- Email: [email protected]
On the nonlinear realization of superconformal symmetry
19 January
1995
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 343 ( 1995) 144-146
On the nonlinear realization of superconformal symmetry Hiroshi Kunitomo
’
Department of Physics, Osaka University, Toyonaka. Osaka 560, Japan Received 15 July 1994 Editor: M. Dine
Abstract The nonlinear realization of superconformal symmetry in two dimensions is considered. Superconformal symmetry is realized by means of the dimension - l/2 Nambu-Goldstone fermion 5 and its dimension 3/2 conjugate 7. A matter coupling of these Nambu-Goldstone fermions reproduces the realization found by Berkovits and Vafa to show the equivalence between the bosonic string and an N= 1 superstring. This indicates that this equivalence is a result of the super Higgs mechanism as a two dimensional field theory, in which the NG fermions become unphysical.
Recently, Berkovits and Vafa showed that bosonic strings may be viewed as a particular class of vacua for N= 1 superstrings [ 11. One of their key observations is that there is a realization of superconformal symmetry by means of fermions (9, Q with dimensions (3/2, - l/2) and the matter stress tensor T,,,. In this realization, the fermion 5 is transformed inhomogeneously and thus plays the role of a Nambu-Goldstone (NG) fermion. In this letter, we rederive their result from the viewpoint of the conventional nonlinear realization of supersymmetry [2]. Superconformal symmetry may be nonlinearly realized by the NG fermion pair (7, 8. Its general matter couplings reproduce the noncritical embedding obtained by Berkovits and Ohta [ 31, which includes the above realization as a critical case. We restrict our attention to the holomorphic sector of the superconformal symmetry. For studying the nonlinear realization, we consider superconformal symmetry as coordinate transformations of holomorphic ’ E-mail: [email protected]. Elsevier Science B.V. SsO10370-2693(94)01421-3
superspace. The infinitesimal try is represented by z’=z -f(z) + Oe(z) , e’=e-&3f(z>e+E(Z)
superconformal
)
symme-
(I)
wheref(z) ( E(Z) ) is an infinitesimal parameter of the (super)conformal transformation. We consider such a nonlinear realization that superconformal symmetry breaks down to conformal symmetry. The nonlinear transformation law of the NG fermion 5 is obtained by replacing 8 with the NG fermion tin ( 1) *: 5’(z’, a= z’=z-f(z)
5(z, 3 - ;V-(z)&z, +5cz, i)dZ)
Thus the infinitesimal KL
l) E(z,
a+
.
e(z) ,
(2) (3)
transformation
of ris given by
3 = s’(z, ?) - 5(z, 3
= e(z) + l(z)S(z, .&Mz, - f@z)%(z,
a +f(z)@xz,
3 ?) ,
(4)
2 Note that we need only one NG fermion while the superconformal symmetry has infinite dimensions. This seems to be a common feature when broken symmetries are generated by moments of one current 141.
H. Kunitomo / Physics Letters B 343 (1995) 144-146
where we note that the NG field 5 is transformed as a dimension - l/2 field under the conformal transformation. One can show that this transformation satisfies the required commutation relations: [SVl, El), @AZ, •2)l~=wlz~
E12)5?
(5)
where
E12
=
-fafiE2+f,aE2-(1*2).
Since 5 has negative dimension - l/2, we cannot obtain a nontrivial conformal invariant theory by using the NG fermion tarone. This is a characteristic feature of the spontaneous breaking of superconformal symmetry. Infinite dimensional unbroken symmetry, conformal symmetry, requires another degree of freedom than the NG fermion 6 The simplest way to construct it is introducing a dimension 312 fermion 77and taking the action (6)
,
$a7je+ ;a2(gan
GNG= q+ q$t-
?a”[+
#a(a2s.
(10)
These satisfy the OPE of superconformal algebra with 11. C NG= We next consider the general matter coupling, where matter fields are linear representations of the conformal symmetry. We denote a current of this matter conforma1 symmetry (stress tensor) as T,. The infinitesimal transformations of the matter fields under superconforma1 symmetry are given by taking (3) as NG field dependent conformal transformations. For instance, the stress tensor itself is transformed as TL(z’) = T(z) - B(z’W
l)Tm(z)
z> T,(z)
,
= T;,(z) -T,(z)
= 2aflm +flT,,
f 2at-@“, + 2Ea5T,, + EtaT,
. (111
Therefore the classical superconformal currents of the coupled system can be written by using the stress tensor T,, as
which implies
T”=TkG+T,,,,
-!-
77(z)5(w) - z-w’
G’=G;,+@-,,,.
Including quantum corrections, currents are written by
The generators of transformations using these fields as
(12) the superconformal
(4) are written by T(z) = TX + T, ,
(13)
where
@dz>
TNG= - ;qag--
145
= 77+ 5ah
f
(9)
These also generate superconformal transformations on q(z) under which the free field action (6) is invariant. By keeping only single contraction, which is equivalent to taking the Poisson bracket, these generators (8) satisfy the classical (centerless) superconformal algebra. To satisfy exact OPE relations, we must replace the classical currents (9) with the quantum ones which have some quantum correction terms. The final form of the nonlinear realization of the superconformal currents is given by
where c, denotes the central charge of the matter stress tensor T,,,. Generators (13) coincide with those given in [3] and satisfy the superconformal algebra with c = c, - 11, which includes the realization obtained in [ 1 ] as a critical c, = 26 case. In this letter, we have reproduced the realization of the superconformal algebra found in [ 1,3] from the viewpoint of the standard nonlinear realization. This indicates that the N = 1 superstring studied in [ 1 ] is in a spontaneously broken phase of superconformal symmetry as a two dimensional field theory. When we consider the superstrings, however, the supersymmetry is a local symmetry and thus the NG fermion becomes
146
H. Kunitomo / Physics Letters B 343 (1995) 144-146
unphysical owing to the super Higgs mechanism [ 51. The physical states, therefore, coincide with those of the bosonic string. The results can be generalized to the other embedding [6-91 in principle. However, we do not have a general procedure yet for obtaining a quantum correction which is an infinite series in some cases [ 31. The author would like to thank Taichiro Kugo, Nobuyoshi Ohta, Makoto Sakaguchi and Akira Tokura for discussions.
References [ 1 ] N. Berkovits and C. Vafa, On the Uniqueness preprint HUTP-93/A031, KCL-TH-93-13, (1993).
of String Theory, hep-th/9310170
[2] D.V. Volkov and V.P. Akulov, JETP Len. 16 (1972) 438. [3] N. Berkovits and N. Ohta, Embeddings for Non-Critical Superstrings, preprint KCL-TH-94-6, OU-HET 189, hep-th/ 9405144 (1994). [4] S. Coleman, Dilitation, in: Properties of the Fundamental Interactions (Editrice Compositori, Bologna, 1973). [ 51D.V. Volkov and V.A. Soroka, JETP Lett. 18 (1973) 312: S. Deser and B. Zumino, Phys. Rev. Lett. 38 (1977) 1433. [ 61 N. Berkovits, M. Freeman and P. West, Phys. Lett. B 325 ( 1994) 63. [7] H. Kunitomo, M. Sakaguchi and A. Tokura, A Universal w String Theory, preprint OU-HET 187, hep-th/9403086 ( 1994). [8] F. Bastianelli, N. Ohta and J.L. Petersen, A Hierarchy of Superstrings, preprint NBI-HE-94-20, hep-th/9403150 ( 1994). [9] H. Kunitomo, M. Sakaguchi and A. Tokura, A Hierarchy of Super w Strings, preprint in preparation.
1995
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 343 ( 1995) 144-146
On the nonlinear realization of superconformal symmetry Hiroshi Kunitomo
’
Department of Physics, Osaka University, Toyonaka. Osaka 560, Japan Received 15 July 1994 Editor: M. Dine
Abstract The nonlinear realization of superconformal symmetry in two dimensions is considered. Superconformal symmetry is realized by means of the dimension - l/2 Nambu-Goldstone fermion 5 and its dimension 3/2 conjugate 7. A matter coupling of these Nambu-Goldstone fermions reproduces the realization found by Berkovits and Vafa to show the equivalence between the bosonic string and an N= 1 superstring. This indicates that this equivalence is a result of the super Higgs mechanism as a two dimensional field theory, in which the NG fermions become unphysical.
Recently, Berkovits and Vafa showed that bosonic strings may be viewed as a particular class of vacua for N= 1 superstrings [ 11. One of their key observations is that there is a realization of superconformal symmetry by means of fermions (9, Q with dimensions (3/2, - l/2) and the matter stress tensor T,,,. In this realization, the fermion 5 is transformed inhomogeneously and thus plays the role of a Nambu-Goldstone (NG) fermion. In this letter, we rederive their result from the viewpoint of the conventional nonlinear realization of supersymmetry [2]. Superconformal symmetry may be nonlinearly realized by the NG fermion pair (7, 8. Its general matter couplings reproduce the noncritical embedding obtained by Berkovits and Ohta [ 31, which includes the above realization as a critical case. We restrict our attention to the holomorphic sector of the superconformal symmetry. For studying the nonlinear realization, we consider superconformal symmetry as coordinate transformations of holomorphic ’ E-mail: [email protected]. Elsevier Science B.V. SsO10370-2693(94)01421-3
superspace. The infinitesimal try is represented by z’=z -f(z) + Oe(z) , e’=e-&3f(z>e+E(Z)
superconformal
)
symme-
(I)
wheref(z) ( E(Z) ) is an infinitesimal parameter of the (super)conformal transformation. We consider such a nonlinear realization that superconformal symmetry breaks down to conformal symmetry. The nonlinear transformation law of the NG fermion 5 is obtained by replacing 8 with the NG fermion tin ( 1) *: 5’(z’, a= z’=z-f(z)
5(z, 3 - ;V-(z)&z, +5cz, i)dZ)
Thus the infinitesimal KL
l) E(z,
a+
.
e(z) ,
(2) (3)
transformation
of ris given by
3 = s’(z, ?) - 5(z, 3
= e(z) + l(z)S(z, .&Mz, - f@z)%(z,
a +f(z)@xz,
3 ?) ,
(4)
2 Note that we need only one NG fermion while the superconformal symmetry has infinite dimensions. This seems to be a common feature when broken symmetries are generated by moments of one current 141.
H. Kunitomo / Physics Letters B 343 (1995) 144-146
where we note that the NG field 5 is transformed as a dimension - l/2 field under the conformal transformation. One can show that this transformation satisfies the required commutation relations: [SVl, El), @AZ, •2)l~=wlz~
E12)5?
(5)
where
E12
=
-fafiE2+f,aE2-(1*2).
Since 5 has negative dimension - l/2, we cannot obtain a nontrivial conformal invariant theory by using the NG fermion tarone. This is a characteristic feature of the spontaneous breaking of superconformal symmetry. Infinite dimensional unbroken symmetry, conformal symmetry, requires another degree of freedom than the NG fermion 6 The simplest way to construct it is introducing a dimension 312 fermion 77and taking the action (6)
,
$a7je+ ;a2(gan
GNG= q+ q$t-
?a”[+
#a(a2s.
(10)
These satisfy the OPE of superconformal algebra with 11. C NG= We next consider the general matter coupling, where matter fields are linear representations of the conformal symmetry. We denote a current of this matter conforma1 symmetry (stress tensor) as T,. The infinitesimal transformations of the matter fields under superconforma1 symmetry are given by taking (3) as NG field dependent conformal transformations. For instance, the stress tensor itself is transformed as TL(z’) = T(z) - B(z’W
l)Tm(z)
z> T,(z)
,
= T;,(z) -T,(z)
= 2aflm +flT,,
f 2at-@“, + 2Ea5T,, + EtaT,
. (111
Therefore the classical superconformal currents of the coupled system can be written by using the stress tensor T,, as
which implies
T”=TkG+T,,,,
-!-
77(z)5(w) - z-w’
G’=G;,+@-,,,.
Including quantum corrections, currents are written by
The generators of transformations using these fields as
(12) the superconformal
(4) are written by T(z) = TX + T, ,
(13)
where
@dz>
TNG= - ;qag--
145
= 77+ 5ah
f
(9)
These also generate superconformal transformations on q(z) under which the free field action (6) is invariant. By keeping only single contraction, which is equivalent to taking the Poisson bracket, these generators (8) satisfy the classical (centerless) superconformal algebra. To satisfy exact OPE relations, we must replace the classical currents (9) with the quantum ones which have some quantum correction terms. The final form of the nonlinear realization of the superconformal currents is given by
where c, denotes the central charge of the matter stress tensor T,,,. Generators (13) coincide with those given in [3] and satisfy the superconformal algebra with c = c, - 11, which includes the realization obtained in [ 1 ] as a critical c, = 26 case. In this letter, we have reproduced the realization of the superconformal algebra found in [ 1,3] from the viewpoint of the standard nonlinear realization. This indicates that the N = 1 superstring studied in [ 1 ] is in a spontaneously broken phase of superconformal symmetry as a two dimensional field theory. When we consider the superstrings, however, the supersymmetry is a local symmetry and thus the NG fermion becomes
146
H. Kunitomo / Physics Letters B 343 (1995) 144-146
unphysical owing to the super Higgs mechanism [ 51. The physical states, therefore, coincide with those of the bosonic string. The results can be generalized to the other embedding [6-91 in principle. However, we do not have a general procedure yet for obtaining a quantum correction which is an infinite series in some cases [ 31. The author would like to thank Taichiro Kugo, Nobuyoshi Ohta, Makoto Sakaguchi and Akira Tokura for discussions.
References [ 1 ] N. Berkovits and C. Vafa, On the Uniqueness preprint HUTP-93/A031, KCL-TH-93-13, (1993).
of String Theory, hep-th/9310170
[2] D.V. Volkov and V.P. Akulov, JETP Len. 16 (1972) 438. [3] N. Berkovits and N. Ohta, Embeddings for Non-Critical Superstrings, preprint KCL-TH-94-6, OU-HET 189, hep-th/ 9405144 (1994). [4] S. Coleman, Dilitation, in: Properties of the Fundamental Interactions (Editrice Compositori, Bologna, 1973). [ 51D.V. Volkov and V.A. Soroka, JETP Lett. 18 (1973) 312: S. Deser and B. Zumino, Phys. Rev. Lett. 38 (1977) 1433. [ 61 N. Berkovits, M. Freeman and P. West, Phys. Lett. B 325 ( 1994) 63. [7] H. Kunitomo, M. Sakaguchi and A. Tokura, A Universal w String Theory, preprint OU-HET 187, hep-th/9403086 ( 1994). [8] F. Bastianelli, N. Ohta and J.L. Petersen, A Hierarchy of Superstrings, preprint NBI-HE-94-20, hep-th/9403150 ( 1994). [9] H. Kunitomo, M. Sakaguchi and A. Tokura, A Hierarchy of Super w Strings, preprint in preparation.