155 research outputs found
About multiplicities and applications to Bezout numbers
Let denote a local Noetherian ring and
an ideal such that for a
finitely generated -module . Let \au = a_1,\ldots,a_d denote a system
of parameters of such that for . It follows that \chi := e_0(\au;M)
- c \cdot e_0(\mathfrak{q};M) \geq 0, where .
The main results of the report are a discussion when resp. to
describe the value of in some particular cases. Applications concern
results on the multiplicity e_0(\au;M) and applications to Bezout numbers.Comment: 11 pages, to appear Springer INdAM-Series, Vol. 20 (2017
Matrix String Theory and its Moduli Space
The correspondence between Matrix String Theory in the strong coupling limit
and IIA superstring theory can be shown by means of the instanton solutions of
the former. We construct the general instanton solutions of Matrix String
Theory which interpolate between given initial and final string configurations.
Each instanton is characterized by a Riemann surface of genus h with n
punctures, which is realized as a plane curve. We study the moduli space of
such plane curves and find out that, at finite N, it is a discretized version
of the moduli space of Riemann surfaces: instead of 3h-3+n its complex
dimensions are 2h-3+n, the remaining h dimensions being discrete. It turns out
that as tends to infinity, these discrete dimensions become continuous, and
one recovers the full moduli space of string interaction theory.Comment: 30 pages, LaTeX, JHEP.cls class file, minor correction
Conifold geometries, topological strings and multi-matrix models
We study open B-model representing D-branes on 2-cycles of local Calabi--Yau
geometries. To this end we work out a reduction technique linking D-branes
partition functions and multi-matrix models in the case of conifold geometries
so that the matrix potential is related to the complex moduli of the conifold.
We study the geometric engineering of the multi-matrix models and focus on
two-matrix models with bilinear couplings. We show how to solve this models in
an exact way, without resorting to the customary saddle point/large N
approximation. The method consists of solving the quantum equations of motion
and using the flow equations of the underlying integrable hierarchy to derive
explicit expressions for correlators. Finally we show how to incorporate in
this formalism the description of several group of D-branes wrapped around
different cycles.Comment: 35 pages, 5.3 and 6 revise
Fibonacci numbers and self-dual lattice structures for plane branches
Consider a plane branch, that is, an irreducible germ of curve on a smooth
complex analytic surface. We define its blow-up complexity as the number of
blow-ups of points necessary to achieve its minimal embedded resolution. We
show that there are topological types of blow-up complexity ,
where is the -th Fibonacci number. We introduce
complexity-preserving operations on topological types which increase the
multiplicity and we deduce that the maximal multiplicity for a plane branch of
blow-up complexity is . It is achieved by exactly two topological
types, one of them being distinguished as the only type which maximizes the
Milnor number. We show moreover that there exists a natural partial order
relation on the set of topological types of plane branches of blow-up
complexity , making this set a distributive lattice, that is, any two of its
elements admit an infimum and a supremum, each one of these operations beeing
distributive relative to the second one. We prove that this lattice admits a
unique order-inverting bijection. As this bijection is involutive, it defines a
duality for topological types of plane branches. The type which maximizes the
Milnor number is also the maximal element of this lattice and its dual is the
unique type with minimal Milnor number. There are self-dual
topological types of blow-up complexity . Our proofs are done by encoding
the topological types by the associated Enriques diagrams.Comment: 21 pages, 16 page
Quantum symmetric pairs and representations of double affine Hecke algebras of type
We build representations of the affine and double affine braid groups and
Hecke algebras of type , based upon the theory of quantum symmetric
pairs . In the case , our constructions provide a
quantization of the representations constructed by Etingof, Freund and Ma in
arXiv:0801.1530, and also a type generalization of the results in
arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic
Cohomology of bundles on homological Hopf manifold
We discuss the properties of complex manifolds having rational homology of
including those constructed by Hopf, Kodaira and
Brieskorn-van de Ven. We extend certain previously known vanishing properties
of cohomology of bundles on such manifolds.As an application we consider
degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex
variables and Complex Geometry. Xiamen. Chin
Verdier specialization via weak factorization
Let X in V be a closed embedding, with V - X nonsingular. We define a
constructible function on X, agreeing with Verdier's specialization of the
constant function 1 when X is the zero-locus of a function on V. Our definition
is given in terms of an embedded resolution of X; the independence on the
choice of resolution is obtained as a consequence of the weak factorization
theorem of Abramovich et al. The main property of the specialization function
is a compatibility with the specialization of the Chern class of the complement
V-X. With the definition adopted here, this is an easy consequence of standard
intersection theory. It recovers Verdier's result when X is the zero-locus of a
function on V. Our definition has a straightforward counterpart in a motivic
group. The specialization function and the corresponding Chern class and
motivic aspect all have natural `monodromy' decompositions, for for any X in V
as above. The definition also yields an expression for Kai Behrend's
constructible function when applied to (the singularity subscheme of) the
zero-locus of a function on V.Comment: Minor revision. To appear in Arkiv f\"or Matemati
Modules of Abelian integrals and Picard-Fuchs systems
We give a simple proof of an isomorphism between the two
-modules: the module of relative cohomologies and the module of Abelian integrals corresponding to a regular at
infinity polynomial in two variables. Using this isomorphism, we prove
existence and deduce some properties of the corresponding Picard-Fuchs system.Comment: A separate section discusses Fuchsian properties of the Picard-Fuchs
system, Morse condition exterminated. Few errors were correcte
Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries
We briefly review the hierarchy for the hyper-K\"ahler equations and define a
notion of symmetry for solutions of this hierarchy. A four-dimensional
hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy
with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden
symmetry if it admits a certain Killing spinor. We show that if the hidden
symmetry is tri-holomorphic, then this is equivalent to requiring symmetry
along a higher time and the hidden symmetry determines a `twistor group' action
as introduced by Bielawski \cite{B00}. This leads to a construction for the
solution to the hierarchy in terms of linear equations and variants of the
generalised Legendre transform for the hyper-K\"ahler metric itself given by
Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of
hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These
metrics are in this sense analogous to the 'finite gap' solutions in soliton
theory. Finally we extend the concept of a hierarchy from that of \cite{DM00}
for the four-dimensional hyper-K\"ahler equations to a generalisation of the
conformal anti-self-duality equations and briefly discuss hidden symmetries for
these equations.Comment: Final version. To appear in the August 2003 special issue of JMP on
`Integrability, Topological Solitons, and Beyond
Finite Temperature Time-Dependent Effective Theory for the Phase Field in two-dimensional d-wave Neutral Superconductor
We derive finite temperature time-dependent effective actions for the phase
of the pairing field, which are appropriate for a 2D electron system with both
non-retarded d- and s-wave attraction. As for s-wave pairing the d-wave
effective action contains terms with Landau damping, but their structure
appears to be different from the s-wave case due to the fact that the Landau
damping is determined by the quasiparticle group velocity v_{g}, which for
d-wave pairing does not have the same direction as the non-interacting Fermi
velocity v_{F}. We show that for d-wave pairing the Landau term has a linear
low temperature dependence and in contrast to the s-wave case are important for
all finite temperatures. A possible experimental observation of the phase
excitations is discussed.Comment: 23 pages, RevTeX4, 10 EPS figures; final version to appear in PR
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