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Register a new userInternal and Super Symmetry in String Bit Models
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Authors
Charles B. Thorn
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We study in a general way the construction of string bit Hamiltonians which are supersymmetric, We construct several quadratic and quartic polynomials in string bit creation and annihilation operators ${barphi}^A_{a_1cdots a_n}$, ${phi}^A_{a_1cdots a_n}$,which commute with the supersymmetry generators $Q^a$. Among these operators are ones with the spinor tensor structure required to provide the lightcone worldsheet vertex insertion factors needed to give the correct interactions for the IIB superstring, whenever a closed string separates into two closed strings or two closed strings join into one.
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In string bit models, the superstring emerges as a very long chain of bits, in which s fermionic degrees of freedom contribute positively to the ground state energy in a way to exactly cancel the destabilizing negative contributions of d=s bosonic degrees of freedom. We propose that the physics of string formation be studied nonperturbatively in the class of string bit models in which s>d, so that a long chain is stable, in contrast to the marginally stable (s=d=8) superstring chain. We focus on the simplest of these models with s=1 and d=0, in which the string bits live in zero space dimensions. The string bit creation operators are N X N matrices. We choose a Hamiltonian such that the large N limit produces string moving in one space dimension, with excitations corresponding to one Grassmann lightcone worldsheet field (s=1) and no bosonic worldsheet field (d=0). We study this model at finite N to assess the role of the large N limit in the emergence of the spatial dimension. Our results suggest that string-like states with large bit number M may not exist for N<(M-1)/2. If this is correct, one can have finite chains of string bits, but not continuous string, at finite N. Only for extremely large N can such chains behave approximately like continuous string, in which case there will also be the (approximate) emergence of a new spatial dimension. In string bit models designed to produce critical superstring at N=infinity, we can then expect only approximate Lorentz invariance at finite N, with violations of order 1/N^2.
We show that planar cal N=4 Yang-Mills theory at zero t Hooft coupling can be efficiently described in terms of 8 bosonic and 8 fermionic oscillators. We show that these oscillators can serve as world-sheet variables, the string bits, of a discretized string. There is a one to one correspondence between the on shell gauge invariant words of the free Y-M theory and the states in the oscillators Hilbert space, obeying a local gauge and cyclicity constraints. The planar two-point functions and the three-point functions of all gauge invariant words are obtained by the simple delta-function overlap of the corresponding discrete string world sheet. At first order in the t Hooft coupling, i.e. at one-loop in the Y-M theory, the logarithmic corrections of the planar two-point and the three-point functions can be incorporated by nearest neighbour interactions among the discretized string bits. In the SU(2) sub-sector we show that the one-loop corrections to the structure constants can be uniquely determined by the symmetries of the bit picture. For the SU(2) sub-sector we construct a gauged, linear, discrete world-sheet model for the oscillators, with only nearest neighbour couplings, which reproduces the anomalous dimension Hamiltonian up to two loops. This model also obeys BMN scaling to all loops.
String bit models provide a possible method to formulate a string as a discrete chain of pointlike string bits. When the bit number $M$ is large, a chain behaves as a continuous string. We study the simplest case that has only one bosonic bit and one fermionic bit. The creation and annihilation operators are adjoint representations of the $Uleft(Nright)$ color group. We show that the supersymmetry reduces the parameter number of a Hamiltonian from 7 to 3 and, at $N=infty$, ensures a continuous energy spectrum, which implies the emergence of one spatial dimension. The Hamiltonian $H_{0}$ is constructed so that in the large $N$ limit it produces a world sheet spectrum with one Grassmann world sheet field. We concentrate on numerical study of the model in finite $N$. For the Hamiltonian $H_{0}$, we find that the would-be ground energy states disappear at $N=left(M-1right)/2$ for odd $Mleq11$. Such a simple pattern is spoiled if $H$ has an additional term $xiDelta H$ which does not affect the result of $N=infty$. The disappearance point moves to higher (lower) $N$ when $xi$ increases (decreases). Particularly, the $pmleft(H_{0}-Delta Hright)$ cases suggest a possibility that the ground state could survive at large $M$ and $Mgg N$. Our study reveals that the model has stringy behavior: when $N$ is fixed and large enough, the ground energy decreases linearly with respect to $M$, and the excitation energy is roughly of order $M^{-1}$. We also verify that a stable system of Hamiltonian $pm H_{0}+xiDelta H$ requires $xigeqmp1$.
We develop the 1/N expansion for stable string bit models, focusing on a model with bit creation operators carrying only transverse spinor indices a=1,...,s. At leading order (1/N=0), this model produces a (discretized) lightcone string with a transverse space of $s$ Grassmann worldsheet fields. Higher orders in the 1/N expansion are shown to be determined by the overlap of a single large closed chain (discretized string) with two smaller closed chains. In the models studied here, the overlap is not accompanied with operator insertions at the break/join point. Then the requirement that the discretized overlap have a smooth continuum limit leads to the critical Grassmann dimension of s=24. This protostring, a Grassmann analog of the bosonic string, is unusual, because it has no large transverse dimensions. It is a string moving in one space dimension and there are neither tachyons nor massless particles. The protostring, derived from our pure spinor string bit model, has 24 Grassmann dimensions, 16 of which could be bosonized to form 8 compactified bosonic dimensions, leaving 8 Grassmann dimensions--the worldsheet content of the superstring. If the transverse space of the protostring could be decompactified, string bit models might provide an appealing and solid foundation for superstring theory.
We provide a formalism to calculate the cubic interaction vertices of the stable string bit model, in which string bits have $s$ spin degrees of freedom but no space to move. With the vertices, we obtain a formula for one-loop self-energy, i.e., the $mathcal{O}left(1/N^{2}right)$ correction to the energy spectrum. A rough analysis shows that, when the bit number $M$ is large, the ground state one-loop self-energy $Delta E_{G}$ scale as $M^{5-s/4}$ for even $s$ and $M^{4-s/4}$ for odd $s$. Particularly, in $s=24$, we have $Delta E_{G}sim 1/M$, which resembles the Poincare invariant relation $P^{-}sim 1/P^{+}$ in $(1+1)$ dimensions. We calculate analytically the one-loop correction for the ground energies with $M=3$ and $s=1,,2$. We then numerically confirm that the large $M$ behavior holds for $sleq4$ cases.
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