790 research outputs found
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
A New 2d/4d Duality via Integrability
We prove a duality, recently conjectured in arXiv:1103.5726, which relates
the F-terms of supersymmetric gauge theories defined in two and four dimensions
respectively. The proof proceeds by a saddle point analysis of the
four-dimensional partition function in the Nekrasov-Shatashvili limit. At
special quantized values of the Coulomb branch moduli, the saddle point
condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain
which coincides with the F-term equation of the dual two-dimensional theory.
The on-shell values of the superpotential in the two theories are shown to
coincide in corresponding vacua. We also identify two-dimensional duals for a
large set of quiver gauge theories in four dimensions and generalize our proof
to these cases.Comment: 19 pages, 2 figures, minor corrections and references adde
Does the complex deformation of the Riemann equation exhibit shocks?
The Riemann equation , which describes a one-dimensional
accelerationless perfect fluid, possesses solutions that typically develop
shocks in a finite time. This equation is \cP\cT symmetric. A one-parameter
\cP\cT-invariant complex deformation of this equation,
( real), is solved exactly using the
method of characteristic strips, and it is shown that for real initial
conditions, shocks cannot develop unless is an odd integer.Comment: latex, 8 page
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
Spectral zeta functions of a 1D Schr\"odinger problem
We study the spectral zeta functions associated to the radial Schr\"odinger
problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the
quantum Wronskian equation, we provide results such as closed-form evaluations
for some of the second zeta functions i.e. the sum over the inverse eigenvalues
squared. Also we discuss how our results can be used to derive relationships
and identities involving special functions, using a particular 5F_4
hypergeometric series as an example. Our work is then extended to a class of
related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we
give a simple method for calculating the related spectral zeta functions. This
method has a number of applications including the use of the ODE/IM
correspondence to compute the (vacuum) nonlocal integrals of motion G_n which
appear in an associated integrable quantum field theory.Comment: 15 pages, version
Identification of observables in quantum toboggans
Quantum systems with real energies generated by an apparently non-Hermitian
Hamiltonian may re-acquire the consistent probabilistic interpretation via an
ad hoc metric which specifies the set of observables in the updated Hilbert
space of states. The recipe is extended here to quantum toboggans. In the first
step the tobogganic integration path is rectified and the Schroedinger equation
is given the generalized eigenvalue-problem form. In the second step the
general double-series representation of the eligible metric operators is
derived.Comment: 25 p
From Marginal Deformations to Confinement
We consider type IIB supergravity backgrounds which describe marginal
deformations of the Coulomb branch of N=4 super Yang-Mills theory with SO(4) x
SO(2) global symmetry. Wilson loop calculations indicate that certain
deformations enhance the Coulombic attraction between quarks and anti-quarks at
the UV conformal fixed-point. In the IR region, these deformations can induce a
transition to linear confinement.Comment: 14 pages, 4 figures, minor corrections, comments and references adde
Exact Superpotentials for Theories with Flavors via a Matrix Integral
We extend and test the method of Dijkgraaf and Vafa for computing the
superpotential of N=1 theories to include flavors in the fundamental
representation of the gauge group. This amounts to computing the contribution
to the superpotential from surfaces with one boundary in the matrix integral.
We compute exactly the effective superpotential for the case of gauge group
U(N_c), N_f massive flavor chiral multiplets in the fundamental and one massive
chiral multiplet in the adjoint, together with a Yukawa coupling. We compare up
to sixth-order with the result obtained by standard field theory techniques in
the already non trivial case of N_c=2 and N_f=1. The agreement is perfect.Comment: 7 pages, v2: typos involving signs fixed; v3: version to appear in
Phys.Rev.
Scattering in the PT-symmetric Coulomb potential
Scattering on the -symmetric Coulomb potential is studied along a
U-shaped trajectory circumventing the origin in the complex plane from
below. This trajectory reflects 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 -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
Effect of Wavefunction Renormalisation in N-Flavour Qed3 at Finite Temperature
A recent study of dynamical chiral symmetry breaking in N-flavour QED at
finite temperature is extended to include the effect of fermion wavefunction
renormalisation in the Schwinger-Dyson equations. The simple ``zero-frequency''
truncation previously used is found to lead to unphysical results, especially
as . A modified set of equations is proposed, whose solutions behave
in a way which is qualitatively similar to the solutions of Pennington et
al. [5-8] who have made extensive studies of the effect of wavefunction
renormalisation in this context, and who concluded that there was no critical
(at T=0) above which chiral symmetry was restored. In contrast, we find
that our modified equations predict a critical at , and an
phase diagram very similar to the earlier study neglecting wavefunction
renormalisation. The reason for the difference is traced to the different
infrared behaviour of the vacuum polarisation at and at .Comment: 17 pages + 13 figures (available upon request), Oxford preprint
OUTP-93-30P, IFUNAM preprint FT94-39, LaTe
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