Nuclear Physics BI30 (1977) 93-104
© North-Holland Publishing Company
CLASSICAL SOLUTIONS IN TWO-DIMENSIONAL SUPERSYMMETRIC
FIELD THEORIES
P. DI VECCHIA
International Centre for Theoretical Physics, Trieste, ltaly
NORDITA, Copenhagen, Denmark
S. F E R R A R A *
International Centre for Theoretical Physics, Trieste, Italy
Laboratori Nazionali di Frascati, Rome, Italy
Received 1 August 1977
Classical solutions of some supersymmetric field theories in two dimensiofis are investigated. These solutions are constructed by solving first-order differential equations in
superspace, which are the supersymmetric extension of the analogous equations used in
the purely bosonic sector.
1. I n t r o d u c t i o n
Non-perturbative solutions of the classical equations of motion have been constructed explicitly in a number of field theories. In Minkowski space the time-independent solutions of the classical equations of motion correspond to new particles
of the physical spectrum [1 ], while in euclidean space they are useful to get approximate expressions for Green functions of the field theory [2]. Their stability is ensured by the existence of topological currents which are conserved independently of
the equations of motion. Classical solutions of the fermion field in the background
field of the instanton have also been extensively studied [3,4].
It has been pointed out [ 5 - 7 ] that in a Jlumber of field theories the minima of
the action can be computed more easily by solving first-order differential equations
instead of the more complicated second-order Euler-Lagrange equations. In general,
a solution of the first-order equations corresponds to an N-instanton solution with
non-interacting instantons.
It has also been recognized [4,8] that if one considers a supersymmetric theory
one can use the supersymmetry of the Lagrangian in order to construct non-trivial
classical solutions for the fermion field in the field of the bosonic classical solution.
* Address after I September 1977: CERN, Geneva, Switzerland.
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P. Di Vecchia, S. Ferrara / Supersymmetric field theories
94
The supersymmetry of the Lagrangian has also been used in the case of the YangMills theory with the fermions in the adjoint representation, to get a relation between
the bosonic and the fermionic zero modes [9].
In this paper we consider a number of supersymmetric field theories in two dimensions and we write first-order differential equations, which are the supersymmetric
generalization of the corresponding equations with only boson fields. These equations
imply that the first-order differential equations for the bosons are not changed by the
presence of the fermions and they give, for the latter, algebraic relations which in
some cases are very useful to find explicitly the classical solution of the fermions in
the field of the instanton.
In sect. 2 we discuss the classical solutions of the supersymmetric extension of the
two-dimensional euclidean non-linear o-model. Sect. 3 is devoted to the supersymmetric euclidean Higgs model. Finally in sect. 4 we consider the supersymmetric extension of the soliton solutions in two dimensions with an arbitrary potential.
2. Euclidean supersymmetric ~-model in two dimensions
The supersymmetric version of the two-dimensional non-linear o-model is given
by the following action:
s
=fd2xd20 ~D~ 7sD~
1
i
i
(1)
with the additional constraint
oi~ i= 1 ,
i = 1, 2, 3 .
(2)
4~i is the following superfield:
Oi(x, O) = Ai(x) + iOxi(x) + ~ i 03,50 Fi(x) ,
(3)
which transforms according to the vector representation of 0(3). The covariant derivative is given by
Da = ~
+ i(3`uO)a 3ta.
(4)
In two dimensions the euclidean 3' matrices can be chosen to be real and they are
given by
l,"
3`0 = _
0
3'1 =
1
(i 1)
0
3'5 = 3'0')'1 =
10)
. (5)
1
This is different from the four-dimensional case where a real representation for the
7 matrices [9] does not exist. The euclidean invariant scalar product between two
Majorana spinors ~0and × is given by ~0~×~.
P. Di Vecchia, S. Ferrara / Supersymmetric field theories
95
The action (1) is invariant under the following supersymmetry transformations:
~c~i = e(~o - iTuO~u) ¢i .
(6)
Starting from the action (1) and taking into account the constraint (2), one gets the
following Euler-Lagrange equations:
(O7sO) q~i = -Oi(OOj~[5OOj) •
(7)
In the case of the conventional non-linear o-model in two dimensions, Belavin and
Polyakov [6] have found the minima of the action by solving the first-order differential equation
DlzAi(x) ¥ eiik euvA] av Ak = 0 .
(8)
The supersymmetric generalization of this first-order equation (8) is given by
Oa(~i(x, O) z- ei]k O](x, O) (TsD)• (~g(x, 0) = 0 .
(9)
Eq. (9) implies the equation of motion (7), as can easily be seen by multiplying (9)
by DTs and using the relation DaD a = O.
In terms of the component fields, from the term which is 0-independent one gets
the following equation (we choose the minus sign in eq. (9)):
Xi
= 6ijk
AjTs Xk -
(1 O)
The terms linear in 0 give the following equations:
eijkAjF k = 0,
(1 la)
1
F i = ~ i eijk(XjXk),
(1 lb)
auAi = eokeu.
(1 ic)
/a.Ak +2 XiTuXk •
Finally, the quadratic term in 0 gives
7UTs3uXa" = eiig[AjTUO,×k + 7"XiOuAg + 2Fi7sXk] •
(12)
The constraint (2) gives the following equations.:
AiAi = 1 ,
(13a)
AiX,. = 0 ,
(13b)
1
-
A i F i - ) l Xi'Y s)(s"
= 0
.
(13c)
Using eq. (10) in (1 lc) one gets for the last term of(1 lc) the following expression:
( 6 i l ~ j m -- ~ i m 6 j l )
XiTuXmAI ,
which is vanishing as a consequence of (13b) and of the identity XiTuXi = O.
(14)
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P. Di Vecchia, S. Ferrara / Supersymmetric field theories
Therefore the presence of femrions in the action (1) does not modif3) the first-order equation (8) for the bosons. It is important to note that the first-order equation
(9) implies for the fermions an algebraic equation (10) together with a first-order differential equation (12). This is analogous to the case of the four-dimensional supersymmetric Yang-Mills theory where eqs. (10), (8) and (12) are replaced, respectively,
by [9]
(l - 7s)~
: 0,
7aDo ~b = O.
(15)
The solution of the system of eqs. (10), (11) and (12) can be obtained using the supersymmetry of the action (1). In terms of the component fields, the transformations
(6) give
8 A i = iex i,
8X i= (7u3~ A i + 75Fi) e ,
8 F i = ieTs 7 u 3 l a X i -
(16)
It is then easy to see that the following expression for ×i:
Xi =
(17)
T"3uAix,
satisfies eq, (10) provided that A i is a solution of eq. (8). If we then insert (17) into
eq. (12) we get that the spinor X nmst satisfy the conformal supersymmetry condition
7vTUavX = 0 .
(18)
The auxiliary field Fi can then be computed inserting (17) into eq. (1 lb). The presence of an arbitraryfunction X in the solution of the equations of motion is a consequence of the invariance of action (1) under conformal supersymmetry in two dimensions. It is now easy to check that the solutions to eqs. (10)-(12) give a vanishing contribution to the spinor current and the energy-momentum tensor. They are
given, respectively, by
}~ = (7v3,u 3 v A i x i ) a
(19)
and
O~tv = 31.tAi3vA i - 1 8~tv~pAi3pA i
+ ¼ i[xi(7~3v + "yv3u)X i
-
8~vxi,'y~/sxi)]
.
(20)
Inserting the solution (1 7) into (19), one gets
j~a : 27v [~vai~tzA i _ 1 8 vts(3p Ai)2]X
,~
v,, boson
= z')' tJu v
X,
(21)
P. Di Veechia, S. Ferrara / Supersymmetric field theories
97
which is vanishing because t~b°s°n=
0 as a consequence of eq. (8). Analogously one
~p.p
can also prove that the fermionic contribution to tile energymonlcntum tensor (20)
is also vanishing as a consequence o f ( 1 7 ) and (18).
If we perform the integration over the variables 0, the Lagrangian corresponding
to the action (1) with the constraint (2) is given by
= _ l(OuAi)2 + 1 i Xi~fl~Ol~Xi + ~1 F~ + F x ( A ~ -
1)
+ 2 A a ( F i A i - l i Xi')'sXi) + 2 i x x x i A i ,
(22)
where A x, Xa and Fx are three Lagrange multipliers corresponding to the three constraints (13). The equation of motion for the auxiliary field Fi is given by
Fi + 2A~,Ai = 0 .
(23)
Multiplying (23) by A i and using the constraints (I 3a) and (13c) one gets
A x = - ~ i Xi'YfXi,
(24)
Fi = ~i X j ' ) ' s x j A i ,
which can be inserted back into (22) to get a Lagrangian without the auxiliary fields
Fi and Ax,
22 = - ½(OuAi) 2 + ~i XiTuOuXi - -~(XiTsXi)2 + F x ( A ]
1) - 2 i A i x i x x .
(25)
It is interesting to note that the elimination of the auxiliary fields has introduced
into the Lagrangian (19) a Thirring-like four-fermion interaction. Eqs. ( 1 0 - ( 1 2 ) can
also be solved using the supersymmetric generalization of the procedure used in ref.
[6]. It is convenient to introduce the complex variable z =~(Xo + ix l) together with
the superfields
q%_= X/~(q51 -+ i4~2)
and
q~3
and the covariant derivative
(26)
D+ = (1 + i T s ) D .
In terms of these new quantities eqs. (9) become
D+(~+ = q~aD+q~+ -
~b+D+q53
,
(27a)
D+~_ = -(/)3D+4)- + q~_D+4)3 ,
(27b)
D+q~3 = -4)+D+~+ + 4)_D+~_ .
(27c)
If we now introduce the superfield
~+
,I,(z, z*, 0+, 0 _ ) = - 1 - q53 '
(28)
it is then easy to prove that eqs. (21)imply that
D+ xP(z, z*, 0+, 0 _ ) = 0 ,
(29)
P. Di Vecchia, S. Ferrara / Supersymmetric field theories
98
where
(30)
0+_ =(1 + & s ) 0 .
If we expand q~ in terms of the component fields one gets
'4,(z, z*, 0+, 0 _ ) =.~ (z, z*) + iO+~ +(z, z*)+ iO_ 4,_(z, z*) + iO÷O_
Z, Z*)
(31)
and we use the fact that
D+ = 2[ (1+ i')'5) ~0_
~-~-+'YlO - O-~
~ 1'
(32)
then eq. (23) implies that
~_=0=7,
b
--4+:-~z*
~z*
~ =0.
(33)
Therefore the equations of motion (10-(13) imply that the superfield ~P(z, z*, 0 + 0 )
is only a function ofz and 0+, i.e.
xI,(z, 0+) = ~ (z) + iO+t~(z) .
(34)
This is the supersymmetric extension of the analogous result of ref. [6] for the field
~ ( z ) (see eq. (12) of ref. [6]).
The supersymmetric invariant expression for the topological number is given by
N = ~1
fd2x d20 D(°i el/k $/D$k
.
(35)
If the equation of motion (9) is satisfied, one gets that (35) is proportional to the action (1):
S : 8~rN,
and it gets contribution only from the boson field A
(36)
i.
3. Euclidean supersymmetrie Higgs model in two dimensions
In this section we consider the two-dimensional euclidean version of the Higgs
model [5]. This model, as far as the bosonic sector is concerned, is known to have
instanton solutions. Moreover, its solutions can be interpreted as three-dimensional
static solutions [ 10] (vortex lines).
The supersymmetric euclidean vector multiplet [1 l] is given by the following
spinor superfield:
V,~(x, O) = ~c~ + (?uO),~Bu + Oc~M + ('Ys0)~L + ½i 07s0p~ .
(37)
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P. Di Vecehia, S. Ferrara / Supers.vmmetric field theories
The (Abelian) gauge transformation is defined as
(38)
6 V~ = i D e A ,
A being a (real) scalar superfield.
In the Wess-Zumino [12] gauge ~ = L = 0, M and ~'u are gauge-invariant fields,
while Bu undergoes the usual gauge transformation
(39)
6B u = 3uA ,
A being the first component of the superfield A(x, 0) = A + .... The properly normalized superfield Lagrangian density for the vector multiplet is
(40)
16 D V D T s D D V .
Note that the gauge-invariant super field D V is the supersymmetric generalization of
the field strength. The Higgs superfield is given by a doublet S i which transforms as
a vector under the 0(2) gauge group
(41)
~S i = g A S i e ii ,
g being the gauge coupling constant and A a superfield gauge parameter. The supercovariant derivative is defined as
(42)
V O = D a 6 i~ + ig Va e O"
and it has the property of being covariant under supersymmetry as well as local gauge
transformations.
The Higgs Lagrangian is constructed in terms of the supergauge field V~, the super.
Higgs field Si, and an additional superfield S, which is an 0(2) singlet but whose introduction is needed in order to have spontaneous symmetry breaking. The final action is given by
I = f d2x d20 [ 1 D V D T s D O V + I OS~fsDS
+ ~(vs)i75 (VS) i + i x s i 2 s + inS] •
(43)
We note that the above action is invariant under (real) euclidean supersymmetry for
any value of X and g (and r/).
In terms of the component fields
S i = A i + i o ~ i + I i OTsOF i ,
(44)
S = N +iOx + ½ i 0 7 5 0 D ,
the complete bosonic potential is
-~(XA~ + ~/)2 + ~ g2 M 2 A 2 _ 2X2 N 2 A 2 "
(45)
Moreover, the kinetic terms of the bosonic fields are
lt'o M) 2
--~.1 d V - I(0/.tN)2 + ~.
~
--
~(OuAi) 2
,
(46)
1O0
P. Di Vecchia, S. Ferrara / Supersymmetrie field theories
which shows that the scalar field M, contained in V~ has opposite sign with respect
to the ordinary bosonic fields. We note also that for N = M = 0 the bosonic Lagrangian just reproduces the usual Higgs Lagrangian.
The Euler-Lagrangian equations are
D , ~ D T s D D V = ig e ij Si75 Vo~ S j ,
1 D T s D S = iXSi2 + it7,
l(VTs rTS) i = 2t'Asis .
(47)
These equations can be solved by means of the first-order equations
3's 7~S" = eq(7~S) i ,
(48)
1 Da V~ = S
(49)
(or the same ansatz with the - sign) provided
g2 = 4X2 .
These first-order equations are just the supersymmetric extensions of the first-order
equations of motion of the Higgs model. Note in fact that eq. (48) implies
~'s q,i = eoq,],
(50)
D u A i = - e u v eq Dv A ] ,
(51)
while eq. (49) implies
m = N,
23's~" = X
(52)
and
Fur = - e u v O t A ] + r/),
(g = 2X).
(53)
Eqs. (51) and (53) are the usual first-order equations of motion of the pure Higgs
model [5].
The value g2 = 4X2, for which the equations can be solved by imposing first-order
differential equations, has a particular meaning because only for this value does the
action have a bigger symmetry *. In fact, when g = 2X, the action is invariant under
extended euclidean supersymmetry with internal symmetry 0(2). [Complex Euclidean
supersymmetry.] Va and S belong to the same irraducible (real) multiplet of complex
euclidean supersymmetry
V = C + ioirl i + i(oi~[50] - ]61i]o I,y501) Ti ]
* From a physical point of view, the valueg 2 = 4X2 is the transition point between superconductivity of first and second type.
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P. Di Vecchia, S. Ferrara / Supersymmetrie field theories
+ ioio/eiiM + ioiTuoieiiBu + ioi750iN
(54)
+ oJ'Y5oJoi~ i ÷ oi~/50ioJ~$oJo ,
in which C = 7/i = Ti] = 0 in the Wess-Zumino gauge and then (M, Bu, ?~2) and
(N, ?~1,D) are, respectively, the component fields of the two real multiplets V~ and
S of the simple euclidean supersymmetry previously introduced. The Higgs field S i
becomes a complex scalar multiplet of complex supersynnnetry
(55)
SI + iS2 = S = A + iOx + ~ i OOF ,
in which
0=01+i02,
A=Al+iA
2,
x=xl+ix
2,
F=FI+iF
2.
The transformation laws of superfields are
~O(x, 0 i) = O(x - iei'yo i, 0 i + e l ) ,
(56)
so that one can easily check the previous identification. For instance,
~ M = 1 i 6i~5~k]e i] ,
6 B u = ½ i e/iei'ys'~xJ.
6 N = - ~1 i ei~ i ,
(57)
It is easy to show that the M and N scalar fields in V have opposite kinetic term
in complex euclidean supersymmetry. This not surprising in view of the fact that the
0(2) euclidean supersymmetry can be obtained by reduction of Minkowski four-dimensional supersymmetry in which z, t are treated as internal coordinates [13]. M
and N are just the space and the time components of a four-dimensional Minkowski
vector *. This is also analogous to the four-dimensional euclidean supersymmetry investigated in ref. [9], in which the two scalar fields also have opposite kinetic terms
because of a similar phenomenon.
We conclude the discussion on the Higgs model by again noting the fact that the
first-order equations for the bosons are not modified by the fermions and that the
supersymmetric extension of the first-order equations implies the Weyl condition for
the fermions together with their first-order equations in the field of the bosonic classical solution.
4. Static classical solutions in two-dimensional Minkowski space with supersymmetry
So far we have considered instanton solutions, i.e. classical solutions in euclidean
two-dimensional field theories. We now turn to soliton solutions, i.e. classical, timeindependent solutions in two-dimensional Minkowski space. These solutions may be re* From this point of view, the previous Lagrangianis nothing but the dimensional reduction [ 13]
of the supersymmetric Minkowski four-dimensional Higgsmodel [14].
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P. Di Vecchia, S. Ferrara / Supersymmetric field theories
garded as one-dimensional instantons just as two-dimensional instantons are related
to three-dimensional vortex solutions.
We consider a bosonic field theory with potential
U(4)) = - ~ V'(4)) ~
(58)
and Lagrangian
~0(4)) : _ ~(0u4))2 + U(4)) •
(59)
If we add to it a fermionic part, then the complete Lagrangian
Z(4), X) = -~(Ou4)) 2 - ½i X~X + V(4)) - ~ i £X Z(4))
(60)
is supersymmetric provided
Z(4)) = V"(4)).
(61)
As a Lagrangian density in superspace it corresponds to
.12(S) = 1 D S D S - i V ( s ) ,
(62)
S = 4) + i-Ox + ~ i O O F .
(63)
with
It is straightforward to show that 8./2 is a total divergence under the variations
84) = i e x ,
8× = ~4)e - V'(4)) e ,
(64)
e being a (constant) anticommuting spinor.
The field equations are
I--14)= V'(4)) V"(4)) + ½ i XX V"(4)),
(ib + V"(4))) × = 0 .
(65)
A soliton solution is a solution of the first-order equation
4)' = + V'(4)),
in which 4)' = dx
d 4)(x).
(66)
As we consider static solutions, then
(l + 71)')(=X+
= 0 ,
X-+ = + V (0) X± •
(67)
Supersymmetry gives
84) = ½ i e . ×~ ,
8X~ = ~ 24)'e~ .
(68)
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P. Di Vecchia, S. Ferrara / Supersymmetric ]ield theories
It is trivial to show that
X = (~q5
V'(0)) e
(69)
is an exact solution of the coupled system of eqs. (65) with 0 still a solution of the
equation
~' = -+ v ' ( 4 0 •
(70)
We also point out that the fermionic constraint in eq. (67) as well as the bosonic
first-order equations are given by the first-order supersymmetric constraint
(I -+ 71) Dq~ = O,
(71)
which gives the equations
X+ = 0 ,
F +- qS' = 0 ,
i.e. ~b' = -+ V'(~).
(72)
The above considerations apply in particular to the sine-Gordon potential V'(q~) =
A cos(~q~ + t3) and to the 4~4 theory with V'(40 = (~2 _ m2). In this way we have
found exact fermionic solutions in the soliton field. They are given by eq. (69), with
q5given by a solution of eq. (70).
5. Conclusions
In the present paper we have investigated a wide class of classical solutions of twodimensional field theories. With the requirement of supersymmetry the first-order differential equations for the bosons have been extended to the fermionic sector. These
supersymmetric first-order relations do not alter the bosonic solution, while they
give an algebraic equation for the fermion, a kind of Weyl condition, together with
their first-order differential equation.
In our case, namely the euclidean o-model, because of superconformal symmetry,
these relations completely determine the fermionic solution in terms of an anticommuting spinor, analytic in the variable z = x o + ixa. This solution is the supersymmetric extension of the corresponding bosonic solution found in ref. [6].
In the case of the Higgs model [5] the critical value gZ = 4)~2, for which the equations of motion can be solved in first-order, has been shown to be related to a higher
euclidean supersymmetry; i.e. to the extended euclidean supersymmetry with 0(2)
as internal symmetry. The rather curious fact that kinetic terms of bosons do not occur with the same sign can be understood simply with the dimensional reduction from
four-dimensional Minkowski supersymmetry, according to the analysis of ref. [13].
This phenomenon is quite analogous to the four-dimensional case [9], which can also
be understood from dimensional reduction from Minkowski six-dimensional supersymmetry [ 13].
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P. Di Vecchia, S. Verrara / Supersymmetric fieM theories
The authors would like to thank Professor Abdus Salam, the International Atomic
Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
Note added in p ro o f
It has been pointed out to us that a supersymmetric version o f the sine-Gordon
equation has been also constructed by J. Hruby, Dubna preprint.
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