- Email: [email protected]
U(1) harmonic equations in the gauge theories
Volume 209, number 4
PHYSICS LETTERS B
l 1 August 1988
O N T H E S O L U T I O N S O F SU(2)/U(1) H A R M O N I C E Q U A T I O N S I N T H E G A U G E T H E O R I E S B.M. Z U P N I K Research Institute of Applied Physics, Tashkent State University, Vuzgorodok, Tashkent 700 095, USSR Received 5 April 1988
Within the harmonic approach a self-duality equation, solutions of superfield constraints in the supergauge theory SYM], and other nonlinear problems in gauge theories may be reduced to universal linear differential equations for the "bridge" matrix h or harmonic connection A (-2). We consider a wide class of explicit solutions for h and A (-2~, which can be obtained by quadratures using a special harmonic parametrization of the analytical prepotential. This method gives a general solution in the case of SYM4.
The harmonic S U ( 2 ) / U ( 1 ) formalism in gauge and supergauge theories can be interpreted as a version o f the twistor approach, in which the S U ( 2 ) / U ( 1 ) harmonics u +, u;- are treated as coordinates o f the sphere S 2 ( u + i u T = l ) . This formalism is closely related to the solutions of superfield constraints of N = 2, D = 4 supersymmetric theories in the framework of the harmonic superspace [ 1 ]. The harmonic approach has also been used in the solution of the self-duality equation and other nonlinear problems in the nonsupersymmetric gauge theories [2]. We shall use the basic notations o f refs. [ 1,2]. The invariant differential operators O( + 2), O( - 2), Oo on SU ( 2 ) / U ( 1 ) satisfy the following relations: [0(+2),0(-2)]----0 °,
[ 0 ° , 0 ( + 2 ) ] = + 2 0 (+2~.
(1)
The operator O° generates the U ( 1 ) charge q. It should be noted that the conformity between the harmonic and the usual complex description ofS 2 [ 3 ] can be obtained with the help o f the relations 1
1
u ~+' = u ~ =
Let us describe here the general features of the harmonic formalism in (super)gauge theories [ 1,2,4 ]. The basic equations or constraints of the (super) gauge theory are formulated in a real (super)space with coordinates z (or x). The gauge r-group G with matrix parameters r ( z ) acts in this real space. The vector or spinor connections A ( z ) are the basic (super) field objects of the r-representation. Using the harmonics u + one can reduce the basic equations to the integrability conditions for the covariant derivatives in special directions. This integrability is equivalent to the covariant conservation of a certain analyticity, which is manifested in a special analytic ( super ) space with coordinates ~(z, u ), u +. The gauge A-group with analytic parameters A (~, u) acts in the analytic basis o f the (super)gauge theory. The "bridge" matrix h (z, u ) ~ G couples the A- and r-bases of the gauge group. This matrix satisfies the following harmonic equation [ 1 ]:
(0(+2~+
0
-~(1 +~4) ' "~~- [ ( 1 + ~ g ? f cs) (~, ~) ] ,
(3)
where q = 2S and S is the spin weight. SU (2) / U ( 1 ) harmonics are pseudoreal under the special involution [ 1 ] 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4)
V(+2))h(z, u ) = 0 ,
(5)
where V(+2)=hO(+2)h-~ is an analytic connection (prepotential). It should be noted that the equation for a nonanalytic harmonic connection A ( 2)=h0( 2)h-I [5] may be used as the basic equation instead ofeq. (5):
O(+2)A(-2)--O(-2)V(+2)-~ [V(+2),A(-2)]=O. (6) 513
Volume 209, number 4
PHYSICS LETTERS B
This equation has a simple perturbative solution [ 5 ]. Now we shall discuss nonperturbative solutions of eq. ( 5 ) for the GL (n, C) and SU (n) gauge groups, which can be obtained with the help of special harmonic parametrization. This method was previously used in refs. [6,7 ] for the analysis of the SU(2) selfduality equation. Let us consider anti-hermitean harmonic 2X2 matrices:
(U+-2)~,=u+ u+-k,
( U ° ) ~ = u + u - k + u z u +~
(7)
A simple harmonic representation of the Lie algebra gl ( 2, C) is of the form F o, = ½ 1 2 + ½ ( U o ) , f~+2) = (U+2),
FO2 = ~ I1z - i ( U 1
fl~-2) =-
(U-2)
,
(8)
F~f,)F}.~ ) =Obc*~ iu'(p+q)ad , SpF~g) =6~b .
(9)
The general prepotential for the SL (2, C) group can be written in the following form: V (+21 =F~2+2)b0({, u) + (F°l - F ° z ) b (+2) (~, u) (10)
For the SU(2) group we must choose real analytical functions/7(q) = b (q). The general harmonic representation for the bridge h is
h(z, u) =F°IX~
det h=ZJZ2 -- ~//(--2)~/'/(+2) = 1 •
b°=b~+2)=0,
b(+4)#0.
(15)
This gauge simplifies the equations for the harmonic components of the bridge ( 1 1 ), which are equivalent to eq. (5); however, these equations cannot be solved by quadratures for the general function b ( +4) ( ~ ) . Let us consider a harmonic representation of the self-dual Yang-Mills (SDYM) equations [ 2,6,7 ], where the following decomposition holds:
b(+4)(x +, u +, u - ) = O{+2)f(+2)(x+, u +, u - ) (16)
Here f(+2) is an arbitrary analytic function and /~+4) is independent of u +. It is evident that the b (+4)-component cannot be gauged to zero in this case. Contrary to the case of SDYM we have the corresponding general decomposition in the analytical N = 2, D = 4 harmonic superspace [ 1 ]:
b (+4~(XA, 0 +, 0 +, U) =O(A+2)f( +2)(XA, 0 +, 0+, U) ,
(17)
(11)
where b ~+4) and f(+2~ are arbitrary analytical superfields. Using this decomposition one can choose the following simple gauge of the SYM 2 prepotential (10):
(12)
b(+4)=0,
+ F°2X2 -{-F (2+2) ~//(-2)
+FI~-2) ~u(+2~ ,
Using this gauge freedom one can choose the following general gauge:
+ 6(+4)(x+, u - ) .
°) ,
+Fl~-2)b(+4)({, u) .
11 August 1988
Now we will show how to exploit the gauge freedom to simplify eq. (5). Consider an infinitesimal gauge representation of V ( + 2). 8 V (+2) ~-,~0( + 2 ) 2 ~ - [ V (+2), 2 ]
b°#0,
b(+2)#0.
(18)
(Note, that for the SDYM prepotential these conditions determine a nongeneral ansatz only. ) Let us impose the subsidiary condition ~,(+2~= 0. Now one can obtain from eq. (5) the following equations for the bridge components in the gauge ( 18 ):
=F~2+2)[ O(+z)z~-2) + 22O+ 2bOZO_ 2b( +2)2 (-2)]
O(+2)ZI + b
<+2)~1 ~.~-0 ,
(19)
+ (F°t --F°22) [0(+2)2°--j (+2~+b(+4)2 ¢-2)
0(+2)X2 --bC+2)X2 = 0 ,
(20)
_bOZ(+2)]
O(+2)q](-2)-b(+2)~(-2)+Xj-X2
+F}yZ)[O(+z)z(+z)+2b(+Z)Z(+2)-2b(+4)2 °] , (13) where 2 (~) e sl (2, C) is a traceless matrix and 2 (q~are its harmonic components, ,~ =F
514
(2+2)~ (-2)
-{_ (FOl i ----22)/~, lW0 "~30._t_ . . t .Z7 (12-- 2)]~', ( + 2) .
(14)
q-b°xt---O .
(21)
Using harmonic distributions [4] one readily obtains the solutions for these equations:
Zl(z,u)=z;~(z,u)=C(z)exp[-B(z,u)l, B(Z,U)=
f
dul (bI+UF)b(+2)(Z, (u+u~_) Ul)
(22) (23)
Volume 209, number 4
PHYSICS LETTERS B
C-2(z)= f du[(l+b°)exp(-2B)](z,u),
~'~-2~(z,u)=z2(z,u)
(24)
f du, (u-u?) (u+u?------~
×{Z2(z, u , ) [ 1 +g°(z, u , ) ] - 1}.
(25)
It is easy to check the condition/~(z, u ) e S U ( 2 ) for this solution in the case o f real functions b °, b(+2): /~(z, u) = e x p [ -
( U°)~+ ( U +2) (1 -l~2)gt (-2) ] , (26)
~(z, u ) = B ( z , u ) - l n C(z) .
(27)
The general S U ( 2 ) solution for the S Y M ] bridge h can be obtained by the gauge transformation ofh. A more simple gauge o f the SYM 2 prepotential (10) corresponds to the case of b ( + 2) = b (+4) P~ 0. The bridge matrix for this gauge has the following harmonic components:
Z2=ZV'=~/I+b(z), ~(-2~=Z1
~'~+2)=0,
(u-u +)
dul ( u + u ? ) [ b O ( z , • l ) - - b ( Z )
b(z) =~ du b°(z, u) .
was solved by quadratures for the case o f a triangular subgroup o f SL(2, C). H a r m o n i c parametrization seems to be more convenient for the solution o f this nonlinear problem. N o w we will consider the generalization of the harmonic representation (10), ( 11 ) for the case o f the gl(3, C) algebra. The representation o f the gl(2, C) subalgebra can be obtained by adding the third rows and columns with zero elements to the matrices (8). The rest of the gl(3, C) harmonic matrices F2(~-~, F~ +), F(-)13 , F(32-) , F°3 have the following nonzero elements only: ( F ( 2 + ) ) -3-_ u + '
(f~+))i3=.u+i ,
(F}f))3=uT,
(F~-))~=-u
-~.
o 3 ~---1 , (F33)3 (33)
The gl (3, C) algebra of matrices F (q) has the standard form (9) with a, b = 1, 2, 3;p, q = 0 , _+ 1, +2. So now we can use this representation for the solution o f eq. (5) in the case of the SU (3) group.
(28) ]
(29)
The author would like to thank cordially A.S. Galperin and V.I. Ogievetsky for stimulating discussions on the harmonic gauge theory.
(30)
The gauge V~+2)=(U+2)b° simplifies the geometric formula for the SYM42 action [ 5 ] which can be transformed into a simple expression:
S = g -2 J d S z { l n [ l + b ( z ) ] - b ( z ) } ,
(31)
containing the real superfield (30). Note, that for the S D Y M equation the solutions (29), (28) are not equivalent to those in ( 2 2 ) - (25). The one-instanton solution corresponds to the following choice [ 6,7 ]:
b°=p2(x't"U+Uk-) -1 , b ( + 2 ) = b ( + 4 ) = 0 . The S D Y M equation in an [ 8,9 ] has been reduced also to tion on S 2, which is analogous plex parametrization (2), (3).
11 August 1988
(32)
alternative formalism a linear Sparling equato eq. (5) in the comThe Sparling equation
References [ 1] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 1 (1984) 469. 12l A. Galperin, E. Ivanov, V. Ogievetskyand S. Sokatchev, preprint JINR E2-85-363 (Dubna, 1985); preprints JINR, E287-263, 264 (Dubna, 1987). [3] R. Penrose and V. Rindler, Spinors and space-time, vols. 1, 2 (Cambridge U.P., Cambridge, 1984). [41 A. Galperin, E. Ivanov, V. Ogievetsyand S. Sokatchev, Class. Quantum Gray. 2 ( 1985 ) 601, [5] B.M. Zupnik, Soy. J. Nucl. Phys. 44 (1986) 512; Teor. Mat. Fiz. 69 (1986) 207; Phys. Lett. B 183 (1987) 175. [6l S. Kalitzin and E. Sokatchev, preprint IC/87/75 (Trieste, 1987). [7] O.V. Ogievetsky, talk at the School on Group-theoretical methods in physics (Varna, 1987). [8] E.T. Newman, J. Math. Phys. 27 (1986) 2797. [91 S. Chakravartyand E.T. Newman, J. Math. Phys. 28 (1987) 334.
515
PHYSICS LETTERS B
l 1 August 1988
O N T H E S O L U T I O N S O F SU(2)/U(1) H A R M O N I C E Q U A T I O N S I N T H E G A U G E T H E O R I E S B.M. Z U P N I K Research Institute of Applied Physics, Tashkent State University, Vuzgorodok, Tashkent 700 095, USSR Received 5 April 1988
Within the harmonic approach a self-duality equation, solutions of superfield constraints in the supergauge theory SYM], and other nonlinear problems in gauge theories may be reduced to universal linear differential equations for the "bridge" matrix h or harmonic connection A (-2). We consider a wide class of explicit solutions for h and A (-2~, which can be obtained by quadratures using a special harmonic parametrization of the analytical prepotential. This method gives a general solution in the case of SYM4.
The harmonic S U ( 2 ) / U ( 1 ) formalism in gauge and supergauge theories can be interpreted as a version o f the twistor approach, in which the S U ( 2 ) / U ( 1 ) harmonics u +, u;- are treated as coordinates o f the sphere S 2 ( u + i u T = l ) . This formalism is closely related to the solutions of superfield constraints of N = 2, D = 4 supersymmetric theories in the framework of the harmonic superspace [ 1 ]. The harmonic approach has also been used in the solution of the self-duality equation and other nonlinear problems in the nonsupersymmetric gauge theories [2]. We shall use the basic notations o f refs. [ 1,2]. The invariant differential operators O( + 2), O( - 2), Oo on SU ( 2 ) / U ( 1 ) satisfy the following relations: [0(+2),0(-2)]----0 °,
[ 0 ° , 0 ( + 2 ) ] = + 2 0 (+2~.
(1)
The operator O° generates the U ( 1 ) charge q. It should be noted that the conformity between the harmonic and the usual complex description ofS 2 [ 3 ] can be obtained with the help o f the relations 1
1
u ~+' = u ~ =
Let us describe here the general features of the harmonic formalism in (super)gauge theories [ 1,2,4 ]. The basic equations or constraints of the (super) gauge theory are formulated in a real (super)space with coordinates z (or x). The gauge r-group G with matrix parameters r ( z ) acts in this real space. The vector or spinor connections A ( z ) are the basic (super) field objects of the r-representation. Using the harmonics u + one can reduce the basic equations to the integrability conditions for the covariant derivatives in special directions. This integrability is equivalent to the covariant conservation of a certain analyticity, which is manifested in a special analytic ( super ) space with coordinates ~(z, u ), u +. The gauge A-group with analytic parameters A (~, u) acts in the analytic basis o f the (super)gauge theory. The "bridge" matrix h (z, u ) ~ G couples the A- and r-bases of the gauge group. This matrix satisfies the following harmonic equation [ 1 ]:
(0(+2~+
0
-~(1 +~4) ' "~~- [ ( 1 + ~ g ? f cs) (~, ~) ] ,
(3)
where q = 2S and S is the spin weight. SU (2) / U ( 1 ) harmonics are pseudoreal under the special involution [ 1 ] 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(4)
V(+2))h(z, u ) = 0 ,
(5)
where V(+2)=hO(+2)h-~ is an analytic connection (prepotential). It should be noted that the equation for a nonanalytic harmonic connection A ( 2)=h0( 2)h-I [5] may be used as the basic equation instead ofeq. (5):
O(+2)A(-2)--O(-2)V(+2)-~ [V(+2),A(-2)]=O. (6) 513
Volume 209, number 4
PHYSICS LETTERS B
This equation has a simple perturbative solution [ 5 ]. Now we shall discuss nonperturbative solutions of eq. ( 5 ) for the GL (n, C) and SU (n) gauge groups, which can be obtained with the help of special harmonic parametrization. This method was previously used in refs. [6,7 ] for the analysis of the SU(2) selfduality equation. Let us consider anti-hermitean harmonic 2X2 matrices:
(U+-2)~,=u+ u+-k,
( U ° ) ~ = u + u - k + u z u +~
(7)
A simple harmonic representation of the Lie algebra gl ( 2, C) is of the form F o, = ½ 1 2 + ½ ( U o ) , f~+2) = (U+2),
FO2 = ~ I1z - i ( U 1
fl~-2) =-
(U-2)
,
(8)
F~f,)F}.~ ) =Obc*~ iu'(p+q)ad , SpF~g) =6~b .
(9)
The general prepotential for the SL (2, C) group can be written in the following form: V (+21 =F~2+2)b0({, u) + (F°l - F ° z ) b (+2) (~, u) (10)
For the SU(2) group we must choose real analytical functions/7(q) = b (q). The general harmonic representation for the bridge h is
h(z, u) =F°IX~
det h=ZJZ2 -- ~//(--2)~/'/(+2) = 1 •
b°=b~+2)=0,
b(+4)#0.
(15)
This gauge simplifies the equations for the harmonic components of the bridge ( 1 1 ), which are equivalent to eq. (5); however, these equations cannot be solved by quadratures for the general function b ( +4) ( ~ ) . Let us consider a harmonic representation of the self-dual Yang-Mills (SDYM) equations [ 2,6,7 ], where the following decomposition holds:
b(+4)(x +, u +, u - ) = O{+2)f(+2)(x+, u +, u - ) (16)
Here f(+2) is an arbitrary analytic function and /~+4) is independent of u +. It is evident that the b (+4)-component cannot be gauged to zero in this case. Contrary to the case of SDYM we have the corresponding general decomposition in the analytical N = 2, D = 4 harmonic superspace [ 1 ]:
b (+4~(XA, 0 +, 0 +, U) =O(A+2)f( +2)(XA, 0 +, 0+, U) ,
(17)
(11)
where b ~+4) and f(+2~ are arbitrary analytical superfields. Using this decomposition one can choose the following simple gauge of the SYM 2 prepotential (10):
(12)
b(+4)=0,
+ F°2X2 -{-F (2+2) ~//(-2)
+FI~-2) ~u(+2~ ,
Using this gauge freedom one can choose the following general gauge:
+ 6(+4)(x+, u - ) .
°) ,
+Fl~-2)b(+4)({, u) .
11 August 1988
Now we will show how to exploit the gauge freedom to simplify eq. (5). Consider an infinitesimal gauge representation of V ( + 2). 8 V (+2) ~-,~0( + 2 ) 2 ~ - [ V (+2), 2 ]
b°#0,
b(+2)#0.
(18)
(Note, that for the SDYM prepotential these conditions determine a nongeneral ansatz only. ) Let us impose the subsidiary condition ~,(+2~= 0. Now one can obtain from eq. (5) the following equations for the bridge components in the gauge ( 18 ):
=F~2+2)[ O(+z)z~-2) + 22O+ 2bOZO_ 2b( +2)2 (-2)]
O(+2)ZI + b
<+2)~1 ~.~-0 ,
(19)
+ (F°t --F°22) [0(+2)2°--j (+2~+b(+4)2 ¢-2)
0(+2)X2 --bC+2)X2 = 0 ,
(20)
_bOZ(+2)]
O(+2)q](-2)-b(+2)~(-2)+Xj-X2
+F}yZ)[O(+z)z(+z)+2b(+Z)Z(+2)-2b(+4)2 °] , (13) where 2 (~) e sl (2, C) is a traceless matrix and 2 (q~are its harmonic components, ,~ =F
514
(2+2)~ (-2)
-{_ (FOl i ----22)/~, lW0 "~30._t_ . . t .Z7 (12-- 2)]~', ( + 2) .
(14)
q-b°xt---O .
(21)
Using harmonic distributions [4] one readily obtains the solutions for these equations:
Zl(z,u)=z;~(z,u)=C(z)exp[-B(z,u)l, B(Z,U)=
f
dul (bI+UF)b(+2)(Z, (u+u~_) Ul)
(22) (23)
Volume 209, number 4
PHYSICS LETTERS B
C-2(z)= f du[(l+b°)exp(-2B)](z,u),
~'~-2~(z,u)=z2(z,u)
(24)
f du, (u-u?) (u+u?------~
×{Z2(z, u , ) [ 1 +g°(z, u , ) ] - 1}.
(25)
It is easy to check the condition/~(z, u ) e S U ( 2 ) for this solution in the case o f real functions b °, b(+2): /~(z, u) = e x p [ -
( U°)~+ ( U +2) (1 -l~2)gt (-2) ] , (26)
~(z, u ) = B ( z , u ) - l n C(z) .
(27)
The general S U ( 2 ) solution for the S Y M ] bridge h can be obtained by the gauge transformation ofh. A more simple gauge o f the SYM 2 prepotential (10) corresponds to the case of b ( + 2) = b (+4) P~ 0. The bridge matrix for this gauge has the following harmonic components:
Z2=ZV'=~/I+b(z), ~(-2~=Z1
~'~+2)=0,
(u-u +)
dul ( u + u ? ) [ b O ( z , • l ) - - b ( Z )
b(z) =~ du b°(z, u) .
was solved by quadratures for the case o f a triangular subgroup o f SL(2, C). H a r m o n i c parametrization seems to be more convenient for the solution o f this nonlinear problem. N o w we will consider the generalization of the harmonic representation (10), ( 11 ) for the case o f the gl(3, C) algebra. The representation o f the gl(2, C) subalgebra can be obtained by adding the third rows and columns with zero elements to the matrices (8). The rest of the gl(3, C) harmonic matrices F2(~-~, F~ +), F(-)13 , F(32-) , F°3 have the following nonzero elements only: ( F ( 2 + ) ) -3-_ u + '
(f~+))i3=.u+i ,
(F}f))3=uT,
(F~-))~=-u
-~.
o 3 ~---1 , (F33)3 (33)
The gl (3, C) algebra of matrices F (q) has the standard form (9) with a, b = 1, 2, 3;p, q = 0 , _+ 1, +2. So now we can use this representation for the solution o f eq. (5) in the case of the SU (3) group.
(28) ]
(29)
The author would like to thank cordially A.S. Galperin and V.I. Ogievetsky for stimulating discussions on the harmonic gauge theory.
(30)
The gauge V~+2)=(U+2)b° simplifies the geometric formula for the SYM42 action [ 5 ] which can be transformed into a simple expression:
S = g -2 J d S z { l n [ l + b ( z ) ] - b ( z ) } ,
(31)
containing the real superfield (30). Note, that for the S D Y M equation the solutions (29), (28) are not equivalent to those in ( 2 2 ) - (25). The one-instanton solution corresponds to the following choice [ 6,7 ]:
b°=p2(x't"U+Uk-) -1 , b ( + 2 ) = b ( + 4 ) = 0 . The S D Y M equation in an [ 8,9 ] has been reduced also to tion on S 2, which is analogous plex parametrization (2), (3).
11 August 1988
(32)
alternative formalism a linear Sparling equato eq. (5) in the comThe Sparling equation
References [ 1] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 1 (1984) 469. 12l A. Galperin, E. Ivanov, V. Ogievetskyand S. Sokatchev, preprint JINR E2-85-363 (Dubna, 1985); preprints JINR, E287-263, 264 (Dubna, 1987). [3] R. Penrose and V. Rindler, Spinors and space-time, vols. 1, 2 (Cambridge U.P., Cambridge, 1984). [41 A. Galperin, E. Ivanov, V. Ogievetsyand S. Sokatchev, Class. Quantum Gray. 2 ( 1985 ) 601, [5] B.M. Zupnik, Soy. J. Nucl. Phys. 44 (1986) 512; Teor. Mat. Fiz. 69 (1986) 207; Phys. Lett. B 183 (1987) 175. [6l S. Kalitzin and E. Sokatchev, preprint IC/87/75 (Trieste, 1987). [7] O.V. Ogievetsky, talk at the School on Group-theoretical methods in physics (Varna, 1987). [8] E.T. Newman, J. Math. Phys. 27 (1986) 2797. [91 S. Chakravartyand E.T. Newman, J. Math. Phys. 28 (1987) 334.
515