We construct a 1+1 dimensional superstring-bit model for D=3 Type IIB
superstring. This low dimension model escapes the problems encountered in
higher dimension models: (1) It possesses full Galilean supersymmetry; (2) For
noninteracting polymers of bits, the exactly soluble linear superpotential
describing bit interactions is in a large universality class of superpotentials
which includes ones bounded at spatial infinity; (3) The latter are used to
construct a superstring-bit model with the clustering properties needed to
define an $S$-matrix for closed polymers of superstring-bits.Comment: 11 pages, Latex documen

C. B. Thorn

Charles B. Thorn

E. Witten

G. 't Hooft

I. Klebanov

O. Bergman

Oren Bergman

R. Giles

W. Siegel

English

arXiv

We construct a 1+1 dimensional superstring-bit model for D=3 Type IIB superstring. This low dimension model escapes the problem encountered in higher dimension models: (1) It possesses full Galilean supersymmetry; (2) For noninteracting Polymers of bits, the exactly soluble linear superpotential describing bit interactions is in a large universality class of superpotentials which includes ones bounded at spatial infinity; (3) The latter are used to construct a superstring-bit model with the clustering properties needed to define an S-matrix for closed polymers of superstring-bits

Bergman, Oren

Thorn, Charles B.

UNT (University of North Texas) Digital Library

a n - U 0 I LL LL 0 F l- a cv c9 0 ii n a, > v) a, C Q I .I .I (3 lnstltute for Fundamental Theory Preprint UFlFT - HEP - 95 - 31 hep-th/95 12 107 RECEIVED O S T I  Universality and Clustering in 1 + 1 Dimensional Superstring-Bit Models* OREN BERGMAN t AND CHARLES B. THORN t Institute for Fundamental Theory Department of Physics, University of Florida Gainesville, FL 5261 1 ABSTRACT We construct a 1+1 dimensional superstring-bit model for D=3 Type IIB superstring. This low dimension model escapes the problems en- countered in higher dimension models: (1) It possesses full Galilean supersymmetry; (2) For noninteracting polymers of bits, the exactly soluble linear superpotential describing bit interactions is in a large universality class of superpotentials which includes ones bounded at spatial infinity; (3) The latter are used to construct a superstring-bit model with the clustering properties needed to define an S-matrix for closed polymers of superstring-bits. * Work supported in part by the Department of Energy under grant number DE-FG05-86ER-40272 and in part by the Institute for Fundamental Theory. Email address: oren@phys.ufl.edu t Email address: thorn@quark.phys.ufl.edu String-bit models are attempts to reformulate string theories in D - 1 space dimensions as field theories of point-like string-bits moving in d 3 D - 2 space dimensions, in which string is composite, not fundamental [l-51. The bits bind together to form long closed polymer chains, which in the continuum limit have precisely the properties of closed relativistic string. String-bits must carry an internal “color” degree of freedom which defines ordering around the chain. This can be achieved if the bits transform in the adjoint representation of the unitary group U(N,), with N,  2 2. A crucial feature of our light-cone approach to string- bit models is that they possess Galilean invariance in d space dimensions, not Poincark invariance in d + 1 space dimensions: the bits enjoy a non-relativistic dynamics. Second-quantization of string-bits employs a string-bit creation operator 4 t ( x ) t ,  where cy and p run over the N, colors and x denotes the d space coordinates. Denote the zero-bit state by IO). A bare closed chain of M bits is then described by: I !P(x l , .  . . , z ~ ) )  = 1 d q  - - - dxM X T r [ Q ( d  . . * 6 ( ~ M ) 1 1 0 ) ~ ( ~ 1 ,  - - * ZM) , (1) implying that @(XI, . . . , x ~ )  is cyclically symmetric. To describe closed chains and their interactions, the Hamiltonian governing string-bit dynamics must allow for their formation and assure their stability. Chain formation requires that string-bits have an attractive interaction between nearest neighbors on a chain, with non-nearest neighbors interacting much’more weakly. It is well- known [6,3,5] that this pattern of interactions arises in a many body system of particles described by N, x N, matrix creation operators using ’t Hooft’s N, +- 00 limit [7]. For an interaction Hamiltonian of the form the limit N, + cx) leads to nearest-neighbor interactions in a bare closed chain of string-bits. For N, finite but large, O(l/N,) effects allow a single bare closed chain to  break into two bare closed chains. Thus 1/N, serves as a chain coupling constant, and ’t Hooft’s 1/N, expansion produces chain perturbation theory. 2 Free light-cone string is recovered for N, + 00 in the continuum limit given by M + 00, m + 0, with mM kept fixed, or equivalently in .the low energy limit given by E << To/m, where To is the string tension and m is the Newtonian mass of a bit. The total Newtonian mass of a chain becomes an effectively continuous P+ of a string. The x- coordinate of string thus emerges dynamically in string-bit models as the conjugate to Newtonian mass. The other light-cone coordinate x+ is identified as time, and its conjugate P- as the bit Hamiltonian. The 0 (l/Nc) chain interactions become string interactions in the continuum limit. Stability of string depends on the ground state energy of a long closed chain, generically given by: The first term is the same for a single chain of M bits and two chains of Ml and Mz bits, with MI + M2 = M .  For long chains, the nature of the true ground state then depends on the second term. If b > 0 then EO,M < Eo,M~ +Eo,M~,  and a long chain is stable. If b < 0 then EO,M > Eo,M~ +Eo,M~, the chain is unstable to decay into two smaller chains, and it will through the O(l/N,) terms alluded to before. Consider a chain of bosonic bits interacting via the nearest-neighbor harmonic potential V(x)  = (w2/2m)z2. The ground state energy for an M-bit chain in d space dimensions is found to be [1,5] and so a long chain of bosonic bits is unstable against decay into two smaller chains. This is just the string-bit manifestation of the tachyonic instability of bosonic string. The negative coefficient of 1/2mM is the mass-squared of the tachyon. This instability is absent in superstring theory which requires the addition of fermionic modes on string in a supersymmetric fashion. For string-bit models the bits are in super- multiplets with a “statistics” degree of freedom distinguishing bosons from fermions. This degree of freedom gives rise to “statistics waves” on long chains, similar to spin waves. Su- persymmetrizing the harmonic string-bit model leads to a cancellation of the contribution 3 to the ground state energy of the coordinate “phonon” waves with that of the “statistics” waves. In fact, the ground state energy is exactly zero for any M [5]. We shall see later that this is a universal property of supersymmetric string-bit models, and not special to the harmonic interaction. Note that recovering the free superstring mass spectrum at M = 03 requires only b = 0, whereas in this model all the terms in (3) vanish. If this were not so, the sign of the first nonvanishing term would determine the stability of finite long chains. If these were unstable except when M = 03, the bit model could not provide a fundamental basis for superstring theory, and at best would only make sense in the continuum limit. We have constructed superstring-bit models in 2 + 1 and 8 + 1 dimensions that underlie D = 4 and D = 10 type IIB superstring theory respectively [5]. In the D = 4 case we could include extra degrees of freedom either as real compactified dimensions, or as additional internal bit degrees of freedom [8] which, on long chains, would produce “flavor waves” playing the role of extra dimensions. The N = 2 spacetime PoincarB supersymmetry of the D-dimensional type IIB superstring requires the corresponding (D - 2) + 1-dimensional superstring-bit model to possess an N = 1 Galilean supersymmetry. The symmetry is Galilean because light-cone variables break the manifest SO(D - 1,l) to SO(D - 2) x SO(1,l). Discretization of P+ breaks this SO(1,l) and also mixes the right- and left- moving supercharges, leaving only an N = 1 supersymmetry generated by the right + left combinations. The Galilean supercharges Q and R transform as spinors of the transverse SO(D - 2) subgroup of SO(D - 1, l), and have opposite chirality in the SO(1,l)  subgroup. Together they build a single supercharge transforming as a spinor of the Lorentz group SO(D - 1, l), and generating the super-Poincar6 algebra. For general d the Galilean supercharges QA, RA must each have d components for a sat- isfactory superstring limit of the string-bit model. The corresponding super-Galilei algebra then reads: {RA,RB) = bABH12, 4 (5) where M is the total number of bits, and P is their total momentum. The superstring-bit models of [5] implement all but the last of these relations: There are additional terms not proportional to 6Ai. For the supersymmetric harmonic model these terms are sub-leading in the l / N c  expansion, so the R-supersymmetry is broken by chain interactions. For other superstring-bit models this happens already at the level of the free chain spectrum. In either case, it might still be that the full Poincar6 supersymmetry can be recovered in the continuum stringy physics. For d = 1, the full superalgebra closes by default, since R and then have only one component each. A 1 + l-dimensional superstring-bit model would underlie D = 3 super- string, the lowest dimensional superstring possible [9]. Specializing the supercharges of [5] to this case gives Q = E/  (x)$(x)  + h.c.1 R=-- J d ~ T r [ e - ~ " / ~ $ ~  (x)$,'(x) + h.c.1 2& J d.dyW(y - 2) 2Nc& + ~Tr[e- ' " /~4t (a)p(y) l l , (s)  +h.c.1 , where @(x)! is the bosonic creation operator, $t(x)! is the fermionic creation operator, and & = [4t4 + $t$]!. The superalgebra is given by: The last equation can be taken as the definition of the Hamiltonian for this system, which in turn implies that the model is invariant under both Q and R. If the function W ( x )  is taken to be odd, the two-bit sector is equivalent to Witten's supersymmetric quantum mechanics [lo], where W ( x )  is the superpotential. In that case the ground state energy of a two-bit closed chain vanishes. For understanding superstring theory, we are interested in a class of superpotentials for which the ground state energy of any length chain vanishes, and for which the gap to excite the chain is finite. Exploring this issue for noninteracting chains, we consider the first-quantized system obtained by acting 5 with R on a bare chain and taking the limit N, + 03. The first-quantized supercharge is then given by where p k  = - 8 / a x k  and 7i-k = d/adk. The summation is understoou to be cyclic, i.e. IC = M + 1 is equivalent to k = 1. The first-quantized M-bit Hamiltonian is then given by We denote states in the first-quantized Hilbert space of an M-bit chain by I - -) to distinguish them from states in the second-quantized bit Fock space, denoted I .-a). The ground state of the chain is then denoted by IO). Consider the linear superpotential W ( z )  = Toz , which defines the supersymmetric harmonic model. The ground state is annihilated by R implying it belongs to a "small" representation of the Galilei superalgebra and has zero energy. In addition its spectrum approaches that of a relativistic superstring with tension To in the continuum limit. Consider a deformation of the superpotential W(z) + W ( z )  + 6W(z) , where 6W(z) is small in the interval 1x1 < L. If L >> 1/a this deformation can be treated perturbatively, since for 1x1 > L the unperturbed wavefunctions are exponentially small. Due to the Galilei superalgebra (7), the exact Hamiltonian can be written as the square of the new supercharge R + 6R. The change in the energy of the ground state to first order is then: 6 which vanishes since RIO) = (OIR = 0. We stress that this holds for any length chain. The spectrum of excitations of the chain is generated by acting on the above ground state with mode raising operators. For the 2 + l-dimensional harmonic model these were derived in [5]. Dropping a dimension and with it the spinor indices gives: where wn = 2To sin n r / M ,  for the coordinate modes raising and lowering operators, and where en = (1/&) (sin nr /2M + cos n r / 2 M )  and qn = (1/&) (sin nr /2M - cos nn/2M),  for the “statistics” modes raising and lowering operators. In the above 2n, en,&, iin are the Fourier transforms of xk, &, p k ,  nk, respectively. Consider an excitation with a single raising operator ALIO). The zero’th order energy of this state is given by The shift in the energy due to the deformation (11) is given to first order by Using the commutation relations and the fact that (oldW’(xk+l - xk)lO) is cyclically invariant we find 2 nr = -(016W‘(x2 - x1)lO) sin - . m M 7 The net effect is then just to shift the string tension so the gap remains finite for all M .  For excitations of the form niAAJO), with more than one raising operator, we have SE = 4(011TAni{R,SR)nA~i10)  i i As we commute the AL,’s to the left we pick up an additional derivative of bW and a factor of 4 q J M  = O(l/M) for each A;, that contracts with SR. The rest contract with some of the Ani’s. The remaining Ani’s are then commuted to the right, all contracting with SR to produce additional derivatives and additional powers of 1/M. Finally, Bnj is contracted with what’s left. The result is a sum of terms of increasing odd number of derivatives of SW multiplied by increasing powers of l/M, the coefficient of SW(2’+1) being proportional to M-2’-1. In the limit M 3 oa only the the I = 0 term contributes, so the analogue of (18) holds for any excitation of the form ni ALJO). The argument for excitations created by products of 23:’s or products of both AL’s and Bi’s is similar. Thus the only effect of the deformation (11) on long chains is to renormalize the string tension as in Eq.(19). We now argue that these results from first order perturbation theory hold to all orders, and indeed should extend to a large universality class of superpotentials. The ground state energy must remain zero as long as the ground state is in a “small” representation of the superalgebra. Clearly the representation can’t change in perturbation theory, but more generally it will remain “small” unless the first excited state becomes degenerate with the ground state, i.e. unless the gap closes. Moreover, the properties of the O(To/mM) excita- tions of long chains have a universal character determined by phonon and statistics waves, which are inevitable collective excitations of stable long chains [4] .  Formation of long bare 8 chains is ensured by a large bond-breaking energy and does not require the infinite range harmonic force. Finally we turn to the issue of interactions between closed chains. By exploiting the N, + 03 limit, we have been able to define a bare closed chain, whose interactions with other closed chains is negligible. Taking N, to be finite will “dress” the bare chains, and will give rise to interactions between different chains. The ionization energy required to break a bond in the closed chain is O(T’/rn). As long as the scattering energy is below threshold for such ionization, the only way for interactions to occur is through bond rear- rangement. A necessary condition for defining a closed chain S-matrix is that the model satisfy a clustering property: An initial state of two spatially separated closed chains must evolve as two noninteracting chains until enough time has elapsed for them to get close to one another. With a linearly growing superpotential as in (lo), this is impossible at finite N,. The supersymmetric harmonic model is thus unsatisfactory for describing chain scat- tering. Asymptotically free chains require a bounded superpotential. In order to keep the ionization energy O(To/rn), however, we must still insist that W + AW, # 0 at spatial infinity. The restricted universality established above indeed allows us to deform the linear superpotent ial into with no change in the continuum properties of free chains. With the bounded superpotential (21) there can still be large (O(To/m)) correlation energies between spatially separated clumps of bits. If the clumps correspond to closed chains, or more generally to singlet states, these correlation energies must vanish. This can be achieved by first noting that the generators for color rotations given by where a! =: [&t - @$t]t :, annihilate singlet states. This motivates replacing p(y) with p(y) - a(y) in Eq. (6) for R. It is easy to check that such a change does not disturb the 9 superalgebra. To see that it gives the desired clustering property, consider the action of the new R on a state we denote by S l S 2 1 0 ) ,  where Sl and S 2  are any color singlet functions of creation operators, such that the locations of all creation operators in 5’1 are more than a distance L from all the locations in S 2 .  Let (TS)~ and ( T S ) ~  be the singlet functions of creation operators defined by RS110) = (TS) l IO)  7 R S 2 1 0 )  = (TS),IO) Then R S I S 2 1 0 )  = ( ~ ~ ) l S 2 ~ 0 )  + ( - )s’s1(Ts)210)  + R I S l S 2 1 0 )  where the action of Rl is defined by requiring one of the two annihilation operators in the two-body term of R to contract with a creation operator in SI, and the other annihilation operator to contract with a creation operator in SZ. Because of the spatial separation of the coordinates in 5’1 and the coordinates in S 2 ,  the superpotential W(z)  can be replaced with its asymptotic value and taken out of the integral. Consequently one is left with the color rotation operator J ( p  - a)  acting on SI or 5’2, either of which gives zero. Thus we conclude that R S l S 2 1 0 )  = ( T S ) l S 2 1 0 )  + ( - )s’s l (Ts)210)  * Applying R once again, using the supersymmetry algebra, and remembering that the su- percharge is odd, we infer: implying that the hamiltonian acts independently on the two singlets. Therefore as long as they remain spatially well separated the two singlets propagate without mutual interaction. 10 REFERENCES [l] R. Giles and C. B. Thorn, Phys. Rev. D16 (1977) 366. [2] I. Klebanov and L. Susskind, Nuc. Phys. B309 (1988) 175. [3] C. B. Thorn, in Sakharov Memorial Lectures in Physics, Ed. L. V. Keldysh and V. Ya. Fainberg, Nova Science Publishers Inc., Commack, NY, 1992; hep-th/9405069. [4] C. B. Thorn, Phys. Rev. D51 (1995) 647. [5] 0. Bergman and C. B. Thorn, Phys. Rev. D52 (1995) 5980. [6] C. B. Thorn, Phys. Rev. D20 (1979) 1435. [7] G. 't Hooft, Nucl. Phys. B72 (1974) 461. [8] R. Giles, L. McLer,ran, and C. B. Thorn, Phys. Rev. D17 (1978) 2058. [9] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theo y, Cambridge.University Press (1987). [lo] E. Witten, Nuc. Phys. 33188 (1981) 513. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 11 

Universality and clustering in 1 + 1 dimensional superstring-bit models

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Universality and Clustering in 1+1 Dimensional Superstring-Bit Models

Universality and Clustering in 1+1 Dimensional Superstring-Bit Models

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