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The θ-structure in string theories: Superstrings
"Voiume i.75, n u m b e r 3
PHYSICS LETTERS B
7 August i 9 8 6
T H E 8-.STRUCTURE 1N STR~NG T H E O R I E S ; S U P E R S T R ~ N G S LI Miao international Centre for Theoretical Pl~vs'ics, 1-34100 Trieste, Daly Received 28 N,.wember !985 Ti;is paper is the contb3uafion of a previous paper in which fl~e #-struclure of bosonic strings was discussed, ~a this paper it is pointed out that the SUSYs are broken, both in open strings and closed strings by introducing non-zero parameters 0. The c,mdusion is ~.hat the compact subspace of space time must ',~o~ be mu!tlply connected uMess the positive energy of the grou-~ad state is reguIarized ~.o be zero,
~n over previous paper we discussed the &structure of strh:gs [i], its appearance is due ~o the multiply connected subspace of space-time, ~he propagators between ~wo string configurafior~s are .:he sums over different i~o:.~otopic classes of paths. Th.e sams can be taken as a set of differentia1 ~-;rms.~ ~,~;-s.~ like .~ha~ in gauge theories. We c o w eluded that the ground state of bosoeic strings can be characterized by a set of parameters 8. If the space time takes the form of M~) ,1× T '~, then there are d parameters 8;,,, where indices m are those of the coordinates of the subspace T J. .'in this paper, the coordinates of M;) ,, × T J wil~ be denoted as { X t'¢)={X", X " } , where X", X " are the coordinates of M~)~, and T '¢, respectively. Moreover, we set ~ = 0 . . . . . 3, m = 4 . . . . . 9 as we discuss the &structure of the superstring model of ref. [2] oMy. The ~-structure in q u a n t u m theories can be described more clear!y under car~oMcal quanfiza.tion formalism. There is the conjugate .momentun TM' p,,~ h~r each dynamic freedom X " in a hamiltonian system, we constrain the opera~ors X M and P,w by commutative relations a m o n g them° namely [ X "~, XN]= [p,,~, p ~ , ] = 0 and [ X at, I)~]= iSM~. Then p,~ can be expressed as - i S / , - 3 X M + f.~;(XM), where ])~(X '~) is an arbitrary function of X M. For any physicai sta~e 1~'}, one can Permanent address: Centre for Astrophysics. University of Science and Tech:x~iogy of Ci~ina, Hefe{, Anhui, People's Repubhc of China.
284
perform
a
transformation
]'~'} = exp[-i ×
EMf,~4f~,4 dX"~]i g'}, thus the operators P~u must be transformed as p,~ -~ - i 3/3XMo [f X M is not compactified, all quantizafio~s of - i 3 / 3 X ~ +J;w are equivalent to - i 3 / 3 X M. B~t if X M is compactified, e~g~ X '~f is the parameter of a circle S z, the case is somewhat different from the triviM one. One can suppose {hat I R ) is a periodic f~mction of X ;~, but e x p [ - i f J ; ~ d X M ] l q ") is usaMiy not, therefore the q~:anfization of P,w = - i ~/3X~* + f,,,~ is not equivMent to that of p,w =~ - i ~ / ~ X M. h " tn string ti.eor.:es, we require that the space T J is flat, so the f;~ are constants. If we take the radii of T a as R,,, hence £ , = ~,,/2~rR .... 8,, ~ (0, 2~r). (0,, } classifies at1 ineq~ivalent quamizafions. O~ae may wonder whether we have to consider the 6~-s~ructure corresponding to the spin structures of spinors in SST. The answer is the following. If we treat superstrhags q u a n t u m mechanicai!y, ~ot by functional field theory, then the spinors S A ( d = t, 2) are defi,~ed on two-dimensional worm sheets. For open strings, there exist no non-trivial spin structures either for the 32co;:aponent spinors S ~ or for the 2-component spinor rSa?; for d o s e d strings, the toDo~ogv of tS2~ • J worId sheets of free strings is S * × R, then there exists a spin structure of the 2D spinor .:S2], which can be classified by the cohomology group HI(S ~, Za) = Z 2 [3]. We will discuss this in de~ai~ ir~ a future work.
0370.-2693/86/$03.50 '~'~ E~sevier Science Publishers B.V. (North-Holland Physics Publishing Divisiot~)
Vo.:mmet75, number 3
PHYSICS LETTERS B
We suppose that the s p a c e - t i m e takes the form of M 2 × T 6, the action of the superstrings of Green and Schwarz [2] in the light cone gauge is
7 August 1986
By this expansion, the supercharges Q" can be expanded as
O ° = i(p
fd~d~[-(1/2~)
~o
G X ' 8~X '
; a,,. + ,i.~p - ~-'~/2 Z (
+ ( i / 4 ~ r ) S E - p ~ 8~S],
-
I ,'2
(2a) _
!
+
-
The c o m m u t a t o r s of Q" are {0<
Q~} = 2 ( h F G ' Y ) " " ,
t / 2 jf0~rr
[
~
,~M",
to the above
Oc,]
(8)
where we have assumed that the Q~ are hermitian under a proper choice of F-matrices, h is a chirai o p e r a t o r such that aS = O• T h e spectrum formula can be derived by consistent conditions: l ,2 = ~ p ~ ' p " * + ~ ~vmass)'
( ~
: -
F-
\
n=!
(2b)
The supercharges corresponding transformations are
(7)
--o0
(1)
where the Rt~o,. ........ g.:h a ' of the strings was taken as ½, the index i denotes aiI transverse directions, and the spinors S/'" are W e y 1 - M a j o r a n a spinors of 10D s p a c e - t i m e and the spinor of 2D world sheets. So A runs throughout 1, 2, and ] . . . . . 32. This action is invariant under the fo!iowing supersymmerry transformations provided X ~ and S ~ satisfy some b o u n d a r y conditions [2]:
(/) s=i(p +) , c
s,,)"
(9) T h e minimal value of the mass M~i, ~ is [t] Io//
Aa
• [ ,'~2--4,1y 2
We first consider .'.he open strings. The supercharges Q"" are not separately conserved while their combinations
O" =
+
(4)
are conserved. According to ref. [1], the m o d e expansions of X ~ and S by taking the 0-structure into account are X t _
- ~ ~ i + ~ bi~ r + i ~
L~_i
n~aO ~
if some parameters ~,, are not zero, then we cannot get a zero-mass spectrum. Therefore the naturat choice of compactified space must not be muhip!ied, but if the c o m p a c t space is not fiat, the . somew" ~..... conclusion is .~erha.~s " different. This difficulty can be overcome by mass regularization, which will be given later. One deduces that the energy of the ground state of open strings is not zero, hence the supers y m m e t r y must be broken• For the ground ~ a t e I0), there is
cos no exp(-in~:), n
(01 p ° I 0 ) = , ( 2 m i n M '22.,/, R ,,~/2} .
(ti)
(sa)
i
/,=p~
=~ 2,
= p'~ + G/2~rP,,
ra = 4 . . . . . 9,
(Sb)
--OO
S z'= ~ S"exp[-in(r+o)].
(6b)
The first level multiplet of the field limit of N = 4 SUSY is no !onger c o m p o s e d of I mass~ess vector, 4 massless M a j o r a n a spinors and 6 massless scalars but of i massive vector, 4 massive M a j o r a n a spinors and 5 massive scalars. T o d e m o n s t r a t e the SUSY breaking, we take the following F-matrices representation F~ = 7~ x e s,
.C,, = r5 x -/~..
(12a,b) 285
Valume 175, number 3
PHYSICS LETTERS B
Q" can be decomposed into 4 independent • Majoraaa s,a{nors QI..~ /~ L = I . . . . . 4 , a = , . . . . 4) of 4D s p a c e - t i m e under the r e p r e s e n t a t i o n of eq. ('..'2). One can derive the anticommutmors a m o n g Q~'"~" from eq. (8) and deduce that E { Q""', O K'~ } = 4 / , ~ ' '
(13)
Hence (0 ,E,~.Om~O ~'" !0) = 2@1,9'; [0) > 0, a~l /v = 4 SUSYs are broken. N o w we cot:sider d o s e d strngs. The m o d e expansions and supercna.ges m the light cone gauge are
-}~ a X , e x p,[ - 2 i e , ( r + e ) ]
,
>=~,2.
(i4a)
X " = x . . . --. . a o~'r + 2 R ,,#N,,,
+.Y., }n ~,c~''~Ap[-2i~(¢-o~ + g~:~'exD [ - 2i n ( r + o )] }, . . .~. . .0/(} . =p'"+~,,,/27rR.,,~,.
~(,
(14b)
m=&..
•,9-
(15a)
oo
S:':
E S':~x~J-2in{r-~)],,
S 2" :
~
~,,~'"e×p[ - 2in ('r + o)]~, i /2
.~
~~b.~_ -'w
+<,) a 0{2 oo
a:;d
Q2a
is g i v e n
the
express{on
of
@]a
with
T : ~ corn . . . . . ators of the supercharges and mass
spectrum formula are
p
+
/H
( ~ 8a)
c~
N = / ~ (.~ \ , a',,
+ 4~g ,,~'-&}, ~
( Sb)
where N is the same expression of N with &, and The energy of the ground state is not zero. By • /: using a decomposition of eq. ~_.2), QA,, can be decomposed into Q~t.,~ ( L = ! . . . . . 4, a = 1 . . . . . 4). 286
where ,',he Q~"~ are 9 independent Majorana spinors of 4D space-time, in this decomposition, we ~ave
E {QA,.,< 0.,,.o } = The
~2'
and -i
a
7 August 1986
would like to thank Professor Strathdee for reminding me to note the heterotic string m o d e l The author would Iike to thank Professor Abdus Sa!am, the International Atomic Energy Agency and U N E S C O for hospitaiity at the International Centre for Theoretical Physics, Trieste. References
[i I M. Li, The &structure in string theories-]: Bosonic strings, ICTP, Trieste, preprint [C/85/i60. [21 MB. Green and J.H. Schwarz, NucL Phys. B181 (198I) 502; Phys. Let~. B109 (1982) 444. [3i J.W. Miinor, FEns. Math. 9 (t963} !98: M.F. Atiyah, Ann. Sci. EcoL Norm. Sup~ a (1971) 47. [4] D. Gross, J. Harvey, E. Marfinec and R. Rohm, Phys. Rev. LetL 54 (1985) 502.
PHYSICS LETTERS B
7 August i 9 8 6
T H E 8-.STRUCTURE 1N STR~NG T H E O R I E S ; S U P E R S T R ~ N G S LI Miao international Centre for Theoretical Pl~vs'ics, 1-34100 Trieste, Daly Received 28 N,.wember !985 Ti;is paper is the contb3uafion of a previous paper in which fl~e #-struclure of bosonic strings was discussed, ~a this paper it is pointed out that the SUSYs are broken, both in open strings and closed strings by introducing non-zero parameters 0. The c,mdusion is ~.hat the compact subspace of space time must ',~o~ be mu!tlply connected uMess the positive energy of the grou-~ad state is reguIarized ~.o be zero,
~n over previous paper we discussed the &structure of strh:gs [i], its appearance is due ~o the multiply connected subspace of space-time, ~he propagators between ~wo string configurafior~s are .:he sums over different i~o:.~otopic classes of paths. Th.e sams can be taken as a set of differentia1 ~-;rms.~ ~,~;-s.~ like .~ha~ in gauge theories. We c o w eluded that the ground state of bosoeic strings can be characterized by a set of parameters 8. If the space time takes the form of M~) ,1× T '~, then there are d parameters 8;,,, where indices m are those of the coordinates of the subspace T J. .'in this paper, the coordinates of M;) ,, × T J wil~ be denoted as { X t'¢)={X", X " } , where X", X " are the coordinates of M~)~, and T '¢, respectively. Moreover, we set ~ = 0 . . . . . 3, m = 4 . . . . . 9 as we discuss the &structure of the superstring model of ref. [2] oMy. The ~-structure in q u a n t u m theories can be described more clear!y under car~oMcal quanfiza.tion formalism. There is the conjugate .momentun TM' p,,~ h~r each dynamic freedom X " in a hamiltonian system, we constrain the opera~ors X M and P,w by commutative relations a m o n g them° namely [ X "~, XN]= [p,,~, p ~ , ] = 0 and [ X at, I)~]= iSM~. Then p,~ can be expressed as - i S / , - 3 X M + f.~;(XM), where ])~(X '~) is an arbitrary function of X M. For any physicai sta~e 1~'}, one can Permanent address: Centre for Astrophysics. University of Science and Tech:x~iogy of Ci~ina, Hefe{, Anhui, People's Repubhc of China.
284
perform
a
transformation
]'~'} = exp[-i ×
EMf,~4f~,4 dX"~]i g'}, thus the operators P~u must be transformed as p,~ -~ - i 3/3XMo [f X M is not compactified, all quantizafio~s of - i 3 / 3 X ~ +J;w are equivalent to - i 3 / 3 X M. B~t if X M is compactified, e~g~ X '~f is the parameter of a circle S z, the case is somewhat different from the triviM one. One can suppose {hat I R ) is a periodic f~mction of X ;~, but e x p [ - i f J ; ~ d X M ] l q ") is usaMiy not, therefore the q~:anfization of P,w = - i ~/3X~* + f,,,~ is not equivMent to that of p,w =~ - i ~ / ~ X M. h " tn string ti.eor.:es, we require that the space T J is flat, so the f;~ are constants. If we take the radii of T a as R,,, hence £ , = ~,,/2~rR .... 8,, ~ (0, 2~r). (0,, } classifies at1 ineq~ivalent quamizafions. O~ae may wonder whether we have to consider the 6~-s~ructure corresponding to the spin structures of spinors in SST. The answer is the following. If we treat superstrhags q u a n t u m mechanicai!y, ~ot by functional field theory, then the spinors S A ( d = t, 2) are defi,~ed on two-dimensional worm sheets. For open strings, there exist no non-trivial spin structures either for the 32co;:aponent spinors S ~ or for the 2-component spinor rSa?; for d o s e d strings, the toDo~ogv of tS2~ • J worId sheets of free strings is S * × R, then there exists a spin structure of the 2D spinor .:S2], which can be classified by the cohomology group HI(S ~, Za) = Z 2 [3]. We will discuss this in de~ai~ ir~ a future work.
0370.-2693/86/$03.50 '~'~ E~sevier Science Publishers B.V. (North-Holland Physics Publishing Divisiot~)
Vo.:mmet75, number 3
PHYSICS LETTERS B
We suppose that the s p a c e - t i m e takes the form of M 2 × T 6, the action of the superstrings of Green and Schwarz [2] in the light cone gauge is
7 August 1986
By this expansion, the supercharges Q" can be expanded as
O ° = i(p
fd~d~[-(1/2~)
~o
G X ' 8~X '
; a,,. + ,i.~p - ~-'~/2 Z (
+ ( i / 4 ~ r ) S E - p ~ 8~S],
-
I ,'2
(2a) _
!
+
-
The c o m m u t a t o r s of Q" are {0<
Q~} = 2 ( h F G ' Y ) " " ,
t / 2 jf0~rr
[
~
,~M",
to the above
Oc,]
(8)
where we have assumed that the Q~ are hermitian under a proper choice of F-matrices, h is a chirai o p e r a t o r such that aS = O• T h e spectrum formula can be derived by consistent conditions: l ,2 = ~ p ~ ' p " * + ~ ~vmass)'
( ~
: -
F-
\
n=!
(2b)
The supercharges corresponding transformations are
(7)
--o0
(1)
where the Rt~o,. ........ g.:h a ' of the strings was taken as ½, the index i denotes aiI transverse directions, and the spinors S/'" are W e y 1 - M a j o r a n a spinors of 10D s p a c e - t i m e and the spinor of 2D world sheets. So A runs throughout 1, 2, and ] . . . . . 32. This action is invariant under the fo!iowing supersymmerry transformations provided X ~ and S ~ satisfy some b o u n d a r y conditions [2]:
(/) s=i(p +) , c
s,,)"
(9) T h e minimal value of the mass M~i, ~ is [t] Io//
Aa
• [ ,'~2--4,1y 2
We first consider .'.he open strings. The supercharges Q"" are not separately conserved while their combinations
O" =
+
(4)
are conserved. According to ref. [1], the m o d e expansions of X ~ and S by taking the 0-structure into account are X t _
- ~ ~ i + ~ bi~ r + i ~
L~_i
n~aO ~
if some parameters ~,, are not zero, then we cannot get a zero-mass spectrum. Therefore the naturat choice of compactified space must not be muhip!ied, but if the c o m p a c t space is not fiat, the . somew" ~..... conclusion is .~erha.~s " different. This difficulty can be overcome by mass regularization, which will be given later. One deduces that the energy of the ground state of open strings is not zero, hence the supers y m m e t r y must be broken• For the ground ~ a t e I0), there is
cos no exp(-in~:), n
(01 p ° I 0 ) = , ( 2 m i n M '22.,/, R ,,~/2} .
(ti)
(sa)
i
/,=p~
=~ 2,
= p'~ + G/2~rP,,
ra = 4 . . . . . 9,
(Sb)
--OO
S z'= ~ S"exp[-in(r+o)].
(6b)
The first level multiplet of the field limit of N = 4 SUSY is no !onger c o m p o s e d of I mass~ess vector, 4 massless M a j o r a n a spinors and 6 massless scalars but of i massive vector, 4 massive M a j o r a n a spinors and 5 massive scalars. T o d e m o n s t r a t e the SUSY breaking, we take the following F-matrices representation F~ = 7~ x e s,
.C,, = r5 x -/~..
(12a,b) 285
Valume 175, number 3
PHYSICS LETTERS B
Q" can be decomposed into 4 independent • Majoraaa s,a{nors QI..~ /~ L = I . . . . . 4 , a = , . . . . 4) of 4D s p a c e - t i m e under the r e p r e s e n t a t i o n of eq. ('..'2). One can derive the anticommutmors a m o n g Q~'"~" from eq. (8) and deduce that E { Q""', O K'~ } = 4 / , ~ ' '
(13)
Hence (0 ,E,~.Om~O ~'" !0) = 2@1,9'; [0) > 0, a~l /v = 4 SUSYs are broken. N o w we cot:sider d o s e d strngs. The m o d e expansions and supercna.ges m the light cone gauge are
-}~ a X , e x p,[ - 2 i e , ( r + e ) ]
,
>=~,2.
(i4a)
X " = x . . . --. . a o~'r + 2 R ,,#N,,,
+.Y., }n ~,c~''~Ap[-2i~(¢-o~ + g~:~'exD [ - 2i n ( r + o )] }, . . .~. . .0/(} . =p'"+~,,,/27rR.,,~,.
~(,
(14b)
m=&..
•,9-
(15a)
oo
S:':
E S':~x~J-2in{r-~)],,
S 2" :
~
~,,~'"e×p[ - 2in ('r + o)]~, i /2
.~
~~b.~_ -'w
+<,) a 0{2 oo
a:;d
Q2a
is g i v e n
the
express{on
of
@]a
with
T : ~ corn . . . . . ators of the supercharges and mass
spectrum formula are
p
+
/H
( ~ 8a)
c~
N = / ~ (.~ \ , a',,
+ 4~g ,,~'-&}, ~
( Sb)
where N is the same expression of N with &, and The energy of the ground state is not zero. By • /: using a decomposition of eq. ~_.2), QA,, can be decomposed into Q~t.,~ ( L = ! . . . . . 4, a = 1 . . . . . 4). 286
where ,',he Q~"~ are 9 independent Majorana spinors of 4D space-time, in this decomposition, we ~ave
E {QA,.,< 0.,,.o } = The
~2'
and -i
a
7 August 1986
would like to thank Professor Strathdee for reminding me to note the heterotic string m o d e l The author would Iike to thank Professor Abdus Sa!am, the International Atomic Energy Agency and U N E S C O for hospitaiity at the International Centre for Theoretical Physics, Trieste. References
[i I M. Li, The &structure in string theories-]: Bosonic strings, ICTP, Trieste, preprint [C/85/i60. [21 MB. Green and J.H. Schwarz, NucL Phys. B181 (198I) 502; Phys. Let~. B109 (1982) 444. [3i J.W. Miinor, FEns. Math. 9 (t963} !98: M.F. Atiyah, Ann. Sci. EcoL Norm. Sup~ a (1971) 47. [4] D. Gross, J. Harvey, E. Marfinec and R. Rohm, Phys. Rev. LetL 54 (1985) 502.