Symmetry patterns in the mass spectra of dual string models

Nuclear Physics B274 (1986) 520-558 North-Holland, Amsterdam

SYMMETRY PATTERNS IN THE MASS SPECTRA OF DUAL STRING MODELS* Thomas L. CURTRIGHT and Charles B. THORN

Department of Ph.vsics, Universit.v of Florida, Gainesville, FL 32611, USA Received 7 February 1986

We develop techniques for analyzing the rotation and gauge group representation content of string models. We explicitly construct all rotation group representations at low mass levels. We then obtain simple approximate formulas for the degeneracy of any given rotation or gauge group representation in the limit of very high mass. We also derive generating functions that give exactly the degeneracy of a representation at any mass level. Finally, we numerically study these results in several cases, and then abstract from the numerical data some simple empirical rules relating different representations.

I. Introduction

Recent theoretical attempts [1] using dual string models to unify gravity with all the other fundamental forces of Nature involve identifying the rest tension of the string with the Planck scale: TO= O(..¢¢~,t,~ek). With such a large tension, studies of the structure of the excited states of these models are of relatively remote phenomenological interest, since such states correspond to particles with J g 2 at least of the order of TO. Nevertheless, there are good theoretical reasons to pursue such studies in order to understand better the mathematical structure of the models. There are a number of important consequences of the rich excitation spectra in dual string models. A characteristic feature of all string mass spectra is an exponential increase in the number of states with the mass. As noted by Hagedorn, the scale of this exponential increase is an ultimate temperature for the string model. Once this temperature is achieved, further heating would create more particles without increasing the temperature. An additional extremely important consequence of the richness of dual string spectra is the Regge behavior of scattering amplitudes. At high energy these amplitudes behave as a momentum-transfer-dependent power of the energy. It is this feature which enables string models to solve the ultraviolet divergence problems of conventional quantum gravity. Indeed it is a spectacular achievement of string * Research supported by the US Department of Energy contract no. DE-AS-05-81ER40008). 0550-3213/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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theories that the cooperative effects of an infinite number of particle states tame short distance behavior in a way consistent with unitarity and causality, i.e. without imaginary couplings (ghosts). An intriguing feature of dual string spectra is a consequence of the no-ghost theorem [2]. In the critical spacetime dimension D (i.e. 26 for the bosonic or 10 for the spinning string) the entire spectrum of physical string states can be built from a collection of "transverse" harmonic oscillator raising and lowering operators which are vectors in D - 2 dimensions. For example, for the bosonic open string in its rest frame, each physical state is described by a polynomial of raising operators a i_ ,, = (al,)*, i = 1. . . . . 24, n = 1,2 . . . . . acting on a ground state 10) which satisfies ai, lO ) = 0, n = 1, 2 . . . . . The mass-squared operator on this space of states is simply ~ 2 = 2,n'T0{ ~ a i_ ,,a,,i - 1 } ,

(1.1)

where [ai,, a~,] = n8,,+,,8 i.i. There is an immediate puzzle associated with this result. If the a ~ are only 24-dimensional vectors, how is the 25-dimensional rotation group realized? This puzzle was resolved long ago [3] by constructing an explicit representation of 0(25) in terms of the transverse operators. This explicit construction worked only for the critical dimension. Nonetheless, it was later demonstrated [4] that apart from the first excited state it is always possible to represent the group O ( D - 1 ) with O ( D - 2) string oscillators. (Of course, if the first excited state has zero rest mass, it is not required by Lorentz invariance to transform as a vector under O ( D - 1), just O ( D - 2).) Since this latter demonstration was never published we shall review the argument in appendix A. However, our main purpose in this paper is to explore in systematic detail the rotation and gauge group symmetry patterns that appear at arbitrary mass levels in the various known consistent string models. We begin in sect. 2 with an elementary analysis of the low mass levels of each model. Then in sect. 3, we discuss generating functions for the symmetry group characters at each mass level for all bosonic and fermionic strings, for arbitrary D. Our discussion includes the generating function for the gauge group characters of the heterotic string. We use these generating functions in sect. 4 to obtain asymptotic formulas for counting representations at a given level, M, in the limit M--* ~ . In sect. 5, we abandon asymptotic approximations in favor of exact calculations, and generalize an earlier result due to Goldstone [5] who obtained an explicit form for the generating function which counts the number of times that a representation of the rotation group 0(3) appears at a given mass level in a simplified model with D = 4. Some aspects of such a general study were considered previously by Nahm [6]. Our exact results are studied numerically for some specific representations in sect. 6, both as a check against and as an extension of the results for low mass levels given in sect. 2. A numerical comparison is also made between the exact degener-

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acies and the asymptotic formulas of sect. 4. Finally, from these numerical studies we abstract some simple empirical rules which relate the generating functions for different representations appearing in the bosonic string. These appear to be exact rules in the limit D ~ m. We close by briefly discussing how to obtain finite D results from these exact D--9 ~ generating functions. Before getting into details, we may summarize our asymptotic results very simply. Let XM[~,] be the "representation degeneracy at level M ", i.e. the number of times that a representation [~] appears at mass level M. In all cases, we find XM[~]

M'~ C°(4~fl)-l/2d[X](fl/M)O+2r)/4e'aBM)W~"

(1.2)

where (13)- 1/2 is proportional to the Hagedorn temperature, d[~] is the dimension of the representation [~], and the power p depends on the symmetry group but not [~]. Similarly, the constant Co depends on the group but not the specific representation. Both C O and p will be given in sect. 4. It is interesting that the Hagedorn temperature associated with each specific representation is the same as the Hagedorn temperature associated with the totality of states. However, the power of M - 1 multiplying the exponential factor in XM[~] is much larger than the corresponding power for the total number of states. In fact, for D - 1 = 2 N + 1,

XM[X]'"

xt~¢]al

"-- d[Xl(N~r2/3M)U'u+l/2)(2~r)NCo .

M~oc

For N = 12 ( D - 26) note that this ratio falls as

(1.3)

1/MlS°!

2. Low-level calculations Consider a bosonic string in D = 2 N + 2 spacetime dimensions. Although a restriction to even D is not necessary in this section, it is convenient to introduce N since it corresponds to the rank of the relevant orthogonal groups, while the interesting, consistent string theories correspond to N = 4 and 12. Let us explicitly construct a few of the lowest mass states formed by acting with polynomials in the transverse oscillators, a~,,, i = 1 . . . . . 2N, n = 1,2 . . . . . on the vacuum, 10), as described in the Introduction. In mass-squared units of 27rT0, we have: 10), a single state at ~ 2 = --1; a i l l 0 ) , 2 N states at J [ 2 = 0 , ai_la~l[0), 2 N ( 2 N + 1),/2 states at J t ' 2 = 1; a'_210 ), 2 N additional states at ..¢t' 2 = 1; and so on. The tachyon at level 0 is a singlet under O(2N). The 2 N states at zero mass transform as a vector under O(2N). The N(2N+ 1) + 2 N = ½ ( 2 N + 1 ) ( 2 N + 2 ) - 1 states at unit mass are more interesting in that they can be combined to form a set of states that transform as a rank two, symmetric, traceless tensor under O(2N + 1). This is precisely as required by Lorentz invariance for the on-shell description of a massive tensor particle.

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The above procedure could be continued indefinitely to arbitrarily high mass levels. For any N, and for any higher mass level, the O(2N) tensors produced by acting with the transverse oscillators on 10> can be combined to obtain irreducible representations of O(2N+ 1) as is appropriate for Lorentz invariant, massive particle states. However, there is an alternative approach to obtain the O(2N + 1) irreps which reduces some of the algebra, Rather than first calculating and then combining various O(2N) representations at each mass level, we may deal directly with representations of O ( 2 N + 1) if we include longitudinal operators in the polynomials acting on 10). To do this, we first note that these longitudinal oscillators commute with the Cartan subalgebra of O ( 2 N + 1), and therefore contribute to the characters of O(2N + 1) representations precisely as though they were O(2N + 1) singlets. Thus, if we define the generating function for the characters of O(2N + 1) representations as the sum over all integer mass levels M, where ,,~2 = 2~.T0(M_ 1), of x M times the character appearing at that mass level, then we see that this generating function is obtained by simply multiplying the similarly defined O(2N) generating function by the multiplicity generating function for a set of integer-moded, rotation group singlet oscillators (i.e. (2.2) below). If we therefore first multiply, and then divide, the O(2N) character generating function by the singlet generating function, we will not change either the total number, nor the characters of the states appearing at any given mass level, but we will have first regrouped the O(2N) states into O(2N + 1) irreducible combinations, and then corrected for the total number of states appearing at the given mass level by adding (or subtracting) O(2N + 1) irreps appearing at lower mass levels. Thus this method is well-suited to a recursive calculation beginning with the lowest mass level, and proceeding upward in mass. The character analysis outlined above is explained further in sect. 3 below. Here it should suffice to illustrate the technique through examples. For the bosonic string with 2N transverse dimensions, the generating function, X(X), which gives the total number of states at the M t h mass level, XM, is obtained from the character generating function described above by setting all rotation angles to zero. The result is well-known to be

XB(X)-- ~ xMxBM=p(x)2N, M=O where

p(x)

(2.1)

is the partition function

p(x)

= f l (1 - x " ) - '

(2.2)

Following the above argument, we may write this as

Xn(X) = p ( x ) - ~ {

p(x)ZU+'},

(2.3)

where the term in braces counts the states constructed from both transverse and

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524

longitudinal oscillators. Any adjustments in the number of O(2N + 1) irreducible representations (i.e. "irreps") are then determined by the prefactor +oc

p(x)-'=

~

(--)~X'3k'-+k'/2=a--x--xZ+xS+x

v ....

(2.4)

k=-oc

To summarize, we obtain the correct number of O(2N + 1) irreps at the M t h level if we construct states as before except that we use polynomials in both transverse and longitudinal raising operators. According to (2.4), we then subtract those O(2N + 1) irreps which were obtained by this same construction applied to the M - lth level, subtract those obtained at the M - 2 t h level, add those obtained at the M - 5 t h level, and so on. The first three levels discussed above are immediately reobtained using this method, with the massless O ( 2 N ) vector given as an O(2N + 1) vector minus a singlet. This simply indicates the removal of the unwanted longitudinal mode from the physical state. A priori, it is not clear whether this method will give O ( 2 N + 1) irreps with negative coefficients at higher mass levels, due to the alternating signs in (2.4). A general argument (due to Goldstone and Thorn) that this does not happen is given in appendix A. Here, it suffices to simply list the results produced for the first eight levels for the bosonic string. We label the irreducible tensor representations of the orthogonal group in the standard way using partitions of the total number of tensor indices, where the nth integer in the partition indicates the number of boxes in the n th row of the Young tableau describing the tensor in question. For example, [2] is a symmetric, traceless, rank-two tensor, and [1,1] is an antisymmetric, rank-two tensor. The first eight levels then contain the O(2N + 1) irreps given in table 2a, for general N, where the left-hand column indicates the level number. Several comments are necessary. As claimed above, no minus signs occur after the first excited level. Representations begin to repeat at the fourth excited level, where another singlet appears to accompany the tachyon, as well as a second massive [2J-tensor. Indeed, irrep repetition occurs at the same mass for the sixth excited level, where there are 2 [2]'s, as well as for all higher levels. Finally, it is clear upon TABLE 2a Bosonic string 0: I: 2:

3: 4: 5: 6: 7:

I0l [ll-[0] [21 I3l + [1, II [41+ I2,11 + [2] + [0] [51 + [3,11 + [31+ [ll + [2, iI + [1,1] [61 + [4,11+ [41+ [2] + [3,11+ [2,11+ [31 + [21 + [11 + [01 + [2, el + [1,1,1l [7]+[5.1]+[5]+[3]+[4,1]+[3,1]+[4]+[3]+{2]+[1]+[3,2]+[2,1,1] +[3.11 + [2,1] + [2,11+ [1,11+ [1,1] + [11

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comparing successive levels in the table that there are Regge recurrences, with the first integer in a given partition incremented by one upon going to the next level. We will say more about the Regge interpretation in sects. 5 and 6. As a check on the results in the table, it is straightforwardly verified that the O(2N + 1) irrep dimensions sum-up to give the correct total number of states as described by the generating function in (2.1). We may extend the above analysis to the case of spinning strings with fermionic oscillators of either the Ramond or Neveu-Schwarz type. The raising operator polynomials which act on the ground state are then composed of the transverse oscillators a i. . and . . d i ,,, for the Ramond sector, and of a~_,, and b_~r, for the Neveu-Schwarz sector. These satisfy the usual relations.

[a:,a'.,]

=

{ d,',, dJ,, } = 3,,+,,3 'j ,

{ be, b/} = 3r+s3 ij,

(2.5)

where i, j = 1. . . . 2N, n, m = + 1, + 2, and r, s = + 5, L + 3 As before, these oscillators contribute to the mass-squared operator an amount proportional to their mode number (i.e. subscript), thus half-integer levels may be populated due to the Neveu-Schwarz oscillators. The 0(2 N) irreps which appear at the various levels may again be combined into O(2N + 1) irreps. As before, the correct result may also be obtained by adding in the longitudinal modes, taking advantage of their singlet character under the Cartan subalgebra of O(2N + 1). Using polynomials in both transverse and longitudinal operators acting on 10), we construct the states, and then adjust the number of O(2N + 1) irreps at any given level as determined by the generating functions. For the Neveu-Schwarz sector,

x N S ( x ) = ~_, ( x l / 2 ) M x N S M = p ( x ) Z N [ q ( x l / Z ) / q ( x ) ] 2 N , M=0 where p ( x ) partitions

(2.6)

is given in (2.2) and q(y) is the generating function for distinct q(y)=

f i (1 + y " ) - - f i (1 - y 2 " - 1 1 - 1 n=I

(2.7)

tI=1

Thus we have p ( x ) q ( x I/2) = p ( x I/2) and (2.6) may be rewritten as xNS(x) = q ( x l / p ( x ' / Z ) { [ p ( x a / 2 ) / q ( x l ] Z U + l } .

(2.8)

The prefactor

q(x)/p(xl/2)

=

Y'. ( - - ) k X { 2 k " - X ) / 2 = I - - x l / 2 - - X 3 / 2 + X 3 + X 5 k=

-

....

(2.9)

oc:

again specifies which O(2N + 1) irreps must be removed from or added to higher

526

T.I,. Curtrtght, C.B. Thorn / Dual strmg models TABLE 2b N c v c u - S c h w a r z sector

I:

[o] [1]-[o1 [1,1] [2] + I1,1,1]

0:

I 1:

3 22 95:

[21+[0]+[2,11+[1,1.1.11 [3]+[1]+[2,1]+[1,1]+[2,1,1]+[1,1,1,1,1] [3]+[1]+[3,1]+[2,1]+2[1,1]+[2,1,1]+[1,1,1]+[2,1,1,1]+[1,1,1,1,1,1]

3:

[4] + 2[2]+ [1] + [0] + [3,1] + [2,2] + 212,1] + [1,1] + [3, 1,1] + [2, 1,1] +

2[1,1,1]+[2,1,1,1]+[1,1,1,1]+[2,1,1,1,1]+[1,1,1,1.1,1,1]

levels, as in the bosonic case above. Explicitly constructing the states for the first eight levels leads to table 2b. We have designated the ground state in table 2b as level " ~," for later convenience in discussions of the superstring. Note that only the integer levels in the table are used in constructing the superstring, as described in ref. [11]. The levels of the Neveu-Schwarz sector have several features in c o m m o n with the purely bosonic string levels listed in table 2a. There is a singlet tachyon, the only minus sign appears at the first excited level which we may interpret as a massless vector, and integer-spaced Regge recurrences are immediately evident. For the R a m o n d sector, the total number of states is given by xR(X)

=

~.~ M~{}

xMxRM~___X R o P ( X )

2N

q(x) 2 N

,

(2.10)

where XRo is the degeneracy of the ground state. This may also be written as x R ( x ) = p( x )-Iq( x )- ' { xRop( x )ZN+ Iq( x )ZN+ ' } ,

(2.11)

with the prefactor specifying the adjustments to be made to obtain the correct O ( 2 N + 1) irreps at the higher levels. I-oo

p(x)

lq(x)-l=

E

( - ) kxk-' = 1 - 2x + 2x 4 . . . .

(2.12)

k=-oo

For the original Ramond sector, before modifications to achieve supersymmetry, the ground state is a pair of opposite chirality O ( 2 N ) spinors, which may be combined into a single O ( 2 N + 1) spinor. Evaluating the Kronecker products of this spinor with the O ( 2 N + 1) tensors constructed from polynomials in the vectors a ~ and d j leads to various irreducible spinor-tensors which we label as s[X], where [~] is again a partition of the number of tensor indices. The results for the first five mass levels are given in the following table.

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527

TABLE2c Ramond sector O: 1: ").

3: 4:

s[o] 2 qI] 2 s[2] + 2 s[1] + 2 s[O]+ 2 s[1,1] 2 s[3] + 4 s[2] + 6 s[1] + 4 s[O]+ 2 s[2,1] + 4 s[1,1] + 2 s[1,1,1] 2 s[4] + 4 s[3] + lO s[2] + 14 s[1] + 8 s[O]+ 2 s[3,1] + 8 s[2,1] + 10 s[1,1] + 2 s[2,1,1] +4 s[1,1,1] + 2 s[1,1,1,1]

O u r methods m a y also be applied to determine the rotation g r o u p irreps present in the heterotic string, with similar reduction in an effort to obtain the massive 0 ( 9 ) representations. However, we shall not explicitly discuss the low-lying levels of the heterotic string, but refer the reader to the recent literature [10] (see also the lectures by R a m o n d [15]). 3. Rotation and g a u g e group characters for strings

3.1. SOME PRIMITIVE CHARACTERS T h e g r o u p characters for all of the interesting string models m a y be written as composites of three basic primitive types: rotational group characters for bosons, rotational g r o u p characters for fermions, and gauge group characters. O u r analysis of these g r o u p characters borrows m a n y mathematical techniques from the b o o k s by Weyl and Littlewood [7]. Every string model has coordinate degrees of freedom which m a y be represented by a set of transverse string oscillators, as given for example in (2.5). Each such oscillator is a vector under O ( D - 2). As explained in the previous section, at any mass-squared level these oscillators m a y be used to construct states which transform irreducibly u n d e r the full rotation group O ( D - 1). In all k n o w n interesting cases, D is even, so we shall restrict our treatment to the case D = 2 N + 2, with N an integer. T h e O ( 2 N + 1) rotation group characters are then functions of the rotation angles for each of the N c o m m u t i n g rotation planes, which we choose to be (1,2), ( 3 , 4 ) . . . . . ( 2 N - 1 , 2 N ) . We label the corresponding angles by 0A, A = 1,2 . . . . . N. String states at each mass-squared level are eigenstates of linear combinations of three possible m o d e n u m b e r operators. R B=

y"

ai

ai II

11 '

nEZ+

R R=

ndi_,,d,i,,

~_, n,EZ+

RNS -_

~ r C l Z - 1/21

r b _i , . b ~i -

~(D

- 2)

El.. Curtright, CB. Thorn / Dual string models

528

_ ={~5,53 .... }. The superscript is of course where Z + = ( 1 , 2 .... }, and IZ - ',] summed over all transverse dimensions, i = 1,2 ..... D - 2. The oscillators obey the canonical relations in (2.5). Consider states made by acting with the bosonic oscillators on 10). The O(D - 1) characters of the resulting states at level R B = M are easily seen to be given by the generating function

XB( X, ( O, ..... ON})= ~ xMXBM( ( Ol ..... 0N}) M= 0

= I-[

fl (1 - x"e'°')-l(1

- x"e-'°")-I

A=I

n=l

I-l

,,=fl, - 2x,,cosiO j +

.

(3.2)

This is the first primitive character. The second primitive character is associated with fermionic oscillators of the Ramond (integer moded) or Neveu-Schwarz (~ integer moded) type. The generating functions for characters of states at level R R = M, o r R N s = M, made by acting with either d, or respectively b, oscillators on I0), are oo

Z(x.(o,

.....

oN))=

E ~ M Z ~ ( ( 0 , ..... oN)) M=0

=

H

ei°~/2 + e

A =1

=

I-I

n= 1

+i x"~"'+l~/Ze~"°~e~°,'/2}

p(x)

,4 ~ 1

m~

oo

=x-N/~p(x) N

~,

m~(Z-

xNS(x, (0, ..... 0N}) =

fi

xMxNSM((O1

x'"/2e '"'e,

(3.3)

1/2) N

.....

ON} )

M=O

=x-N~8 l--I A=I

=x

- N/8 I-[

+ x'ei°A)( 1 + x'e-i°A I"

A=I

=x-N/~p(x)N

x m:/2eim0A}

.o(~) n l ~ --oo

~ m~Z N

xm:/2e ''''°,

(3.4)

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529

Dud strrrtg models

where p(x) is again the partition function (2.2). Notice that the sums in the final forms run either over the root lattice or the spinor weight lattice of O(2N + 1). The final type of primitive character we shall use is that of a gauge group G, of rank K, that arises from compactification of K of the transverse dimensions [9, lo]. Let $A be the angles specifying a group element, g, generated by the Cartan subalgebra { H,, . . . , H, ). That is, g = exp( - iIZ:,+AHA).

(3.5)

Then the generating function for the gauge group character string is

of the compactified

(34 where r is an appropriate weight lattice associated with G. For the interesting models, r is an even self-dual lattice. 3.2. GROUP CHARACTERS FOR THE INTERESTING STRING MODELS

We now list the group characters for the more interesting string models in terms of the primitive characters defined above. (I) Open bosonic string xB=xB(x,{~1,....~12}).

(3.7)

(2) Open spinning string x;-E;‘s =

XB(X,(6

,,.‘.,

e,>)x”*““(x, { 8,t..., 0,)).

The open superstring [ll] character is easily obtained powers in xNS and multiplying xR by 5. x$X, = Xa(X,{4,..., X

(3-g)

by deleting the i integral

e,))~{xNs(X,(e,....,eN))+XNs(Xe2~;,{e,,...,e,})},

F&r=xBb, {4,..., 4>):xR(X, {e,7-..74>).

(3.9)

The closed string states are most efficiently thought of as elements of the tensor product of two open string spaces. Strictly speaking the total mode numbers of each factor are linked by a constraint. However it is simpler to deal with the character of the unconstrained tensor product space in which each factor can have independent total mode number.

530

T.L. Curtright. C.B. Thorn /

Dual strmg models

The constraint L = M, or L = M + 1, or whatever, is easily imposed afterward by identifying the appropriate powers of x and y.

(3) Closed bosonic string

(3.11)

X B = xB(x, (01 ..... 012 })xB(T, { 01 . . . . . 012 ))-

(4) Closed spinning string Xspin =XB( X, {01 . . . . . 04})xB(y, {01 . . . . . 04})

×xR'NS(x,{O, ..... 04))xR'NS(y,{O, .....

04) ).

(3.12)

The closed superstring [11,12] character is obtained from this by deleting the integral powers of x, y in xNS(x, {01 . . . . . 04}), xYS(y, {01 . . . . . 04}) and multiplying xR(x, { 0 t . . . . . 04}), xR(y, {01 . . . . . 04} ) each by _~. The rotation and gauge group characters of the heterotic, closed string theory [10] are obtained as a product of the open superstring character with a bosonic open string character with 16 compactified dimensions, e.g.

(5) Heterotic string hcterotic XE, xE,---'xB(x,{O,,

" -- , 0 4 } ) x B ( y , {

01 . . . . .

04

, R (X' ))_~X

(01,"

.., 04})

(3.13)

XXE,(Y, (@1 . . . . . ~4 })XE,(Y, {q"~. . . . . @&}), hcterotic = x B ( x , { 0 1 X spin(32)/z,

, • • •, 0 4 } ) x B ( y ,

XXspin(32)/z2(y,

( 0 1 '" " • ' 0 4 } ) ~ x R ( x ,

{ ~1 . . . . .

~8 } ) "

{01 ' " " " ' 04})

(3.14)

4. Symmetry patterns at high mass in this section we compute the number of times each representation occurs at very high mass levels. Our method is to project each of the character generating functions onto a given representation [2,] by multiplying by XIXl(g) and integrating over the group. For open strings,

x ( x , [hi) =

fdgxtXl(g)x(x, OA).

(4.1)

The number of times [2,] appears at level M is then 1

2~ri

d-

(4.2)

where this formula assumes X(x,[X]) is a power of x times a power series in x.

T.I.. Curtright, C.B. Thorn /

531

Dual string models

When X(x,[X]) is a power series in x ~/2 one must apply this formula separately to the even and odd terms in this series. The large M behavior is then obtained through a stationary phase evaluation. In all cases the stationary value of z is near z = 1. Writing z = e -2~°, X(Z) has the generic behavior

X(Z) ~ C(2¢ro)Pe B/2~'°.

(4.3)

o~0

T h e stationary phase point for large M is then

2

o0_ M.__~c~ m

'j2

independent of p, so that

XMM~ ~

C(4./r/~ ) - 1/2( /~ )(3+2p)/4 -~ e2~.

(4.4)

T o evaluate the x ~ 1 behavior of the characters one simply carries out the Jacobi transformation o ---, 1 / o on the characters (which are all elliptic functions) making use of the following formulae:

,=1

(1 - 2 e - 2,,,cos 0 + e - 4,~o) = exp - - - + 6o 6 x

- sin{0

1 - e - 2~"/°cosh- + e

n=l

~.] e-'°~''+~':e'a"' 1 -~ - 7~-

-4qro

,

O"

~ •

( exp i a ( 2 e r n - O ) - - -

~'(

n-

0 ) 2)

(4.6)

, . = 1 ( 1 - e -2 .... ) = ~ - o e x p

---

12o

+-~-)

1 "y~l"

(

2¢rn/o), (4.7)

exp - -- "y* -

e - ~°~2eiV'* = l e i . e j l - 1 / 2

0 K/2

.tt l(1-e

•y* E l ' *

O

~

"

(4.8)

72L. Curtright, C.B. Thorn / Dual string models

532

The range of O, 0 can always be taken so that only one term survives as o -o 0:

N~r XB( x, { O, ..... ON) )o_ ° exp( 60 xNS'R(x,

Noro 6

( 0 1 . . . . . ON ) ) o 4 0 e N~,/12oi-i

X~;(x'%)o~olei'efl-1/2exp

-o

0) ] YI sin10"4 + ~ 4-~o ] A s h ( 0 a / 2 o ) '

exp ( - i

½ 0A - o ~ 21r ] ]'

= exp - ,4 ~ ~ ] ] F self-dual

0

(4.9)

(4.10)

~-~] ]. (4.11)

In all cases we see that the group integration will be confined to the neighborhood of the identity:

IOAI < % = O ( M b - 1 / 2 ) , lea[ ~< a~o = O ( M - 1 / 4 ) .

(4.12)

Thus it is clear that all representation dependence will be contained in a factor Xta](I) = dim([•]) =- d TM.

(4.13)

It is also clear that the value of fl will be universal for each model and coincident with the value of fl for the totality of states; for the open strings, f l b o s o n / c = ½ N ~ r 2 --~ 4~r 2, N~12 flspinningstring

1 2 iNor ~ N~4

2 ~ 2.

(4.14)

For the closed strings, there is in principle a different fl for left- and right-moving modes. Thus for the heterotic case we have /~L heterotic ~_. 2~r 2 , •R heter°tic ~---4~r 2.

(4.15)

The power of M multiplying the exponential in eq. (4.3) is determined by the power of o in the small o behavior of the generating functions together with the small angle behavior of the group volume element. If all the angles are scaled by 0 --o }~8, the volume element obviously scales as dg -o xdimGdg.

(4.16)

T,L. Curtright. C.B. Thorn / Dual string models

533

For the rotation group integrals the O's are .5.<% = O(M -1/z) whereas for the gauge group integrals the ~'s are < ~00 = O(M -~/4). In addition, the bosonic part of the rotation group character contributes an extra power Fl A sin~_0A -IqA(~OA) which scales as )~N. For the spinning string the major effect of the fermionic variables is to change the temperature. In the NS sector, the occurrence of the ~ integer levels give rise to a further factor of _{ in addition to the change in temperature: this is because one must write 1 NS ( x , O ) + x N S ( x e > " , O ) ) + { _ _ ( x N S ( x , O ) - - x N S ( x e 2 " , O ) ) , xNS(x, O)= ~(x

(4.17) the first term containing the even G-parity sector and the second the odd-G parity sector, and asymptotically the terms xNS(x e2% 0) are suppressed. In the Ramond sector this does not happen and one simply changes the temperature. So define (4~r)N M3/4

~b~](fl, N) - ~Ju

e2~.

(4.18)

Then the open string asymptotic degeneracies are given by ~,~1(4~r2,12) for the 26-dimensional bosonic model, q~l(2~r z, 4) for the Ramond sector of the 10-dimensional spinning string, 5qa ~ M [xl (2~r2,4) for the Neveu-Schwarz sector, and 5q~M~ txl(2~r 2,4) for both sectors of the GOS superstring. The muitiplicative constant Ju is given by the determinant Jl J2

"'" "'"

Ju-I JN

Jui-l Ju

"'"

J2N-2

jN =

Jl

Jo

where

j,,=

1

r ~ 03+2"d0

~"+4Jo

sh~

(4.19)

22''+4 -- 1 =

2n+4

IBz,,+41 ,

(4.20)

and B2/,. are the Bernoulli numbers. For closed strings, we define X(x,

y, [X])

=

fdg×m(g)x(x,y, 0 , ) ,

xlXl(2rri)2~z--D-~z,-7WTFX(Z,Z',[X]). 1

2~t =

,.

dz

,.

dz'

(4.21)

(4.22)

534

7"I.. Curtright. C.B. Thorn / Dual striH,g modeiv

For large M one does a stationary phase evaluation for both z and z', and for the heterotic case the critical points are different: p'/2

exp(2( flV/-~- + f l V / ~ ) ) .

(4.23)

Similarly to eq. (4.18) the non-heterotic closed strings may be described in terms of a slightly different universal asymptotic form

(j~ I 1/2

d [X]

1

(2,rr)2NM 3 / 2

~blaJ(fi, N)=- ~HN ¢r2 J

(~)N2+3N/2 -

~_~

e"f~ ,

-

(4.24)

where ~!~(4~r 2 12) counts the 0(25) representations [~.] for large M in the bosonic model, ¼tpi, 4M(2vr ) _ 2, for N S - N S sector of the spinning string model l.t,t, tXl2V2M~.tg,,,2,4),, for the NS-R or R-NS sectors and q4%(2~r2,4) for the R-R sector. For the GOS superstring, a4'M t ix) (2*r2,4) counts the asymptotic number of [~,] in all sectors. The coefficient H N is given by the determinant

ho hi

hi h2

"'"

•. .

hN

hN_l

hN

...

h2u_ 2

hN-1

HN

(4.25)

with 1

~ 04+2"d0

h,,-- ,:,,+, ~,,/" 7hT0- = IB2,,+,I.

(4.26)

The heterotic string, being a hybrid closed string has a less explicit asymptotic form. The representation and M dependence is, of course, still given by the elementary considerations described at the beginning of this section, so we have

21/4d IX]d b,]

B,; M-- ~ -

4M 3/2

2---~-} exp(2rr( 2 ~ M + 2vrM)), (4.27)

where d IAI is the dimension of the spin representation [~.] and d [~] is the dimension of the gauge group representation. The constant B(; may be expressed in terms of integrals over the groups:

Be; = ~HV~;,

(4.28)

535

T. I,. Curtright, C.B. Thorn / Dual string models

where H is the determinant

ho

hi

h2

h3

2~ 22 2~ 2~

H=

(4.29)

h2 h3 h4 h5 h3

h4

h5

h6

with 1

2.

f ~ 04+2"d0

(4.30)

~2.+4 )o sh0sh0v~ '

the difference between 2. and h,, being explained by the presence of two fl's. The constant 1/"(; involves the gauge group and may be expressed VG= f°~dl6* U dO r~G

Ir'¢'le-¢-/2/fd'61~U /"G

11 -

ei"*l

(4.31)

•

r~G

For Spin(32) we have the explicit formula

Vspin(32)= 2( 1

8

(4.32)

where

K=

I ko kl

kl k2

.•..- .

k15 k16

kls

k16

•. .

k3o I

k. = (2n - 1)!!

(4.33)

(k o = 1).

(4.34)

For E s × E s we don't have the explicit formula but we can write VG =

"'/'1 e-.2/2

dS~ 1-I I 1 - e i " * [ /

Es

•

(4.35)

r~E 8

5. Closed form generating functions In this section we derive closed form generating functions for the representation degeneracies, i.e. functions which count the number of times a given representation occurs at a given mass level. We seek to generalize a formula obtained long ago by Goldstone [5] which gave the degeneracies X k[ J] of an arbitrary 0(3) representation

536

T.I.. Curtright, C.B. Thorn / Dual string models

J in the context of a simplified model of a string moving in two transverse dimensions. Goldstone's result is

x'xk[:] =p(x) 2 E x"'+"'"-':(-)"-l(1- :)2. /~ = J

(5.1)

n = 1

An interesting interpretation of this formula is that each n value represents a class of Regge trajectories whose leading member is

J,,(M 2) = M2/n + ½(1 - n).

(5.2)

These were termed "sister trajectories" and their presence in higher point functions [13] confirms their physical relevance. In addition to this Regge interpretation, however, formulas such as (5.1) give a powerful tool for analyzing the representation content for a given level. Since this is what concerns us in the present article, we shall construct analogous generating functions for the general case. We first recall the derivation of (5.1). The 0(3) bosonic string character is oo

x~(x,e) = FI ( 1 - 2 : c o s e + . , : " )

-~

n=l

=p(x)

-1,

2 ~.~ ( -- ) n - l ( 1 -- X " ) ( l -- X 2 n ) X "0' -')/2(1 - 2 x " c o s e + x 2")

(5.3) where the last form follows from partial fractions. Now we can expand (5.3) in terms of 0(3) characters [7], using (1 - x 2")

( 1 - x " ) ( 1 - 2 x " c o s a + x 2")

= ~.~ x"Ssin[(2J + 1 ) ~ e ] / s i n [ ~ e l .

.,=o

(5.4)

Projecting xB(x, 0)

onto a definite 0(3) character then yields (5.1). To generalize, we apply the above procedure to expand xB(x, {01 ..... 8N} ) into 0(3) characters for each angle OA.

x"(x.{6, ..... eN})=p(x) ~

E

E

F I : : ' + " ' ~ " ~ - ' : ( - ) ~'-'

{~}~lZl N {,,}~z~ A

x(1 - x"~)2sin[(2kA + 1)~_8A]/sin[½6A ] .

(5.5)

537

7",l.. Curtright, C.B. Thorn / Dualstring models

Similarly. we expand the fermionic characters x N S ( x ' {01 . . . . . ON)) = x - N / g p ( x ) N

E H x m ] / 2 ( 1 --Xm*+l/2) {.,}~ iZl N A

x sin[(2m A + 1)~04]/sin[~_Oa], xR(x. (o~ . . . . .

ON})=X-N/~p(x) N

~. 1-Ix
_x.,A+,/2)

× sin[(2m a + 1)½0A]/sin[½0A],

(5.6)

where Z + = {1,2 .... }, IZl = {0,1,2 . . . . }, and I Z - {_1 = {½,{ . . . . }, as before. It is now a simple matter to also expand the composite characters for the spinning and closed strings into 0(3) characters, using very elementary group theory. We list the results for all interesting models at the end of this section. We have now reduced the problem to the one of calculating [we take ?~ >/?~2 >/h3 >/ . . - > ?'N] 1 N sin(kA +5)0A

f dgxtXJ(g) aI-I =1 1

sin ½OA

~

N sin(kA + ½)0A sin!0A

rrN ~ ~ dOAo(Oa)xtxl(OA)A1-I =I 1

1

1

•r N 2 N N!

~,

yondOal2sin(N-s+½)O'l 12sin(N+)%-s+½)O'l

N sin(kA +

×FI A=I

1 5)OA

sin~0a

1 1 1 ,,. ~rN 2 N N, ~ ~ dOAp,Y'.p' e v v , 2 s i n ( N -

× ' - " 2sin(

N - PN + ½) 0N2 sin( N

Pl "+-½ ) 0 , 2 s i n ( N - P2 + ~)02 "'"

+ XPi' -- etl + 3)011 1

×

N sin(kA + 5)OA "''2sin(N+Xp;~-Pfv+~-)ONI-IA~I sinl0A

(5.7)

Now we recognize that each 0 integral is the 0(3) group integral of the product of 3

538

T.L. Curtright. C.B. Thorn /

Dual string models

0(3) characters. The result of doing these integrals is therefore X sin(kA +

f dgx'Xl(g) ,41-] =1 N!

~_)0A

sin i0A

p

A

12'4- A+P~I

tk?~

(5.8)

'

where we have introduced the symbol

o,7'(1,) =

{1,o,

n ~ k ~< m otherwise.

(5.9)

Since this answer is to be inserted into a sum over the k ' s which is completely symmetric, we may drop the (1/N!)Ep, and set P ' = I. The factor to be inserted may then be represented in determinantal form

O2'<+~,-2(k,) hi I h 2 - 1 I"

OIAi ~ "++'l~ , - ~ ( k ~ ) IA2I

O[ hm . f-t r".1N ]~ a- + Nx + III,

--.

"v i"a+l ~+ ,N- '-

11~k,) t

7t~- 2 + N ~, 21

"""

O ~ ,,. ( k N

(5 .lO)

)

This determinant is in fact efficiently calculated (it is either + 1 or - 1 ) and for the general case we have not tried to simplify further. However for the open bosonic string the sum over the k ' s is particularly simple since it just involves multiplying the determinant by x " k . The sums can be taken inside the determinant and in the case 7tsv>1 N - 1, the determinant can be explicitly evaluated yielding

FI~ °"`~"-A+'> I-I A

B
[x"<-x""lll-x"<+""].

(5.11)

Thus, for the bosonic open string in D = 2 N + 2 dimensions, we obtain the compact formula

x"(x,[Xl)=p2N(x) E I-I{x""(~"-A+x)+""(""-'>/2(1-x"")2(-)"~-'} n~zN.

x rl

A

[ x . . - x,,.][1 - x . . + , , . ] ,

(5.12)

A
where we have assumed 2k I ~ ~%2~/ ° " " ~ / XN ~/ N - 1. However for N -- 2, we have shown that this formula gives the correct answer for all 7tI >/2t 2 >/0. We conjecture that it is similarly valid without restriction on [X] for general N.

T.L. Curtright, C.B. Thorn / Dual.string models

539

We make a brief comment about the trajectory interpretation of eq. (5.12). In the limit where all ?'A are large, the smallest values of n are selected. But note the exclusion principle associated with the factor

H

A
Ix . . . . x " , ' l = o ,

if for any pair ( A , B), n.4 = n B. Thus the large )~A limit singles out the terms with

( . , , .2 . . . . . . ~ ) = ( P~. e2 . . . . . eN), where P is any permutation of (1,2 . . . . . N):

x~(x,[~])

-

all ,k.4 ~ oc

×

p2N(x) Y"p I-IA ( xe.,~.- A+l,+e.,,~.-,,/2(l _ x,~.)2(_ ),'.-~ } FI [~"~- xPA][a - xP"+"]

A
N

- - p 2 N ( x ) I-I (1 - - X " ) 2 I--[ (1 -- X " - A ) ( 1 - - X B+A) A
n=l

X E E p X (N+I)N(N-I)/3+~'.4Pa()~-4-A+I) P

So we have a family of leading trajectories labeled by the permutations (1,2 . . . . . N ) : 2~rTo(mass)2 + 1 = Y ' ~ P A ( X A - A + I ) + ~ ( N + I ) N ( N - 1 ) .

of

(5.13)

A

Since )~ >~ h 2 >/ • • • >/)~N, the highest of these trajectories corresponds to P = I: )-"~A()k A - A + 1) + ~ ( N + 1 ) N ( N - 1) = Y~AJkA, A

A

or

Y'~AX A = (27rTo)(mass) 2 + 1.

(5.14)

A

Finally, we complete our analysis of the O ( D - 1) content of all the interesting string models by expanding the various required composite characters in 0(3) characters.

E 1.. Curtright, C.B. Thorn / Dual strDLgmodels

540

5.1. THE OPEN SPINNING STRIN(}

In the product of XB with

X NS'R w e

encounter

sin(k a + ½)0A sin(mA + ½)0A ]

FI

sinl0A

A

J

sin ~20A

~, +,,,,

aN+,,,N

E

/,= Ik,-,,,, I

..-

sin(l a + {)0A

E

FI

6,= Ixx-,, ~F a

(5.15)

sin½0A

Using this result we easily obtain (doing the sums over the kA): sin(k A + {_)0A

NS

kc IZl N A X

E E I-I{(-) .*~ IZl N ncZ~' A

sin_{0A '"-lx{''-;+''~{''"

1})/2(1-x"')( 1 - x ....+1/2)

X(x,,41k.~- .....I_x,,A{*~+ .... + ' ) ) } .

x.,%°( x, o) = x - ~/~p~"( x )

x

E

FI

k~IZ_I/2[N

A

E

(5.16)

sin(k A + ½)0A sin~BA

E 1-I{(-)"~-lx'"'~+"'""-1))/2(1-x'~)

m e l Z _ I / 2 I N nEZ~+' A

X(1-x

.... +l/2)(x,,.,Ik,- .... r_x,,,{a,+

.....+D)}.

(5.17)

5.2. T H E CLOSED BOSONIC STRING

The same steps lead to XB(x,y,0)=

Y'~ I-I *EtZl N A

sin(kA +

~)0A p2N(x)p2N(y) 1

sin,!OA

{y,.{*.~+l)(1 +x,,,,)_ x,,..,*A+l,(1 X t~z~ Y2 n~z+ Y2N U (1-x""Y&)(Y '~-x'A)

+y'A)

,,~+'~

(-)

X X '1A('1' -l)/2y 14(la - 1 7 / 2 ( 1 -- X ' " )2(1 - - y ' 4 ) 2 } .

(.5.18)

T.I.. Curtright. C.B. Thor, / Dual string models

541

5.3. T H E C L O S E D S P I N N I N G S T R I N G

In this case take the product of (5.16) or (5.17) with variable x with (5.16) or (5.17) with variable y, and then use eq. (5.15). But because of the absolute value in the powers, doing any remaining geometric sums seems messy and unilluminating, so we leave them undone. We only quote the NS-NS case:

xNS'NS(x,y,O)=x-N/Sy-N/~p3N(x)p3N(Y)

Z k~

k,4 +

E

X

I-I

IZl" A

sin( k A + 1 ) OA sin~0A

J~A

E

52

52

E

E

J~ ~ IZl ~' ,&.4 = IhA -Ji.4 I mj E IZl x nt ~Z~' rne~ IZl ~ n2~Z'~

x

H I ( -- ) . . . . .+n-4X(IHi4+ll14{lll.4 . "-~,/2v<,,,L+,:~

,,,--~,/2

x ( 1 - x ..... )(1-~-',.,+'/~)

X(y":~lJ'-"-""-"l-y

''-''J'-~+ ......

+'))}.

(5.19)

The other cases involve the obvious change of summation ranges. 5.4. T H E H E T E R O T I C S T R I N G

We take the product of (5.9) with argument y and (5.16) or (5.17) with argument x and again use (5.15), with N = 4. X h~'~°tic --

XG(Y, ~b)XU(Y, O) X,,pi,;NSR (X,O)

=pt2(x)pS(y)xG(Y,~) x

Z

H

sin( k A + ½) 0,4

k~(i z_)zl'l/21,]] A

x

.,.,.~(

E

Izl'

E ],,.~zl

sin ½0A

1 - I ( - ) '~+ .....v',", "/-'x"~'"-,-"/: A

I, I Z - l / 2 1 4 ]

X I-I (

y/~lS~-

k.41_ yt,(S4 + k~ + 1))( x".4l"" - ..... I _ x"~(J" +'''~ + " )

A

Xx "d/2(1-x'',)(1-yt,)(1-x

.... +1/2).

(5.20)

To analyze the O( D - 1) representations, in all cases one simply replaces HA(sin(kA + ',)04~sin ~04) by the appropriate determinant of eq. (5.10).

542

T. 1.. C o'tright, C.B. Thorn /

Dual string models

In the heterotic case we can analyze the gauge group representation by projecting X(;(Y, ¢P) onto a given G representation. In the E, × Es case, X(~ factors XE. xE~(Y, ~, qb') = XE~(Y, qb)XE.(Y, qb'), so each factor can be analyzed independently. Since we are somewhat unfamiliar with the theory of characters for Es, we treat explicitly here the Spin(32)/Z 2 case, which is very similar to our analysis of O(2N + 1). For a given character XI~'l(ff) of SO(2N) or Spin(2N) we need to evaluate

f d g x l , ' * l ( , , : b ) e ' " *N!2 =- 1 u-1

f==H I tg?"-Sle ***, d,:/,AI<,,,+N_, --

1 (2,.a-) N

(5.21)

where the matrix Ct~ is given by 2 cos seo, , 1,

Gs=

s 4=0 s=O.

Introducing the symbol 1

r+,

G-~,(r)

= G

f

,, _~

d~, c,"c,'e','o,

1,

Ivl = r + s

1,

l v l = [r -

I

_

sl * o [~'t = [ r - s [ = 0 otherwise

2. O,

we find 1

f dgxt"j('~)e'~*- N !

1

-2- N - 1

E ,.., S'e p, p

(5.22)

, , ]-'I~2N+;~A--A--IA 1..~-,,+,,,, (v,,~).

(5.23)

A

Inserting this result into f d g X b't(g)Xspin~2u)(y, ~), we obtain (dropping ( 1 / N !)Y.p, and setting P' = I), 1 Xwi.,2N,(Y,[#]) = p N ( y ) E yC/2 2 N-1 ~1"

X

%~,,,+2N- 2(Vl)

~,,, +2N-

~,u~+2N-

~ # . ~ + 2 N - 4/'_ . ~Id21 ',I'2

,;2-1,

'(v2)

~ u , v + N - 1 /~_ X I # N - N + I[~ ]'N ]

vg', +1

"""

3 ( ..~1)

)

...

"'"

8,,,+N-1~_ ,at + N - 1

~. }'1)

. ~ p q + N - 2/'.. "~ ut~l + N - 2 ~ . t 2 !

~#N('yN ) #u x

(5.24)

T.L. Curtright, C.B. Thorn / Dual strmg models

543

The generating function, which counts the number of times the representation [X] of SO(9) and [/~1 of Spin(32)/Z 2 appears, is obtained by replacing X(Y, if) in eq. (5.20) 1 " 1 by (5.24) with N = 16 and by replacing l-IA(sin(k A + ~)0_ ,4/sins0_ A) by the determinant in (5.10) for N = 4.

6. Numerical studies and empirical rules First, we sect. 5 with asymptotic numbers as

compare the exact results for representation degeneracies obtained in the asymptotic formulas obtained in sect. 4. The leading terms in the expansions are not particularly accurate numerically even for level large as 103, with the corrections to the given results behaving like 0(1/M1/2). Nevertheless, one can discern from the numerical data for relatively low M (-- 100, e.g.) that there is a smooth increase towards the asymptotic values. It suffices here to consider the case N = 1, for both the bosonic and the spinning strings, to illustrate the accuracy and rate of approach of the asymptotic results. We present the exact degeneracies up to the seventy-second level for the cases J = 0 and J = ¼ in table 6a, along with the ratio: R - X . . . . t/Xasymptotic" Next, we numerically evaluate the exact generating functions given in sect. 5 for several rotation group representations, including those irreps which appeared in the low-level calculations of sect. 2. This serves as a check on the results of sect. 5, as well as extending the low-level results. The results are presented in tables 6b, 6c, and 6d for the case of O(3), or N = 1, and in tables 6e, 6f, and 6g for the case of 0(9), or N = 4. (However, as will be explained below, the N = 4 data given for the bosonic string is also correct for all higher values of N. In particular the representations and degeneracies listed in table 6e are exact for the bosonic string in 26 dimensions, with N = 12 and rotation group 0(25). In all these tables, the level number is given in the left-most column, while the other columns have headings given by the partitions labeling the irreps, and contain the corresponding irrep degeneracies. For the Neveu-Schwarz sector, we have also split the data into odd G-parity (i.e. M half-odd integer) and even G-parity (i.e. M integer) subsectors. The tables provide a graphic confirmation of the Regge trajectory interpretation of the formulas of sect. 5. Several trajectories with " u n i t slope" are easily visible. F o r example, in the 0(3) bosonic string case, table 6b, the uppermost entries in each column lie on a trajectory, J(m2), for which A J / A m 2 = 1, in mass-squared units of 2~rT0. In fact, from the same table, we see that the ( J + 1)th entries at the top of the column for angular momentum J (i.e. at levels J through 2 J ) are repeated at the top of the columns for all angular momenta greater than J, thus clearly revealing the presence of several trajectories with unit slope. The intercepts (i.e. values at M = 1, or rn 2 = 0) of these trajectories range from one, for the highest trajectory, to (1-J) for the "lowest" trajectories clearly revealed in the column for angular m o m e n t u m J.

T.I.. Curtright, C.B. Thorn / Dual strmg models

544

TABLE 6a Approach to asymptotic forms ( N = 1 )

Bosonic M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

X it [0]

Neveu-Schwarz R

x~t. w2[O]

R

xM[O] - 1

1

- 1 0 0 1 0 2 0 3 1 5 2 10 5 16 13 28 24 50 46 84 87 141 153 241 266 396 459 653 766 1070 1267 1725 2075 2762 3342 4397 5330 6918 8432 10814 13188 16784 20468 25836 31529 39505 48169

- 0.650 0,000 0.000 0.782 0.000 0.934 0,000 0.781 0.193 0.717 0.213 0.792 0.295 0.707 0.431 0700 0.453 0,716 0.502 0,699 0.555 0.691 0.578 0.704 0,603 0.698 0.631 0.702 0.646 0.709 0.662 0.711 0.678 0.716 0.689 0.722 0.699 0.726 0.710 0.731 0.717 0.736 0.725 0.741 0.733 0.745 0.739

0 1 0 3 1 8 6 19 21 48 57 117 150 268 366 609 840 1338 1866 2856 4004 5961 8332 12163 16938 24278 33666 47577 65571 91584 125469 173394 236150 323376 437738 594606 800199 1078910 1443867 1933749 2573802 3425923 4536306 6003668 7910381 10413618 13656094

0.000 0.247 0.000 0.369 0.080 0.407 0.195 0.396 0.282 0.419 0.326 0.442 0.376 0.450 0.415 0.469 0.442 0.484 0.468 0.498 0.489 0.513 0.507 0.526 0.523 0.538 0.538 0.550 0.550 0.560 0.562 0.571 0.573 0.580 0.582 0.589 0.592 0.597 0.600 0.605 0.608 0.612 0.615 0.619 0.622 0.626 0.629

1 0 2 1 6 4 16 15 38 46 93 118 220 294 496 687 1101 1533 2371 3315 4969 6960 10194 14213 20469 28407 40277 55610 77871 106847 148046 201906 277092 375678 511260 689036 930727 1247241 1673120 2229937 2972595 3941078 5223210 6890352 9082567 11924529 15639111

Ramond R - 0.126 0.260 0.000 0.302 0.099 0.382 0.163 0.416 0.251 0.412 0.325 0.432 0.363 0.451 0.405 0.463 0.437 0.480 0.461 0.496 0.484 0.509 0.503 0.523 0.519 0.535 0.534 0.547 0.547 0.558 0.559 0.568 0.570 0.578 0.580 0.587 0.589 0.595 0.598 0.603 0.606 0.611 0.614 0.618 0.621 0.624 0.627 0.631

XM[~]

0 2 4 6 12 22 36 62 104 166 268 426 660 1022 1564 2358 3540 5266 7756 11362 16524 23854 34252 48890 69368 97942 137588 192314 267628 370798 511524 702886 962084 1311886 1782548 2413678 3257276 4381678 5875860 7855730 10472268 13920894 18454404 24399594 32176956 42327006 55543764

R

0.000 0.123 0.183 0.184 0.239 0.280 0.293 0.323 0.350 0.363 0.383 0.402 0.414 0.429 0.443 0.454 0.466 0.477 0.486 0.496 0.505 0.513 0.521 0.529 0.536 0.543 0.549 0.555 0.562 0.567 0.573 0.578 0.583 0.588 0.593 0.598 0.602 0.606 0.610 0.614 0.618 0.622 0.626 0.629 0.633 0.636 0.639

545

T.L. Curtright, C.B. Thorn / Dual string models" TABLE 6a (continued)

Bosonic M

XM[0]

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

60033 73092 90597 110203 135920 165058 202754 245774 300707 363914 443621 535858 651146 784992 950930 1144321 1382199 1660103 1999927 2397448 2880884 3447194 4132342 4935551 5903269

Neveu-Schwarz R 0.750 0.745 0.754 0.750 0.758 0.755 0.762 0.760 0.766 0.765 0.770 0.769 0.774 0.773 0.777 0.777 0.780 0.780 0.783 0.784 0.787 0.787 0.789 0.790 0.792

XM_ 1/2[0]

R

17888489 23352915 30447782 39578338 51375759 66508311 85974068 110861898 142743466 183376556 235226145 301102986 384859612 490953120 625378214 795152824 1009564653 1279586158 1619548831 2046498210 2582451947 3253736918 4094034405 5143777023 6454277008

0.632 0.635 0.638 0.641 0.644 0.647 0.649 0.652 0.655 0.657 0.659 0.662 0.664 0.666 0.669 0.671 0.673 0.675 0.677 0.679 0.681 0.683 0.685 0.687 0.689

Ramond

XM[0] 20439046 26679227 34716207 45112943 58459910 75647055 97639493 125840647 161811057 207754531 266172692 340511826 434747356 554248666 705284400 896183140 1136763687 1439888841 1820836458 2299390760 2899179531 3650500957 4589694006 5762961369 7225890409

R 0.634 0.637 0.640 0.643 0.645 0.648 0.651 0.653 0.656 0.658 0.661 0.663 0.665 0.668 0.670 0.672 0.674 0.676 0.678 0.680 0.682 0.684 0.686 0.688 0.690

XM[121

R

72715210 94976104 123774610 160953756 208855914 270453860 349511878 450789804 580297794 745609996 956258238 1224221420 1564527890 1996006228 2542214614 3232583292 4103830398 5201707676 6583142494 8318879372 10496725282 13225522700 16640028142 20906885900 26231922982

0.643 0.646 0.649 0.652 0.655 0.657 0.660 0.663 0.665 0.668 0.670 0.673 0.675 0.677 0.680 0.682 0.684 0.686 0.688 0.690 0.692 0.694 0.696 0.698 0.700

By performing a simple analysis of the data in the tables, "sister trajectories" may be uncovered with lower slopes of ~, ~ . . . . . For example, using table 6b again, we calculate a difference table of "degeneracy differences" for two adjacent columns, after compensating for the unit shift in mass-squared at the start of the two columns. Define AXo=I

,

AXM=--X3M[J=MI-x3M_I[J=M--1],

M>~I.

(6.1)

From table 6b for the N = 1 bosonic string, this gives a sequence {AxM, M = 0,1 . . . . } = { 1 , 1 , 1 , 3 , 5 , 7 , 1 2 , 1 8 , 2 7 , 4 1 , 6 0 , 8 6 . . . . }. Next, note that the first J + l terms in this difference sequence occur as the differences between adjacent columns for all angular momenta greater than or equal to J. That is, {AxM, M = 0 . . . . . J } =

{X2j[J + I] - X z j _ l [ J + I - 1], X2J+I[J + I] X Z j [ J + I -- 1] X3j[J + I] - X 3 . / - ~ [ J + I - 1]}, for all 1>/0. This is just the effect of the first set of sister trajectories with A J / A r n 2 = 7l These particular sister trajectories are "subtractive", i.e. their contributions at a given angular momentum and mass-squared are to be -

.

.

.

.

.

546

T.L. Curtright. C.B. Thorn / Dual string models TABLE 6b Bosonic string (N = 1) [01

[1]

[21

[3]

[4]

0:

1

1:

- 1

1

2: 0 3: 0 4: 1 5: 0 6: 2 7: 0 8: 3 9: 1 10: 5 11: 2 12: 10 13: 5 14: 16 15: 13 16: 28 17: 24 18: 50 19: 46 20: 84 21: 87 22: 141 23: 153 24: 241 25: 266 26: 396 27: 459 28: 653 29: 766 30: 1070 31:1267 32: 1725

0 1

1 0

1

0

2

0

1

2 1 4 3 8 7 15 15 28 30 51 58 92 108 163 196 285 348 490 605 833 1034 1396 1740 2313 2887 3789 4730

1 3 3 7 6 14 15 26 31 51 60 95 116 171 215 308 385 541 683 932 1183 1591 2012 2673 3381 4429 5599 7266

2 2 4 4 10 10 19 23 38 47 75 92 140 179 257 329 466 595 821 1055 1426 1828 2442 3117 4112 5244 6836 8685

0 2 2 5 5 11 13 24 29 48 61 96 122 182 236 339 440 617 800 1099 1422 1920 2479 3302 4244 5587 7157 9327

[51

[6]

[7]

[8]

[91

1 0 2 2 5 6 12 14 27 34 55 72 112 145 217 285 407 536 750 980 1348 1757 2369 3080 4102 5298 6980 8977

1 0 2 2 5 6 13 15 28 37 60 79 124 162 242 322 461 611 858 1128 1552 2038 2752 3593 4796 6220 8205

1 0 2 2 5 6 13 16 29 38 63 84 131 174 260 348 500 667 938 1243 1712 2258 3058 4010 5360 6982

1 0 2 2 5 6 13 16 30 39 64 87 136 181 272 366 527 707 996 1325 1832 2425 3290 4332 5803

1 0 2 2 5 6 13 16 30 40 65 88 139 186 279 378 545 734 1037 1384 1916 2547 3462 4571

[101

1 0 2 2 5 6 13 16 30 40 66 89 140 189 284 385 557 752 1064 1425 1976 2632 3586

[11]

[12]

[13]

1 0 1 2 0 1 2 2 0 5 2 2 6 5 2 13 6 5 16 13 6 30 16 13 40 30 16 66 40 30 90 66 40 141 90 66 190 142 90 287 191 142 390 288 192 564 393 289 764 569 394 1082 771 572 1452 1094 776 2017 1470 1101 2692 2044 1482

[14]

1 0 2 2 5 6 13 16 30 40 66 90 142 192 290 395 573 779 1106

subtracted from the earlier slope-1 trajectories. Indeed, the total number of subtractions due to such trajectories, at level 2J + 1 + K for angular momentum J, is easily seen to be EM=0 ..... ~AxM. So long as K<~J+ 1, this entirely accounts for the deviation of the irrep degeneracies from those given by the slope-1 trajectories. For larger level numbers, i.e. M >/3J + 3, there is an additional contribution beyond that of the slope-1 and slope-½ trajectories. This additional contribution is additive, and is due to slope-.~ trajectories. Similarly, trajectories with slopes ~1, ~1 , . . . , appear in the numerical data. All these lower slope effects may be understood by examining the explicit formulas of sect. 5, which of course were used to prepare the numerical tables. For

547

T.I.. Curtr ght, C.B. Thorn / Dual string modeL~" TABLE 6c Neveu-Schwarz sector ( N = 1} [0] - 1,/2: 1 ,/2 :

3/2:

/2: /2: 9/2: 11/2: 13/2: 15/2: 17/2: 19/2: 21/2: 23/2: 25/2: 27/2: 29/2: 31/2: {1: 1: 2: 3: 4: 5: 6: 7: 8: 9: 1(1: 11: 12: 13: 14: 15:

[1]

[2]

[3]

1 {1 1

1 0

2

{1 3

3 3

1 5

2 2

1

9

7

7

8 11 17 11 6 26 25 25 19 36 52 38 21 71 77 78 48 102 143 122 57 183 218 219 117 268 371 344 150 450 564 579 268 661 920 894 366 1(159 1380 1446 609 1554 2168 2198 -I 1

1 0

1

(1 2

2 2

1 4

1 1

1

7

4

6

[41

I51

[6]

2 2 2 8 2 2 13 8 2 31 14 8 49 33 14 98 55 34 158 111 57 285 182 117 454 330 195 766 535 356 1198 906 584 1938 1435 997

1 1

[7]

[8]

[9]

[10]

[11]

[121

[13]

[14]

2 2 8 14 34 58 119 201 369 610

2 2 2 8 2 14 8 34 14 58 34 120 58 203 120 375 204

2 2 8 14 34 58 120

2 2 8 14 34 58

2 2 8 14 34

2 2 8 14

2 2 8

1 1 6 9 24 42 88 149 281 466

1 1 1 6 1 9 6 24 9 42 24 88 42 151 88 285 151

1 1 6 9 24 42 88

1 1 6 9 24 42

1 1 6 9 24

1

6 8 13 7 6 1 4 20 19 18 9 6 16 27 39 29 22 9 15 56 59 59 36 24 38 79 112 92 74 40 46 145 169 171 120 82 93 212 294 267 220 137 118 361 448 457 351 255 220 530 735 709 602 411 294 858 1110 1155 947 709 496 1260 1757 1763 1544 1129

1 1 6 9 24 42 86 145 273 448 776

1 1

1

6 9

1 6

example, Goldstone's formula (5.1) may be written as

x*x,[j] = f. x"J+"{"-"/2(-)"%(x) k =J

n= 1

=XJ'rl(X)--X2J+I'r2(X)+X3J+3"r3(X)

-

+ "'" ,

(6.2)

where ~h(x) gives the distribution of trajectories with unit slope, '7"2the distribution of the subtractive slope-½ trajectories, etc. For higher rank groups, N >/2, Regge recurrences and lower-slope trajectories, including subtractive trajectories, are also

548

T. 1.. Curtright. C.B. Thorn / Dual string models TABLE 6d Ramond sector ( N = 1) [1/2]

0:

1

1:

0 2 4 6 12 22 36 62 104 166 268 426 66(1 1{)22 1564 2358 354{) 5266 7756 11362

2: 3: 4: 5: 6: 7: 8: 9: 10: 11 : 12:

13: 14: 15:

16: 17:

18: 19: 20:

[3/21

[5/21

I7/21

[9/21

[11/2]

[13/21

[15/2]

[17/21

[19/2]

~ 4 10 18 32 58 98 164 274 442 7(/4 1114 1730 2660 4058 6114 9136 13554 19930

2 4 8 16 30 56 100 172 290 480 780 1248 1970 3068 4724 7200 10862 16240 24080

2 4 10 22 40 76 138 238 408 682 1112 1792 2844 4444 6872 10510 15896 23834

2 4 10 24 46 88 164 288 496 840 1384 2244 3590 5648 8778 13496 20510

2 4 10 24 48 94 176 316 552 942 1570 2568 4136 6556 10256 15854

2 4 10 24 48 96 182 328 580 1000 1678 2768 4490 7160 11272

2 4 10 24 48 96 184 334 592 1028 1736 2878 4696 7528

2 4 10 24 48 96 184 336 598 1040 1764 2936 4806

2 4 10 24 48 96 184 336 600 1046 1776 2964

present, as revealed in the tabulated data for N = 4 in tables 6e, 6f, and 6g. Trajectories with slopes of 1, ~ may be extracted from that data using the difference methods described above. The presence of such trajectories may also be understood upon examination of the explicit formulas of sect. 5, as for the 0(3) case. In particular, trajectories for higher rank groups result from incrementing the irrep labels by an integer, e.g. [~kl, ~2 . . . . . ~kN] ~ [hi q'- l, ~2 . . . . . ~'N]" By forming difference sequences, such as in (6.1), from the numerical data in the tables, we have noted some empirical patterns which suggest a relation of the generating function for the degeneracies of an arbitrary representation to the generating function of the O(2N + 1) singlet for the bosonic string, in the limit N --, ~ . We summarize those results in the form of a conjecture, which we have not yet established analytically from the results of sect. 5.

xpIX]

C°wecture:

XB( x, [?~i ..... X,,]) = xB(x, [0]) i-i( 1 _ xhOOk(it )

,

(6.3)

i

where p[~.] = E~=~k?~,, and the product in the denominator is over all the boxes of the Young diagram represented by [~.], with the usual definition for the hook of the ith box, hook(i).

,..,

o

P

"4

t,._~

g

t..

,.%

c~

ka

._=

,.-i [..-,

od

,-[

,-.-

I

#-

I,., 0

Z

i

r%

,.,%

,':,i F-i F,i /-i e-:4 F-i F-i ¢:,i ': /-i ~; ,-:'/ ,~ ~

[2.2] 72: U/2: 11/O: 13/2: IS/?: 3: 4: 5: 6: 7:

1 9 12 39

2 4 12 27

2 4 15

2 4

2

1 2 8 17

1 2 9

1 2

1

I5.4

1 1 5 9 25

12.1.11

[3.1.1]

[4.1.1]

[5,1.11

[6.1.1]

0

2 2 6 11 2-l 47 1 0 2 3 10 15 39 [l.l.l.ll

3/2: 5/2: l/2: 9/2: 11/2: 13/2: 15/2: 2: 3: 4: 5: 6: 7:

i4.21

2

p.1.11 3/2: 5/2: l/2: 9/2: 11/2: 13/2: 15/2: 1: 2: 3: 4: 5: 6: 7:

[h.21

13.21

1 0 3 4 14 21 54 1 1 4 7 1X 33

1 3 7 15 36 15

1 4 9 24 53

1 4 10 26

1 4 10

1 4

1 1 5 9 26 49

1 1 1 13 39

1 1 7 15

1 1 7

1 1

P.1.1.11

[3.1,1,1]

[4,1.1,1]

1 2 6 14 34 14

1 2 9 19 53

1 2 9 22

2 3 10 21 52

2 4 14 31

2 4 15

[5.1.1,1]

2 4

552

T.L. Curtright, C.B. Thorn / Dual string models TABLE 6g Ramond sector ( N = 4)

O: 1: 2: 3: 4: 5: 6: 7: 8: 9:

2: 3: 4: 5: 6: 7: g: 9:

3: 4: 5: 6: 7: 8: 9:

4: 5: 6: 7: 8: 9:

.qo]

,ql]

s[2]

s[3]

s[a]

1 0 2 4 8 16 32 60 114 212

2 2 6 14 30 62 126 246 472

2 4 10 24 54 116 242 488

2 4 12 30 68 152 326

2 4 12 32 74 168

sill]

s[21]

s[31]

s[411

2 4 10 24 54 116 240 484

2 8 20 50 120 268 576

2 8 24 64 158 368

2 8 24 68 172

s[lll]

s[211]

2 4 12 30 66 148 314

2 8 24 64 156 360

[11111

s[2111]

2 4 10 28 62 136

2 8 22 60 148

553

T.I.. Curtright, C.B. Thorn / Dual string models

In particular, we may consider those cases of this general relation which apply to the representations appearing in table 6e.

×B(x,[,,]) =X B(x, [0])x " ( I ( 1 - x * ) - ' ,

(6.4)

A=I

[v']) = x"(x, [o])x "{''*''/2

k=l

(1 -

(6.5)

XB(x,[n 1 ] ) = x B ( x , [ O ] ) x "÷2 (1 --X") 'i_ti' (1 -- x * ) - ' , '

(1 - x " ) ( 1

- x ''+') ,,+2

XB(x'[n'l'l]) =xB(x'[O])x"+5 (1-------X-~---X-Z) k=lI-I(1 --Xk) -L, xB(.x', IF/ 2 ] ) = X B ( x , [ O ] ) x

,

(6.6)

71--7)

''+4

(1 - - x n - l ) ( a

--X ''+2) n-}-2

FI

(6.7)

(6.8)

It is straightforward to check the consistency of these relations with the data given in table 6e. However, before making a comparison, it is necessary to consider the effects of N being finite. The major distinction between N finite and N ~ oe is the possibility of associate diagrams, which lead to equivalent representations under the proper rotation group. For example, for 0(3) the irrep [13] is equivalent under rotations to [0]. It is easy to see that these associate diagrams first appear at a minimum mass-squared of order N 2. Thus in the limit N ~ oe, they do not contribute to the degeneracy of any finite level. In particular, for the N = 4 degeneracies in table 6e, associate diagrams are not contributing to the irreps listed. Hence the degeneracies shown in that table are exactly the same for all larger values of N. By taking into account associate diagrams, in all possible ways (including traces of nonassociate diagrams), one can in fact extract finite N degeneracies from the infinite N results, or vice versa, up to any given level number. Similar N --* oe conjectures should apply to the case of the spinning string, but we have not yet obtained such simple forms as (6.3) for those results. Some of this work has been known but unpublished for several years. We should like to thank Edward Witten for a provocative conjecture which inspired its completion and publication. Also, one of us (C.B.T.) would like to thank Jeffrey Goldstone for a stimulating conversation, and for communicating his results to us.

T.L. Curtright. C.B. Thorn / Dual string models

554

Note added After the completion of this work we learned that J. Goldstone independently derived the representation counting formula (eq. (5.12)) for the bosonic string with general N. He also discovered a way of rewriting that formula, which we find is very useful in understanding our N ~ m conjectures of sect. 6 [16]: xB( x' [~])

=

E ;11 . < t / 2 <

(--)Y--'(n.4--A)xY'A"A(nA --1)/2-

~A(2A --

N-I,nAp2N(x

)

"
-''

XH(1--X"A) 2 H A

[(1--X"A--"S)2(]--X"A+"A)]

B
x"', x"' . . . . . where X v,, is the U N character for irrep [k,]. The exclusion principle which forces all the n A to be different was explained in sect. 5. Because of this exclusion principle, it is not hard to see that, in the large-N limit, the only terms which survive are those in which only the n A with A - N depart from the compact " F e r m i sea" distribution n~ = A. N o w think of the XuN as the ratio of the determinants of N × N matrices, Xu, = DIAl/D[°] with xniXi

xnl(k2 -1)

. . .

x n I ( A N - N +1)

D IM = x,,X~

x,12(x,- 1)

. . .

X n 2 ( A N - N + I)

X t ,vAl

xnx(h2 -1)

. . .

XnN(X,,,. - N + I )

As N - - , oo with all non-zero XA << N, the " b i g " entries of this matrix are in the right-hand columns, the [h] dependence is in the left-hand columns, and, as we have said, only the rt A in the bottom rows are different from n A ---A. Thus, we expect D IM to factorize:

x)F{.}(x),

D IM - D~([)~], so that

xoN- xul[X],x,

x2 .... ),

independent of {n}. Now, with kA = 0 for A > K, K<< N, X u , ( [ X ] , x, x 2. . . . .

=

xE~A~'4

H

A
xN) (1--X<-X~+B-A) (1 - x A-A)

A~
= x.l~l H A=I

(1 -x*)}(1 + O(xN)).

(1 - x ~,-~,,~+*) k=l

1/~

k=l

T.I,. Curtright, C.B. Thorn / Dual string models

555

Combining all of these results we obtain

Xu(X ,[Ol) U~ X%([?, ] , x , x 2. . . . ) = 1-i( l_xhook,,, ) , i

in accord with eq. (6.3). By changing variable to k A - n N + l _ A - - ( N + 1 - A ) , one can also show [17] that the expression for XB(x, [0]) simplifies for large N: X B ( x , [ O ] ) - f i (1 - x2"+1)-3"(1 - x2") 1-3" tl ~ 1

£

( __ ) Y'Ak4 xP[k]+5"Aka(kA -1)/2

×

k, k2 k,

-..

1-I (1 -

/[,q

= (1 - x ) f i (1 - x 2 " + 2 ) - " ( 1 - x2"+3) -''. n=l

Appendix A In this appendix, we lay to rest a perhaps unsettling feature of some formulas in sects. 2 and 5: there are minus signs (e.g. in (2.4) and (5.1)) which raise the possibility that an O(D - 1) representation might occur a negative number of times. Indeed, the singlet representation does occur - 1 times at the first excited level for the bosonic string, and for the Neveu-Schwarz sector of the spinning string. This is not a disaster because there is also an O ( D - 1 ) vector at that level, and if the mass-squared of the level is taken to be zero (a leading Regge trajectory with unit intercept) the negative singlet simply removes the zero helicity gauge particle, thereby making the vector transverse. However, any additional negative coefficients at other mass levels would indicate a severe consistency problem. Happily, it was shown some time ago, by Goldstone and Thorn [4], that additional negative signs do not occur. The argument is reviewed in the following paragraphs. We start with the fact that in an enlarged oscillator space in which each oscillator is an O( D - 1) vector, there are a non-negative number of any O(D - 1) representations occurring at each level. Out of these D - 1 dimensional oscillators we construct Virasoro operators. +OC

L,, = -'_, E

.' a k a . _

" k .,

(A.1)

with a oi = 0. If we subject the enlarged state space to the set of "physical state

556

T.I,. Curtright, C.B. Thorn / Dual string models

conditions" L,,lphys ) = 0,

n > 0,

(A.2)

the physical subspace will have a non-negative number of O ( D - 1) representations at each mass level since the L,,'s are O ( D - 1) scalars. If we can show that, apart from the first excited level, the physical subspace has both the same size and representation weights, level by level, as the space generated by D - 2 dimensional oscillators, we will have established that there is a unitary representation of O(D-1) on the state space of D - 2 dimensional oscillators. For D = 4, for example, this establishes that there are no negative coefficients in eq. (5.l) beyond the first two levels. In the original form of the argument, the methods of Brower and Thorn [14] were used to establish the linear independence of the states (A .3)

La21LX2-2 ... L~',,10),

for a oi 4: 0. But since we are dealing with a positive definite state space, the linear independence of (A.3) is equivalent to the nonvanishing of the determinant of the contravariant form matrix M "{ X ' } , { x } ~

,2

=

.--

L Xt.,: / ) - l'

.

.

.

(A.4)

.

Now this matrix is uniquely determined by the properties of the Virasoro algebra and of the state 10)" L,,I0 ) = 0 ,

n > 0;

Lo[0 ) = t 2aolO).

(A.5)

Therefore, we may infer from the linear independence of (A.3) for a~ 4~ 0, that the set LX21LX_"2... L~',,[ V)

(A.6)

is also linearly independent provided [V) satisfies L,, I V) = 0,

n > 0;

Lol V) = hi V ) ,

h * o,

(A.V)

even when a o = 0. In particular, for a *o -= 0, the states

aC,lo)

(A.S)

satisfy (A.7) with h = 1. For ( D - 1) odd, we now choose a Cartan subalgebra of O ( D - 1) corresponding to c o m m u t i n g rotations in the ½(D - 2) planes (1,2), (3,4) . . . . . ( D - 3, D - 2). Then the state aDi-'[0)

(A.9)

T.I,. Curtright C.B. Thorn / Dualstring models

557

has zero weight (i.e. zero eigenvalues for the rotation generators in our chosen Cartan subalgebra). Although for a0'-- 0 the set (A.3) is not linearly independent (e.g. L ~_110) = 0 for all X > 0), nevertheless the set Lx--'2.--Lx",,]0), _

L x'-1 L x, -'2 . . . .Lx,, ,,a n-x- l l o )

(A.IO)

is linearly independent for D >~ 3, as can be established by the methods of Brower and Thorn [14]. The set of states (A.10) can obviously be put in one to one correspondence with the set aLX~a Lx,-

l x,,

(A.11)

-1 _ 2 . . . a : , , 1 0 ) ,

where at: ,,--- a~ 7, t. Furthermore the states in (A.10) and (A.11) all have zero weight under our chosen Cartan subalgebra. Thus in the standard process of constructing the physical subspace level by level through orthogonal complementation [14], the state counting and weight patterns of the physical subspace will be identical to those of the oscillator space generated by a i_,,, i = 1 . . . . . D - 2, for every physical subspace of level higher than one. For example, at level two the physical subspace is the orthogonal complement of the space generated by L_,ai_llO),

L_210 ),

i = 1,2 . . . . . D - 1,

(A.12)

which are in 1-1 correspondence with the states a°-i- t a ~ -,I 0 )

,

a D _

21[ 0)

.

(A.13)

Only for the first excited level does the argument break down because then the set (A.10) reduces to the single state a°Tl[0),

(A.14)

and the physical subspace at this level is not required to be orthogonal to (A.14).

References [1] J. Schcrk and J.H. Schwarz, Nucl. Phys, B81 (1974) 118 [2] R. Brower, Phys. Rev, D6 (1972) 1655; P. Goddard and C.B. Thorn, Phys. Lett. 40B (1972) 235 [3] P. Goddard, C. Rebbi and C.B. Thorn, Nuovo Cim. 12A (1972) 425 [4] J. Goldstone and C.B, Thorn, unpublished (1972) [5] J. (;oldstone. private communication (1976) [6] W. Nahm, Nucl. Phys, B120 (1977) 125 [7] H. Weyl, The classical groups (Princeton University Press, 1946); D. Littlewood, The theory of group characters (Oxford University Press, 1958)

558

T.L. Curtri?ht. C.B. Thorn / Dual string models

[8] P. Ramond, Phys. Rev. D3 (1971) 2415: A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86: A. Neveu, J.H. Schwarz and C.B. Thorn, Phys. Eett. 35B. (1971) 529: A. Neveu and J.H. Schwarz, Phys. Rev. D4 (1971) 1109: CB. Thorn, Phys. Rev. D4 (1971) 1112 [9] I.B. Frenkel and V.G. Kac, Inv, Math. 62 (1980) 23: P. Goddard and D. Olive, in Vertex operators in mathematics and physics, ed, J. kepowsky S. Mandelstam, and I.M. Singer (Springer, 1985) [10] D.J. (}ross, J.A. Harvey, E.J. Martinec, and R. Robin, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys 1:{256 (1985) 253 [11] F. Gliozzi, J. Scherk and D, Olive, Phys. Lett. 65B (1976) 282: Nucl. Phys. B22 (1977) 253 [12] M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502: Phys. Lett. 109B (1982) 444 [131 P. Hoyer, N.A. Tornquist and B.R. Webber, Phys. Lett. 61B (1976) 191: P. Hoyer, Phys. Lett. 63B (1976) 50 [14] R.C. Brower and C.B. Thorn, Nucl. Phys. B31 (1971) 163 [151 P. Ramond, in Proc. Yale Summer School (1985) [16] J. Goldstone, private communication [17] T.L. Curtright, J. Goldstone and C.B. Thorn, in preparation