163 research outputs found
Exact results in a N=2 superconformal gauge theory at strong coupling
We consider the N = 2 SYM theory with gauge group SU(N) and a matter content consisting of one multiplet in the symmetric and one in the anti-symmetric representation. This conformal theory admits a large-N ’t Hooft expansion and is dual to a particular orientifold of AdS5 × S5. We analyze this gauge theory relying on the matrix model provided by localization à la Pestun. Even though this matrix model has very nontrivial interactions, by exploiting the full Lie algebra approach to the matrix integration, we show that a large class of observables can be expressed in a closed form in terms of an infinite matrix depending on the ’t Hooft coupling . These exact expressions can be used to generate the perturbative expansions at high orders in a very efficient way, and also to study analytically the leading behavior at strong coupling. We successfully compare these predictions to a direct Monte Carlo numerical evaluation of the matrix integral and to the Padé resummations derived from very long perturbative series, that turn out to be
extremely stable beyond the convergence disk || < of the latter
Divergent Perturbation Series
Various perturbation series are factorially divergent. The behavior of their
high-order terms can be found by Lipatov's method, according to which they are
determined by the saddle-point configurations (instantons) of appropriate
functional integrals. When the Lipatov asymptotics is known and several lowest
order terms of the perturbation series are found by direct calculation of
diagrams, one can gain insight into the behavior of the remaining terms of the
series. Summing it, one can solve (in a certain approximation) various
strong-coupling problems. This approach is demonstrated by determining the
Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling
constants. An overview of the mathematical theory of divergent series is
presented, and interpretation of perturbation series is discussed. Explicit
derivations of the Lipatov asymptotic forms are presented for some basic
problems in theoretical physics. A solution is proposed to the problem of
renormalon contributions, which hampered progress in this field in the late
1970s. Practical schemes for summation of perturbation series are described for
a coupling constant of order unity and in the strong-coupling limit. An
interpretation of the Borel integral is given for 'non-Borel-summable' series.
High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD
Test of Universality in the Ising Spin Glass Using High Temperature Graph Expansion
We calculate high-temperature graph expansions for the Ising spin glass model
with 4 symmetric random distribution functions for its nearest neighbor
interaction constants J_{ij}. Series for the Edwards-Anderson susceptibility
\chi_EA are obtained to order 13 in the expansion variable (J/(k_B T))^2 for
the general d-dimensional hyper-cubic lattice, where the parameter J determines
the width of the distributions. We explain in detail how the expansions are
calculated. The analysis, using the Dlog-Pad\'e approximation and the
techniques known as M1 and M2, leads to estimates for the critical threshold
(J/(k_B T_c))^2 and for the critical exponent \gamma in dimensions 4, 5, 7 and
8 for all the distribution functions. In each dimension the values for \gamma
agree, within their uncertainty margins, with a common value for the different
distributions, thus confirming universality.Comment: 13 figure
Nonlinear Analysis and Optimization with Applications
Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world
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