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We initiate a novel formalism for computing correlation functions of trace operators in the planar N=4 SYM theory. The central object in our formalism is the spin vertex, which is the weak coupling analogy of the string vertex in string field theory. We construct the spin vertex explicitly for all sectors at the leading order using a set of bosonic and fermionic oscillators. We prove that the vertex has trivial monodromy, or put in other words, it is a Yangian invariant. Since the monodromy of the vertex is the product of the monodromies of the three states, the Yangian invariance of the vertex implies an infinite exact symmetry for the three-point function. We conjecture that this infinite symmetry can be lifted to any loop order.
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We develop superstring bit models, in which the lightcone transverse coordinates in D spacetime dimensions are replaced with d=D-2 double-valued flavor indices $x^k-> f_k=1,2$; $k=2,...,d+1$. In such models the string bits have no space to move. Letting each string bit be an adjoint of a color group U(N), we then analyze the physics of t Hoofts limit $N->infty$, in which closed chains of many string bits behave like free lightcone IIB superstrings with d compact coordinate bosonic worldsheet fields $x^k$, and s pairs of Grassmann fermionic fields $theta_{L,R}^a$, a=1,..., s. The coordinates $x^k$ emerge because, on the long chains, flavor fluctuations enjoy the dynamics of d anisotropic Heisenberg spin chains. It is well-known that the low energy excitations of a many-spin Heisenberg chain are identical to those of a string worldsheet coordinate compactified on a circle of radius $R_k$, which is related to the anisotropy parameter $-1<Delta_k<1$ of the corresponding Heisenberg system. Furthermore there is a limit of this parameter, $Delta_k->pm 1$, in which $R_k->infty$. As noted in earlier work [Phys.Rev.D{bf 89}(2014)105002], these multi-string-bit chains are strictly stable at $N=infty$ when d<s and only marginally stable when d=s. (Poincare supersymmetry requires d=s=8, which is on the boundary between stability and instability.)
We study the behavior of a simple string bit model at finite temperature. We use thermal perturbation theory to analyze the high temperature regime. But at low temperatures we rely on the large $N$ limit of the dynamics, for which the exact energy spectrum is known. Since the lowest energy states at infinite $N$ are free closed strings, the $N=infty$ partition function diverges above a finite temperature $beta_H^{-1}$, the Hagedorn temperature. We argue that in these models at finite $N$, which then have a finite number of degrees of freedom, there can be neither an ultimate temperature nor any kind of phase transition. We discuss how the discontinuous behavior seen at infinite $N$ can be removed at finite $N$. In this resolution the fundamental string bit degrees of freedom become more active at temperatures near and above the Hagedorn temperature.
The state space of a generic string bit model is spanned by $Ntimes N$ matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group $U(N)$ (or $SU(N)$ if the matrices are traceless). We consider a system of $b$ species of bosonic bits and $f$ speciesof fermionic bits. The string, emerging in the $Ntoinfty$ limit, identifies $P^+=mMsqrt{2}$ with $M$ the bit number operator and $P^-=Hsqrt{2}$ with $H$ the system Hamiltonian. We study the thermal properties of this string bit system in the case $H=0$, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function ${rm Tr} e^{-beta mM}$ can be identified, putting $x=e^{-beta m}$, with a generating function $chi_0^{bf}(x)$, for which the coefficient of $x^n$ in its expansion about $x=0$ is the number of color singlets with bit number $M=n$. This function is a purely group theoretic object, which is well-studied in the literature. We show that at $N=infty$ this system displays a Hagedorn divergence at $x=1/(b+f)$ with ultimate temperature $T_H=m/ln(b+f)$. The corresponding function for finite $N$ is perfectly finite for $0<x<1$, so the $N=infty$ system exhibits a phase transition at temperature $T_H$ which is absent for any finite $N$. We demonstrate that the low temperature phase is unstable above $T_H$. The lowest-order $1/N$ asymptotic correction, for $xto1$ in the high temperature phase, is computed for large $N$. Remarkably, this is related to the number of labeled Eulerian digraphs with $N$ nodes. Systematic methods to extend our results to higher orders in $1/N$ are described.
We discuss the homological aspects of the connection between quantum string generating function and the formal power series associated to the dimensions of chains and homologies of suitable Lie algebras. Our analysis can be considered as a new straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of $SL(2,{mathbb Z})$) to the partition functions of Lagrangian branes, refined vertex and open string partition functions, represented by means of formal power series that encode Lie algebra properties. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras and in the role of Selberg-type spectral functions of an hyperbolic three-geometry associated with $q$-series in the computation of the string amplitudes.
We propose boundary conditions on a two dimensional 6-vertex model, which is defined on the lightcone lattice for an open string worldsheet. We show that, in the continuum limit, the degrees of freedom of this 6-vertex model describe a target space coordinate compactified on a circle of radius R, which is related to the vertex weights. This conclusion had already been established for the case of a 6-vertex model on the worldsheet lattice for the propagator of a closed string. This exercise illustrates how the Bethe ansatz works in the presence of boundaries, at least of this particular type.
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