294,994 research outputs found
ADE Spectral Networks
We introduce a new perspective and a generalization of spectral networks for
4d theories of class associated to Lie algebras
, , , and
. Spectral networks directly compute the BPS spectra of 2d
theories on surface defects coupled to the 4d theories. A Lie algebraic
interpretation of these spectra emerges naturally from our construction,
leading to a new description of 2d-4d wall-crossing phenomena. Our construction
also provides an efficient framework for the study of BPS spectra of the 4d
theories. In addition, we consider novel types of surface defects associated
with minuscule representations of .Comment: 68 pages plus appendices; visit
http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that
generates spectral networks; v2: version published in JHEP plus minor
correction
On evaluation of the Heun functions
In the paper we deal with the Heun functions --- solutions of the Heun
equation, which is the most general Fuchsian equation of second order with four
regular singular points. Despite the increasing interest to the equation and
numerous applications of the functions in a wide variety of physical problems,
it is only Maple amidst known software packages which is able to evaluate the
Heun functions numerically. But the Maple routine is known to be imperfect:
even at regular points it may return infinities or end up with no result.
Improving the situation is difficult because the code is not publicly
available. The purpose of the work is to suggest and develop alternative
algorithms for numerical evaluation of the Heun functions. A procedure based on
power series expansions and analytic continuation is suggested which allows us
to avoid numerical integration of the differential equation and to achieve
reasonable efficiency and accuracy. Results of numerical tests are given
Cycle expansions for intermittent maps
In a generic dynamical system chaos and regular motion coexist side by side,
in different parts of the phase space. The border between these, where
trajectories are neither unstable nor stable but of marginal stability,
manifests itself through intermittency, dynamics where long periods of nearly
regular motions are interrupted by irregular chaotic bursts. We discuss the
Perron-Frobenius operator formalism for such systems, and show by means of a
1-dimensional intermittent map that intermittency induces branch cuts in
dynamical zeta functions. Marginality leads to long-time dynamical
correlations, in contrast to the exponentially fast decorrelations of purely
chaotic dynamics. We apply the periodic orbit theory to quantitative
characterization of the associated power-law decays.Comment: 22 pages, 5 figure
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
We address the nonperturbative structure of topological strings and c=1
matrix models, focusing on understanding the nature of instanton effects
alongside with exploring their relation to the large-order behavior of the 1/N
expansion. We consider the Gaussian, Penner and Chern-Simons matrix models,
together with their holographic duals, the c=1 minimal string at self-dual
radius and topological string theory on the resolved conifold. We employ Borel
analysis to obtain the exact all-loop multi-instanton corrections to the free
energies of the aforementioned models, and show that the leading poles in the
Borel plane control the large-order behavior of perturbation theory. We
understand the nonperturbative effects in terms of the Schwinger effect and
provide a semiclassical picture in terms of eigenvalue tunneling between
critical points of the multi-sheeted matrix model effective potentials. In
particular, we relate instantons to Stokes phenomena via a hyperasymptotic
analysis, providing a smoothing of the nonperturbative ambiguity. Our
predictions for the multi-instanton expansions are confirmed within the
trans-series set-up, which in the double-scaling limit describes
nonperturbative corrections to the Toda equation. Finally, we provide a
spacetime realization of our nonperturbative corrections in terms of toric
D-brane instantons which, in the double-scaling limit, precisely match
D-instanton contributions to c=1 minimal strings.Comment: 71 pages, 14 figures, JHEP3.cls; v2: added refs, minor change
Quasi locality of the GGE in interacting-to-free quenches in relativistic field theories
We study the quench dynamics in continuous relativistic quantum field theory,
more specifically the locality properties of the large time stationary state.
After a quantum quench in a one-dimensional integrable model, the expectation
values of local observables are expected to relax to a Generalized Gibbs
Ensemble (GGE), constructed out of the conserved charges of the model.
Quenching to a free bosonic theory, it has been shown that the system indeed
relaxes to a GGE described by the momentum mode occupation numbers. We first
address the question whether the latter can be written directly in terms of
local charges and we find that, in contrast to the lattice case, this is not
possible in continuous field theories. We then investigate the less stringent
requirement of the existence of a sequence of truncated local GGEs that
converges to the correct steady state, in the sense of the expectation values
of the local observables. While we show that such a sequence indeed exists, in
order to unequivocally determine the so-defined GGE, we find that information
about the expectation value of the recently discovered quasi-local charges is
in the end necessary, the latter being the suitable generalization of the local
charges while passing from the lattice to the continuum. Lastly, we study the
locality properties of the GGE and show that the latter is completely
determined by the knowledge of the expectation value of a countable set of
suitably defined quasi-local charges
Two-particle irreducible effective actions versus resummation: analytic properties and self-consistency
Approximations based on two-particle irreducible (2PI) effective actions
(also known as -derivable, Cornwall-Jackiw-Tomboulis or Luttinger-Ward
functionals depending on context) have been widely used in condensed matter and
non-equilibrium quantum/statistical field theory because this formalism gives a
robust, self-consistent, non-perturbative and systematically improvable
approach which avoids problems with secular time evolution. The strengths of
2PI approximations are often described in terms of a selective resummation of
Feynman diagrams to infinite order. However, the Feynman diagram series is
asymptotic and summation is at best a dangerous procedure. Here we show that,
at least in the context of a toy model where exact results are available, the
true strength of 2PI approximations derives from their self-consistency rather
than any resummation. This self-consistency allows truncated 2PI approximations
to capture the branch points of physical amplitudes where adjustments of
coupling constants can trigger an instability of the vacuum. This, in effect,
turns Dyson's argument for the failure of perturbation theory on its head. As a
result we find that 2PI approximations perform better than Pad\'e approximation
and are competitive with Borel-Pad\'e resummation. Finally, we introduce a
hybrid 2PI-Pad\'e method.Comment: Version accepted for publication in Nuclear Physics B. 31 pages, 16
figures. Uses feynm
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