58 research outputs found
Analytic continuation of residue currents
Let be a complex manifold and f\colon X\to \C^p a holomorphic mapping
defining a complete intersection. We prove that the iterated Mellin transform
of the residue integral associated to has an analytic continuation to a
neighborhood of the origin in \C^p
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
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
Weighted integral formulas on manifolds
We present a method of finding weighted Koppelman formulas for -forms
on -dimensional complex manifolds which admit a vector bundle of rank
over , such that the diagonal of has a defining
section. We apply the method to \Pn and find weighted Koppelman formulas for
-forms with values in a line bundle over \Pn. As an application, we
look at the cohomology groups of -forms over \Pn with values in
various line bundles, and find explicit solutions to the \dbar-equation in
some of the trivial groups. We also look at cohomology groups of -forms
over \Pn \times \Pm with values in various line bundles. Finally, we apply
our method to developing weighted Koppelman formulas on Stein manifolds.Comment: 25 page
Introduction to Integral Discriminants
The simplest partition function, associated with homogeneous symmetric forms
S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int
e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if
e^{-S} in the integrand is substituted by arbitrary function f(S), i.e.
integral discriminant is a characteristic of the form S itself, and not of the
averaging procedure. The aim of the present paper is to calculate J_{n|r} in a
number of non-Gaussian cases. Using Ward identities -- linear differential
equations, satisfied by integral discriminants -- we calculate J_{2|3},
J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant
appears to be a generalized hypergeometric function. It depends on several
SL(n) invariants of S, with essential singularities controlled by the ordinary
algebraic discriminant of S.Comment: 36 pages, 19 figure
On mass corrections to the decays P \to l^+l^-
We use the Mellin-Barnes representation in order to improve the theoretical
estimate of mass corrections to the width of light pseudoscalar meson decay
into a lepton pair, . The full resummation of the terms
and to the
decay amplitude is performed, where is the lepton mass and
is the characteristic scale of the
form factor. The total effect of mass corrections for
the channel is negligible and for the channel its order
is of a few per cent.Comment: 10 pages, 1 figure; one figure is adde
Crystal Melting and Wall Crossing Phenomena
This paper summarizes recent developments in the theory of
Bogomol'nyi-Prasad-Sommerfield (BPS) state counting and the wall crossing
phenomena, emphasizing in particular the role of the statistical mechanical
model of crystal melting. This paper is divided into two parts, which are
closely related to each other. In the first part, we discuss the statistical
mechanical model of crystal melting counting BPS states. Each of the BPS state
contributing to the BPS index is in one-to-one correspondence with a
configuration of a molten crystal, and the statistical partition function of
the melting crystal gives the BPS partition function. We also show that smooth
geometry of the Calabi-Yau manifold emerges in the thermodynamic limit of the
crystal. This suggests a remarkable interpretation that an atom in the crystal
is a discretization of the classical geometry, giving an important clue as to
the geometry at the Planck scale.In the second part we discuss the wall
crossing phenomena. Wall crossing phenomena states that the BPS index depends
on the value of the moduli of the Calabi-Yau manifold, and jumps along real
codimension one subspaces in the moduli space. We show that by using type
IIA/M-theory duality, we can provide a simple and an intuitive derivation of
the wall crossing phenomena, furthermore clarifying the connection with the
topological string theory. This derivation is consistent with another
derivation from the wall crossing formula, motivated by multi-centered BPS
extremal black holes. We also explain the representation of the wall crossing
phenomena in terms of crystal melting, and the generalization of the counting
problem and the wall crossing to the open BPS invariants.Comment: PhD thesis, 129 pages, 39 figures, comments welcome; v2: typos
corrected, references added, now in IJMPA format; v3: figures correcte
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