51 research outputs found
Multiple Mellin-Barnes Integrals as Periods of Calabi-Yau Manifolds With Several Moduli
We give a representation, in terms of iterated Mellin-Barnes integrals, of
periods on multi-moduli Calabi-Yau manifolds arising in superstring theory.
Using this representation and the theory of multidimensional residues, we
present a method for analytic continuation of the fundamental period in the
form of Horn series.Comment: 18 pages, AMS-tex + 3 postscript figures, to be published in Theor.
Math. Phys. Russi
The analytic structure of 2D Euler flow at short times
Using a very high precision spectral calculation applied to the
incompressible and inviscid flow with initial condition , we find that the width of its analyticity
strip follows a law at short times over eight decades. The
asymptotic equation governing the structure of spatial complex-space
singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113,
761--781) is solved by a high-precision expansion method. Strong numerical
evidence is obtained that singularities have infinite vorticity and lie on a
complex manifold which is constructed explicitly as an envelope of analyticity
disks.Comment: 19 pages, 14 figures, published versio
Residue currents associated with weakly holomorphic functions
We construct Coleff-Herrera products and Bochner-Martinelli type residue
currents associated with a tuple of weakly holomorphic functions, and show
that these currents satisfy basic properties from the (strongly) holomorphic
case, as the transformation law, the Poincar\'e-Lelong formula and the
equivalence of the Coleff-Herrera product and the Bochner-Martinelli type
residue current associated with when defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In
particular, corrected and clarified some things in Section 5 and 6 regarding
products of weakly holomorphic functions and currents, and the definition of
the Bochner-Martinelli type current
Amoebas of complex hypersurfaces in statistical thermodynamics
The amoeba of a complex hypersurface is its image under a logarithmic
projection. A number of properties of algebraic hypersurface amoebas are
carried over to the case of transcendental hypersurfaces. We demonstrate the
potential that amoebas can bring into statistical physics by considering the
problem of energy distribution in a quantum thermodynamic ensemble. The
spectrum of the ensemble is assumed to be
multidimensional; this leads us to the notions of a multidimensional
temperature and a vector of differential thermodynamic forms. Strictly
speaking, in the paper we develop the multidimensional Darwin and Fowler method
and give the description of the domain of admissible average values of energy
for which the thermodynamic limit exists.Comment: 18 pages, 5 figure
Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom
We consider natural complex Hamiltonian systems with degrees of freedom
given by a Hamiltonian function which is a sum of the standard kinetic energy
and a homogeneous polynomial potential of degree . The well known
Morales-Ramis theorem gives the strongest known necessary conditions for the
Liouville integrability of such systems. It states that for each there
exists an explicitly known infinite set \scM_k\subset\Q such that if the
system is integrable, then all eigenvalues of the Hessian matrix V''(\vd)
calculated at a non-zero \vd\in\C^n satisfying V'(\vd)=\vd, belong to
\scM_k. The aim of this paper is, among others, to sharpen this result. Under
certain genericity assumption concerning we prove the following fact. For
each and there exists a finite set \scI_{n,k}\subset\scM_k such that
if the system is integrable, then all eigenvalues of the Hessian matrix
V''(\vd) belong to \scI_{n,k}. We give an algorithm which allows to find
sets \scI_{n,k}. We applied this results for the case and we found
all integrable potentials satisfying the genericity assumption. Among them
several are new and they are integrable in a highly non-trivial way. We found
three potentials for which the additional first integrals are of degree 4 and 6
with respect to the momenta.Comment: 54 pages, 1 figur
Dual conformal constraints and infrared equations from global residue theorems in N=4 SYM theory
Infrared equations and dual conformal constraints arise as consistency
conditions on loop amplitudes in N=4 super Yang-Mills theory. These conditions
are linear relations between leading singularities, which can be computed in
the Grassmannian formulation of N=4 super Yang-Mills theory proposed recently.
Examples for infrared equations have been shown to be implied by global residue
theorems in the Grassmannian picture. Both dual conformal constraints and
infrared equations are mapped explicitly to global residue theorems for
one-loop next-to-maximally-helicity-violating amplitudes. In addition, the
identity relating the BCFW and its parity-conjugated form of tree-level
amplitudes, is shown to emerge from a particular combination of global residue
theorems.Comment: 21 page
Residues and World-Sheet Instantons
We reconsider the question of which Calabi-Yau compactifications of the
heterotic string are stable under world-sheet instanton corrections to the
effective space-time superpotential. For instance, compactifications described
by (0,2) linear sigma models are believed to be stable, suggesting a remarkable
cancellation among the instanton effects in these theories. Here, we show that
this cancellation follows directly from a residue theorem, whose proof relies
only upon the right-moving world-sheet supersymmetries and suitable compactness
properties of the (0,2) linear sigma model. Our residue theorem also extends to
a new class of "half-linear" sigma models. Using these half-linear models, we
show that heterotic compactifications on the quintic hypersurface in CP^4 for
which the gauge bundle pulls back from a bundle on CP^4 are stable. Finally, we
apply similar ideas to compute the superpotential contributions from families
of membrane instantons in M-theory compactifications on manifolds of G_2
holonomy.Comment: 47 page
On the Classification of Residues of the Grassmannian
We study leading singularities of scattering amplitudes which are obtained as
residues of an integral over a Grassmannian manifold. We recursively do the
transformation from twistors to momentum twistors and obtain an iterative
formula for Yangian invariants that involves a succession of dualized twistor
variables. This turns out to be useful in addressing the problem of classifying
the residues of the Grassmannian. The iterative formula leads naturally to new
coordinates on the Grassmannian in terms of which both composite and
non-composite residues appear on an equal footing. We write down residue
theorems in these new variables and classify the independent residues for some
simple examples. These variables also explicitly exhibit the distinct solutions
one expects to find for a given set of vanishing minors from Schubert calculus.Comment: 20 page
Multidimensional residues and their applications
The technique of residues is known for its many applications in different branches of mathematics. Tsikh's book presents a systematic account of residues associated with holomorphic mappings and indicates many applications. The book begins with preliminaries from the theory of analytic sets, together with material from algebraic topology that is necessary for the integration of differential forms over chains. Tsikh then presents a detailed study of residues associated with mappings that preserve dimension (local residues). Local residues are applied to algebraic geometry and to problems connected with the investigation and calculation of double series and integrals. There is also a treatment of residues associated with mappings that reduce dimension--that is, residues of semimeromorphic forms, connected with integration over tubes around nondiscrete analytic sets
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