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Instanton solutions in supersymmetric chiral field theories
148
Nuclear Physics B (Proc. Suppl.) 6 (1989) 148-150 North-Holland, Amsterdam
INSTANTON SOLUTIONS IN SUPERSYMMETRICCHIRAL FIELD THEORIES Sultan CATTO and Yusuf GURSEY Baruch College of the City U n i v e r s i t y of New York, New York, NY i0010 and NSLS, Brookhaven National Laboratory, Upton, Long I s l a n d , NY 11973 * Instanton s o l u t i o n s in three and f o u r dimensional supersymmetric c h i r a l f i e l d presented. Possible a p p l i c a t i o n s of the underlying theory is discussed.
I. INTRODUCTION
theories are
described by the bosonic f i e l d
In two dimensions there exists supersymmetric
@ and fermionic
f i e l d ~ in n dimensions (n=3, 4 in our case) are given by
equations depending on an arbitrary function f(S) of the scalar supermultiplet S given by
(3) (I)
S: @+ of ~ +1/2 Of @ F,
72 ~ = f ( ~ ) f ' ( ~ )
+ i/2 f"(~)
~+~
0: - i a 2 @ (4)
i~ ~ ~ = f ' ( ~ ) ' ~
where @ and ~ are Majorana spinors, @ is a real scalar
and F is the a u x i l i a r y
scalar f i e l d .
The
where ~ = I . . . . . n,
and
f u n c t i o n f(S) is given by (5) (2)
f(S) = f(@) + f'(@) ~'~ 1/2
Gto{f'(~)
{~ ,Vv} =2 6 ,j
+
F+1/2 f " ( ~ )
~+~}.
(In Minkowski space ,~-t_> # = ~f¥4, 6~v-> ~ v ") Solutions are assumed to be, in t h e i r general
In three dimensions conformally i n v a r i a n t forms
fo rm
of s i m i l a r equations have been w r i t t e n 1'2 Recently i t was shown3 t h a t there e x i s t s a
(6)
= mn/2-1 ( i + x2k2x2)-~
(7)
= a ml-6/2(1+ipky x) ~ n
g e n e r a l i z a t i o n in three and four dimensions i n v o l v i n g an a r b i t r a r y f u n c t i o n f ( S ) . P a r t i c u l a r choice of f(S) = kS3 in three dimensions gives the conformally i n v a r i a n t case 4 considered in
w~th y.x=y
x ~, nTrl = 1 and
references 1,2 and 4. In four dimensions, only the choice of f(S) = kS2 y i e l d s the conformally
(8)
f'(~):
b #P,
i n v a r i a n t case 1. n = constant fermionic spinor. Number of 2. INSTANTONS I t was shown3 t h a t the equations of motion t h a t are s a t i s f i e d f o r a supersymmetric theory
dimensions of space is n. The f a c t o r m has dimension - I ,
and exponent is obtained from
dimensional a n a l y s i s . U t i l i z i n g
*Research supported in part by the PSC-CUNY Research Award No. 6-67354. 0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Eq.(7) one
S. Catto, Y, Gi~rsey ] Instanton solutions in supersymmetric chiral fieM theories
149
One can now solve for y in terms of n. In other
obtains
words, one w i l l find an appropriate potential (9)
i y.B~ = -a pk m1-B/2 (n-2By) @B_ 2a(~_x2/~)Bkml-~/2-1/y(n/2-1)~I/Y+B _2k(x/p)Bym-i/y(n/2-1) @i/y.
Here, n is the dimension of space, n= y~yP .
for any dimension as well as the appropriate amplitude for the Fermionic f i e l d s , a. Substitution into Eq.(|) yields (15)
v2@= k2m-2/~(n/2-1) ¢1+2/y _ I/2 lal2kn/~y, m1/y(n/2-1)-n/21@n/Y-1-
Since
kn/~. m-1/y(n/2-1).¢ 1/Y .C (10)
iy.B~= b @Pv , and also from Eq.(2)
for these to be consistent one must satisfy (11)
By = n/2, x2 = p2, or
~ =±~ .
(16) v2~ = 4k2m-(n/2-1)1~y(y+1-n/2)~ 1+I/Y 4k2m-2/y(n/2-1}y(y+l)~ I+2/Y .
Once the above equations are satisfied, ~ is
Again n is the number of dimensions, in this
redundant so one may set for convenience ~ = I,
case arising from @.x = n. Searching for
and ~ = ±I. Comparing Eq.(9) with Eq.(10), and
solutions that reconcile the above equations,
u t i l i z i n g Eq.(11).
one sees that necessarily
(12a)
b = -(k/~)nm -I/Y(n/2-1) ,
(17)
(12b)
p = i/y
,
C = O.
One possibility is that the following equations are satisfied:
(12c)
f'(@) = -(k/~) nm-1/Y(n/2-1) ~I/y. (18)
n/y-I = 1 + I / y
The fields become and
(13)
= mn/2-1(1+ k2x2)-Y (19)
n2y/y+l = -4y(y+1).
and These equations do not lead to positive integer (14)
= aml-n/2y(n/2-1)(l+i~ky x) n/2y
solutions for n and must be discarded. We are then led to the condition
where
= ±1, and ntn
= 1. From Eq. (12c) one
has (13)
(20) f,,:_(k/yp)m-1/y(n/2-1)n@I/y-1
(14)
which clearly leads to the equations (21)
and f = _(k/~)m-i/y(n/2-1)ny/(y+l).@i/y+1+C.
y+l-n/2 = 0
v2~ = -k2m-2n2y/y+l 1/2 lal 2 kn/~y, m1-n/2Y ~i+2/~
150
S. Catto, Y. Giirsey /Instanton solutions in supersymmetric chiral field theories
and (22)
and V2¢ = -4k2m-2y(y+l) @1+2/y
(32)
~ = 2v~vk/m. ( l + i k y . x )
2
.
U t i l i z i n g the above three equations, we have
The Lagrangians f o r both of these cases are
(23)
and hence conformal invariant.
supersymmetric, scale invariant, r e l a t i v i s t i c la[ 2 = 8k~m-2+2/(n-2)(n/2-1) 2
The four-dimensional second-order supersymTherefore u = I . Thus, in general, provided
metric equations involving an a r b i t r a r y
n ~ 2,
function f(S) of an unconstrained chiral superfield discussed above and shown in
(24)
f(@) = _2km-1(n/2_1) ¢(n/n-2)
The fields, then, are given by (for n> 2)
reference I , could arise as chiral fermion theories (involving left-handed spinors only) which might be derived from effective Lagrangians approximating supersymmetric grand
(25)
~ = mn/2-1(1+k2x2)-(n/2-1),
unified theories or weak interaction theories. We shall explore this aspect as well as the
(26)
~ =(n-2)J2Vk/m . (l+iky x)~n/n~2
applications to the high temperature limits in subsequent p u b l i c a t i o n s . 5
The Lagrangian leading to equations (1) and (2) is supersymmetric for n = 2, 3, and 4. Thus, the solutions in three and four dimensions are of particular interest. For n = 3,
ACKNOWLEDGEMENTS We thank F. GUrsey, H-C. Tze. Y. K. Ha, D. Chesley, D. Hylton, M. Sidran, M. Soto, R. Mirman for stimulating discussions, and Dean Norman Fainstein for his enthusiasm and
(27)
(28)
f(~) = _km-I ~3 = Jm (l+k2r2) - I / 2 , (-~'=~-~, - ~ ' . ~ = r 2)
and (29)
financial support. REFERENCES 1. E. Witten, Nucl. Phys. B185 (1981) 513. 2. C. Fronsdal, Phys. Rev. D26 (1982) 1988.
~ = J2Jk m-1(l+iko.--~T-.-.-.-.-.-.-.~¢ 3
3. S. Catto and F. G~rsey, Phys. Rev. 29 (1984) 653.
where ~ ' = ~ a r e dimensions.
the Pauli matrices in three 4. I. Affleck, J. Harvey, and E. Witten, Nucl. Phys. B206 (1982) 413.
S i m i l a r l y , for n = 4, 5. S. Catto and Y. G~rsey, to be published. (30) (31)
f(~) = _2km-1 ~2 ~ : m(l+k2r2) " I
Nuclear Physics B (Proc. Suppl.) 6 (1989) 148-150 North-Holland, Amsterdam
INSTANTON SOLUTIONS IN SUPERSYMMETRICCHIRAL FIELD THEORIES Sultan CATTO and Yusuf GURSEY Baruch College of the City U n i v e r s i t y of New York, New York, NY i0010 and NSLS, Brookhaven National Laboratory, Upton, Long I s l a n d , NY 11973 * Instanton s o l u t i o n s in three and f o u r dimensional supersymmetric c h i r a l f i e l d presented. Possible a p p l i c a t i o n s of the underlying theory is discussed.
I. INTRODUCTION
theories are
described by the bosonic f i e l d
In two dimensions there exists supersymmetric
@ and fermionic
f i e l d ~ in n dimensions (n=3, 4 in our case) are given by
equations depending on an arbitrary function f(S) of the scalar supermultiplet S given by
(3) (I)
S: @+ of ~ +1/2 Of @ F,
72 ~ = f ( ~ ) f ' ( ~ )
+ i/2 f"(~)
~+~
0: - i a 2 @ (4)
i~ ~ ~ = f ' ( ~ ) ' ~
where @ and ~ are Majorana spinors, @ is a real scalar
and F is the a u x i l i a r y
scalar f i e l d .
The
where ~ = I . . . . . n,
and
f u n c t i o n f(S) is given by (5) (2)
f(S) = f(@) + f'(@) ~'~ 1/2
Gto{f'(~)
{~ ,Vv} =2 6 ,j
+
F+1/2 f " ( ~ )
~+~}.
(In Minkowski space ,~-t_> # = ~f¥4, 6~v-> ~ v ") Solutions are assumed to be, in t h e i r general
In three dimensions conformally i n v a r i a n t forms
fo rm
of s i m i l a r equations have been w r i t t e n 1'2 Recently i t was shown3 t h a t there e x i s t s a
(6)
= mn/2-1 ( i + x2k2x2)-~
(7)
= a ml-6/2(1+ipky x) ~ n
g e n e r a l i z a t i o n in three and four dimensions i n v o l v i n g an a r b i t r a r y f u n c t i o n f ( S ) . P a r t i c u l a r choice of f(S) = kS3 in three dimensions gives the conformally i n v a r i a n t case 4 considered in
w~th y.x=y
x ~, nTrl = 1 and
references 1,2 and 4. In four dimensions, only the choice of f(S) = kS2 y i e l d s the conformally
(8)
f'(~):
b #P,
i n v a r i a n t case 1. n = constant fermionic spinor. Number of 2. INSTANTONS I t was shown3 t h a t the equations of motion t h a t are s a t i s f i e d f o r a supersymmetric theory
dimensions of space is n. The f a c t o r m has dimension - I ,
and exponent is obtained from
dimensional a n a l y s i s . U t i l i z i n g
*Research supported in part by the PSC-CUNY Research Award No. 6-67354. 0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Eq.(7) one
S. Catto, Y, Gi~rsey ] Instanton solutions in supersymmetric chiral fieM theories
149
One can now solve for y in terms of n. In other
obtains
words, one w i l l find an appropriate potential (9)
i y.B~ = -a pk m1-B/2 (n-2By) @B_ 2a(~_x2/~)Bkml-~/2-1/y(n/2-1)~I/Y+B _2k(x/p)Bym-i/y(n/2-1) @i/y.
Here, n is the dimension of space, n= y~yP .
for any dimension as well as the appropriate amplitude for the Fermionic f i e l d s , a. Substitution into Eq.(|) yields (15)
v2@= k2m-2/~(n/2-1) ¢1+2/y _ I/2 lal2kn/~y, m1/y(n/2-1)-n/21@n/Y-1-
Since
kn/~. m-1/y(n/2-1).¢ 1/Y .C (10)
iy.B~= b @Pv , and also from Eq.(2)
for these to be consistent one must satisfy (11)
By = n/2, x2 = p2, or
~ =±~ .
(16) v2~ = 4k2m-(n/2-1)1~y(y+1-n/2)~ 1+I/Y 4k2m-2/y(n/2-1}y(y+l)~ I+2/Y .
Once the above equations are satisfied, ~ is
Again n is the number of dimensions, in this
redundant so one may set for convenience ~ = I,
case arising from @.x = n. Searching for
and ~ = ±I. Comparing Eq.(9) with Eq.(10), and
solutions that reconcile the above equations,
u t i l i z i n g Eq.(11).
one sees that necessarily
(12a)
b = -(k/~)nm -I/Y(n/2-1) ,
(17)
(12b)
p = i/y
,
C = O.
One possibility is that the following equations are satisfied:
(12c)
f'(@) = -(k/~) nm-1/Y(n/2-1) ~I/y. (18)
n/y-I = 1 + I / y
The fields become and
(13)
= mn/2-1(1+ k2x2)-Y (19)
n2y/y+l = -4y(y+1).
and These equations do not lead to positive integer (14)
= aml-n/2y(n/2-1)(l+i~ky x) n/2y
solutions for n and must be discarded. We are then led to the condition
where
= ±1, and ntn
= 1. From Eq. (12c) one
has (13)
(20) f,,:_(k/yp)m-1/y(n/2-1)n@I/y-1
(14)
which clearly leads to the equations (21)
and f = _(k/~)m-i/y(n/2-1)ny/(y+l).@i/y+1+C.
y+l-n/2 = 0
v2~ = -k2m-2n2y/y+l 1/2 lal 2 kn/~y, m1-n/2Y ~i+2/~
150
S. Catto, Y. Giirsey /Instanton solutions in supersymmetric chiral field theories
and (22)
and V2¢ = -4k2m-2y(y+l) @1+2/y
(32)
~ = 2v~vk/m. ( l + i k y . x )
2
.
U t i l i z i n g the above three equations, we have
The Lagrangians f o r both of these cases are
(23)
and hence conformal invariant.
supersymmetric, scale invariant, r e l a t i v i s t i c la[ 2 = 8k~m-2+2/(n-2)(n/2-1) 2
The four-dimensional second-order supersymTherefore u = I . Thus, in general, provided
metric equations involving an a r b i t r a r y
n ~ 2,
function f(S) of an unconstrained chiral superfield discussed above and shown in
(24)
f(@) = _2km-1(n/2_1) ¢(n/n-2)
The fields, then, are given by (for n> 2)
reference I , could arise as chiral fermion theories (involving left-handed spinors only) which might be derived from effective Lagrangians approximating supersymmetric grand
(25)
~ = mn/2-1(1+k2x2)-(n/2-1),
unified theories or weak interaction theories. We shall explore this aspect as well as the
(26)
~ =(n-2)J2Vk/m . (l+iky x)~n/n~2
applications to the high temperature limits in subsequent p u b l i c a t i o n s . 5
The Lagrangian leading to equations (1) and (2) is supersymmetric for n = 2, 3, and 4. Thus, the solutions in three and four dimensions are of particular interest. For n = 3,
ACKNOWLEDGEMENTS We thank F. GUrsey, H-C. Tze. Y. K. Ha, D. Chesley, D. Hylton, M. Sidran, M. Soto, R. Mirman for stimulating discussions, and Dean Norman Fainstein for his enthusiasm and
(27)
(28)
f(~) = _km-I ~3 = Jm (l+k2r2) - I / 2 , (-~'=~-~, - ~ ' . ~ = r 2)
and (29)
financial support. REFERENCES 1. E. Witten, Nucl. Phys. B185 (1981) 513. 2. C. Fronsdal, Phys. Rev. D26 (1982) 1988.
~ = J2Jk m-1(l+iko.--~T-.-.-.-.-.-.-.~¢ 3
3. S. Catto and F. G~rsey, Phys. Rev. 29 (1984) 653.
where ~ ' = ~ a r e dimensions.
the Pauli matrices in three 4. I. Affleck, J. Harvey, and E. Witten, Nucl. Phys. B206 (1982) 413.
S i m i l a r l y , for n = 4, 5. S. Catto and Y. G~rsey, to be published. (30) (31)
f(~) = _2km-1 ~2 ~ : m(l+k2r2) " I