Complex WKB Analysis of a PT Symmetric Eigenvalue Problem

The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].Comment: 22 pages, 13 figures. Added references, minor revision

Excited state g-functions from the Truncated Conformal Space

In this paper we consider excited state g-functions, that is, overlaps between boundary states and excited states in boundary conformal field theory. We find a new method to calculate these overlaps numerically using a variation of the truncated conformal space approach. We apply this method to the Lee-Yang model for which the unique boundary perturbation is integrable and for which the TBA system describing the boundary overlaps is known. Using the truncated conformal space approach we obtain numerical results for the ground state and the first three excited states which are in excellent agreement with the TBA results. As a special case we can calculate the standard g-function which is the overlap with the ground state and find that our new method is considerably more accurate than the original method employed by Dorey et al.Comment: 21 pages, 6 figure

On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P.Comment: 9 pages, no figure

Scattering of Giant Holes

We study scalar excitations of high spin operators in N=4 super Yang-Mills theory, which are dual to solitons propagating on a long folded string in AdS_3 x S^1. In the spin chain description of the gauge theory, these are associated to holes in the magnon distribution in the sl(2,R) sector. We compute the all-loop hole S-matrix from the asymptotic Bethe ansatz, and expand in leading orders at weak and strong coupling. The worldsheet S-matrix of solitonic excitations on the GKP string is calculated using semiclassical quantization. We find an exact agreement between the gauge theory and string theory results.Comment: 13 pages. v2: minor corrections, references adde

Exact Mesonic Vacua From Matrix Models

We investigate in detail the structure of mesonic vacua of N=1 U(Nc) supersymmetric gauge theory with Nf flavors from the matrix model. We show that the Witten index from the matrix model calculation agrees with a result from field theoretical analysis. We also discuss the relationship between a diagrammatic summation and direct matrix integration with insertion of a variable changing delta function. Using this formalism, we obtain the quantum moduli space and evidence of the Seiberg duality from the matrix models.Comment: 14 pages, 1 figure, typos corrected and note on the quamtum moduli space adde

Exactly solvable model of the 2D electrical double layer

We consider equilibrium statistical mechanics of a simplified model for the ideal conductor electrode in an interface contact with a classical semi-infinite electrolyte, modeled by the two-dimensional Coulomb gas of pointlike $\pm$ unit charges in the stability-against-collapse regime of reduced inverse temperatures $0\le \beta<2$. If there is a potential difference between the bulk interior of the electrolyte and the grounded interface, the electrolyte region close to the interface (known as the electrical double layer) carries some nonzero surface charge density. The model is mappable onto an integrable semi-infinite sine-Gordon theory with Dirichlet boundary conditions. The exact form-factor and boundary state information gained from the mapping provide asymptotic forms of the charge and number density profiles of electrolyte particles at large distances from the interface. The result for the asymptotic behavior of the induced electric potential, related to the charge density via the Poisson equation, confirms the validity of the concept of renormalized charge and the corresponding saturation hypothesis. It is documented on the non-perturbative result for the asymptotic density profile at a strictly nonzero $\beta$ that the Debye-H\"uckel $\beta\to 0$ limit is a delicate issue.Comment: 14 page

Scattering in the PT-symmetric Coulomb potential

Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written in closed analytic form and it is shown that similarly to other ${\cal PT}$-symmetric scattering systems the latter exhibit handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.Comment: Journal of Physics A: Mathematical and Theoretical 42 (2009) to appea

Non-Abelian Meissner Effect in Yang--Mills Theories at Weak Coupling

We present a weak-coupling Yang--Mills model supporting non-Abelian magnetic flux tubes and non-Abelian confined magnetic monopoles. In the dual description the magnetic flux tubes are prototypes of the QCD strings. Dualizing the confined magnetic monopoles we get gluelumps which convert a "QCD string" in the excited state to that in the ground state. Introducing a mass parameter m we discover a phase transition between the Abelian and non-Abelian confinement at a critical value m=m_* of order of Lambda. Underlying dynamics are governed by a Z_N symmetry inherent to the model under consideration. At m>m_* the Z_N symmetry is spontaneously broken, resulting in N degenerate Z_N (Abelian) strings. At m<m_* the Z_N symmetry is restored, the degeneracy is lifted, and the strings become non-Abelian. We calculate tensions of the non-Abelian strings, as well as the decay rates of the metastable strings, at N >> 1.Comment: Reference [45] corrected. Final version, to appear in Phys. Rev.

Adding flavor to Dijkgraaf-Vafa

We study matrix models related via the correspondence of Dijkgraaf and Vafa to supersymmetric gauge theories with matter in the fundamental. As in flavorless examples, measure factors of the matrix integral reproduce information about R-symmetry violation in the field theory. The models, studied previously as models of open strings, exhibit a large-M phase transition as the number of flavors is varied. This is the matrix model's manifestation of the end of asymptotic freedom. Using the relation to a quiver gauge theory, we extract the effective glueball superpotential and Seiberg-Witten curve from the matrix model.Comment: 15 pages, harvmac; improved analysis of the healing of cuts, added calculation of superpotential, improved referencing and notatio

Giant Magnons and Singular Curves

We obtain the giant magnon of Hofman-Maldacena and its dyonic generalisation on R x S^3 < AdS_5 x S^5 from the general elliptic finite-gap solution by degenerating its elliptic spectral curve into a singular curve. This alternate description of giant magnons as finite-gap solutions associated to singular curves is related through a symplectic transformation to their already established description in terms of condensate cuts developed in hep-th/0606145.Comment: 34 pages, 17 figures, minor change in abstrac