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Classification of dual model gauge algebras
Volume 64B, number 1
PHYSICS LETTERS
CLASSIFICATION
30 August 1976
OF DUAL MODEL GAUGE ALGEBRAS*
P. RAMOND** and J.H. SCHWARZ California Institute of Technology, Pasadena, Calif. 91125, USA Received 25 June 1976 A complete classification of infinite graded algebras containing a Virasoro subalgebra and additional generators transforming with conformal spin ½, 1, or ~ is derived.
One of the primary areas of research in the field of dual strings is the search for new models. This is obviously an important subject to pursue if one believes that a string model should describe the hadrons, especially as the existing ones possess some unrealistic features. It has also been a frustrating pursuit as each new trick that was tried failed to bear fruit. Quite recently the first fundamentally new development in nearly five years was reported [1 ]. The model that results [2] is not more realistic than the previous ones, but it does help to focus attention on possibilities that were overlooked in the earlier attempts. In this letter we systematically examine a large class of infinite algebras of the type suggested by the work of refs. [1] and [2]. Our attitude is that even though there may not be a dual model associated with infinite algebra of the type to be considered, every dual model should possess a characteristic infinite gauge algebra. Of course, our results need not be the last word on the subject since we cannot exclude the possibility that there is a dual string model whose gauge algebra is not of the type considered here, although our assumptions do include all the known examples. Our most important assumption is that the gauge algebra contain a Virasoro subalgebra
matics. We also allow our algebra to contain additional generators of certain specified conformal spins. A set of generators (XJm} is said to have conformal spinJ if and only if
[L m, Ln] = (m - n ) L m + n + c l m ( m 2 - 1)6m,__ n
i" t The possibility o f also including additional J' -- 2 generators has been investigated. We find that no other irreducible algebras can be obtained in this way. "i"2 The association o f conformal spin and statistics that we make in eqs. ( 3 ) - ( 5 ) is only essential if one insists that the algebra be canonically representable. A peculiar algebra not satisfying "spin and statistics" consists o f J = 1 generators i D - - } a n d J = 2 generators iBm) with (B m, Bn}= (Dm, D'n~ = 0, {Bm, Dn}= Lm+n, [Lm, Lnl = (m - n)Lm+n. Jacobi identities do not aflow the inclusion o f c-number anomaly terms in this algebra.
m , n = 0,-+1, +2, .....
(1)
This appears to be a fundamental feature of any string model as it is intimately connected with string kine* Work supported in part by the U.S. Energy Research and Development Administration under contract E(11 - 1 ) - 6 8 . ** Yale Junior Fellow on leave from Yale University.
[Lm, X J] = [m(J - 1) - n] XJm+n .
(2)
The only possibilities that we choose to considert ~ are J = 1/2 generators {P/m } with anticommutation rules
i , i = J , 2 ..... e,
(3)
J = 1 generators {~m } with commutation rules [T a , T b ] = iCabcTC+n + c3mSm,_nSab a , b = 1,2 ..... N ,
(4)
and J = 3/2 generators {G m } with anticommutation rules 13 a f G m , G n } = 28,,oLm+ n - 2 k ( m - n)X~7~m+n
+c4(m 2 - 1 / 4 ) 6 m , _ n 6 c , #
~ , / 3 = 1 , 2 ..... K .
(5)
Repeated indices are always to be summed. We also assume that the subscripts on the fermionic generators (F's and G's) assume half-integral values whereas the ones on the bosonic generators (L's and T's) assume integral values? 2 . The form of eqs. ( 3 ) - ( 5 ) is completely
75
Volume 64B, number 1
PHYSICS LETTERS
general when one allows for the possibility of forming suitable linear combinations of generators and demands consistency with the Jacobi identities obtained from commuting both sides with Lp. In fact, with these understandings one may also assume that Cab c is totally antisymmetric and that c 4 = 4c 1 . The rest of the algebra, arrived at in the same way, is (6)
IT a , p i n ] = ~~ ^I.'i]-~m+n ar/ 2 ,,gay m+n
{rim,C
mL,,ialm+ n
} = Eia. Tin+ a n .
(7) (8)
Note that the N generators T(~ form a Lie algebra with structure constants Cab c. We do not assume 1 anything about this algebra. The matrices 7 pa and }h a provide P and K dimensional representations of the algebra respectively. (This follows from the Jacobi identities obtained by commutings eqs. (6) and (7) with Tpb.) These representations may be reducible, but we" insist that they be self-conjugate, which implies that pa and h a are antisymmetric matrices. The form of eq. (7) or (8) implies that the P representation is contained in the Kronecker product N ® K. The cons t a n t s / ~ and E a represent the appropriate grouptheoretical coefficients. In fact, the Jacobi identity involving T's, G's, and r ' s implies that c3 E a = c2 D a .
(9)
We choose the normalization of the P's so that c 2 = c 3 and Daa = E~a are separately realized. The procedure now is to systematically study all the Jacobi identities. When this is done one deduces that i
r
_
a
a
-
r
(1o)
and ¥ ~ k = P~'~,~
(11)
are totally antisymmetric in their subscripts. Also a a + h~.r~ a a~ ) _- 280r8 a6 - 8~8.r6 - 8 o r r S ~ , (12) k(~a~h~ D~ Dlq = 8o(~Sii- khaa~& ,
(13)
D iaa D ib# + D laa D iba-- 28a#Sab _ k{)¢a , hb }or#,
(14)
mc -- Da]oLt.,,l/ ^b _ Di#h#a a b . - 2 i ," ~abc~,iot
(15)
The only other constraints are c 4 = k c 3 = kc 2 = 4c 1 . 76
30 August 1976
Contracting a pair of indices in eqs. (12)-(14) gives k(xaha)a # = ( K - 1 ) S a # ,
(16)
D a D a =PSa~
(17)
ia i~
D ict a D i[3 a =NS~-
k(kaka)~
(18)
Combining these, we deduce that for K #: 0 K+P=N+I
.
(19)
This is a remarkable conclusion that greatly restricts the possibilities to be considered. A special case for which the reasoning is particularly easy is P=0. In this case eq. (14) implies that the Kdimensional ha's form a Clifford algebra. Since a Clifford algebra requires matrices with dimension of at least 2N-I/2 for N o d d and 2 N/2 for Neven, the only possibilities consistent with eq. (19) are N= 0, 1,2, 3, 4, 5, 7. Moreover, one has det ka :/:0, and hence the Lie algebra must be simple. Therefore there are only three possible solutions for P= 0 and K:/: 0. ( 1 ) N = 0 and K = 1 ; this gives the gauge algebra of the dual pion model [3]. ( 2 ) N = l and K=2; this gives the gauge algebra of the U(1) model of refs. [1] and [2]. (3)N= 3 and K= 4; this gives the SU(2) gauge algebra discussed in refs. [1] and [2]. In this case the G's belong to the reducible representation D (1/2) • D (1/2). This is the complete list of known dual models aside from the original Veneziano model (which has K = 0). Let us now return to the general analysis with P nonvanishing. In this case the dimension of the X matrices is not so readily bounded, so a different approach is called for. We note that our infinite graded algebra always contains a finite graded subalgebra consisting of the N + 2K+'3 generators T~0, G~I/2, L 0, L±I. A complete classification of the finite algebras has recently been worked out [4] and therefore can be scanned for potential candidates. There are relatively few cases to check, and most of these can be excluded by the requirement that N ® K D P = N + I - K . The three exceptional graded algebras, for example, require T~ to be the generators of SU 2 × SU 2, G 2, or SO 7. None of these cases is acceptable. There are also two infinite sequences in which T(~ are the generators of SO n or U n . Of these only one, namely the SO 3 case, satisfies the N ® K D P requirement. The infinite graded algebra with this subalgebra was also considered in ref. [1 ] ! It is characterized by K= 3, P= 1, and the conditions Cabc = eabc, h~a= --2iea~c,, Da = ~a~, k = -i/2, and ~=0.
Volume 64B, number 1
PHYSICS LETTERS
We have shown that the only infinite graded algebras containing the Virasoro algebra and additional generators with J = 1[2, 1 , 3 / 2 are the few previously known ones listed above. Strictly speaking, we have assumed k ~ 0 so that all the generators can be obtained by repeated anticommutation and commutation of G's. Based On our experience with dual models, this appears to be a natural assumption. Our analysis could in principle be extended to allow additional generators with other conformal spins. Indeed a class of such algebras is given in ref. [1]. However, only algebras with the types of generators considered here appear to have a chance o f being represented canonically in terms o f a - a n d b-type oscillators .3 . Only canonically representable algebras would seem to have a chance to give rise to ghost-free dual string models*4. t 3 In fact, even among these there is one, namely the SO2 algebra with K = 3 and P = 1, that cannot be represented canonically. t 4 One of the canonically representable algebras, namely the SU2 algebra with K = 4 and P = 0, does not give a ghostfree dual model.
30 August 1976
Therefore we conclude that unless there is some possibility that we have not considered (such as gauge algebras without Virasoro subalgebras, for example), no dual string models with new gauge algebras remain to be found.
References [1] M. Ademollo et al.,Phys. Lett. 62B (1976) 105. [2] M. Ademollo et al.,Dual stringwith U(1) colour symmetry,
Torino Preprint IFTT304, 1976. [3] A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; J.H. Schwarz, Phys. Reports 8C (1973) 271. [4] M. Scheunert, W. Nahm and V. Rittenberg, Bonn University Preprint BONN-tIE-76-7, 1976; V.G. Kats, Functional analysis and applications 9 (1975) 91; P.G.O. Freund and I. Kaplansky, JMP 17 (1975) 228.
77
PHYSICS LETTERS
CLASSIFICATION
30 August 1976
OF DUAL MODEL GAUGE ALGEBRAS*
P. RAMOND** and J.H. SCHWARZ California Institute of Technology, Pasadena, Calif. 91125, USA Received 25 June 1976 A complete classification of infinite graded algebras containing a Virasoro subalgebra and additional generators transforming with conformal spin ½, 1, or ~ is derived.
One of the primary areas of research in the field of dual strings is the search for new models. This is obviously an important subject to pursue if one believes that a string model should describe the hadrons, especially as the existing ones possess some unrealistic features. It has also been a frustrating pursuit as each new trick that was tried failed to bear fruit. Quite recently the first fundamentally new development in nearly five years was reported [1 ]. The model that results [2] is not more realistic than the previous ones, but it does help to focus attention on possibilities that were overlooked in the earlier attempts. In this letter we systematically examine a large class of infinite algebras of the type suggested by the work of refs. [1] and [2]. Our attitude is that even though there may not be a dual model associated with infinite algebra of the type to be considered, every dual model should possess a characteristic infinite gauge algebra. Of course, our results need not be the last word on the subject since we cannot exclude the possibility that there is a dual string model whose gauge algebra is not of the type considered here, although our assumptions do include all the known examples. Our most important assumption is that the gauge algebra contain a Virasoro subalgebra
matics. We also allow our algebra to contain additional generators of certain specified conformal spins. A set of generators (XJm} is said to have conformal spinJ if and only if
[L m, Ln] = (m - n ) L m + n + c l m ( m 2 - 1)6m,__ n
i" t The possibility o f also including additional J' -- 2 generators has been investigated. We find that no other irreducible algebras can be obtained in this way. "i"2 The association o f conformal spin and statistics that we make in eqs. ( 3 ) - ( 5 ) is only essential if one insists that the algebra be canonically representable. A peculiar algebra not satisfying "spin and statistics" consists o f J = 1 generators i D - - } a n d J = 2 generators iBm) with (B m, Bn}= (Dm, D'n~ = 0, {Bm, Dn}= Lm+n, [Lm, Lnl = (m - n)Lm+n. Jacobi identities do not aflow the inclusion o f c-number anomaly terms in this algebra.
m , n = 0,-+1, +2, .....
(1)
This appears to be a fundamental feature of any string model as it is intimately connected with string kine* Work supported in part by the U.S. Energy Research and Development Administration under contract E(11 - 1 ) - 6 8 . ** Yale Junior Fellow on leave from Yale University.
[Lm, X J] = [m(J - 1) - n] XJm+n .
(2)
The only possibilities that we choose to considert ~ are J = 1/2 generators {P/m } with anticommutation rules
i , i = J , 2 ..... e,
(3)
J = 1 generators {~m } with commutation rules [T a , T b ] = iCabcTC+n + c3mSm,_nSab a , b = 1,2 ..... N ,
(4)
and J = 3/2 generators {G m } with anticommutation rules 13 a f G m , G n } = 28,,oLm+ n - 2 k ( m - n)X~7~m+n
+c4(m 2 - 1 / 4 ) 6 m , _ n 6 c , #
~ , / 3 = 1 , 2 ..... K .
(5)
Repeated indices are always to be summed. We also assume that the subscripts on the fermionic generators (F's and G's) assume half-integral values whereas the ones on the bosonic generators (L's and T's) assume integral values? 2 . The form of eqs. ( 3 ) - ( 5 ) is completely
75
Volume 64B, number 1
PHYSICS LETTERS
general when one allows for the possibility of forming suitable linear combinations of generators and demands consistency with the Jacobi identities obtained from commuting both sides with Lp. In fact, with these understandings one may also assume that Cab c is totally antisymmetric and that c 4 = 4c 1 . The rest of the algebra, arrived at in the same way, is (6)
IT a , p i n ] = ~~ ^I.'i]-~m+n ar/ 2 ,,gay m+n
{rim,C
mL,,ialm+ n
} = Eia. Tin+ a n .
(7) (8)
Note that the N generators T(~ form a Lie algebra with structure constants Cab c. We do not assume 1 anything about this algebra. The matrices 7 pa and }h a provide P and K dimensional representations of the algebra respectively. (This follows from the Jacobi identities obtained by commutings eqs. (6) and (7) with Tpb.) These representations may be reducible, but we" insist that they be self-conjugate, which implies that pa and h a are antisymmetric matrices. The form of eq. (7) or (8) implies that the P representation is contained in the Kronecker product N ® K. The cons t a n t s / ~ and E a represent the appropriate grouptheoretical coefficients. In fact, the Jacobi identity involving T's, G's, and r ' s implies that c3 E a = c2 D a .
(9)
We choose the normalization of the P's so that c 2 = c 3 and Daa = E~a are separately realized. The procedure now is to systematically study all the Jacobi identities. When this is done one deduces that i
r
_
a
a
-
r
(1o)
and ¥ ~ k = P~'~,~
(11)
are totally antisymmetric in their subscripts. Also a a + h~.r~ a a~ ) _- 280r8 a6 - 8~8.r6 - 8 o r r S ~ , (12) k(~a~h~ D~ Dlq = 8o(~Sii- khaa~& ,
(13)
D iaa D ib# + D laa D iba-- 28a#Sab _ k{)¢a , hb }or#,
(14)
mc -- Da]oLt.,,l/ ^b _ Di#h#a a b . - 2 i ," ~abc~,iot
(15)
The only other constraints are c 4 = k c 3 = kc 2 = 4c 1 . 76
30 August 1976
Contracting a pair of indices in eqs. (12)-(14) gives k(xaha)a # = ( K - 1 ) S a # ,
(16)
D a D a =PSa~
(17)
ia i~
D ict a D i[3 a =NS~-
k(kaka)~
(18)
Combining these, we deduce that for K #: 0 K+P=N+I
.
(19)
This is a remarkable conclusion that greatly restricts the possibilities to be considered. A special case for which the reasoning is particularly easy is P=0. In this case eq. (14) implies that the Kdimensional ha's form a Clifford algebra. Since a Clifford algebra requires matrices with dimension of at least 2N-I/2 for N o d d and 2 N/2 for Neven, the only possibilities consistent with eq. (19) are N= 0, 1,2, 3, 4, 5, 7. Moreover, one has det ka :/:0, and hence the Lie algebra must be simple. Therefore there are only three possible solutions for P= 0 and K:/: 0. ( 1 ) N = 0 and K = 1 ; this gives the gauge algebra of the dual pion model [3]. ( 2 ) N = l and K=2; this gives the gauge algebra of the U(1) model of refs. [1] and [2]. (3)N= 3 and K= 4; this gives the SU(2) gauge algebra discussed in refs. [1] and [2]. In this case the G's belong to the reducible representation D (1/2) • D (1/2). This is the complete list of known dual models aside from the original Veneziano model (which has K = 0). Let us now return to the general analysis with P nonvanishing. In this case the dimension of the X matrices is not so readily bounded, so a different approach is called for. We note that our infinite graded algebra always contains a finite graded subalgebra consisting of the N + 2K+'3 generators T~0, G~I/2, L 0, L±I. A complete classification of the finite algebras has recently been worked out [4] and therefore can be scanned for potential candidates. There are relatively few cases to check, and most of these can be excluded by the requirement that N ® K D P = N + I - K . The three exceptional graded algebras, for example, require T~ to be the generators of SU 2 × SU 2, G 2, or SO 7. None of these cases is acceptable. There are also two infinite sequences in which T(~ are the generators of SO n or U n . Of these only one, namely the SO 3 case, satisfies the N ® K D P requirement. The infinite graded algebra with this subalgebra was also considered in ref. [1 ] ! It is characterized by K= 3, P= 1, and the conditions Cabc = eabc, h~a= --2iea~c,, Da = ~a~, k = -i/2, and ~=0.
Volume 64B, number 1
PHYSICS LETTERS
We have shown that the only infinite graded algebras containing the Virasoro algebra and additional generators with J = 1[2, 1 , 3 / 2 are the few previously known ones listed above. Strictly speaking, we have assumed k ~ 0 so that all the generators can be obtained by repeated anticommutation and commutation of G's. Based On our experience with dual models, this appears to be a natural assumption. Our analysis could in principle be extended to allow additional generators with other conformal spins. Indeed a class of such algebras is given in ref. [1]. However, only algebras with the types of generators considered here appear to have a chance o f being represented canonically in terms o f a - a n d b-type oscillators .3 . Only canonically representable algebras would seem to have a chance to give rise to ghost-free dual string models*4. t 3 In fact, even among these there is one, namely the SO2 algebra with K = 3 and P = 1, that cannot be represented canonically. t 4 One of the canonically representable algebras, namely the SU2 algebra with K = 4 and P = 0, does not give a ghostfree dual model.
30 August 1976
Therefore we conclude that unless there is some possibility that we have not considered (such as gauge algebras without Virasoro subalgebras, for example), no dual string models with new gauge algebras remain to be found.
References [1] M. Ademollo et al.,Phys. Lett. 62B (1976) 105. [2] M. Ademollo et al.,Dual stringwith U(1) colour symmetry,
Torino Preprint IFTT304, 1976. [3] A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; J.H. Schwarz, Phys. Reports 8C (1973) 271. [4] M. Scheunert, W. Nahm and V. Rittenberg, Bonn University Preprint BONN-tIE-76-7, 1976; V.G. Kats, Functional analysis and applications 9 (1975) 91; P.G.O. Freund and I. Kaplansky, JMP 17 (1975) 228.
77