Z. Phys. C 58, 513-518 (1993)

ZEITSCHRIFT FORPHYSIKC 9 Springer-Verlag 1993

Extensions of string theories R. Amorim*, J. Barcelos-Neto** Instituto de Fisica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro 21945, Brasil Received 25 February 1993

Abstract. With the motivation that critical dimensions D , 4 might be suggesting that string theories have not been completely formulated, we study more general alternatives. We first consider a direct extension in the world-sheet formulation with Ns bosons and N F fermions and analyze the conditions for canceling the anomaly in all possible combinations of N B, N F and D. Later on we incorporate degrees of freedom of antisymmetric tensors to the previous model. The only possibility to cancel the anomaly in this case is with N ~ = N F = 1 and the our everyday spacetime dimension D = 4.

I Introduction It is usually desirable that a physical theory is free of anomaly. An interesting and important point is that the appearance of anomalies in a theory can be suggesting either the theory is not completely formulated or that there are some restrictions to be imposed on it. There are many examples we can mention where this occurs. The Glashow-Salam-Weinberg theory of the electroweak interactions is free of the axial anomaly if the number of quarks and leptons doublets are the same. That is why is crucial to detect the quark t. The so-called chiral Schwinger model does not exhibit the axial anomaly if a Wess-Zumino term is added to the Lagrangian [1 ]. Other interesting examples are found in string theories [2]. As it is well-known the bosonic string theory is free of conformal anomaly if the spacetime dimension D is 26. However, the presence of tachyons in the bosonic string spectrum means that the theory is not completely formulated. In fact, on its supersymmetric extensions, both in the world-sheet and spacetime, tachyons can be ruled out but the critical dimension turns to be D = 10.

Until now we have lived together with D = 10 and trying to reach D = 4 by means of (still not clear) compactification mechanisms. The question we place is if D - - l0 does not mean that the supersymmetric string theory is still incomplete. The purpose of the present paper is to stress this possibility. (It may be opportune to mention that supersymmetric extensions others than N = 1 do not lead to any interesting physical results [2].) We shall first consider an extension of string theory containing Ns bosons and N e fermions (this does not necessarily coincide with usual superconformal theories) and study the cancellation of the anomaly for all possible values of N B, N e and D. Later on, we consider an extension of this model where N B bosonic and N F fermionic antisymmetric tensors are included. The interesting consequence is that the only consistent contribution free of anomalies occurs just for D = 4 and Ns = NF = 1. In this paper we work in a covariant formulation by using the B R S T / B F V formalism [3]. In order to avoid the problem of covariant separation of first and second class constraints of superstrings [4] we work in the worldsheet superspace. We also only deal with closed strings and develop the calculations in a formulation compatible with the limit of zero tension [5, 6]. This will permit us to see if the results are dependent of this limit.

2 Strings with an arbitrary number of bosons and fermions The usual spinning string theory is a supergravity theory in two dimensions. After eliminating some degrees of freedom (gauge choice) we arrive at a simpler Lagrangian which gives the following set of constraints [2, 7] (/)3- = ~ 2 . . ~ T2 (X, 2 + igQ,5 I p " ) ~ 0

~b II = ~ - X ' * Electronic mail: ift01001@ufrj ** Electronic mail: ift03001@ufrj

+ 2 Tq/- q~' ~ 0

S~= ~ ' ~ u , + T ( y s ~ u L . X ' ~O

(2.1)

514 where ~ u is the canonical momentum conjugate to the bosonic coordinate X u. qtu is a Grassmannian element of a two components world-sheet spinor (~ = 1, 2) and also a spacetime vector. T is the string tension and prime means derivative with respect to the space parameter a. For more details and convention see [8]. We extrapolate the constraints above by introducing coordinates X u and q / u where a = l ..... N B and A = 1,..., N F. The set of constraints then turns, to be 3_ = ~_~2 .}_ T2(X,2

~11 = ~ . X '

Since we are considering closed strings, all fields variables are periodic in rr. We thus can write the following decomposition in Fourier modes (considering 7r the period) 1

1

(r) e2ina

+itu~,s r

+ 2 TO/. r

~0

(2.2)

~uu, (rr, r ) = ~ -

l

(2.) e 2 i n o.

n~ ~U~an

(2.6)

4~ ~," (rr, r) = ~2 Z q~ff' II ('t') e 2incr n

where in the cases of ~b3_ and ~bII there are implicit sums in the bosonic and fermionic indices a, A, respectively. Using the fundamental nonvanishing brackets

S~a (G, 7~)=--TE n Z2 S~a cr n

{x~ (~, ~), ~ [ (~', ~)} = G~.~v 6 (~ - ~')

where

{r

(2.3)

~)n3- _~_1 ~, [~n--m" "-~m - 4 T 2 m ( n - m ) m

(~, ~), ~'~ (~', O} -

i T 6~g~tlu~g(rr-rr')

(2.4)

one obtains the diffeomorphism algebra*

xXn_m. X m - 2 Z2m~mYsq/n_m] q~/I = ~-~ m

i~n_m'Xm--y~Cn_m'~llm

(2.7)

m

{~ 3_(a), ~ 3_(a')} = 4 T ~ (q~ II(a) + ~ It(a')) ~

O

s ~ o . = Z (89~o,.-~- ~4m

6 (a - a')

m

+iTm(ys~A,,,-~)

{~ 3_(a), ~" (~')} &

~'X

,,,,,.

Using these decompositions in (2.3)-(2.5) we obtain that the fundamental nonvanishing brackets modify to

=(~3_ (a)+q~ 3- (a')) Tg~ 6 ( a - a ' )

{~" (o),~" (o')}

{ XalLn, ~ r n } = 17ttv 6 ab 6n+m,O

=(~" ( a ) + ~ " (a')) ~

&

6 ( a - ~')

(2.8)

i

(2.5)

{ s~o (o), s L (~')} = --iOABOab(O~

('/7) e 2 i n a

{~'~.n, ~4pm} = - - ~ '1~ ~ . 6 ~ ' L + m , 0 and the algebra becomes

3- +2y~/~b II))g (o-- o-') {~b~ ,q~m ~ } = 4 i T2(n -- m ) Cbnl[+m

{~b ~ (a), S~o(a')}

{d?~ , ~b ll,,}= i ( n - m )dp ~+m

= 2 Ty~ e (Sda (0") ~- 1 S g a (a t)) ~

a (a - o;' )

{S~an, adbm}

{~b" (,~), S~o (r =(S~a(O-)~-l S~a(O-t)) ~

{~b~r,~b~} = i ( n - m)q~nll+m

6(o---a')

(~ABOab(2 ~

(2.9)

~)nq-m ~- 5 tt"nq-ml

{qb~, S~am} = i Ty~ p (n - - 2 m ) S f,,,,+ m where repeated indices imply summation. We mention that (2.4) is a Dirac bracket corresponding to the elimination of the usual second-class constraint relating momentum and coordinate of spinorial fields. * From now on it will be understood that brackets are always taken at the same time parameter r

{,~lf ~ ~=i( 89 "Fn , S AamJ To write down the corresponding quantum algebra we first observe that when Xa~, ~aU~ and ~u~, become operators (which we shall denote them by X~,, ^u ~al, ^u and r we have that only ~0II is not well-defined. Thus, the quantum algebra has the following general form

515 The quantities U are the structure functions related to the diffeomorphism algebra. From (2.5), we get

[q~ ,~m~] = --4 TZ(n --rn)~ ~l+m+ A j" (Fl)(~n+m,0 [~,~]=

--("--m)~OnJ-+m

g •177 = 4 T 2 ( d ( a " - a ) + d ) ( a " - - a ' ) )

[~/I ,~mll ] = _ (n _ m ) ~pll+m + A II (n ) dn+~, ~ [S~an, S~bm]=dAB(~ab(l

d~fl~nj-+m~l-y~5fl~t+m

(2.10)

+ Y~l~A(n)dn+m,O)

d

• U•

fi(a--a')

gl[j'j" =glllltl

[4)~, ~ a m ] = -- T(n -- 2m) y;~ 5~A~,,+ m

a ( G " - o.')) ~ d a ( o . - o.')

= (a (G" - a ) +

= - ( I n - m)S~Aa, n+m"

E~ 2~, ~r

U A~'~'j" = - i d A ~ d ~ d (o. -- o.') d (o. -- o.")

The calculation of the anomalous terms A • AII, and A follows the usual procedure [2, 5, 6]. The result is

3

=

- -

2 i OA~ y ; p d (O. _ O., ) d (O. _ O..)

U . ,Ao:,Bfl z Y,,~ AB .uezfl --v I 5

D D A II ( n ) = N B 6 ( n 3 - - n ) + N F 1 2 (n3+2n)+2o~n 1

uAcx,B~, II

d x (2 d (o. - o." ) + 5 ( o . ' - o . " )) ~o. 6 ( ( 7 - o . ' )

(2.17) d ~aa d ( o . _ o . , )

UA~, j.,Bp = T d A , y~sp

O

_D6 ( N B + s N F ) n + 6 ( N F - N'3)n+ 2c~n

•

(2.11) A j" ( n ) = 0

(o.-o.")+2d(o.'-o."))

UII ,A~,B p = dAB d~ p

A(n)=0

•

( o . - o.")+ 89

- o.")) ~

d

a ( o . - o.')

where ~ is defined in q~0II [0> = ~ [0>.

(2.12)

uA~, II,B/~= 6AB d~/~

• (89 (o.- o.") + a(o.'- o.")) ~

d

a(o.- o.').

We see that only [q~21,~2 ] has anomalous term. This commutator reads The combination of (2.15)- (2.17) leads to the expression of the BRST charge = -(n-m)(Of+m4-~[(N~+ 89

3

+ (NFD - N ~ D + 12a) n] d~+m, 0 .

f2 =ff do. (t/j" 4) j" +1/II ~b II -.[-y~taS~a (2.13) + 4 T2r/j" ~d~j" a a E II q-r/

The next step is the calculation of the BRST charge. This is achieved by introducing a pair of ghosts for each constraints, i.e.

dq j"

dr/ll do.

__89

and let (~j',~rj'), (ffll,r~ll) and (p-~,,p~,) be their respective canonical momenta. The general expression of the BRST charge reads Ji- ~(1)

q-r/

II dr/j" ~ o . ~j"

dy a~

(2.14)

S~a-+(Y~a, Y3a)

~r = ~r

~ j.

dr/II d y A~ q_r/ll ~ _ fill + 2 T y ; a ~a_a/yAa r/j-

q~ j" -,(,~ j', O j" ) 4) It--, (r/II, rTII)

j. r ~

(2.15)

i yA~yA~ffj.). 2

Using the Fourier expansions (2.6) as well as the following ones

where

r/II, j" = 1 Z r/ll,J- e2ina n

O (~ = 5 do. (r/j" ~ j" + r/II ~b II -}-Y]a S~Ao)

fill,j-

2i :g Z ~11' J- e 2ina "

n (1)= - 89 ~ do.' do" do." (r/1 ( a ' ) r/j" (o.)

y ~ t a = l Nn

X U j" j" II ( o . , a ' , o . " ) f f II (o.")

~ ~2inoYAan ~

n

+ n j" ( a ' ) ~ II ( a ) ull j. j. (o., o.,, o.,, ) ~ j. (o.,, ) + . . . ) .

(2.16)

(2.18)

2i

-~ - 7 PAa--

~ PAa. -~ ' ~ n

(2.19)

516 we get (r/) q~ •

g2 = ~

q~II

Since D, N B and N e have to be non-negative integers, we obtain the following set of possibilities

+yff~ sA_~,)

+ ~, [(n+m)(2Z2rtllm-nrl~rl~-m n,m

• - - n ~ n• t7--II m~LT~ll ,~11 r/~r n "~-~m - - 2 m--nHn

+ (m + 89 (2

nn

+r~A~ 1 • r m - n ' ,nllyA_~n)_l n -- ~7~rn_nY nA~ ya_~m].

(2.20)

AS one observes, terms containing ff Iml_ , ~//I, ffm ~ _ . r/n~ A~ a n d p- m _ , y nA a will not be well-defined under quantization. The next step in the B R S T formalism is to introduce the quantities [2]

D

NB

NF

1 1 2 2 2 4 10 13 13 26 26 52

26 0 13 1 0 0 1 2 0 1 0 0

0 52 0 2 26 13 1 0 4 0 2 1

2 9/2 2 0 9/2 9/2 0 2 9/2 2 9/2 9/2

GnII = ~, (n + m) ffnll_mhimI n

G ) = ~, (n + m) ff)-m r/2

(2.21)

m

F = Z ( m + ~1n ) p n-A~ - m yam~" m

The classical algebra of these quantities reads { G~II , G~ } = - i (n - m ) Gnll+,~ { G ) , G~ } = - i (n - m ) G)+

m

As one observes, the particular cases o f D = 10 and D = 26 are reobtained in this general formulation by m a k i n g N F = 0, N B = 1 and N F = 1, N B = 1, respectively. W e also notice that there is a possibility of having the critical dimension D = 4 but with an unpleasant situation of a string with 13 fermions and no bosons. E v e n t h o u g h we have included the conditions for D = 1, it does not m a k e sense a spacetime dimension lower than the e m b e d d e d world-sheet space.

(2.22)

{Fn,Fm} = - - i ( n - - m ) F ~ + m

3 Strings with tensorial degrees of freedom

other brackets are zero. T h e corresponding q u a n t u m algebra is [5, 6]

Let us next consider a direct extension of what was done in the previous section. W e assume that strings, besides space-time vectorial degrees of freedom, also have antisymmetric space-time tensorial ones. We denote these ,uv quantities by y f v and ;~a~, belonging respectively to the bosonic and fermionic sectors of the extended model. We also assume that there are N s and N~- of such quantities, respectively (this particular assumption would not be necessary at first) and that the set o f new constraints reads*

[(~nII , (~lml] = (n - m ) ~/l+m ~- 1 (n - 13 n 3) (~n+m,O [ ~ ) , ~m~ ] = ( n - - m )

an•

+ ~ ( n - - 13 n3)(~n+m,O

[pn,Pm]=(n__m)Pn+ m

(2.23)

+~NFNB(lln3--2n)O,,+m,O 9

In order to see the cancellation of the a n o m a l y in (2.13) we introduce the operator/S/I defined by [2] /SnH =

~2+

~2+

T2(X,2+

y,2

+ i ~u~5 q/' + i x Y s X ' ) ~ 0

H] +

= _ ~ / I + 6/I + 6n~ + p , .

(2.24)

This leads to

[C, -

q~•

cb = ~ . X ' "

+ ~ . Y"

i i "X" + ~ T q / . q / ' +-~ TX ~0

(3.1) S~Aa = ~ a " ~'~,~ + r

]

X~A + T(rsXA)

- m) C +

.

+ I [ ( N ~ D + I N F D + 11 N ~ N F - 2 6 ) n 3

where ~ a uv is the canonical m o m e n t u m conjugate to y u v . Using the additional fundamental brackets

+ ( N e D - N ~ D - 2 N ~ N F + 12~ + 2) n] 6.+m, o .

{ (2.25)

(o,)} = 1 (~ab ( n I~p ~ v 2 -- .ltA ~ v p ) (~ ( ~ - - ~ t )

(3.2)

We observe that the algebra is free of a n o m a l y if (N~+ 89

11 N ~ N F - 26 = 0 (2.26)

( N n - NF) D + 2 N n N F - 1 2 4 - 2 = 0 .

* We shall keep the same letters to represent the new constraints. We mention that constraints (3.1) could be directly obtained from an appropriate Lagrangian [9] by using the machinery of constrained Hamiltonian systems

517 I~v {x~= (a), x ~pA (a')}

i

2 T OancS=P(]'lltP nv2 -- rll~Z n v p ) (~ ( a -- a t ) (3.3) we find the same diffeormorphism algebra (2.5) and the same Fourier decompositions given by expressions (2.9) and (2.10). The calculation of A ", A II and A are obtained as before. They could be inferred from (2.11) by just making the replacement

D ~ D + 1D (D-- 1) = 1D (D + 1).

(3.4)

Consequently, the anomalous algebra [q~ If,q~ II] reads

[~n~',~]

this problem can be circumvented to obtain the diffeomorphism algebra for null spinning strings. This (classical) algebra reads [ 10]

{0" (IT), O" (IT')} = 0 {r

(IT),r H(IT,)} = (r

{r (IT),r (IT,)} = ( r

{r

+ ( N F - - N B ) D ( D + 1)n+ 24o~n]c~,+m,o .

(3.5)

Since the introduction of the new tensor variables does not change the number of constraints and the diffeomorphism algebra, we have that the number of ghosts and the structure functions are precisely those previously given. This means that the expression of the BRST charge does not change and that the ghost sector related to the operators d/I, dn~ a n d / e is still the same. Hence, we have the same quantum algebra (2.23). Introducing the operator s II as in (2.24) and considering that [r II, c~ tl] is given by (3.5), we find

m

+ ~A~[(((NB+I NF)D (D + 1) + 22NBN F - 52) n 3 + ( ( N F - - N B ) D ( D + 1) -- 4NBNF+ 24~ +4)n]O.+m,o.

(3.6)

The algebra is free of anomalies if

(NB+ 89

(4.1)

=

(S = (IT) + ~1s a (IT')) ~a

a(iT-a')

which is not the limit of zero tension of (2.5). The generalization of the algebra above for the degrees of freedom we have introduced is trivial and has the same general appearance. We notice that the only difference between the correct (classical) null algebra and the one obtained from (2.5), or the one including the antisymmetric fields, resides in the bracket {S, S}. However, the corresponding anticommutator does not exhibit any anomaly. So, all conclusions on critical dimensions we have obtained are still true for the null case.

5 Conclusion

[in", t~] = (n -- m ) s

~(IT-IT')

(IT), s 9 (IT')} = 0

{~) H( a ) , S a (lTt)}

D ( D + 1)n 3

(IT,)) Tda ~ (IT- IT')

( a ) + r LI(IT,)) !~IT a(iT_iT,)

{s= (IT), sp (IT')} =ia=~ r

= --(n--m)dpll+m + ~A~[(NB+89

(IT) + r

1)+ 22NBNF-- 52=O

(3.7)

( N B - - N F ) D ( D + 1) + 4 N B N F - 24~ 4 4 = 0 . The interesting point here is that there is just one possibility for not having anomalies (excluding the nonsense case with D = 1). This occurs at N B = N e = 1 and D = 4 (and ~ takes the value 1/4).

4 L i m i t of zero tension

In virtue of the brackets (2.4) and (3.3) we see that the limit of zero tension is not a well-defined process. This problem is related to the fact that the usual linking between momentum and coordinate of fermionic fields is lost when T = 0. In the paper of [8] it was shown how

We have studied some general extensions of superstring theories with the motivation that critical dimensions D :e 4 might be suggesting that usual string theories are not completely formulated. The first extension we have proposed is to consider N s bosons and N F fermions. We have seen that there is a possibility of critical dimension D = 4 but with 13 fermions and no bosons. It does not appear that this may be the solution of the problem initially formulated. We have also displayed a table with all possible values of N B, NF and D. Later on we have included degrees of freedom coming from antisymmetric tensors added to the previous model. Surprisingly, the cancellation of the anomaly just occurs with Ns = NF = 1 (usual superconformal symmetry) and the natural spacetime dimension D = 4. This is the main result of our work. The 10 bosonic dimensions where the usual spinning strings live are here splited in 4 spacetime coordinates and 6 degrees of freedom belong to the additional tensorial fields. One might argue that spacetime does not have tensorial degrees of freedom. On the other hand, it also does not have fermionic degrees of freedom and these are accepted in constructing superstring theories. We think that this may justify our procedure. The nice point is that the anomaly cancellation is achieved for the physical spacetime dimension just when these antisymmetric tensors are included.

518 W e have also shown, generalizing the conclusion already obtained for usual string theories [5, 6], that the cancellation of anomalies does not depend of the string tension as was initially believed [11 ]. We mention that although we have here constructed the quantization strongly based on the superdiffeom o r p h i s m algebra, the same results could have been obtained starting f r o m a Lagrangian formalism [9]. O f course, there are m a n y questions to be answered. The first one concerns the spectrum. To to this we need to construct the convenient vertex operators consistent with the extended model here introduced. Probably, it will be f o u n d massless antisymmetric tensorial particles. These could be interpreted as gauge fields o f (closed) strings interactions [12]. A n o t h e r point is related to the supersymmetric spacetime version o f this model. We are presently working in these problems and possible results shall be reported elsewhere [9].

Acknowledgment. This work was supported in part by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico - CNPq (Brazilian Research Agency.

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