a turning point where it stops, changes the direction, and finally rolls down to u = --oo. In the latter case, the “particle” starting at some finite position, either rolls down to u = --oo directly, or moves to a turning point, changes the direction, and rolls down to u = --o;). Let us look at the solutions more closely. Near T = 0, the condition for Ai to be nonsingular forces a(O) = 0. First, when n # 0, we must have f(0) = 0. When n = 0, cxE f(0) can be arbitrary. The behaviour of the solution near T = co can be also read off from the conditions f’(m) = a’(m) = 0; ,D= f(m) = 0 for (1,s < 0) near T = 00, arbitrary y = u(o0) and V. Putting f(r) = f,@, a(r) = y+u,rs we find 1 = y - n, s = 2y + 2 - 2n for V # 1. Since I, s < 0, we have the consistency
278
P. OH
conditiony
0
-0.5 -1 -1.5
u(r) -2 -2.5 -3
‘. ---__
Fig.
----_
-3.5 -
l
I
I
I
0
1
2
3
4
1. a(r)
as a function
---_____________--
of T for n = 1 with various dashed
line, and (-1,1.5)
I
I
I
5
6
7
---8
(n, V): (0,O)for the solid for the dotted
line
---9
line,
(-1,O)
10 for the
BOGOMOL’NYI
0.8
I
SOLITONS AND HERMITIAN
I
I
I
I
2
3
4
5
SYMMETRIC
SPACES
279
I
I
I
I
6
7
8
9
.:
:: : : : : ., : : : ., : : : ‘, : ‘,
0.7 0.6 0.5 f(r)
0.4 0.3 0.2 0.1 0 0
Fig. 2. f(r)
1
10
r as a function of T for K. = 1 with various (71,V): (0,O)for the solid line, (-1,0) dashed line, and (-l,l.S)
for the
for the dotted line
nite point (Y. However, it cannot exist for the following reason; the initial “velocity” of the particle is given by u’(0) cx f’(0) = 0 (f’(r) 0: T from (a-ii)). Hence the particle does not carry enough kinetic energy to return to its starting point in this dissipative system with a conservative potential. Note that when n # 0, even though the initial velocity is in general equal to 0 except for n = -1, the solutions are possible because of the impact term at r = 0. In the opposite case cx > cash-’ V, it is similar to (a-ii) and only rolling down the hill is permitted. A detailed numerical study given in Figs. 1 and 2 indeed confirms the existence of these solitons. Note that there do not exist any topological lump solutions because nn(M) = 0. And the topological vortices do not exist because there is no bump in the effective potential where the particle can stop at the top. In the solutions, the magnetic flux is given by Cp = 27ry, and the energy is saturated by the topological charge; E = 4nlTI = 2x17(1 - V)l. The system also carries a nonvanishing angular momentum. Let us define J =
s
d2xEijZiDOQADjQBvAB.
(3.8)
By a simple calculation using the Gauss law (2.11) (with the plus sign in the righthand side due to the E factor), and the self-dual equations (2.14) and (3.6), we find J = m((+y-r1)~1~~). Thus the solitons in general carry a fractional angular momentum, representing anyons. For the nontopological solitons it is simply J = rny2.
280 4.
P. OH
Conclusions
We have shown that the coadjoint orbit approach for the relativistic NLSM coupled with a CS gauge field leads to a class of new self-dual field theories on the target space of HSS which contain the previous O(3) and CP(N - 1) models, and a new noncompact SU(1, 1) model. We have also found an explicit expression of the Bogomol’nyi potential when the maximal torus subgroup is gauged, and we have shown that the noncompact NLSM admits self-dual soliton solutions which are saturated by the Bogomol’nyi bound, and gave a complete description of the rotationally symmetric solutions. There remains, several further issues to be discussed. First, note that the identity (1.6) and its gauged version on HSS is essential for the existence of self-duality. In this respect, it would be an intriguing problem to extend the above formalism to other non-HSS coadjoint orbits, and also to higher noncompact groups. Quantization of the model is another problem to be addressed. Second, it would be interesting to see whether there exists a well-defined procedure in which the nonrelativistic NLSM of the generalized CS Heisenberg ferromagnet system defined on the coadjoint orbits [6] could emerge as a nonrelativistic limit of the present relativistic NLSM. This would require finding a connection between the symplectic structure of HSS [6] for the non-relativistic NLSM and the phase-space structure of relativistic NLSM.
Acknowledgements
I thank D. Chae, Y. Kim, K. Kimm, K. Lee, Q-H. Park, and C. Rim for useful discussions, Sung-Soo Kim for his invaluable help, and Prof. A. Strasburger for his hospitality at XVI Workshop on Geometric Methods in Physics. This work is supported in part by the Korea Science and Engineering Foundation through the project number (950702-04-Ol-3), and by the Ministry of Education through the Research Institute for Basic Science (BSRI/97-1419). REFERENCES [ll P. Oh and Q-H. Park: Phys. Len. B 383 (1996), 333. 121 M. Lakshmanan: Phys. Len. A 61 (1977), 53; L. A. Takhtajan: Phys. Let?. A 64 (1977), 235; J. Tjon and J. Wright: Phys. Rev. B 15 (1977), 3470. 131 L. D. Faddeev and L. A. Takhtajan: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin 1987. [41 A. P. Fordy and P. P. Kulish: Commum. Math. Phys. 89 (1983), 427. 151 P. Oh: /. Phys A: Math. Gen. 31 (1998), L325. [61 P. Oh and Q-H. Park: Phys. Lett. B 400 (1997), 157; (E) 416 (1998), 452. [71 Y. Kim, P. Oh and C. Rim: Mod. Phys. Lett. A 12 (1997), 3169. 181 G. Nardelli: Phys. Rev. Left. 73 (1994), 2524; B. J. Schroers: Phys. Left. B 356 (1995), 291; J. Gladikowski, B. M. A. G. Piette and B. J. Schroers: Phys. Rev. D 53 (1996), 844; K. Kimm, K. Lee and T. Lee: Phys. Rev. D 53 (1996), 4436; K. Arthur, D. H. Tchrakian and Y. Yang: Phys. Rev. D 54 (1996), 5245; P. K. Ghosh and S. K, Ghosh: Phys. Left. E 366 (1996), 199; Y. M. Cho and K. Kimm: Phys. Rev. D 52 (1995), 7325; K. Kimm, K. Lee and T. Lee: Phys. Lett. II 380 (1996), 303; P. K. Ghosh: Phys. Lett. B 384 (1996), 185.
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[9] We will only consider the simplest noncompact case SU(l,l). Here E = -1 is necessary to render the energy to be nonnegative in the representation in which the element K is anti-Hermitian. Higher orbits of noncompact group need a more careful treatment. [lo] A. M. Perelomov: Whys. Rep. 146 (1987), 135. [ll] K. Kimm, K. Lee and T. Lee: Whys. Len. B 380 (1996), 303. [12] S. lhkeno: Progr. Theor. Whys. 66 (1981), 1250; .I. Gruszczak: J. Phys. A: Math. Gen. 14 (1981), 3247. [13] A. A. Belavin and A. M. Polyakov: JETP Lett. 22 (1975), 245.