422 research outputs found
Matrix models and stochastic growth in Donaldson-Thomas theory
We show that the partition functions which enumerate Donaldson-Thomas
invariants of local toric Calabi-Yau threefolds without compact divisors can be
expressed in terms of specializations of the Schur measure. We also discuss the
relevance of the Hall-Littlewood and Jack measures in the context of BPS state
counting and study the partition functions at arbitrary points of the Kaehler
moduli space. This rewriting in terms of symmetric functions leads to a unitary
one-matrix model representation for Donaldson-Thomas theory. We describe
explicitly how this result is related to the unitary matrix model description
of Chern-Simons gauge theory. This representation is used to show that the
generating functions for Donaldson-Thomas invariants are related to
tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The
matrix model also leads to an interpretation of Donaldson-Thomas theory in
terms of non-intersecting paths in the lock-step model of vicious walkers. We
further show that these generating functions can be interpreted as
normalization constants of a corner growth/last-passage stochastic model.Comment: 31 pages; v2: comments and references added; v3: presentation
improved, comments added; final version to appear in Journal of Mathematical
Physic
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
We show that for a class of two-loop diagrams, the on-shell part of the
integration-by-parts (IBP) relations correspond to exact meromorphic one-forms
on algebraic curves. Since it is easy to find such exact meromorphic one-forms
from algebraic geometry, this idea provides a new highly efficient algorithm
for integral reduction. We demonstrate the power of this method via several
complicated two-loop diagrams with internal massive legs. No explicit elliptic
or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde
Multiple reflection expansion and heat kernel coefficients
We propose the multiple reflection expansion as a tool for the calculation of
heat kernel coefficients. As an example, we give the coefficients for a sphere
as a finite sum over reflections, obtaining as a byproduct a relation between
the coefficients for Dirichlet and Neumann boundary conditions. Further, we
calculate the heat kernel coefficients for the most general matching conditions
on the surface of a sphere, including those cases corresponding to the presence
of delta and delta prime background potentials. In the latter case, the
multiple reflection expansion is shown to be non-convergent.Comment: 21 pages, corrected for some misprint
Systems Structure and Control
The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc
12 loops and triple wrapping in ABJM theory from integrability
Adapting a method recently proposed by C. Marboe and D. Volin for =4 super-Yang-Mills, we develop an algorithm for a systematic weak coupling
expansion of the spectrum of anomalous dimensions in the -like sector of
planar =6 super-Chern-Simons. The method relies on the Quantum
Spectral Curve formulation of the problem and the expansion is written in terms
of the interpolating function , with coefficients expressible as
combinations of Euler-Zagier sums with alternating signs. We present explicit
results up to 12 loops (six nontrivial orders) for various twist L=1 and L=2
operators, corresponding to triple and double wrapping terms, respectively,
which are beyond the reach of the Asymptotic Bethe Ansatz as well as
L\"uscher's corrections. The algorithm works for generic values of L and S and
in principle can be used to compute arbitrary orders of the weak coupling
expansion. For the simplest operator with L=1 and spin S=1, the Pad\'e
extrapolation of the 12-loop result nicely agrees with the available
Thermodynamic Bethe Ansatz data in a relatively wide range of values of the
coupling. A Mathematica notebook with a selection of results is attached.Comment: 31 pages, 1 figure. A Mathematica notebook with a selection of
results is attached (please download the compressed file "Results.nb" listed
under "Other formats"). v2: typos corrected; more precise checks of the
results; an earlier incorrect version of the figure was replaced. Published
in JHE
Quantum Systems with Hidden Symmetry. Interbasis Expansions
This monograph is the English version of the book "Quantum systems with
hidden symmetry. Interbasis expansions" published in 2006 by the publishing
house FIZMATLIT (Moscow) in Russian. When compiling this version of the book,
typos and inaccuracies noted since the release of the Russian edition have been
corrected
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