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SISTER TRAJECTORIES IN STRING THEORY By STEVEN L. CARBON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 ACKNOWLEDGEMENTS I owe many thanks to Professor Charles Thorn, my thesis advisor, for many enlightening discussions, and for his encouragement. I would also like to thank the University of Florida particle theory group for presenting a stimulating atmosphere in which to work. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . . ii ABSTRACT . . . iv 1 INTRODUCTION ....................... 1 2 SISTER TRAJECTORIES ................... 5 Tree Level SixParticle Scattering . . 9 Tree Level EightParticle Scattering . 14 3 DOUBLELOOP FOURTACHYON SCATTERING 18 4 OPEN STRING SISTERS IN CLOSED STRING SCATTERING 29 5 OSCILLATOR REPRESENTATION OF SISTER TRAJECTORIES 37 6 CONCLUDING REMARKS ................. 50 APPENDIX: WEIGHT DIAGRAMS AND LAX OPERATORS 52 Review of Weight Diagrams . . 53 Standard Construction of Lax Operators . 58 Diagrammatic Construction of Lax Operators . 61 Proof of Diagrammatic Scheme. . .. 70 Supersymmetric Lax Operators . . 75 Discussion . . . 80 REFERENCES . .. . .82 BIOGRAPHICAL SKETCH .................... 85 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SISTER TRAJECTORIES IN STRING THEORY By STEVEN L. CARBON May 1993 Chairman: Charles B. Thorn Major Department: Physics It is shown that by using higherorder corrections that include sister trajec tories, it may be possible to restore the CerulusMartiii bound in string theory, which would resolve an objection to locality. In the singleRegge limit, we review the existence of the first sister trajectory in the sixpoint function and then exhibit the second sister in the eightpoint function. New work includes demonstrating that the first sister enters the fourpoint function at two loops, and that it can be seen across the intermediate open string propagator which appears as a higher order correction to closed string fourpoint scattering. We then introduce a procedure for determining the state representation of the sister trajectories, am(t), for spacelike momentum transfer squared t. These sister states are obtained by analytically continuing from the physical states, and involve reciprocal oscillators signalling the sisters unplihyical nature. We consider both tree level and higher order scattering. CHAPTER 1 INTRODUCTION A major hope of string theory has been that it would describe physics at the Planck scale. This entails understanding its short distance behavior and considering the related issue of locality. The latter is important because if the theory is nonlocal at a fundamental level then acausality may result, which is probably unacceptable. The high energy behavior at tree level seems to suggest that it is not local, although Gross and Mende1 claim that it still obeys causal ity because the interaction of strings is local. The tree level objection to locality in string theory is that in the limit of high center of mass energy, s + 00, for fixed scattering angle sin2(0/2) a _, it does not behave as do theories for nonextended objects. Cerulus and Martin2(CM) found that for general theo ries, under certain assumptions which include locality, the scattering amplitude obeys the rigorous lower bound  A(s, t) I> evlIn sc(9). String theory, on the other hand, has the tree level fixedangle behavior I A(s, t) I esf(0), which was pointed out even in Veneziano's original paper.3 Gross and Mende1 and later Mende and Ooguri,4 attempted to determine if stringy perturbative cor rections could restore the CM bound, but were unable to control the higher order corrections to reach a decisive conclusion. The CM bound was derived using the assumptions of unitarity, existence of a finite mass gap, and polynomial boundedness. Offshell, covariant string theory must introduce FadeevPopov ghosts to be unitary. However, unitarity can be established onshell in the critical dimension, for instance, by breaking 2 manifest Lorentz invariance and fixing to the lightcone gauge. String theory also does not have a finite mass gap since it has massless particles. But this probably does not lead to the violation of the CM bound. The final assump tion of uniform polynomial boundedness states that, for fixed t, the amplitude I A(s, t) I is bounded by sN where N does not depend on s or t. String theory does not obey this condition either for t > N since its fixed t behavior goes as st. Nor does this behavior obey even the weaker condition where N ~ 0(1 t 12) which Martin showed also gives the CM bound.5 Furthermore, in quantum field theory, polynomial boundedness is a consequence of locality. As a result, it is generally thought this power behavior of s leads to the CM bound violation.1'6 Although restoring the CM bound is necessary for a local string theory, it is not sufficient. Nevertheless, resolving this issue may be important for future development of string theory. In this thesis we will show how stringy corrections can slow the exponen tial falloff of the scattering amplitude for t  oo. However, unlike Gross and Mende who examined the high energy behavior that dominates over the entire moduli space, we will focus on particular processes that dominate just a small region. Our analysis will expose an overall subdominant behavior that is consistent with the lower limit of the CM bound. This suggests that the more dominant behavior considered by Gross and Mende may actually exceed the lower bound. However, we have yet to sum the perturbative series to ver ify if in fact the bound is obeyed. By considering subdominant behavior we hope that in performing the perturbative sum it may be possible to avoid the uncontrollable corrections which obstructed Gross and Mende. The plan is as follows. In Chap. 2 we review the emergence of the lin ear Regge trajectories, called sisters, which have more gradual slopes than the 3 standard a(t) Regge trajectories at large negative transfer momentum squared t. Rather than working directly in the fixed angle limit, we find it more conve nient to take s * oc with t held fixed. We show that the complete set of sister trajectories is consistent with the CM bound. We then present a slightly new approach for obtaining the sister contributions and discuss sixpoint and eight point tree level scattering. The generalization to Npoint scattering should then be apparent. Since the sister trajectories occur in the bosonic sector, our results equally apply to the Superstring and Heterotic string. In Chap. 3 we will show that sisters first appear in the open string four point scattering amplitude at the doubleloop level. We focus on this process because fourpoint scattering is the simplest case which must be shown to obey the CM bound. An interesting result suggested from our analysis is that the sisters also occur in the nonplanar case defined when twists are placed on both loops while they are separated by the intermediate sister propagator. Thus, in Chap. 4 we are led to consider the case of four interacting closed strings with an intermediate open string propagator. We will find that the open string propagator supports the sister trajectory, which is degenerate with the dilaton trajectory that may appear on the connecting closed string propagators. In Chap. 5 the focus is shifted to the the state interpretation of the sisters. We will determine the oscillator representation of the sisters by isolating the appropriate propagator, and then saturating it with string oscillators. We then analytically continue to the unphysical sister state. This procedure also affords us the opportunity to confirm the amplitudes derived in the high energy analysis. In Chap. 6 we give some concluding remarks and discuss a possible physical interpretation of the sisters. 4 In the appendix we discuss the weight diagram construction of Lax oper ators, which have no connection with sister trajectories. Lax operators have gained recent popularity in their application to the theory of matrix models, which have been shown to be related to low dimensional string theories.7,8,9 Each Lax operator can be associated with a particular representation of an affine Lie algebra, and generates a corresponding KdV equation. These KdV equations, in turn, generate integrable systems which can reproduce matrix model results. Our notation is as follows. The standard Regge trajectory is given by a(t) = a't + ao, where we choose the open string slope a' = 1 and intercept a0 = 1. This leads to a tachyon mass of m2 = 1. In the same units, for the closed string we have a(t) = t + 2. The trajectory a(ti) is associated with the momentum transfer squared ti across the propagator zi. Finally, the trajectory t(sij) is defined with respect to the energy sij = (pi + Pi+1 + + Pj)2. CHAPTER 2 SISTER TRAJECTORIES A central feature of the dual resonance theory3 was that in the high energy limit s + co the scattering amplitude scales as A(s, t) oc sa(t), for fixed t. Until the mid70's, it was thought that the Regge trajectories a(t) were linear and parallel, i.e., a(t) = a't+a0, differing only in their intercept ao. Then, in 1976, Hoyer, T6rnqvist and Webber10 discovered that the theory also predicted a new "sister" trajectory with a slope half that of the leading Regge trajectory. They were led to this result by a careful examination of the six particle scattering tree amplitude of Fig. 1. Hoyer et al. argued as follows. In the limit s +* o, the sixpoint amplitude factorizes as follows: A6 D(aa)V(aa, ab)D(ab)V(ab, ac)D(ac). (2.1) Here, the propagators D(a) have zeros for a = 1,2,.... On the other hand, the vertices V(aa, ab) have unphysical poles for aa, ab = 1,2,..., which have the undesirable properties of negative spin(nonsense) and wrong signature. Now, for ab there is only one zero coming from the central propaga tor, while there are two poles coming from the adjacent vertices. This leaves an unphysical pole, which does not appear in the exact expression for the scatter ing amplitude. For the theory to be consistent there must be some mechanism to cancel this unwanted pole. 2 3 4 5 X 1zl z2 z3 1 1 6 S Figure 1. The first sister f(t) requires twists on both zj and z3. At that time, most people worked with fourdimensional models. Although string theories are simplest in critical spacetime dimensions of 10 and 26, we can still adopt this restriction in the high energy limit by permitting momentum to grow large in only four dimensions. In this case, for six particle scattering, there are eight kinematic degrees of freedom yet nine free parameters. Thus, one must apply a fourdimensionality constraint.11 For the higher dimension string theories, one would have to consider a higher Npoint function with corresponding dimensionality condition. The following discussion holds in any case. To examine the high energy behavior of the amplitudes in a way that makes sense requires that one first analytically continue the energy into the complex plane, e.g., s  ioc.12 What Hoyer et al. observed was that previous analyses had imposed the dimensionality condition only after the energy had been analytically continued back to the physical plane. The effect of not main taining the constraint throughout the calculation is that some critical point remains hidden. Fixing this oversight and imposing the constraint before the back continuation, allows a factorization to occur in the amplitude, which then 7 exposes the critical point. Integration about this point then leads to the be havior I A(s,t) 1+ (t) where /3(t) = a(t) 1 is the first sister trajectory, for 03(t) = 1,0, 1,... or equivalently a(t) = 1, 1,3,.... Furthermore, at a(t) = 1, it was explicitly shown that the pole due to the sister trajectory 03(t) precisely cancels the remaining unphysical pole coming from the a(t) tra jectory. In addition, the sister trajectory has associated daughters which cancel the poles at a(t) = 2, 3,.... Ref. 10 also noted that the sister trajectory had been elusive in the past because at each vertex it can not couple to more than one onshell state. This decoupling can easily be understood by considering the factorization (2.1). In the high energy limit, the end vertices that couple an intermediate propagator to two onshell states are represented by factors of unity, which obviously do not have poles. Shortly after the discovery of the first sister in the sixpoint amplitude, working in the helicitypole limit, Hoyer13 showed that the eightpoint tree amplitude predicts a second sister trajectory y(t) = a(t) 1. The purpose of this second sister is to cancel unphysical poles occurring on the /(t) sister trajectory for 03(t) = 2, 3,.... He then proposed the generalization, 1 1 am(t) = a(t) (m 1), (2.2) m 2 where the mth sister first appears at tree level for 2m + 4 interacting particles. Hoyer et al. then showed that the first sister 3(t) could be obtained using the more general singleRegge limit,14 as opposed to the helicitypole limit used in Ref. 10. Other work followed which examined the sister trajectories under the, even less general, multiRegge limit.15'16 Sisters were subsequently found in the NeveuSchwarz sector of the NSR superstring,17 and related phenomenological implications were discussed.18 0(t) 0(t) Figure 2. Plot showing the leading a(t) Regge trajectory and the first two sisters, 3(t) and 7(t). Quir6s showed that the sister 3(t) appears in the singleloop sixpoint diagram, and that it renormalizes the corresponding tree level sister.19 Further, several papers also considered the closed bosonic string which found that, as in the open string, sisters appear at tree level when there are at least six interacting particles.20,21,22 The first two sister trajectories are shown in Fig. 2 along with the leading Regge trajectory. Due to successively more gradual slopes, the net behavior of the sisters is clearly not linear as t + oo. We can find the asymptotic behavior by considering the intersection of two neighboring curves am(t) and am+i(t) and then letting m  oo. Equating these using (2.2), we easily find that to lowest order a(t) m2. Comparing this with (2.2) gives am(t) m = V2a(t) (2.)t. (2.3) 9 For fixed angle scattering we conclude sam(t) * sv', which is the CM bound. This short calculation also demonstrates why locality may be responsible for the CM bound violation. In string theory, the fundamental length scale is defined by I = VaV. Further, the limit m * oo is completely equivalent to a'  0. Thus, the length scale I associated with each sister approaches zero as the order of the sister increase. This means that each successive sister appears more local than the previous one, and in the asymptotic limit we reach point like behavior. The possibility for restoring the CM bound in fourpoint scattering exists if we can show that the entire set of sisters, am(t), is present. Since, at each vertex, sisters do not couple to more then one onshell state, we must consider higher order corrections. We make an initial step in this direction by showing that the first sister 0(t) couples at the doubleloop level. Because t is to be held fixed, and in order to work under the most general conditions, we apply the singleRegge limit s * oo. However, since the original approach found in Appendix B of Ref. 14 requires a priori knowledge of any twists, we modify the calculation to remove the need for their explicit presence in the initial expression of the amplitude. This has the advantage of allowing us to consider many cases simultaneously, which significantly reduces the amount of work evaluating higher order functions. Note that in the multiRegge limit, one can only determine the need for twists by first inserting them, and then computing the final result to see if sisters appear. Tree Level SixParticle Scattering The sister trajectory /(t) is seen in the high energy limit of the sixpoint function only if twists are placed on both of the adjoining propagators as shown 10 in Fig 1. We begin, however, with the corresponding untwisted amplitude. This is easily calculated in the FubiniVeneziano formalism from A6 = (0,P1 I V(p2)AV(p3)AV(p4)AV(p5) I 0,P6). (2.4) In general, passing the vertex operators through each other produces factors of the form exp [2pi pZ (1 z)2piPj, (2.5) n=1 where z is the product of coordinates zi which are associated with the propa gators connecting the vertices. Final expressions for the amplitude are usually written in terms of the righthand factors. The lefthand form is more conve nient, however, for locating critical points in the high energy limit s + ioo. Consequently, we use the lefthand side of (2.5) if one of the connecting propa gators sees the energy s, and the righthand side for nonoverlapping quantities. In the particular case of Fig. 1, the complete exponential factor is then easily found to be oo n exp [2 z(P2z P3) (P4 + P5z) (2.6) n=1 Substituting in the momentum scalar products, the full amplitude becomes A 1 dz dz2dzz a(t) z1a(t2) 1a(t3) A6 , dzadz2az3z1 z2 23 x (1 zl)1a(s23)(1 z3)1a(s45) (2.7) x exp z nz(s24 34) + zzs +s 34 + z( (335 34) ) n=l where we have defined s= 34 + S61 s24 s35. (2.8) In writing (2.7), we have also dropped terms in the exponential which can be safely neglected in the high energy limit. 11 We are now in a position to impose the fourdimensionality constraint, which, in the high energy limit, reduces to s35s24 1. (2.9) s34s61 Applying this constraint to (2.7) allows the argument of the exponential to be factorized, giving A6 01dzldz2dz3z1a(ti) 1a(t)3 1a(t3)(1 l)1a(23) 0 0 n )] (2.10) x (1 z3)1a(45) exp[ E 2 1  Xl)(Z2  n=1 where 1 = l ) 1, X3 _ 1 (2.11) s35 S24 To discuss the high energy limit we must let s + ocei6 where the real part of s is held fixed, and 6 is such that the real part of s is in the strip of convergence.14 The result is a Fourier integral whose asymptotic behavior is dominated by its critical points.12 For (xl,x3) to be a useful critical point it must fall within the integration region, 0 < z1, z3 < 1. Critical points taken at the boundaries do not produce sisters. Since the boundary of the integration region is not included, the factors in (2.7), other than the exponential, can be ignored during integration. To recover the proper limit s + ooe i, we obtain a double critical point by choosing the phases s34, 61 * ooei, (2.12) s24, 35 + cei This is completely equivalent to twisting the propagators corresponding to zi and z3 since energies that overlap an odd number of twisted propagators change sign. In other words, the role of the twists here is to place the critical point inside the integration region. 12 To obtain the leading sister trajectory by evaluating (2.7) about the critical points, we keep only the lowest order terms in the exponential and integrate about Iz x1 5e, I z3 x3 1< e, (2.13) for e small. Choosing higher powers of z1 and z3 would lead to daughter trajectories. After shifting z1 and z3, we obtain A6 ~ l 1a(ti) x1a(t6)(1 l)1'a(s2)(1 x3)10a(s45)6, (2.14) where 16= dz2z2 )ez dz1dz3 exp [z2z1z31, (2.15) and 1 c = zX3(x1 1)(x3 1). (2.16) Setting y = iez2z3s, gives 6 = i()1 dz22a(t2)e f d 0 dy exp yz), (2.17) where yo = iE2z2s. Integrating over z1 we easily find 6 1 1 2(t2) zjc y exp(iy) exp(iy) I6 = 1 dz2 z2 e P )f0dy (2.18) o Y0 y where the y integral is symmetric. If z3 were not critical, taking the limit e  0 now would give L6 = 0 (use dy ~ e). This demonstrates the need for a double critical point. Now, define z = sz2c, which gives I6 = _(c)a(t)+1 1 dzza(t2)ez 0Y dexp(iy) exp(iy) (2.19) For us to consistently write z = y2e4g1c we must have 1 > e > g4 to reach the lower limit z + 0 for fixed y. Consquently, in the high energy limit 13 s t ioo, yo + 00 and so the integration over y gives iwr. Next, the z integral gives r(1 a(t2)) which is valid only for a(t2) < 1. Thus, the complete amplitude is .1+ )1 X 1 1 \ 1a(ti) 1a(ta) A6 c 2 1 3 (2.20) x (1 x1)1a(s23)(1 x3)1a(45). We now analytically continue the energy back by making the replacement A ei'. Defining f3(t2) = a(t2) 1, which corresponds to the first sister trajectory, and simplifying, we finally arrive at A6 ~Zirei7r(t2)2l(t2)10(t2)F(/3(t2) 1)X(t2)0a(1) 4(t2)a(t3) 3 (2.21) x (1 x1),(t2)a(S23)(l x3)(t2)a(s4). Since each of the energies comprising 9 overlaps with s34, the Regge behavior )(Wt2) shows that the central propagator in Fig. 1 sees the sister. Using the fourdimensionality constraint (2.9) we can write s = 834 + 61 s24 35 = s61(s35 861)(s24 s61), (2.22) and easily recover Eq. B.19 of Ref. 14. Examining the F function in (2.21), we see that the poles of the sister trajectory are for 3(t2) = 1,0,1,.... Our approach makes it particularly easy to determine the signature r of these poles. Twisting the sister propagator t2 changes the sign of all overlapping energies. Although both the numerator and denominator of x1 and x3 change sign in Eq. (2.11), the signs of the energy ratios remain unchanged. Thus, the twisted and untwisted diagrams can be added together giving an overall factor r + 1. Therefore, the poles of 0(t2) have pure positive signature. Since these poles correspond to odd values of spin, i.e., a(t2) = 1,1,3,..., they have unphysical wrongsignature. For the existence of the sister it was necessary that the argument of the exponential factorize, producing a 2tuple critical point. Integrating over both 2 3 4 5 6 7 \P(t) y(t) 0(t) / z1 z2 z3 z4 z5 1 8 Figure 3. At tree level, the second sister 7(t) first appears in eightpoint scattering. Concurrently, z2 and z4 see #3(t). coordinates, in effect, removed the linear power of the propagator variable 22 from the exponential. In general, integrals of the form I= 1 a1n exp(czrn) (2.23) in the limit c + oo integrate to _I (  ) ~ for a < n. (2.24) m mm Thus, sisters do not appear in 4 or 5pt scattering since both retain the linear power of z. Furthermore, to produce the second sister "7(t), both the linear and quadratic powers of z must be integrated away, leaving the cubic power. This occurs when the critical point is a 4tuple, which first arises in the eightpoint scattering amplitude. Tree Level EightParticle Scattering In this section we will expose the second sister, "(t), in the open string tree diagram of Fig. 3, where the sister appears across the propagator with z3.16 In the corresponding amplitude, we isolate the relevant terms by including, in the exponential only quantities which overlap the central propagator. We gather the other terms into a function f(zi, z2, z4, zs), whose exact form can 15 be ignored since, as shown in the last section, the sisters depend only on the exponential factor. The advantage of using the singleRegge limit over past approaches becomes more apparent in this example. From Fig. 3(without twists), we immediately write down 1 5 AS = 0 j dzif(zl,z2,Z4,z5)z3at3) 0i= 0 (2.25) x exp [2 n(P2zz p3z2 p4) (P5 + P6z + P7z z ). n=1 Substituting in the high energy limit values of the momentum scalar products gives the eighttachyon amplitude 1 5 A8 f 0 dzif(z,'z2,z4,z5)z31 i=1 00 n x exp[ znz(s25  s35) + z1zz4 (s26 s25 + s35 s36) n=1 (2.26) + 45 + 2z z2 z4 + 5 2(s35 845) + z4 (s46 S45) + z (s47 s46) + 4zz (836 s35 + S45 s46) + z2z4z5(s37 s36 + s46 S47)) where, now, = (s81 826 + s36 s37). (2.27) Applying the fourdimensionality constraints s81836 8135 = 1, 8146 = 1, 81845 = 1, (2.28) s26s37 s25s37 s26847 s25847 factorizes the argument of the exponential yielding 1 51a(t3) AS 0 dzif(zl,z2, z4, z5)z3 ( i= l (2.29) x exp [ (z X)(Z x2)(z x4)(z x5)], n=1 where s36 s37 + s47 s46 s25 35 l 4= s s26 s25 + s35 s36 (2.30) S47 s46 25 s26 + s36 s35 X2 = 5 5 s37 s36 + s46 47' For the critical point (xl, X2, X4, x5) to be inside the integration region, we must place twist on each of the associated propagators, and apply an additional fourdimensionality constraint: 3645 1. (2.31) s35s46 Consequently, due to the twists we have the sign changes 826, 35, 37, s46 + ooe. (2.32) To remove the first two powers of z3 in the exponential in (2.29), and to obtain a leading trajectory, we will integrate around z1 x1 1e, z2 x2 15 E, (2.33) z4 4 < E, I25 Vx5 1 E Clearly, this is just one of many critical points that we could have chosen. By writing z x4 = (z4 V/4)(z4 + v'4) ~ 2/f4 (z4 ViT), etc., and shifting the z's, we find A8 ~ f(x1, x2, vx4, /i) dz3z31O(t3)esz3c x dzl dz2 exp z3zlz24(v/4 4)(v5 x5) (2.34) x dz4 dz5 exp[2z2z4z5s. (x2 X1)(12 x2)], where 13 3 3 c = 3( l)(X2 x2)(x4 x4)( x5). (2.35) The last four integrals in (2.34) can be done in pairs, resulting in A8 ~ 47r2(2.i2x,(x2 l1)(2 x2)(1 V )(1 i5))1 (2.36) x f(x1, x2, V/5, v/i dz34Q))ea3 c 0 Using (2.24) then gives A8 ~ f(x1,x 2,v)2(sc) ( ) 1) 31 (2.37) x (22x4x5(x al)(a2 x2)(1 v/4)(1 V/ )) Again, we analytically continue back by replacing 9 with e"9. Thus, we find the Regge behavior A8 oc 9 a(ta)1 g(t3), which corresponds to the second sister trajectory. The first pole at q(t3) = 2 cancels the pole of the first sister trajectory at /(t3) = 2. The daughters of 7(t3) cancel the other poles at 3(t3) = 3, 4, .. When the central propagator in Fig. 3 carries the second sister 'Y(t), the adjacent propagators, z2 and z4, see the first sister 3(t). Each of these sisters can easily be computed by constructing the exponential term in (2.25) from the appropriate overlap quantities, taking the corresponding high energy limit and then integrating over a 2tuple. Finally, if we had chosen to initially integrate over a 2tuple critical point for z3, then we would have found the first sister trajectory, /(t). CHAPTER 3 DOUBLELOOP FOURTACHYON SCATTERING We now adopt our procedure to handle loop corrections. As in the tree level case, we must first isolate terms in the corresponding amplitude which overlap the appropriate propagator. In particular, to search for sisters in the double loop fourpoint amplitude, we consider the limiting situation where the two loops are sufficiently separated such that they and the connecting propagator can be treated as individual objects. Two such topologies are shown in Fig. 4. Both may be constructed by sewing together two singleloop diagrams. For this, we use the formalism from appendix D of Di Vecchia et al. 23 where the open string Npoint multiloop vertex has the form V(N;g) oc dDp exp [i7Tp r 1 + p B + C], (3.1) and where 7r is the period matrix. Completing the square and integrating over the loop momentum p gives V(N;g) c(det D/2 exp [rB 71 B + Cl. (3.2) The factorized fourtachyon doubleloop amplitude is then 1 4 1 2 A(4;2) = dz dzi f d(0,p,p2 exp 2 + CL i1 (3.3) L(<'2 < r 1 (BR')2 + t x 2 exp +[ ]0,p3,P4), where the subscripts L and R refer to the left and right loop, resp., and the superscript on Lc) labels the leg connecting the loops. The period matrix 18 2 3 \ / 2 3 S (t) j3(t) 1 4 14 (a) (b) Figure 4. Two distinct topologies for producing the 3/(t) sister in double loop fourpoint scattering. has been reduced to the singleloop case r = 2ri ln k, where k will be defined below. The details of the measure dtz, which is a function of k1 and k2, may be suppressed in the analysis below as long as we avoid the boundaries of the integration region. In the multiloop case the coefficient BY in (3.1) is given by(with a' = 1) / .am) qm( n Tt (z)) ^ Ta (zo) B 'a= \/2 I) In (3.4) i=1 m=O P Ta(Vi(z)) P Ta(zo)) z=O' (3.4) where zo, irp, and p are fixed points, and a product of Schottky group elements is defined by Ta=S SS S r=1,2,...,g; nieZ/{O}; pi Ipi+1, (3.5) where g is the genus number. Also, (P) Ec, means that the sum is over all elements of the Schottky group except that the leftmost element in Ta can not be S,. In the singleloop case Ta = S' and S'(y) = kny, where k is the multiplier and related to the radius of the loop. Here, however, the sum restriction leaves just the identity. Finally, for one loop o1  oo, i71 + 0. Thus, dropping the loop index, B = V2 1 m In (3.6) i=1 m=O MzO 20 where the projective transformation is explicitly given by T(z) = zil(zi zi+l)z + zi(zi+l zi1) alz + a2 (zi zi+l)z + (zi+1 zi1_l) a3z +4 (3.7) To reduce (3.3) we will need the commutator OO [B. a()] = 12 ( lnVc(z)) (3.8) [ c (m 1)! z= ( m=l Partial derivatives of the projective transformation can easily be taken giving ala4 a2a3 (zi zi+)(zi1 zi) (a3z + a4)2 z=0 (zi+1 zi_) (3.9) or, more generally, m!(a3)mn(ala4 a2a3) (a3z + a4)m+l z= =_ mlm! (zi zi+l)m(zi1 zi) (Zi+1 zi1) The singleloop threepoint diagram is constructed by sewing together two legs of a fivepoint diagram, and then fixing three of the projective coordinates. For that case, following Di Vecchia et al. 24 we sew together legs 3 and 4 and then choose z3 = k, z4 = oo, and z5 = 1. In the present case, we will associate the coordinate z5 with the connecting leg coordinate zc. This gives 9zVc(z = 0) = 21 1, along with Oz'Vc(z = 0) = 0 for m > 2. Thus, the commutator (3.8) becomes \n+l(\rl(z)\m [B, = c) ()m E (z))m z= = 2 E(1 zl)m. (3.11) m= 1 m= 1 Next, the coefficient C in (3.1) is given by 3 00 C i) (i) 1 ln[i'(z)]z=0 i=1 m=0 3 o0 (i) (j) + 2 E > n! rm! ln[(y) Vj(z)]y= z=0 (3.12) i i,j=n O Vm=O) y=z=O where the prime form is defined by wi \z T/ (w) w T (z) E(z,w) = (z w) z T() w T,(w) (3.13) Sz To,(z) w Tc,(w)' and the indicates the the identity is not included. For a single loop the prime form reduces to 00 nw w kz E(z,w) = (z w) I z k w k (3.14) z knz w knw" n=l Below, we will need the commutator 00 (0 [Cc)] O,pi) = ( 1)! mln[Ve(z)]zo m=l 1 Oo +2 (Pi mln[zi Vc(z)]z=O i c m= 1 +2E Pi 9 zi knVc(z) Vc(z) knz = Sm=1 = (m 1)!z n zi knzi Vc(z) k Vc(z) z=0' (3.15) where pc = p. Due to momentum conservation we can neglect the second denominator in the last term. Further, in the high energy limit s  oo, we have Pl P3 + s/2, Pl P4 * s/2, P2 P3  s/2 and P2 P4 + s/2. These imply, P Pi = (P1 +P2)Pi + 0. (3.16) Consequently, some of the terms in (3.15) do not survive the high energy limit in (3.3). This permits us to drop the entire first term, and the i = c term in the last sum. Rearrangement then yields 00 (0 I[C, aMIc)] O,p) =2 E (m )! i c m=1 00 00 x [in [zi knV (z)] [Vc(z) krz] z= n=O r=l (3.17) Taking the derivatives gives 00 (0 [C, c)] 0,pi) =2 pi[ ic m=1 n=O kmn(Vc(z))m (zi knVc(z))m (3.18) + (_)+l(Vi(z))m S(V(z) knz1)m =0' m,n=l which simplifies to (0 l [C, a,)] lO, Pi) kmn(1 z)m + (k Y E m,n=l ( Z1)m (knzi)mJ (3.19) We also need the singleloop result exp 12 + C 0, P10,P2) = ,' 2 2 I 0, p,p2), (3.20) where f arises in planar loop amplitudes and can be expressed in terms of the Jacobi theta function. Substituting (3.11) and (3.19) into (3.3) then gives A(4;2)" 1j dzz1a(t) Jd 1j dazi122p P2V,2'p4 i= 1 ep2 zm(1 zl)m(1 z3)m Pi .j In zi In zj x exp 2 n m=1 ij 1 k kmn I:P (k 2 zj)m n=0 2 1 kmn in Z1 [E _I PInk2 L (k ) zi)m n=0 1 i j5c EEPi i#c j + E pi Pj kmr X 2 = (kr z)m r  2 O n= 1 1 S (1 kfzi)m n=1 00 1 +E(1 knz)m) n=1 00 1 Z=1(1 kyJ)m)) (3.21) where i and j correspond to the different loops, and we have dropped a mo mentum independent factor which can be ignored in the high energy limit. 0C =2Ep [E ic m=l1 n=0 (0 k"mn = (k" zi)m n=0 1 I 23 Replacing the momentum scalar products by their high energy limits allows us to factorize the argument of the exponential to get A(4;2) dzz1(t) dp dz( ','12'34)1ra(t) JO zm(1 zi)m( z x exp s z zl)( z3)mgm(zl, z2, kl)gm(z3, z4, k2) m=1 (3.22) where we have defined 9in y, k) = (km (1 )m < x + y >. (3.23) n=O n=1 The function gm(x, y, k) is for orientable planar loops and is essentially the mth derivative of in Thus, we can immediately write down the expression in the nonorientable case: o In x (k)mn 00 1 = Ink ((k)n )m (1 (k)nx)m < > n=0 n=1 (3.24) and for the nonplanar case: SIn (_ )mkmn 00 1 gmn(x, y, k) = ( (1+ knx)m n=0 n=1 Now, we search for critical points which do not reside on the boundary of the integration region. Unfortunately, due to its complicated form, one must numerically search for zeros in gm(x, y, k). It is found that gm(x, y, k), for all m, does indeed possess zeros that are exclusively within the integration range. These zeros generate the criticalpoint curve x = P(y, k), for some function P(y, k) which satisfies gm(P(y, k), y, k) = 0. In addition, numerically analysis indicates that both nonorientable and nonplanar cases also possess critical point curves. In all these cases the zeros do not seem to be confined to any particular region of integration space. 24 This case differs from the tree calculation in two respects. First, to factorize Eq. (3.21) it was not necessary to impose a dimensionality constraint. Clearly, this is due to the fact that there are only four interacting particles, and not due to the loops. Second, unlike the tree amplitudes, the presence of twists is not significant. In the former case, the twists were necessary to change the sign of some of the energies to place critical points inside the integration region. In the loop amplitudes, the signs change as a result of the periodicity of the Jacobi theta function. Continuing with the calculation, in the limit s * ooei6 (3.22) becomes A(4 2) ~ dzz1a() d dzi(0 12 34)1(t)esZh22 ) J O i=1 (3.26) x exp [sz(1 z1)(1 z3)g1(zl, z2, kl)gl(z33, z4, k2)] where h2 = (1 21)2(1 z3)2g2(zl, z2, kl)92(z3, z4, k2) (3.27) 2 We will evaluate about the critical curve I 21 P(z2, k) 1 e, z3 P(z4, k2) 1< (3.28) Expanding the gl's about this curve, and then shifting z21 and 23, gives A(4;2) ~ 1 dzz1a(t) / 1 dz2dz4(12 034)1a(t)esz h2 Jo J Jo (3.29) x dz1dz3 exp[szzlz3h ], where hl =(1 P(z2, kl))2(1 P(z4, k2))2 (3.30) x gj(zl = P(z2, kl), z2, kl)g(z3 = P(z4,k2),z4, k2), and h2, '12, and 034 are now evaluated on the critical curve. The integration of zi and z3 proceeds as before, giving A(4;2) irs1 dzz2a(t) f d (3.31) O (3.31) x j dz2dz4(12 34)la(t)hlesz2h2. J0 25 Similarly, the z integration is also easily done giving A(4;2) ~ iTreir(t) (t) n(f(t) 1) I dy 11 (3.32) x dz2dz4(12034)1(t)h1h 0(t)+1, which exhibits the first sister trajectory /(t). Since the integrands involve derivatives of the Jacobi theta functions, we are unable to complete the cal culation showing explicitly that the sister does not decouple. For the planar diagram, however, in the special case 0(t) = 1, it can easily be shown that the signs of each of the integrand factors are the same over the entire integra tion region. On the other hand, to show that decoupling does not occur in the nonorientable and nonplanar cases is more difficult, although the results of the next chapter indicate that the sister survives the latter case. The existence of the the second sister requires that two of the g's share the same critical point. Using (1 + x)(r+l) (1 + X)r = xe, (3.33) it follows that 00 gr(x, y, k) gr+1(x, y, k) = (knxeknz knyekY) n=O O (3.34) SE x/k /kn n0 e /" kne n=0 Since the difference is independent of the index r, for any given critical point either one gr vanishes, resulting in a single sister, or they all vanish simultane ously. In the latter case, (3.22) results in the form A(4;2) j1dzz1a(t) dl dz2dz4(012'34)1a(t) Sf (3.35) x dzdz3 exp sziz3 Z zmem]. rm=1 2 3 1 4 (a) 2 3 1 4 (b) Figure 5. General 2mloop fourpoint diagrams for generating the mth sister am(t). Integrating over z1 and z3, we obtain A(4;2) i7rs1 1 dzz1a(t) fdp 1 0 1 (3.36) x dz2dz4(012034)1a(t) zmem) )J m=1 The right factor gives a z1 in leading order. Consequently, the z integral generates a leading pole at a(t) = 1, whereas the second sister requires a(t) = 2. Presumably, the 7*(t) trajectory is present if there are at least two loops on both sides of the propagator. We suspect that, in this case, there would be a factorization of the form Gmn(x, y, k, k2) = gm(x,y, kl)gm(x,y, k2) (3.37) where k1 and k2 correspond to sameside loops. In Fig. 5. we display two distinct possible multiloop topologies for producing the higher order sisters. s s (a) (b) Figure 6. The Regge cut behavior is across the dotted lines. In both cases the central propagator may allow up to the mth sister if there are at least m loops on either side. However, evaluating Fig. 5a is not practical since the Schottky representation of the prime form (3.13) is much too formal when two or more unfactorized loops are present. On the other hand, since Fig. 5b completely factorizes the loops it requires no more than the techniques presented in this chapter. The sister trajectories may also appear across propagators which are em bedded in an irreducible diagram. An example is the doubleloop diagram displayed in Fig. 6. The sister here may be across one of the horizontal propa gators. Such diagrams are, however, dominated by the behavior of Regge cuts. In the present case, the cut in Fig. 6a gives '(t In s e2 A (~ (3.38) (ln s)P ' for some p at fixed t. The cut has the same Regge slope as the first sister, yet its a(t)intercept is higher. In general, the nth cut occurs at the same order as that of the n4h sister, but with a trajectory lying above the sister. This implies that the collective behavior of the cuts would actually exceed the 28 CM bound. A high energy analysis of the entire moduli space, such as that of Gross and Mende, would be dominated by the cuts. This is supported, in part, by their proposal that the fixed t behavior have the form (t Ins A ~ +1 (3.39) (In S)12g where g is the genus number. The singleloop amplitude, computed first in the fixed angle limit, was shown explicitly to reduce to (3.39), for p = 1, in the fixed t limit 0  0. CHAPTER 4 OPEN STRING SISTERS IN CLOSED STRING SCATTERING An unexpected result of the last chapter is uncovered by considering the nonplanar diagram in Fig. 7. The central propagator that carries the sister f(t) is that of the open string, while the nonplanar loops on either side contain closed string poles. This raises the interesting possibility of open string sisters coupling to closed string propagators as in the diagram shown in Fig. 8. Below we show that this is in fact the case. In the case of the Heterotic string, however, the diagram in Fig. 8 decouples since the open string propagator can not accommodate the achiral boundary conditions required by the closed string propagators. The amplitude for fourtachyon closed string scattering with an intermedi ate open string propagator, takes the form AG2 =(2) d2zld2z2cl(O,p4 V't(p3,z l,211) x o(O I T(At,a) I0)c (41r), 1J<1 x Ao c(0 I T(A, a) I 0)o x V(p2, z2,2) I 0, p)c. (4.1) Among the many expressions appearing in the literature for the transition op erator T between the open and closed string state, we will use that of Shapiro and Thorn.25 We will ignore here the ghosts terms given in their explicit expres sion for T. These give a nontrivial contribution only if loops are present. Even then, the ghosts can be ignored since they have no bearing on the calculation which focuses on the exponential contributions away from the integration Figure 7. Nonplanar doubleloop fourpoint diagram. The loops contain closed string poles. 0o(t) Figure 8. Fourpoint string propagator. closed string interaction with an intermediate open boundary region. The transition operator is then given by n,m=0 E C)As As  2 nm n Am n,m=0O (n,m)?(0,0) (4.2) where, (1) (_)n+m nm 2m 2n + 1 and 721 I_2I 2  n m 00 + (2) + nm2nI 2m1 n,m=0 1 00 a2c m=1 (2) 1 ()n+m 1 11 , nm4n+m+l[n m (3) ()n+m _ n+m n m (4.3) (4.4) (4.5) 31 The sine and cosine oscillators of the closed string are given by 1 i Ac = (Ar + Ar), As = (Ar Ar), (4.6) where Ar and Ar correspond to the left and right movers, respectively. In terms of the sine and cosine modes, the closed string vertex operator is written V(P2,z,z) 0, Pl)cl =ei(p+P)x z 12a( exp[p ( n(zn + ) n=1 As (zn )n 0)c,, n /i (4.7) where the first two factors are the zero modes. In Eq. (4.7) we have written the closed string trajectory a(t) acl(t). To eliminate confusing notation, we will write the open string trajectory also in terms of a(t), i.e., ao(t) = 2a(t) 3. Then, we have zLo2 0)o = zp22 0)o = zt2 0)o = z22a(t) 0)o. (4.8) Pushing T to the right, Tt to the left, moving the propagator to the right, and then using momentum conservation to eliminate part of the zero modes, Eq. (4.2) becomes AG2 =()2 dzz22a(t) < d2z1d2z2 zlz2 2a(t <47r ) o 1 zj<1 1 0 + i x o(0 I exp [/ P3 (a2n( 1) n=1 00 + i Cn a2m+l(  m=0 0O x exp [ C $ma2n+1 '2m+1l (4.9) x exp [ (a2nl a 2mz2(n+m+1) n,m=0 x exp [P2 (E 2nz2n (z + 42 ) V'_ n=1 o nV2 Z(1) 2m+l n _z2 n \) + j CnmaQ2m1z2m1(4 2 0)o. m=0 We can easily move the even oscillators through to the vacuum states since they only appear at the far left and far right. This produces the factor exp [P2 P3 z 2nzl + 2i)(' + (.) n=1 (4.10) = (1 z2zlz2)(1 z2zl12) p2P3, which can be pulled outside the vacuum states. Next, pushing the quadratic terms past each other produces the factor 00 exp 4(2(k + )C (Ca42na 2m+2(n+k+l)], (4.11) enk km 2nlOa2m+1 k,n,m=0 and an oscillator independent exponential which we can neglect since it will not survive in the high energy limit. Moving the quadratic terms to the vac uum states will produce no other permanent effect as they pass by the vertex 33 operators. Pushing the factor (4.11) past the right vertex then results in AG2 )2 j1 dzz22a(t) jz 1 d d2z22 zlz22a(t) 2 2 2p2IP3 x (1 z2zlz2)(1_ z2z12) 2p2 p3 00 x o(0 exp[iVzp3 Cnma2m+1i(z  n=1 m=0 00 x exp[ivP2 E C( 2m2z2m+1(z2 i) n=1 m=0 (x + 4(2m + 1) (2k + I)a2j1 .C2)Cz2(j+k+1) 0)o. j,k=0 (4.12) Again, the quadratic term will not leave any permanent imprint after moving to the lefthand side. Finally, AG2 )211 dzz22a(t) j d2z1d2z2 ZlZ2 2a(t) x (1 z2zlz2)(1 z2zl2) 2p2.p3 00 xexp [2p2 P3 E(2m + 1)z2m+1kmnm(zk kl)(z 2 k,n=l m=0 (4.13) To perform the sums in the second exponential, we go to the limit s  oceib and keep only the term linear in z. The sums can now be done by noting 2 2.7 (4.14) n 1 2n (4.14) Thus, 00 00 1 L znO E 1 2n = (1 z)I 1. (4.15) n=1 n=1 Eq. (4.13) then becomes AG2 (1)2 1 dzz22a(t)z<1 d2zd2z2 I zlz22a(t) x (1 z2zlz2)(1 Z2Z12) s8 (4.16) x exp [sz((1 z 1) (1 2i) (1 z2)5 (1 z2))] As in the doubleloop case, since there are only four interacting particles, the argument of the exponential appears in a factorized form without resorting to a dimensionality constraint. Examining the second exponential term in (4.16), we see that there is a critical point when z1 = \ or z2 = 52. Writing z = pei', implies 0 = 0 or 7r. To integrate (4.16) about these points, we return to the Taylor series expansion in (4.13) and (4.10), i.e., AG2 ~ dzz22a(t) dpdP2(P1P2)1a(t) 2 d01 d02 Jo Jo Jo 47r 47r x exp [2sz2P1p2 cos 01 cos 02 (4.17) 00 x exp [4sz E CkkC)p ^sin(k91) sin(n02)] k,n=l We expand by setting sin(nO) w nO for 0 = 0, and sin(nO) ; ()nnO for 0 = 7r. The p sums for 0 = 0 can easily be carried out as follows: 00 00 (_pn 1 1 , P n'nCO( P 2n = p(1 p). (4.18) n=l n=l1 With a similar expression for 0 = r, eq. (4.17) becomes AG2 ~2 dzz22a(t) dpdp2(P1P2)1(t) fd0 d02 O JO J 47r 47r x (exp [sz90192P1P2(1 PI)(1 P2) + 2sp1p2z2 (4.19) + exp sz9Ol2PlP2(1 + Pl) (1 + P2) 2splP2z2), where the first exponential is for 91 and 02 expanded around the same value, and the second for the converse case. Integration over 01 yields AG2 8 dzzl2a(t) dpidP2(PIP2 )2a(t) dy [eY e' x ((1 pi)(1 p2)2 exp [2splP2 21 + (1 + pl)1(1 + p2)1 exp 2spiP2z2]), (4.20) where the exact expression for yo is not needed. The integration over y gives 2i7r. Unlike the previous examples, the sister is not necessarily the dominant behavior. This requires that we extend our considerations to higher orders. By Taylor expanding (1 p)2 and (1 + p)2, we obtain many terms which may indicate the presence of the open string sister, o. To make a firm determina tion requires some care since the O3o trajectory is degenerate with the dilaton trajectories that may appear across the adjacent closed string propagators. There is no doubt, however, when (4.20) generates a triple pole. The form of the required solution is suggested by the partial wave analysis term /+ sat In2 s, (4.21) C (ao(t) t)(acl() t)(acl(t) t) sa ln2 21) for the case ao(t) = acl(t). Eq. (4.20) yields this result if we select the p2 expansion terms for both pl and P2. This gives, AG2 2T 1 dzzl2(t) dpidp2(plP2)a(t) 28s7r 0Jo 0 (4.22) x (exp [2splp2z2 + exp 2sp1P2z21. Let w = P1P2. Then AG2 2 1 dzzl2a'(t) dwwa(t) e2swz+2swz2) dP2P 1 ,2 7 1Q Jo (p2P (4.23) 36 The P2 integral easily gives In w. Next, defining w = z2y, we integrate over z to obtain S [1 (22sy e2sy) AG2 210s 1 dyya(t) In2 y e2sy 2sy i d2 dyya(t) e 2sy +e 2sy Finally, the end result is AG2 ir1 (1 eirflo(t) )29+ (t)I(1 /o(t))so(t) In2 s, (4.25) where the open string sister 03o(t) = a(t) 2. One possible concern that may arise in the above calculation is that in writing Eq. (4.17) we have discarded the term [(1 z2zlz2)(1 z2zlz2)(1 z2 lz2)(1 z2z12)] 2 (4.26) When tachyons are present this may diverge at the critical points in the neigh borhood z = z = I z2 = 1. Fortunately, the sister trajectory emerges from the other end of the integration region where these quantities approach zero. In place of Fig. 8b, we could also represent the open string propagator as a disk that is cut out of a plane which parametrizes the worldsheet.26 In principle, we can recover the situation discussed in this chapter if we impose Neumann boundary conditions on the hole, and then factorize by restricting the locations of the vertices. An alternative case is when the hole obeys Dirichlet boundary conditions. In this case, the open string propagator is physical only when there is zero momentum across it. Since the sister trajectories occur in the limit of large s momentum transfer squared, we can rule out their existence in the Dirichlet theory. CHAPTER 5 OSCILLATOR REPRESENTATION OF SISTER TRAJECTORIES In this chapter we will determine the state representation of the sister tra jectories. The basic nature of sister states will differ from the states associated with the standard Regge trajectory a(t) since the corresponding sister poles are not physical. In the spacelike t region, the poles have the manifest unphysical characteristic of nonsense, i.e., negative spin J. The timelike resonances are not physical either. In exact expressions, the residues associated with physi cal resonances can always be written as polynomials in the energies.10 At tree level, we can see from the final expressions (2.21) and (2.37) that the energies overlapping the sister propagator are not in this form. However, this is not the case in the doubleloop expression (3.32) where only a single energy appears. Here, repeating the argument given for the sixpoint case in Chap. 2, we rely on the fact that the sister has wrongsignature. Constructing the sister states will give us another means for uniformly displaying the unphysical nature of the poles in all regions of t. There exists, yet, a second motivation for being interested in the state representations of the sisters. At about the same time sisters were found in the high energy analysis, they were noticed in an entirely different context by Goldstone27 who was investigating the problem of counting physical states at each mass level.28,29 This is a nontrivial problem because in D dimensions the physical states transform under the group O(D 1), whereas, due to gauge invariance, string states fill multiplets of the transverse group O(D 2). For 38 the case of four spacetime dimensions, the counting problem was solved in 1976 when Goldstone presented the generating function 00 00 x(x, J) = [E(1 n2] XrJ+r(rl)/2()r(l(l xr)2. (5.1) n=1 r=1 This has since been generalized to higher dimensions, and for the Superstring and Heterotic string.29'30'31 Expanding out (5.1), the coefficient of x counts the number of 0(3) representations of spin J, while the exponent is the corre sponding mass level. The connection to sisters can be made if in (5.1) one sets the x exponent in the second factor equal to M2, i.e., M2 1 Jr(M) = (r 1). (5.2) r 2 Since poles in the a(t) plane are labeled by (M2, J), we can identify (5.2) with (2.2). The state analysis of Goldstone is in the timelike t region, while the high energy analyses exposes dominant behavior in the spacelike region. Determin ing the state representation of the sister trajectories will provide a more direct link between these two approaches. The unifying feature of pole cancellation can be seen in Fig. 9 which, for a(t) > 0, displays the lowest mass levels obtained from Goldstone's formula (5.1). The figure shows how the various trajectories conspire to form the physical states(solid dots) and remove some of the pure gauge states(crosses), and that the 3(t) trajectory in both regimes enters with the opposite sign to the a(t) and *y(t) trajectories. The state representations of the physical states, defined at the poles of the standard Regge a(t) trajectories and its daughters, are well known. The first three states of the leading trajectory displayed in Fig. 9 are given by the tachyon  0), the "photon" a_  0), and the massive spin two symmetric state a1[iai 0), where the transverse index i = 0, D 2. Suppressing the spacetime index, the general leading state is given by an 0), n > 0. It is J a(t) Cx cc X S(t) a"02a2 (t) Figure 9. Lower mass states in the open bosonic string. Dots denote phys ical states, and crosses denote pure gauge states. The coefficients indicate con tributions from the various leading and daughter trajectories. Nonsense poles are from high energy scattering analysis, and sense poles are from Goldstone's formula. important to note that the mode number of the states along the a(t) trajectory differ by one. This implies that by varying nr in the general open string state na n a nr 0), (5.3) we move along a path in Fig. 9 that parallels the rth sister trajectory. Al though the poles of the rA sister trajectory do differ by mode number r, the corresponding sister states can not be represented by the physical states (5.3). Instead, by analogy with the high energy analysis, we must analytically continue away from the states defined by (5.3). To proceed, we will work directly with the factorized scattering amplitude. This isolates the appropriate propagator which allows us project onto it all possible classes of physical states. We are then free to select the states which lead to the sisters. For the sixpoint 40 diagram of Fig. 1, again ignoring the twists, we project the physical states onto the central propagator. To preserve unitarity, we insert the corresponding identity operator on adjacent sides of the propagator, i.e., A6 = dzld30,p I V(p2,zl)V(p3,)Io 1IV(p4,1)V(p5,z3) 0,P6), JQ zlz3 L 1 (5.4) where n00= 00 I = E ansla) 0) E 2, 1 2 '( n=0 .n2=0 The normalizations in (5.5) are fixed by the projector condition 12 = I, and the commutation relations [a4a, 4] = mrnm+nrl7'. (5.6) When we project I onto the central propagator, we easily obtain 1 00 00 1 1 ,I I I n, J0) ( anr r (57) S 1 rn nr! 1 n ljy a(t), (5.7) r=l nr=0 where t t2. Substituting this into (5.4), using the fourdimensionality con straint (2.9), and then taking the high energy limit s34 + oo, yields Ag j dz1dz3zz 1a(tl) z 1(t3)(1 zl)1a(s23)( z3)1a(s4s) r=1 n,= where x 1, x3 and 9 are as before. Note, by using the integral representation jnj_ a(t) jdzzr1=,(1)I (5.9) we can replace the sums in (5.8) by exponential functions to get A6 1 dzdz3dzz1(t)Z 1a()(l zl)la(s23)(1 z)lQ(S4) x z1a(t)exs X ~ exp I X 3 x3 r=51 (5.10) 41 Thus, we have completely recovered Eq. (2.10). In fact, at virtually each step of our computations below, there is a parallel step using the exponentiated form. This provides a useful check on our results, and allows us to be brief in much of the derivation. As in Chap. 2, to obtain the leading /3(t) trajectory we require that (xl, x3) be a critical point. In this case, not only do we twist the propagators zl and z3, but the index n1 must be analytically continued to a negative value. The standard procedure is to replace the infinite sum by a SommerfeldWatson contour integral, and then push back the contour exposing the poles on the negative real axis. To generate the necessary pole in n1 we first evaluate (5.8) at the critical point. About (xl, x3), the amplitude (5.8) is approximately A6 ~ x c(txl)X3 1a(t3)(1 Xl)l+a(s23)(1 x3)+a(s45)I6, (5.11) where, after shifting z1 and z3, the integral becomes r00 00 r Or=2 n=O (5.12) x 1 1 dzldz3(gzlz3)nl nl=0 n1 j a(t) The double integral is easily performed, giving 16 =2N o1 11 1 [K X) (r x1]3 r r=2nr=O (5.13) x y+ 1 1 (1)n+1 ( n= nl! = l =1jnj a(t) (nl + 1)2 where yo = e2.. As we will see shortly, it is crucial for the sister that n1 appear as a double pole. As a prelude to replacing the discrete variable ni by a continuous one, we must replace n1! by its gamma function representation F(nI + 1). To analyt ically continue to the pole at nI = 1 we must be careful since Eq. (5.13) 42 vanishes for odd n1 because of the numerator in the last factor. Using the SommerfeldWatson transformation to convert the sum over n into a contour integral, we get 1 1/E \llr r=2 n.=0 (5.14) S dx yz+1 1 1 (1)x+1 x sinxr(x+l)x+E=2jnja(t) ( + 1)2 ' where we have separated out, and displayed, the odd n contributions. To continue back to the pole, we must first signaturize the last factor by setting 1 = e". Pushing back the contour then exposes the double pole with residue 16 =291 J00 00 n X1)Xr X3)] r=2 nr=0 (5.15) d[ 1 y+ 1 eir(x+l) dx sinrxr(x+1)x + jnj a(t) x=1 Since this expression vanishes when we set x = 1 in the last factor, we only need to differentiate this term. The result is ir. The reason why we require a double critical point should now be clear. If only z\ or z3 were critical, a single nl pole would result whose residue vanishes. Next, in the limit x + 1, sin x7rx(x + 1)  1. (5.16) Thus, 00 00 1 r X Xr n I6 = 2i7r1 11 E rnrnr! ) 3 r r=2 n,=0 (5.17) 1 1 + Ej=2 IJnj a (t) In the special case nr = 0 for r = 2,3,..., we have the nonsense pole at a(t) = 1, i.e.., 6 = 2i 1 a(t) (5.18) 43 Since the analytical continuation was along the curve described by an"1 0), it is clears that this pole is generated by the leading Regge trajectory a(t). The analysis above shows that this pole is given by "(a_1)1  0)". The inverse oscillator indicates the unphysical nature of this state. Now, we must explicitly show that the pole (5.18) is canceled by a cor responding pole on the leading first sister trajectory, /3(t). To analytically continue to this pole along /3(t), we must convert the n2 sum in (5.13) to a contour integral 1 10 n+ 1 (l)n+1 16 = g E c rZ =O nl! (ni + 1)2 n=O "(5.19) f dx 1 1+2 2 A _ C sin rx 2r(x + 1) 13 n + 2x a(t)' where we have set nr = 0 for r = 3,4,.... Picking up the pole at x = 2a(t)  nn1 gives 21 Y 1+1 1 (1)n1+1 1 0 n1! (nl + 1)2 sin r(a(t) ) (l) nl=0 (5.20) rF(a(t) 1ni + 1)) x3)] Repeating the steps for n and evaluating at the pole nl = 1, we finally obtain 16 = _ir25/(t) 1 1 2 Xl (2 x3)](t)+x (5.21) sin 7r(f(t)+ 1) r(f(t)+ 2) X1)X3x3) (5.21) That the first pole at f3(t) = 1 cancels the amplitude (5.18) can be seen by writing P(fo(t) + 2) = (f/(t) + 1)F(/3(t) + 1), and then canceling the pole coming from the sine function against the zero in r1(/(t) + 1). The remaining poles are represented by the states (a_)la on 0) and cancel unphysical poles generated by the daughters of the a(t) trajectory. 44 Repeating the sixpoint computation for a general critical point defined by z4 = x\ and z = x3 produces the (r 1)h daughter of the /(t) sister. The complete set of states corresponding to /(t) and all its daughters is given by (a1)ma" 2 I 0), for m > 0 and n2 > 0. The set can only exist in totality, and results from a complete saturation of the propagator with the oscillator a1. In the present context, we see that the 0/(t) sister does not appear in either the four or fivepoint function, or on the zi and z3 propagators of the sixpoint function, because coupling the propagator to two onshell states at any vertex prevents total saturation. This analysis suggests that to obtain the second sister trajectory, Y(t), we must first saturate the appropriate propagator with the oscillator ai, permitting 03(t) to exist, and then with a2. To verify this, we again consider the eightpoint diagram of of Fig. 3. Inserting the identity operator on adjacent sides of the propagator z3, and using the fourdimensionality constraints, yields the form A8g dzldz2dz4dz5f(zl, z2, z4, z5) oo oo 1 1 (5.22) Z1 rnrnr! an (lj (t)(Hr r=1 nr=0 where t t 3, and Hr = (z xl)(z x2)(z4 x4)(zr x5). (5.23) and where the x's are as before. Recall, we can obtain the leading 7(t) trajectory if we twist all the noncen tral propagators and expand about the point z1 = xl, z2 = x2, z4 = VI, and z5 = V/px. For (xi, x2) and (/4, v/i) to represent double critical points will also require we continue both nl and n2 to negative values. Assuming this to be case, we have A8 f(x1,x2,' /,vV')0 0 E 1 (IHr)nr dzldz2dz4dz5 r=3 nr=0r r  x [gzlz2(v4 X4)(x5 x5)] 0 1 x 2(x _)( 2 )]n 1 n2=0 2n2 (5.24) Each pair of integrals is the same form as in (5.12). The first pair generates a double pole at nl = 1 and moves us onto a 3(t) trajectory. Subsequently, the second pair gives a double pole at n2 = 1 which now transfers us to a 7(t) trajectory. Integrating and analytically continuing to nl = 1 we immediately find Ag ~ 4i7r (22x4x5(x x)(2 X2)(1 V/f4)(1 V 1)) 00 00 1 x f(xl1,x2, vX, v ) Hr)n (5.25) r=3 nr=0 00 2+1 1 (1)n2+1 1 Sn! (n+ 1)2 1 +  n2=0 where the exact form of vo is not important. We can approach the final state (a_la_2)1 10) by moving along either the leading /(t) or 7(t) trajectories. The nonsense poles obtained in two cases must cancel. For the first path, we set nr = 0 for r = 2,3,..., and perform a SommerfeldWatson transformation on n2, to get A8 47r2 22x445(x2 x1)(}x x2)(1 v/)(1 1)) 1 / (5.26) x f (xl, x 2 x4, V x 3 t) t ' Similarly, for n3 : 0, sliding down the (t) trajectory we obtain the result 4 2 x o \1 A8 7r 2(2x4x5(x2 x)(x2 x2)(1 Vi)(1 V/)) 1 S 1 (5.27) x f(x1, x2, V 4, ) H)+ )()2 1 1 3) 3 sin r(7(t)t + 2) r(7(t) + 3) 46 where, for 7(t) = 2, this cancels the pole (5.26). Extending these results to the most general case, suggests that the rth sister trajectory forms when the propagator becomes successively saturated by the oscillators am, starting with m = 1, and eventually reaching m = r1. In another words, to get to the trajectory am(t) we begin by moving down either a leading or daughter a(t) trajectory curve to either a leading or daughter 0(t) trajectory, which we reach by analytical continuation, etc. The resultant sister and its daughter trajectories are represented by open string states of the form (a_1)ml ... (ar+1)m (ar)"r 10), for m1, ,mr1 > 0,nr > 0, (5.28) where the leading trajectory is given by mr = ... mr1 = 1, nr = 0. By analogy we can immediately write down the corresponding closed string sister states by replacing the open string oscillator a_ with the closed string oscillators a_i&_i everywhere. An important point that needs to be stressed here for applying the proce dure we have presented, is that it be possible to completely isolate the sister propagator. In the case of the doubleloop fourpoint interaction, this crite rion adds justification to our approach in Chap. 3 where we factorized the amplitude so that we could treat as individual objects the two loops and the connecting propagator. In the state analysis approach, projecting the physical states onto the connecting propagator gives 1 1 01 4 A(4;2) = H E rnrnr! ,nj a(t) d dzi r=1 n=0 i=1 x (0 exp[ + CL a(c n, ,(c n 0) (5.29) x (0 (a ) p CI) ,o> 47 After simplifying, the calculation leading to the sister proceeds exactly as in the sixpoint case given earlier in this chapter, and reproduces the results of Chap. 3. Exposing the open string sister in the fourpoint closed string diagram of Fig. 8 presents a new difficultly, however, since we must look for a triple Regge pole that also is not a leading order term. The expression AG2 =( 2 ) 1d2zld2z2cl(0,p4 Vf(p3,z l,1 ) x o(O I T(At,a) i )c, x IAoI x c(0 T(A,at)  0)o x V(p2,z2, 2) 0,Pi)ci. (5.30) reduces, after some algebra, to 00 00 AG2 12 nr 1 d2z1d2z2 47 rnnr! = jnj + 3 2a(t) ,1< r=1 nr= 0 I zlz2 I2(t) [2p2 p3(z/2 + /2)( r/2 + r/2)ven (5.31) x [2P2 P3r2 Cr n z )( z 2 k,n=1 m=0 We used the fact here that the even and odd oscillator parts can be treated separately. To obtain the open string /3o(t) trajectory we set nr = 0 for r = 3, 4, ..., which allows the sums to carried out. In the high energy limit, we find 1 2 1 1 1 d2zld2z2 AG2 ) ( n1 n! 2n2n2 n1 + 2n2 + 3 2a(t) J< d1d x I z12 2a(t) [s(z1 + )(z2 + z2)] n2 x [s ((1 zl) (1 )) ((1 z2) (1 2))]n (5.32) Of course, using the integral representation for the propagator we can easily recover the corresponding expression of Chap. 4. Now set z = pei'. Expanding about the critical points at 0 = 0 and r gives Z 1 2n2 1 1 AG2 =2 + 2n + 3 (t) dpd2(P1P2)n+n21at ni=O n2=0 sni+n2 E d01 d02(9) (1 P 2)( 2 ] + [(1 + pI)0(1 + p2)1()n1n2), (5.33) where the first term in the last factor is for 01 and 02 being expanded about the same value, while the second term is the converse case. The 0 integrals are executed as before, giving (1 + ()nl)262nl+2/(n1 + 1)2. Each p integration produces a factor B(n1 +n2 at, 1 n1). Combining the terms then produces the factor (1 + ()n1+n2). This leads to the result Ak=2 = 1 1 2n2 1 sni+n2 47r2 n2 n1 + 2n2 + 3 2a(t) n=0o n2=0 1 2 e2n"+2 x B(n + n2 (t), 1 l)2 + (n2)(1 + ()nl) E 12, 2 (n1 + 1)2' (5.34) where we have used (1 + ()nl)2 = 2(1 + ()nl). Utilizing the SommerfeldWatson transformation, the residue due to the double pole at n1 = 1 yields the result Ak=2 1r) 2n2 1 + ()n2 B( 1 + n2 a(),3/2)21+n2 472 =(i2! 2 + 2n2 2(t) n2=0 (5.35) Writing the Beta function in terms of F functions, and pulling out the first three poles from one of the F's gives Ak=2 =i 2n 1+()fn i+n2 87 n 2!(1 + n2 a(t))3 n2=0 (5.36) x [(1 r(2 + n2 a(t))r(3/2) ] 2 (1 + n2 a(t))(n2 a(t))F( + n2 a(t))J 49 Thus, we have recovered the triple pole at n2 = a(t) 1. Computing the residue by taking the second derivative of the energy factor, gives the final result i9 1_ _+__8_ _ Ak=2 = 9+(t 1 e 1 Io(t) in2 s. (5.37) 7r sin 7r(,3o(t) + 1) LF(lo(t) + 2) To show that these poles cancel, we again start with (5.34) but now pick up the single pole at n1 = 2a(t) 3 2n2 to get k=2 1 0 2n2 F(3 n2 + a(t))r(5/2 a(t) + n2) 2 4 n2 2 r(1/2) S2a(t)3n2( ()n2)(1 (_)2a()32n) 4a(t)44n2 (2a(t) 2 2n2)2 (5.38) We must take the residue of the quadruple pole at n2 = a(t) 1, which will give a factor of 1 For this, we take two derivatives of the energy factor and, to get a nonvanishing result, one derivative of the factor (1 + ()2a(t)32n2), which can be done 3 ways. This leads to Ak=2 31 1 1 + ei(a()) 1 r F(3/2) 2n2 s 3! 8 7r2 sin7r(a(t) 1) r(a(t)) [(1/2)J (2s)a(2(2)ln2s' (5.39) which reduces to the negative of (5.37). CHAPTER 6 CONCLUDING REMARKS In this thesis we have indicated that the CerulusMartin bound may not be violated in string theory if one includes higher order corrections to the tree diagram. To complete the proof requires that the entire perturbation series be summed to determine if the coefficients of the amplitudes have any effect upon the result. It is not clear, however, how to take the fixed angle limit in the high energy analysis employed above. Instead, the proper approach may be to adopt the techniques used by Gross and Mende.4 Basically, for closed strings, this means searching for saddle points on an N sheeted Riemann sur face defined by an appropriate algebraic curve. To have the sisters produce the dominant behavior would require that we consider the limiting situation where the Riemann surface is divided in two, separated by a thread representing the sister propagator. The hope is that the uncontrollable higher order corrections which plagued the work of Ref. 4 would now be absent. Although this may remove one of the objections to locality, it should not imply that in any way have we shown that string theory is, or can be, a local theory. There still remains serious objections which may be more difficult, if not impossible, to overcome. For example, in string field theory, Eliezer and Woodard6 note that the cubic formulation of the field theory produces an infinite number of Abelian solutions.32 This causes a breakdown of the initial value problem since it requires an infinite amount of initial data. They show that attempts to restore this loss of predictability result in acausal behavior, 51 which, again, leads to nonlocality. Another problem in string field theory, is that the individual elements of the perturbative Smatrix still violate the CM bound. Possibly, with the help of the sisters, one can find a local field theory where strings are produced nonpertubartively and appear as bound states. This would probably resolve most of the locality problems in the field theory. We should point out, though, that it is not even clear if field theory should be the fundamental formulation for strings. Finally, let us comment on a physical interpretation for the sister states. Recall, the Regge slope a' is related to the string tension T, or energy per unit length, by a (6.1) 7rT This shows that, for instance, the first sister 03(t) has twice the tension of a(t). We visualize this occurring by bending over the string once to create a double strand, giving a 'folded' string. This picture is in accord with a reduction of the fundamental length scale I = vW1. The notion of folded strings originally dates back to the early 1970's where it was noted that pure states of the form anr 1 0) have Regge slope reduced by a factor of 1. Thus, the state analysis of Chap. 5 furthers the identification of the sister trajectories with folded strings. In conclusion, the motivation for studying sister trajectories is that they may eventually lead to a useful description of the short distance behavior of string theory. Presently, there is a growing belief that the current version of string theory is nonlocal at a fundamental level. The sister trajectories, and their interpretation as folded strings, may be the necessary ingredient for reformulating string theory to produce a local theory. APPENDIX WEIGHT DIAGRAMS AND LAX OPERATORS Recently, matrix models have received a great deal of attention as nonper turbative descriptions of string theory.7,8,9 Since initial advancements, progress has proceeded in many different directions. In particular, Douglas33 has shown that the limited number of known matrix model solutions can be derived from the Lax pair formalism usually associated with the KdV equations. This iden tification with integrable systems greatly increases the number of classifiable matrix models since it was shown a long time ago34 that Lax operators are associated with affine Lie algebras. For example, the models discussed by Douglas are related to the canonical representations of A(1). More recently, Di Francesco and Kutasov35'36 have discussed D(1) based matrix models which the standard matrix techniques37,38,39 have yet to solve. Thus, it may be worthwhile to focus on the integrable systems approach. Several approaches to constructing the Lax operators have been developed. The matrix procedure discussed by Drinfel'd and Sokolov34 defines first a ma trix eigenvalue equation. The system incorporates knowledge of the Cartan subalgebra and root system of some embedding affine Lie algebra g. Starting with an affine Lie algebra facilitates the construction of an integrable system from the resulting Lax pair operators. To fix the gauge invariance in the matrix system, the gradation conventions of Drinfel'd and Sokolov require that one of the simple roots, say the mth must be removed from the affine system. The resulting system is denoted by (, cm,). This is equivalent to deleting the mth Dynkin vertex. For the most part, Drinfel'd and Sokolov choose the "canoni 53 cal" gauge in which to express the coordinate dependent terms. In this gauge, Lax operators generate the regular KdV hierarchy equations. The modified KdV(mKdV) equations can be generated by expressing the coordinate term q(x) in the "diagonal" gauge. The canonical Lax operators can then be recovered using the wellknown Miura transformations. The diagonal gauge is technically simpler than the canonical gauge. Furthermore, the final Lax operator is in a factorized form which has been used to quantize the theory.40 In this appendix our focus will be on the explicit construction of the Lax (pseudo)differential operators in the diagonal gauge using a simple diagram matic technique.41 In most cases this technique arrives at these operators much quicker than a direct application of the scheme of Drinfel'd and Sokolov. Fur thermore, the scheme also applies to higher representations of the embedding affine Lie algebra. In the first section we briefly review the construction of weight diagrams corresponding to representations of affine and nonaffine Lie algebras. From there we review the matrix method of Drinfel'd and Sokolov for building Lax operators. Next is a presentation of our method, which re places the matrix procedure with a scheme utilizing cyclic weight diagrams of representations of affine Lie algebras. We then present a proof that the diagrammatic algorithm produces the correct Lax operator. Finally, we dis cusses the generalization to Lax operators based on supersymmetric affine Lie algebras. Review of Weight Diagrams As noted in the introduction, each Lax operator can be associated with a representation of some affine Lie algebra. Thus, in this section, we give a 54 brief review for constructing weight diagrams corresponding to these represen tations. Recall, one can associate uniquely to every irreducible representation of a basic Lie algebra a highest weight vector.42 For each highest weight one can construct a weight diagram which encodes all relevant information concerning the particular representation studied, e.g., from it one can build explicit matrix representations of the generators of the Cartan subalgebra as well as the various raising and lowering operators. The level of a weight is the number of lowering operators applied to the highest weight which produces that weight. Finally, the height 7 of the weight diagram is the level of the lowest weight. Weight diagrams are generated by subtracting rows of the Cartan matrix initially from the highest weight vector written in the Dynkin basis. Rules of construction can be summed up as follows: 1. Subtract the iii row of the Cartan matrix n times from a weight vector whose iAh component has a positive value n. 2. When weight vectors have more than one positive component, subtract all possible permutations of the appropriate Cartan rows. A theorem due to Dynkin43 states that the final weight diagram is always "spindle shaped". In other words: i) the number of weight vectors at the level k is equal to the number at level y7 k, ii) the number of weights at level k + 1 is greater than or equal to the number at level k for k < i. For an explicit example consider the algebra A2. Though this is almost a trivial case, the results will be useful for the next section. The Dynkin diagram is given by 1 0 0 55 where, recall, the single bar represents 1200. The Cartan matrix is then easily found to be A=(2 1). (A.1) The highest weight vector of the fundamental representation is (10). 44 Since a positive one appears in the first place, we subtract the first row of the Cartan matrix one time. This gives the weight (11). Now, due to the one in the second place, we subtract the second row of the Cartan matrix once to get (0 1). This completes the process since no positive components remain. The result is the height two weight diagram (10), 11 (11)2 (A.2) (01)3 where the subscripts on the weight vectors indicate a counting of the vectors. The ones adjacent to the arrows represent the normalization factors of the cor responding negative simple roots. These values are fixed by the commutation relations of the Lie algebra. To simplify our diagrams, we will not display val ues of unity. Later, we will see that the procedure for building weight diagrams is slightly modified in the supersymmetric case. For an affine Lie algebra,45 since there exists a linear combination among simple roots, weight diagrams of affine representations generally have infinite extent. However, some affine representations give cyclic weight diagrams of finite extent. In fact, it is these cyclic cases that are crucial to the scheme below. To produce a cyclic weight diagram, the affine component which is appended to the highest weight vector of the underlying nonaffine Lie algebra, unlike the nonaffine weight components, may have to be assigned a negative value. 1 2 1 1 A 1 1 2 (a) (b) (1 1 0 ) , (0 1 1 ) 4, I ( 1 0 1) (c) Figure 10. The (110) representation of A'). (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrow is the deleted root. As an explicit example, consider the nontwisted affine algebra A1). To generate the cyclic weight diagram corresponding to the canonical representa tion, start with the weight vector (110), where 1 corresponds to the affine root. Figure 10 gives the Dynkin diagram and subsequent Cartan matrix which then generates the displayed resultant weight diagram. This particular cyclic weight diagram can further be thought of as the affine extension of the highest weight diagram based on the fundamental representa tion of the basic Lie algebra A2. This is easy to see by removing everywhere the component due to the affine root. However, this is not always the case. For example, Fig. 11 displays the cyclic weight diagram constructed with the weight vector (211), where now the affine component is 2. Although (11) 57 (2 1 1) /I \ (11 2) (1 2 ) \ L4 / I I I L4   (1 1 2) (1 2 1) (2 11) Figure 11. Cyclic weight diagram of A(') from the weight (211). The dashed arrows are the deleted root. (1 1) (1 2 ) ( 2 1) (0 ) (0 0o) 4, ,4, ( 1 2) (2 1 ) (11) Figure 12. Highest weight diagram of the adjoint representation of A2, (11). generates the highest weight diagram of the adjoint representation of A2, we see by comparing with Fig. 12 that the affine extension contains an extra zero weight (000). 58 In general, the affine component in the affinely extended vector, associated with the highest weight vector of a basic Lie algebra, will always be negative. However, we will give an example below showing that some supersymmetric cases require positive affine components. Standard Construction of Lax Operators The Lax operator46 L(x, t) is defined to be linear and Hermitian. Further more, it satisfies the characteristic equation L(x,t)o(x,t) = yo(x,t), (A.3) where the eigenvalue p is required to be constant under nonlinear evolution. In other words, the nonlinear behavior of the eigenfunctions O(x, t) are governed by some operator A(t), which may be nonlinear, via the equation t A(t)(x,t). (A.4) Futhermore, A(t) enters into the differential scalar Lax equation OL( t) = [A(t),L(x, t)], (A.5) which generates the integrable KdV equations. The matrix construction of Lax operators utilizing generators of some em bedding affine Lie algebra ?(k), reviewed by Drinfel'd and Sokolov,34 begins with a matrix operator of the form L = I + A + q(x), (A.6) ax where I denotes the N x N dimensional unit matrix, and to simplify notation we have suppressed the argument t. The third term is discussed below. The second term is generated by the negative simple roots Ei of the embedding 59 affine Lie algebra. In the gradation conventions of Drinfel'd and Sokolov, we have the circulant matrix r A = ciEi (A.7) i=O The procedure is then to reduce the system of linear equations given by the kernel matrix equation L(x) = 0, (A.8) where O = (N',... N), to the linear differential eigenvalue equation (A.3), where the vacuum solution 4 is a function of the components of the eigen function b. Drinfel'd and Sokolov show that such reduction is possible if one removes a simple root, say the mth from the affine root system. They denote this situation (O(k), cm), which is in the homogeneous or standard gradation.47 The coefficients in (A.7) are then assigned the values ciom = 1 and Cm = A, where A is a constant function of the spectral parameter z. Removing an element from the simple root system is equivalent to deleting the corresponding vertex from the Dynkin diagram. Thus, when an extremal vertex is deleted, the system (G(k), Cm) represents a single residual basic Lie algebra. Removing the affine vertex obviously gives G. On the other hand, deleting internal vertices splits the Dynkin diagram into two sections, corre sponding to a pair of basic Lie algebras. For example, splitting A(2) at the mrh vertex gives Lax operators in the Bnm and Cm series. Furthermore, the (pseudo)differential operator associated with the Dn series is derived using the embedding algebra D() In both situations, a pair of (pseudo)differential operators is found whose product gives the Lax operator L of Eq.(A.5). To determine the exact form of the vacuum solution 4 recall that negative simple roots are lowering operators on system eigenstates. Further, removal of a root in the affine system produces the simple root system of a nonaffine 60 Lie algebra. Thus, due to the linear combination among the roots of the affine system, one root must be singled out to act as a conventional state raising operator. This role is given to the removed root. Thus, the vacuum eigenstate will be annihilated by a vacuum projection operator A defined by r A = Eiom + AEm = A + AEm. (A.9) i=O This requirement fixes the scalar vacuum solution 0 by setting it equal to a linear combination of the components of such that A= 0, (A.10) is satisfied. A direct relation between the scalar operator L and the matrix operator will be given in the next section. For the kernel equation (A.8) to produce a unique solution, we require that the number of independent degrees of freedom equal the rank of the embedding affine Lie algebra O(k), or equivalently the residual system ( (k), Cm). The extra degrees of freedom generate gauge invariance. To fix the gauge invariance, one must find a matrix operator S(x) that enforces the gauge transformation 0 = ead S, (A.11) where ad denotes the adjoint mapping. The gauge freedom in Eq.(A.8) al lows one freedom in determining the form of the coordinate dependent term qo(x),i. e., C O = I + A + qo(x). (A.12) Drinfel'd and Sokolov find the sufficient condition that S E CX(R rl), where rT is generated by the positive simple roots Fi, i / m. Many authors, including Drinfel'd and Sokolov, work most frequently in the "canonical" gauge. However, in this paper we choose to work in their "diagonal" gauge which has the form /q 0 ... 0 0 0 q2 ... 0 0 qdiag(x) = : .. : (A.13) 0 0 ... qN1 0 0 0 ... 0 qN This gauge leads to the convenient form r qdiag = E vi( q2",., qN)Hi, (A.14) i=0 which is in the canonical or principal gradation.47 Here, Hi are the generators of the Cartan subalgebra and the functions vi are linear combinations of the elements qi. In this gauge, the gauge term qdiag associated with (g(k), Cm) is the special case where the sum excludes i = m. The Lax operator Ldiag generates the mKdV equations, and is related to Lcan via the wellknown Miura transformations. Diagrammatic Construction of Lax Operators To exploit gauge invariance of the Lax operators, one should choose a q(x) gauge most suited to ones needs. Here, we are interested in developing a diagrammatic scheme for constructing L. In this regard, the diagonal gauge proves more useful than the other choices. In this section, we will demonstrate how the diagonal gauge allows one to build Lax operators directly from cyclic weight diagrams of representations of affine Lie algebras. To motivate the algorithm, we first review the construction of L by solving the matrix system L = 0. For the present discussion, it will be sufficient to consider embedding algebras of the form (g(k), co) where the affine vertex is deleted. Thus, the diagonal gauge simply reduces to the form r qdiag(x) = i(x)Hi, (A.15) i=l1 62 where we have excluded H0 from the sum. Consider again the canonical representation of the embedding affine Lie algebra (A21, co) presented in Fig. 10. Matrix representations of the Cartan matrix can be readoff from the cyclic weight diagram. The matrix element (Hi)jj is extracted from the ith element of the j1th weight vector, while the off diagonal elements are set to zero. The matrix entry of the negative simple root (Ei)jk is assigned its normalization factor if the kth weight vector branches into the jth weight vector as a result of subtracting the iA row of the Cartan matrix in the process. The other entries are by default zero. Thus, the matrix representations of the simple roots are easily found to give A = (1 0 (A.16) 0 1 0 where the effect of the affine root, indicated in Fig. 10 by the dashed arrow line, is assigned the value A. Plugging these values into the kernel equation (A.8) produces the system of equations [09+ vl]b1 = A3, [0 v + v.'L, = i01, (A.17) [0 V.] 3 = 2 Here, on the righthand side we have placed the terms due to the matrix A. The vacuum condition (A.10) determines the scalar function to be 0 = ,'3. Thus, we must solve by starting with the last equation. First, we multiply this equation through by [9 v1 + v2], and then eliminate ('.? using the second equation. Then, multiplying through by [0 + vi] and using the top equation gives the scalar Lax eigenvalue equation L(A2 o) = [0 + [ V1 + v21][ 2]3 (A.18) (A.18) = A3, 63 where the spectral parameter is given by p = A. Imposing the field redefini tions qI = vi, q2 = v2 v1, (A.19) gives the standard form L(A(,co) = [9 + q1][9 + q2][( 1 q2]. (A.20) This example exhibits a common feature relevant for our scheme below. When the vacuum condition (A.10) requires the scalar eigenfunction to be given by a single component of the eigenfunction, say = ,, then the resulting characteristic equation satisfies Ly, = p ,. (A.21) Consequently, the system reduction must start with the ith equation in the matrix system, and proceed upward till the top equation is reached. If i < N the process continues with the bottom equation and moves upward until the ith equation is reached again. We shall refer to this case as trivial since the cor responding cyclic affine weight diagram is linear, containing no branch points. A second feature brought out in this example, is that the number of factors in the resultant Lax operator (A.20) is equal to the number of weights in the weight diagram. Unfortunately, this is valid only for trivial cases. Nevertheless, this last observation is key to our scheme. To highlight one more property of the general procedure we turn to a non trivial example. For this, we require a representation of an affine Lie algebra whose cyclic weight diagram has at least one branching point. Thus, consider the canonical representation of the affine algebra (D1), co). Fig. 13 presents the Dynkin diagram, Cartan matrix and corresponding cyclic weight diagram ( 2 0 1 0 2 1 1 1 2 0 0 1 0 0 1 0 0 o 0 0 1 1 i; 2 0 0 2 " (1 1 0 0 0) I 4, S(1 1 1 0 0)  S^4 (0 0 1 1 1) (0 0 0 1 1) (0 0 0 1 1) (0 0 1 1 1) (1 110 0) 1 0 0  (1 1 0 0 O) (c) Figure 13. The (11000) representation of D4. (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrow is the deleted root. which has two branch points. Reading off from the weight diagram gives 000 000 100 010 0 0 0 001 000 000 0 0 0 (A.22) The branch points have manifested themselves by placing more than one non zero entry in the second and sixth rows. Now, further reading off the elements 65 of the Cartan matrices gives the system of equations [8+ v+] 0 = A07, [8 v1 + V2 ]2 = 01 A08, [8 V2 + V3 + v4]03 = 02, [8 v3 + v4]4 = 3, (A.23) [8 + v3 V4]s = 03, [8 + v2 6 = 05, [8 + v1 v2]07 = 06, [a Vl]0,8 = b7. The vacuum condition (A.10) produces two distinct solutions, '8 and the linear combination 04 bs. Here, we consider the first case. Proceeding as before, we eliminate 46 and 07 in the last two equations to get [8 + v2 va v4][8 + vi v2][8 vi]08 = 04 05. (A.24) Now, we encounter a wellknown technical problem not found in the trivial case. The components 04 and 05 can not both be simultaneously eliminated since the expressions [8 v3 + v4] and [8 + v3 v4] do not commute. This dilemma is directly linked to the fact the corresponding cyclic weight diagram has a branch point connecting the fourth and fifth weights to a single weight located below them. To overcome this obstacle the pseudodifferential operator p1 must be introduced. Its operation on any function f(x) is given by the expansion 81lf(x) = (1)if(i)(x)8l1i. (A.25) i=0 Utilizing the pseudodifferential operator, we rewrite the fourth equation in (A.23) as 04 = [0 V3 + V411 (A.26) 66 Thus, the combined effect of the bottom five equations is 03 ={[9 v3 + v41 + [ + v3 v4]1}[ + v2 v3 V41 (A.27) x [0 + v1 v21[Q v1]8 A helpful identity we use repeatedly is {A1 + B}1= {A[A + B]Bl}1 = B[A + B]'A. (A.28) When applied to Eq.(A.27), a cancellation occurs among the via's appearing in the curly brackets. This simplifies the expression to 1 03 = [v3 +v4]1 [O +v3 v4] [ +2 v3 4][+VI v2][v]s. (A.29) 2 Continuing, incorporating the next two equations in (A.23) requires a second application of the relation (A.28). Finally, the Lax operator based on (D1), co) with vacuum V)8 is L =1[9 + V1 ][9 V1 + V2[19 V2 + V3 + V4][O V3 + V4] 4 (A.30) 4 1[10 + V3 V4][9 + V2 V3 V41[+V V1 ][ V]. Using the field redefinitions qi = vi, q2 = V2 v, (A.31) q3 = V2 + V3 4, q4 = v3 + v4, we get L= 1 l[+q ][9+q2 [O+q3[a+q4l [9q4][Oq3]['q2][9ql]. (A.32) which is proportional to the standard result. We have chosen these two examples because they introduce the techniques needed to generate Lax operators associated with even the most complicated algebraic systems. Furthermore, they show how closely the structure of cyclic affine weight diagrams is linked with the construction of general Lax operators. 67 As a result, we propose a set of four steps which allows one to construct Lax operators associated with cyclic representations of affine Lie algebras. First, we propose that to every weight vector of an affine cyclic weight diagram one can associate an operator as follows: Step 1: (aoala2...) ) [Ox + alvi(x) + a2v2(x) +..]. (A.33) The coefficient ao does not appear on the righthand side as it corresponds to the deleted vertex. Next, we introduce a step which is designed to facilitate the construction of Lax operators when branch points exist in the corresponding weight diagram. Essentially, this step reduces more complicated nontrivial cases to a sum of manageable trivial cases by reducing the branched weight diagram to a sum a linear subdiagrams. Step 2: Replace branching weight diagrams by the sum of linear subdiagrams, each representing a vertical route (A.34) beginning with, and ending on, the vacuum weightss. For example, Fig. 14 presents the four linear subgraphs associated with the canonical representation of D(1) In drawing cyclic weight diagrams, it is important that the arrows gener ated by the deleted vertex are distinguished from the others. Our convention is to use dashed lines. Furthermore, the direction of the arrows must also be noted. The Lax operators associated with each subdiagram are then con structed as follows: Step 3: Circulate around the loop beginning with the vacuum solution, such that the flow is opposite most of the arrows. If a weight vector is approached by an arrow's a) tail, append its weight factor to the operator's left side, b) head, append the weight factor's inverse to the operator's left side For weights at the tail end of both connecting arrows, do nothing. Multiply by the product of the corresponding normalization factors. (A.35) The loop is to be circulated in a direction opposite most of the arrows so that the leading term of the Lax operator L = (n + ... has positive exponent. i.e., n > 0. For trivial cases, this completes the computation of L. However, for non trivial cases with branching weight diagrams we can not naively build the final Lax operator from a sum of its constituent linear subgraphs.Instead, as we shall prove in the next section, they are added together analogously to how one computes total resistance of resistors in parallel. Step 4: The Lax operator is given by the inverse of the sum of (A.36) the inverses, of the operators resulting from step three. For example, in the nontrivial case (D4 co). with 0 = 5 s as before, we build four operators corresponding to the linear subdiagrams in Fig. 14., L1 =[9 v 1[(9 + V1] [ V1 + V][(9 t"'. + V3 + V4][9 V3 + V41 (A.37) X [O + v2 v3 v4][O + v1 v2][O v1], L2 =[9 v1]l[a + v1][O v1 + v2[][ v2 + v3 + v4][a + v3 v4] (A.38) x [Q + vi v3 v\][o + v1 V,][Q V1], L3 =[a v, + V_][o v2 + V3 + v4][o V3 + v4][9 + v2 V3 v31 (A.39) x [o + v, v]}[9 v,], r  (1 1 0 0 0) (1 1 1 0 0) < (0 0 1 1 1) (0 0 0 1 1) (0 0 0 1 1) (0 0 1 1 1 ) (1 11 0 0) (1 1 0 0 0) (1 1 0 0 0) ,4, (1 1 1 0 0) <, (0 0 1 1 1) 4, = (0 0 0 1 1) 4, (0 0 1 1 1 ) (1 1 1 0 0) (1 1 0 0 0) < (1 1 1 0 0) (1 1 0 0 0) (1 1 0 0) (1 1 1 0 0) < (1 1 1 0 0)(I (11 1 0 0)< 4, ... (0 01 1 1) (0 0 1 1 1) (0 0 1 1 1) + (0 0 1 1) + (0 0 01 1) + (0 0 0 1 1) (0 0 1 1 1) (0 0 1 1 1) (0 0 1 1 1) (1 1 1 0 ) (1 1 1 0 0) (1 1 1 0 0) (1 1 0 0 0) (1 1 1 0 0) (11 0 0 0) (11 0 0 0) Figure 14. Subdiagram of the cyclic weight diagram of D 1). The dashed arrows are the deleted root. and L4 =[0 v1 + v2][( v2 + V3 + v4][O + v3 v4][O + v2 va v3] x [9 +v1 v2][a Vi]. Factoring out common terms, we find 4 L1 = (Li)1 i=1 = [ v111[0 + V1 v21] [a+ v V3 V4]1 X {[O9 v3 + v4]1 + [ + v3 v411 }[ V2 + V3 + V]1 x [Ov,1+v2] [+v1l]l{1 +[v 1['+vV,]}. (A.40) (A.41) 70 By taking the reciprocal, and simplifying, we reproduce the previous result Eq.(A.30). To end this section we consider the alternative vacuum choice 13 04. It should be obvious that since we are dealing with cyclic weight diagrams, Lax operators associated with other vacuum states can be achieved by cyclically permuting factors in the primary Lax operator. Thus, this second vacuum choice immediately gives the Lax operator L = 1[oq3][Oq2][Oq]9l[O+q1][O+q][9+q93][Oq4]1[Oq4]. (A.42) 4 Proof of Diagrammatic Scheme. To prove the equivalence between the matrix system C0 = 0 and the diagrammatic algorithm, we begin by rewriting the former as 7D(x) = A (x), (A.43) where, to simplify notation, we have defined D A = I + q(x). (A.44) ox The structure of the associated cyclic weight diagram is encoded entirely in the matrix A. Specifically, recall that the general matrix element Aij is pro portional to A if the difference between the ith weight and the connecting jth weight equals the eliminated root of the embedding affine simple root system. All other connecting weights Ai are proportional to 1. Otherwise, the matrix element is assigned the value 0. In all cases the proportionality constant is the normalization factor of the connecting root. We construct the proof in stages. For the first stage we consider the trivial case, i.e., a single entry in each row and column of A. Removing a root produces 71 only a single vacuum state. This stage corresponds to weight diagrams with no branch points and only one arrow associated with the eliminated root. Clearly, we can rearrange the matrix equations in L = 0 such that the vacuum state equation appears last. Furthermore, it can be arranged such that A is lower triangular with ones located along a diagonal once removed from the main diagonal, except for the eliminated root whose coefficient A appears in the upper righthand corner, i.e., A is a circulant matrix. Thus, the Lax eigenvalue equation becomes LV4N(x) = pON(X). (A.45) Since D9 is diagonal, the kth equation in (A.43) can be written N Dkk Akii, k = 2,3,... ,N. (A.46) i=1 Clearly, since A is a circulant matrix as specified above, the inequality i < k holds for k 0 1. Next, by repeatedly replacing the function 0j, appearing on the righthand side, with the it matrix equation we eventually reach the expression Ak1 1, i.e., N NN = N ANb,. i=l 1 NN = D, AN i A j (A.47) i=1 j=1 (A.47) N N = (_)7 ... AT1iAZ ... D k Akll i=1 k=l1 where 7 is the height of the cyclic weight diagram. Due to the successive applications of the state lowering operators A1y with i > j, this equation is interpreted as taking the highest state 71P and lowering it to the vacuum state ON Replacing 01 through S0, = ANo 4,,, (A.48) where we have used A1N = .A0 which excites the level of the state since 1 < N, gives N N DVNN = ()IA'A ".. ANMD A, A DA AklDI 'N. (A.49) i=1 k=1 where No is the normalization factor of the affine root. Finally, moving terms to the lefthand side we recover (A.45) where N N L = {N0o ... D 1ANiD 1 ... Ak,) }1, (A.50) i=1 k=1 and the spectral parameter is given by P = ()A. (A.51) Since each row and column of A contain only one entry the sum will generate a single term, i.e., L1 = {NDV 1D_ D.. }1, (A.52) where N is the product of the normalization factors. Now, each D is a weight factor as defined in step 1. Thus, there is a direct mapping between the order of the weight factors and their location in the corresponding weight diagram. Now, suppose we permit multiple row entries in A, i.e., branch points in the weight diagram. First consider the case where such multiple entries occur above the Nth row. As before, there is a single vacuum state, and the constant A is located in the upper righthand corner of A. Therefore, the constraint i < k, for k : 1, remains in effect for Eq.(A.46). Hence, the derivation leading to (A.50) follows through unchanged. Now, each new entry in A causes an additional final term in (A.50). Clearly, per step 4 of the diagrammatic algorithm, the 73 final Lax operator is obtained by taking the reciprocal of the sum of terms generated by (A.50). Next, suppose the multiple row entries in A, due to the branch point, occur in the NLh row. The vacuum condition (A.10) shows that this is equivalent to a degenerate vacuum state with, say, degeneracy d. Subsequently, this row will be associated with the eliminated root, and the d integers will be assigned the value A. In fact, A appears only in this row. Clearly, in the weight diagram the d weights share the same level. Let us first discuss the case where the coefficients A occur in the first row. Thus, Eq.(A.46) remains valid, keeping intact the constraint i < k, for k 4 1. Furthermore, the scalar eigenfunction O(x) is now a linear combination of the components ON, wN1,... v. ,, and the equation for 1 becomes N Di1 = E An, ,. (A.53) s=Nd+l Consequently, Eq.(A.49) is modified to N N N E E 'A.,' i=1 k=lj=Nd+l where f is the number of field replacements performed. The characteristic equation is obtained by multiplying both sides by Ai, and then summing over s, i.e., N N N N SA,. = () ... A,D,'Ai ... s=Nd+l s=1 k=l j=Nd+l (A.55) x )1Akl D1 AljPj. Note, the sum over s on the righthand side has been extended to the entire range for convenience. 74 Each term in A1, contains the factor A, which can then be factored out. As a result, the scalar eigenfunction is found to be N A= A A, L,, (A.56) s=Nd+l and the Lax operator, N N L = A{5 ... Y A,, TA, 1 ...AklD1}1. (A.57) s=1 k=1 Clearly, this has the same interpretation as the nondegenerate branching case. For the last stage of the proof, we relax the condition that multiple occur rences of A must all be in the first row of A. In the weight diagram this means not all the arrows associated with the eliminated root point to the bottom level. Recall from the discussion surrounding Eq.(A.9), the eliminated root with coefficient A acts as a state raising operator. Thus, every occurrence of A will appear in the upper triangular portion of A, and the unit coefficients of the state lowering roots are in the lower triangular portion. For A in the kth row of A, k : 1, Eq.(A.46) is modified to N Dj 4' = 1 Ao,',, (A.58) nAk where, since A corresponds to the state raising operator, k > j. Consider the case where j is the largest such index to satisfy this equation. Then, allowing degenerate vacuum states, we have N N : .. ED() "1 A,i)l. Aj1 LA ' i=1 j=1 (A.59) N N N = () ... )7 DIAiEI .AjEI[ ,.\(.'. + E Aj,(.], i=l j=l n k where, again, j is the number of field replacements performed. The effect of the factor in front of C" is to first, due to A, raise this state to ij and then 75 to lower it till the vacuum state 0i, is reached. Now, since the corresponding weight diagram is cyclic, there must exist some factor that will circulate ', back to gO. First, as was the case with (D(1), co), consider the situation where bk is an intermediate state in (A.59), i.e., N N = (_) ... D E AsDi1 Ajkk. (A.60) i=1 k=l1 This gives N N . = ()a{j ... )1A I 1 },,. (A.61) i=1 k=1 Thus, per step 3 of the diagrammatic algorithm, the factor Dk, associated with the weight vector at the tail end of both connecting arrows does not appear. Further, proceeding from higher weights to lower weights in the weight diagram contributes factors of )1 in the operator defined in step 2 for the linear subdiagram. Finally, if ik does not appear as an intermediate state of the vacuum state , in (A.59), then it must occur as an intermediate state for one of the other vacuum states. Again, since the weight diagram is cyclic, there is some closed path going from ', to each of these other vacua. However, to write down a final expression is too unweilding. Nevertheless, it should be clear that the general rules of the diagrammatic algorithm are complete and provide an accurate mapping between weight diagrams and the scalar Lax equation. Supersymmetric Lax Operators In this section we generalize the diagrammatic scheme to supersymmetric affine Lie algebras. The classification of all possible supersymmetric exten sions of the basic Lie algebras has been given by Kac.48 In addition to the 76 bosonic simple roots of the basic Lie algebra, the simple root system of the supersymmetric algebras contains two distinct kinds of fermionic roots. The Dynkin symbol of the first type is sometimes given by a shaded vertex repre senting a nonzero norm. The second fermionic root type has zero norm whose Dynkin symbol is given correspondingly by a crossed out vertex. As always, the bosonic root is denoted by a white vertex. A new feature occurring in the supersymmetric Lie algebras is that they may have several nonequivalent simple root systems, corresponding to differ ent Dynkin diagrams and Cartan matrices. In other words, the different root systems can not be transformed into each other through standard Weyl rota tions. Instead, they are obtained by performing the "Weyl" transformation with respect to the nilpotent fermionic root. For more details, see Frappat et al. 49 which also presents a large collection of Dynkin diagrams associated with all of the classical contragradient supersymmetric cases, those of the affine and twisted affine supersymmetric algebras. Nonequivalent simple root systems which represent the same supersym metric Lie algebra differ in the distribution of bosonic and fermionic roots. However, here we are interested in considering a natural extension of the Drinfel'dSokolov procedure to the supersymmetric case. This restricts the possible choices for the simple root system used for building the supersym metric Lax operators.50 Recall, in the bosonic case the mKdV Lax operator constructed with the gradation choice of Drinfel'd and Sokolov generates Toda lattice models.34,51,52 For supersymmetric algebras it has been shown53 that Toda lattices are possible only for simple root systems composed purely of fermionic roots. Supersymmetric Lie algebras with purely fermionic root sys 77 teams have been given by Leites et al. 54 SL(n + 1 I n), OSp(m I 2n) (m = 2n, 2n + 2, 2n 1), D(2 11; a). (A.62) Furthermore, the infinitedimensional affine supersymmetric Lie algebras with purely fermionic simple root systems are SL(n I n)(1), OSp(2n + 2  2n)(1), D(2  1; a)(1), (A.63) while the infinitedimensional twisted affine cases are SQ(2n + 1)(2), SL(n I n)(2), OSp(2n  2n)(2). (A.64) The supersymmetric extension of the KdV equations was first discussed in Manin and Radul. 55 They suggested replacing the bosonic derivative 0x by its supersymmetric analog, i.e., x  D = + x (A.65) Note that D2 = '. The system of matrix equations of Drinfel'd and Sokolov can then be generalized to50 Cb(x, ) =[D + Q(x, 0) + A]h(x, 0) = 0, (A.66) where A is generated by the purely negative fermionic roots, and Q(x, 0) is a Grassmann odd fermionic superfield which can be expanded as r Q(x,9) = H iJi(x, 0), (A.67) i=1 where now Hi are elements of the CartanKac subalgebra. The vacuum con dition is as before, AO(X, 0) = 0. (.8 (A.68) 78 Since the second type of fermionic root is nilpotent, they deserve special treatment when constructing cyclic weight diagrams. To illustrate how this comes about, consider the fundamental representation of the supersymmetric algebra OSp(2 I 2). The Dynkin diagram of the purely fermionic root system is given by 1 1 where both fermionic roots are denoted as having zero norm, and where we have indicated the choice (11) for a highest weight vector. The Cartan matrix is then easily found to be A=(02 2). (A.69) To construct the highest weight diagram we proceed as before. Since a positive one appears both in the first and second places we have two permutations of subtraction to perform. In particular, we can start by subtracting the first row of the Cartan matrix giving (13), and then subtracting the second row resulting in (33). However, unlike the bosonic case, we may not subtract the first row of the Cartan another time from the weight (13). This is because here the fermionic weight vectors are nilpotent and subtracting any Cartan row twice gives a decoupled state. Similarly, we can start by subtracting the second Cartan row once(and only once) and then the first row giving (33). Thus, we find the weight diagram with height two: (11)i (13)2 (31)3 (A.70) (33)4 The decoupling which occurs when constructing a cyclic weight diagram for an affine supersymmetric algebra is almost as straightforward. For example, in Fig. 15 we display the partially decoupled weight diagram of (SL(2 I 2)(1), co) (1 1 0 2) 2  b bb (0 1 1 2) 4  0 (1 0 0 0) <I Jb3 (0 1 1 0)3 b (0 1 1 0)5 (1 1 0 0)6  Figure 15. The (1001) representation of SL(2 I2)(1). (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrows are the deleted root. where states were decoupled as we went from top to bottom. There are several ways to decouple the remaining weights since the lowering operators b0, b2 and b3 still appear more than once. The only way for a cyclic weight diagram to emerge is by decoupling the weights outside the box. To see that this is also consistent, note that all paths leading from weight 5 to weights 2 or 4 require two applications of b0. To construct a superLax operator let us take the vacuum solution 4 = 4. We easily find the superLax operator to be (A.71) 80 As in the nonsupersymmetric case, Lax operators corresponding to the other three vacua are obtained through cyclic permutations of the above operator. Discussion In this appendix we have shown how one can read off from cyclic weight diagrams, associated with representations of affine Lie algebras, Lax operators in the diagonal gauge. This method is most useful when tables of matrix representations are not at hand and must be generated by weight diagrams anyway. Furthermore, this procedure can easily be implemented on computer by virtue of the fact that computer generated algorithms currently exist for building highest weight diagrams.56 With minor modifications, these programs can be adapted for cyclic weight diagrams. It remains to be seen whether higher representations lead to any new physics. If so then a program of categorizing these results might be pursued to identify redundant solutions. This might be easier to answer for supersym metric algebras since nilpotency projects out decoupled weight vectors. What is clear though, at least for the nonsupersymmetric cases is that these higher representations lead to integrable systems. Recall, to prove the integrability of KdV systems Drinfel'd and Sokolov found the necessary infinite set of con served currents to be given by the coefficients of the Laurent expansion of C in the affine parameter A. Our conclusion follows from the fact that every representation of a basic Lie algebra has an affine extension, and that defining properties of affine Lie algebras are representation independent. Finally, it would be interesting to see if our procedure could be modified to directly generate Lax operators in other gauges. 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Math. Phys. 98 (1985) 65 56. W. G. McKay, J. Patera, and D. W. Rand, SimpLie User's Manual and Software (CRM, Montreal, 1990) p. 1 BIOGRAPHICAL SKETCH Steve Carbon was born in Evanston, Illinois, on May 21, 1958. It was while was attending high school near Chicago that he was fascinated by his first physics course. Later, he graduated from the Miami Public School system and enrolled at the University of Central Florida in Orlando. He obtained B.S. degrees in both math and physics while at UCF before obtaining a job with RCA at the Kennedy Space Center as a software analyst. While with RCA, he earned an M.S. degree in math at UCF and started on an M.S. degree in physics. He then left his job with RCA to complete his physics masters. Af terwards, he came to the University of Florida and subsequently began doing research under the supervision of Professor Charles Thorn. His research inter ests have included Skyrmions, matrix models, covariant quantization of string string, and the high energy behavior of string theory. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles B. Thorn, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate. in scope and quality, as a dissertation for the degree of Doctor of Philosophy. re Ramond Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard D. Field Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Christopher Stark Associate Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1993 Dean, Graduate school UNIVERSITY OF FLORIDA 3 I1262 08553 9244 3 1262 08553 9244 