Quantum Mechanics, Random Matrices and BMN Gauge Theory

We review how the identification of gauge theory operators representing string states in the pp-wave/BMN correspondence and their associated anomalous dimension reduces to the determination of the eigenvectors and the eigenvalues of a simple quantum mechanical Hamiltonian and analyze the properties of this Hamiltonian. Furthermore, we discuss the role of random matrices as a tool for performing explicit evaluation of correlation functions.Comment: 16 pages, contribution to proceedings of Workshop on Random Geometry, Krakow, May, 200

The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz

Using the quantum string Bethe ansatz we derive the one-loop energy of a folded string rotating with angular momenta (S,J) in AdS_3 x S^1 inside AdS_5 x S^5 in the limit 1 << J << S, z=\lambda^(1/2) log(S/J) /(\pi J) fixed. The one-loop energy is a sum of two contributions, one originating from the Hernandez-Lopez phase and another one being due to spin chain finite size effects. We find a result which at the functional level exactly matches the result of a string theory computation. Expanding the result for large z we obtain the strong coupling limit of the scaling function for low twist, high spin operators of the SL(2) sector of N=4 SYM. In particular we recover the famous -3 log(2)/\pi. Its appearance is a result of non-trivial cancellations between the finite size effects and the Hernandez-Lopez correction.Comment: 18 pages, one figure, v2: footnote changed, v3: reference added, typo correcte

Three-spin Strings on AdS_5 x S^5 from N=4 SYM

Using the integrable spin chain picture we study the one-loop anomalous dimension of certain single trace scalar operators of N = 4 SYM expected to correspond to semi-classical string states on AdS5 ×S 5 with three large angular momenta (J1,J2,J3) on S 5. In particular, we investigate the analyticity structure encoded in the Bethe equations for various distributions of Bethe roots. In a certain region of the parameter space our operators reduce to the gauge theory duals of the folded string with two large angular momenta and in another region to the duals of the circular string with angular momentum assignment (J,J ′,J ′), J&gt; J ′. In between we locate a critical line. We propose that the operators above the critical line are the gauge theory duals of the circular elliptic string with three different spins and support this by a Recent development, triggered by the pp-wave/BMN correspondence [1], has led to new insights on the duality between string theory and gauge theory and ha

A Holographic Quantum Hall Ferromagnet

A detailed numerical study of a recent proposal for exotic states of the D3-probe D5 brane system with charge density and an external magnetic field is presented. The state has a large number of coincident D5 branes blowing up to a D7 brane in the presence of the worldvolume electric and magnetic fields which are necessary to construct the holographic state. Numerical solutions have shown that these states can compete with the the previously known chiral symmetry breaking and maximally symmetric phases of the D3-D5 system. Moreover, at integer filling fractions, they are incompressible with integer quantized Hall conductivities. In the dual superconformal defect field theory, these solutions correspond to states which break the chiral and global flavor symmetries spontaneously. The region of the temperature-density plane where the D7 brane has lower energy than the other known D5 brane solutions is identified. A hypothesis for the structure of states with filling fraction and Hall conductivity greater than one is made and tested by numerical computation. A parallel with the quantum Hall ferromagnetism or magnetic catalysis phenomenon which is observed in graphene is drawn. As well as demonstrating that the phenomenon can exist in a strongly coupled system, this work makes a number of predictions of symmetry breaking patterns and phase transitions for such systems.Comment: 38 pages, 7 figures, references adde

Higher Genus Correlators for the Complex Matrix Model

We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential.Comment: 17 page

Matrix Model Calculations beyond the Spherical Limit

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.Comment: 44 page

The Concept of Time in 2D Quantum Gravity

We show that the ``time'' t_s defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d_h(s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this definition of Hausdorff dimension breaks down. Numerical measurements are consistent with these results. The same definition leads to d_h(s)=16 at the critical point when applied to flat space. The fractal dimension d_h(s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d_h(g), which is based on the use of the geodesic distance t_g as ``proper time''. There seems to be no simple relation of the kind t_s = t_g^{d_h(g)/d_h(s)}, as expected by dimensional reasons.Comment: 14 pages, LaTeX, 2 ps-figure