Chain-Free String Constraints (Technical Report)

We address the satisfiability problem for string constraints that combine relational constraints represented by transducers, word equations, and string length constraints. This problem is undecidable in general. Therefore, we propose a new decidable fragment of string constraints, called weakly chaining string constraints, for which we show that the satisfiability problem is decidable. This fragment pushes the borders of decidability of string constraints by generalising the existing straight-line as well as the acyclic fragment of the string logic. We have developed a prototype implementation of our new decision procedure, and integrated it into in an existing framework that uses CEGAR with under-approximation of string constraints based on flattening. Our experimental results show the competitiveness and accuracy of the new framework

Three-Point Functions in N=4 SYM Theory at One-Loop

We analyze the one-loop correction to the three-point function coefficient of scalar primary operators in N=4 SYM theory. By applying constraints from the superconformal symmetry, we demonstrate that the type of Feynman diagrams that contribute depends on the choice of renormalization scheme. In the planar limit, explicit expressions for the correction are interpreted in terms of the hamiltonians of the associated integrable closed and open spin chains. This suggests that at least at one-loop, the planar conformal field theory is integrable with the anomalous dimensions and OPE coefficients both obtainable from integrable spin chain calculations. We also connect the planar results with similar structures found in closed string field theory.Comment: 34 pages, 9 figures, harvmac; references adde

Single spike solutions for strings on S2 and S3

We study solutions for rigidly rotating strings on a two sphere. Among them we find two limiting cases that have a particular interest, one is the already known giant magnon and the other we call the single spike solution. The limiting behavior of this last solution is a string infinitely wrapped around the equator. It differs from that solution by the existence of a single spike of height theta that points toward the north pole. We study its properties and compute its energy E and angular momentum J as a function of theta. We further generalize the solution by adding one angular momentum to obtain a solution on S3. We find a spin chain interpretations of these results in terms of free fermions and the Hubbard model but the exact relation with the same models derived from the field theory is not clear.Comment: LaTeX, 20 pages, 3 figures. v2: Refs adde

On the anatomy of multi-spin magnon and single spike string solutions

We study rigid string solutions rotating in $AdS_5\times S^5$ background. For particular values of the parameters of the solutions we find multispin solutions corresponding to giant magnons and single spike strings. We present an analysis of the dispersion relations in the case of three spin solutions distributed only in $S^5$ and the case of one spin in $AdS_5$ and two spins in $S^5$. The possible relation of these string solutions to gauge theory operators and spin chains are briefly discussed.Comment: 45 pages, the presentation rearranged in 3 sections, results unchanged, references added, some typos correcte

Boundary Conditions as Dirac Constraints

In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are noncommutative.Comment: 18 pp, Latex, minor changes, typos correcte

Optimal shapes of compact strings

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved that the answer for infinite systems is a face-centred-cubic lattice. This simply stated problem has had a profound impact in many areas, ranging from the crystallization and melting of atomic systems, to optimal packing of objects and subdivision of space. Here we study an analogous problem--that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. We find that, in cases where boundary effects are not dominant, helices with a particular pitch-radius ratio are selected. Interestingly, the same geometry is observed in helices in naturally-occurring proteins.Comment: 8 pages, 3 composite ps figure

Super spin chain coherent state actions and AdS5×S5AdS_5 \times S^5 superstring

We consider a generalization of the leading-order matching of coherent state actions for semiclassical states on the super Yang-Mills and the superstring sides of the AdS/CFT duality to sectors with fermions. In particular, we discuss the $SU(1|1)$ and $SU(2|3)$ sectors containing states with angular momentum $J$ in $S^5$ and spin in $AdS_5$. On the SYM side, we start with the dilatation operator in the $SU(2|3)$ sector having super spin chain Hamiltonian interpretation and derive the corresponding coherent state action which is quartic in fermions. This action has essentially the same ``Landau-Lifshitz'' form as the action in the bosonic SU(3) sector with the target space $CP^2$ replaced by the projective superspace $CP^{2|2}$. We then attempt to relate it to the corresponding truncation of the full $AdS_5 \times S^5$ superstring action written in a light-cone gauge where it has simple quartic fermionic structure. In particular, we find that part of the superstring action describing $SU(1|1)$ sector reduces to an action of a massive two-dimensional relativistic fermion, with the expansion in the effective coupling $\lambda/J^2$ being equivalent to a non-relativistic expansion