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1.
Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This
function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc., and it can also be interpreted
as correlation functions on integrable -modules of level one. Such -correlation functions at higher levels were then calculated by Cheng and Wang.
In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of of type B, C, D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B, C, D and some fermionic type q-identities. 相似文献
2.
The classical linking number lk is defined when link components are zero homologous. In [15] we constructed the affine linking
invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the
study of causality in Lorentzian manifolds.
Let M
m
be a spacelike Cauchy surface in a globally hyperbolic space-time (X
m+1, g). The spherical cotangent bundle ST
*
M is identified with the space of all null geodesics in (X,g). Hence the set of null geodesics passing through a point gives an embedded (m−1)-sphere in called the sky of x. Low observed that if the link is nontrivial, then are causally related. This observation yielded a problem (communicated by R. Penrose) on the V. I. Arnold problem list [3,4]
which is basically to study the relation between causality and linking. Our paper is motivated by this question.
The spheres are isotopic to the fibers of They are nonzero homologous and the classical linking number lk is undefined when M is closed, while alk is well defined. Moreover, alk if M is not an odd-dimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold
dangerous tangencies. If (X,g) is such that alk takes values in and g is conformal to that has all the timelike sectional curvatures nonnegative, then are causally related if and only if alk . We prove that if alk takes values in and y is in the causal future of x, then alk is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x.
We show that x,y in a nonrefocussing (X, g) are causally unrelated if and only if can be deformed to a pair of S
m-1-fibers of by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact. 相似文献
3.
4.
For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f
0, f
1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f. 相似文献
5.
S. Lievens N. I. Stoilov J. Van der Jeugt 《Communications in Mathematical Physics》2008,281(3):805-826
It is known that the defining relations of the orthosymplectic Lie superalgebra are equivalent to the defining (triple) relations of n pairs of paraboson operators . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of . Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the
present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra of plays a crucial role. Our results also lead to character formulas for these infinite-dimensional representations. Furthermore, by considering the branching , we find explicit infinite-dimensional unitary irreducible lowest weight representations of and their characters.
NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity
Attraction Poles Programme (Belgian State – Belgian Science Policy).
An erratum to this article can be found at 相似文献
6.
This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of
standing wave for the system:
where and . Firstly, by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds
of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution to the Cauchy problem for
(DS) provided . Secondly, by using the scaling argument, we show how small the initial data are for the global solutions to exist. Finally,
we prove the strong instability of the standing waves with finite time blow up for any ω > 0 by combining the former results.
This work is supported by Sichuan Youth Science and Technology Foundation(07ZQ026-009) and The Institute of Mathematical Sciences
at The Chinese University of Hong Kong. 相似文献
7.
For a (co)monad T
l
on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z
*. Our construction is built on a distributive law of T
l
with a second (co)monad T
r
on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation
that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called ×
R
-Hopf algebras, is introduced. In the particular example when T is a module coring of a ×
R
-Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z
* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T
S
is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild
and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the
group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies
of a groupoid. The latter extends results of Burghelea on cyclic homology of groups. 相似文献
8.
We continue our study of the collision of two solitons for the subcritical generalized KdV equations
Solitons are solutions of the type where c
0 > 0. In [21], mainly devoted to the case f (u) = u
4, we have introduced a new framework to understand the collision of two solitons , for (0.1) in the case (or equivalently, ). In this paper, we consider the case of a general nonlinearity f (u) for which , are nonlinearly stable. In particular, since f is general and c
1 can be large, the results are not perturbations of the ones for the power case in [21].
First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation
term whose size is explicitly controlled from above: after the collision, , where is close to c
j
(j = 1, 2). Then, we exhibit new exceptional solutions similar to multi-soliton solutions: for all , there exists a solution such that
where (j = 1, 2) and converges to 0 in a neighborhood of the solitons as .
The analysis is split in two distinct parts. For the interaction region, we extend the algebraic tools developed in [21] for
the power case, by expanding f (u) as a sum of powers plus a perturbation term. To study the solutions in large time, we rely on previous tools on asymptotic
stability in [17,22] and [18], refined in [19,20].
This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN). 相似文献
9.
We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, , the two-dimensional extension, , of the gas cloud in the trap, the thickness, of the trap, and the scattering length a of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premises that the confining energy, ~ 1/h
2, is much larger than the internal energy per particle, and a/h→ 0, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length a
2D = hexp(−(const.)h/a). In the parameter region where , with the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter ~ Na/h. If the coupling parameter is and thus independent of a. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded. 相似文献
10.
David Gérard-Varet 《Communications in Mathematical Physics》2009,286(1):81-110
We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous
random process, with typical size . In the parent paper [8], we derived a homogenized boundary condition of Navier type as . We show here that for a large class of boundaries, this Navier condition provides a approximation in L
2, instead of for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this
boundary layer are deduced from a central limit theorem for dependent variables. 相似文献
11.
Liang Kong 《Communications in Mathematical Physics》2008,283(1):25-92
Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over
a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The
Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of
the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for
1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field
algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case. 相似文献
12.
A Negative Mass Theorem for the 2-Torus 总被引:1,自引:1,他引:0
K. Okikiolu 《Communications in Mathematical Physics》2008,284(3):775-802
Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δ
g
. We define the Robin mass m(p) at the point to be the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off, and we define trace . This regularized trace can also be obtained by regularization of the spectral zeta function and is hence a spectral invariant
which heuristically measures the total wavelength of the surface.We define the Δ-mass of (M, g) to equal , where is the Laplacian on the round sphere of area A. This scale invariant quantity is a non-trivial analog for closed surfaces of the ADM mass for higher dimensional asymptotically
flat manifolds.In this paper we show that in each conformal class for the 2-torus, there exists a metric with negative Δ-mass. From this it follows that the minimum of the Δ-mass on is negative and attained by some metric . For this minimizing metric g, one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and an Onofri-type inequality.We remark that if the flat
metric in is sufficiently long and thin then the minimizing metric g is non-flat. The proof of our result depends on analyzing the ordinary differential equation which is equivalent to h′′ = 1 − 1/h. The solutions are periodic and we need to establish quite delicate, asymptotically sharp inequalities relating the period
to the maximum value.
The author was supported by the National Science Foundation #DMS-0302647. 相似文献
13.
Jonathan Weitsman 《Communications in Mathematical Physics》2008,277(1):101-125
We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures
are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum
field theory. We give three concrete examples of our construction. The first example is a family of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family of measures on a space of maps from to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of -valued three-forms on M.
We show that these measures are positive, and that the measures are Borel probability measures. As an application we show that formulas arising from expectations in the measures reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar
computation for the measures , where M is a homology three-sphere, will yield the Casson invariant of M.
Dedicated to the memory of Raoul Bott
Supported in part by NSF grant DMS 04/05670. 相似文献
14.
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation 总被引:2,自引:0,他引:2
We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first
prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve and the set of non-characteristic points. We show that I
0 is open and that T(x) is C
1 on I
0. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for .
This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185. 相似文献
15.
We introduce a newfamily of C
2-cofinite N = 1 vertex operator superalgebras , m ≥ 1, which are natural super analogs of the triplet vertex algebra family , p ≥ 2, important in logarithmic conformal field theory. We classify irreducible -modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible characters. Finally, we contemplate possible connections between the category of -modules and the category of modules for the quantum group , , by focusing primarily on properties of characters and the Zhu’s algebra . This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008).
The second author was partially supported by NSF grant DMS-0802962. 相似文献
16.
In this paper we introduce Baxter integral -operators for finite-dimensional Lie algebras and . Whittaker functions corresponding to these algebras are eigenfunctions of the -operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the
manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions,
which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental
integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump
and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter -operator acting on Whittaker functions with local Archimedean L-factors. The Baxter -operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter
operator in the spherical Hecke algebra , K being a maximal compact subgroup of G. Finally we stress an analogy between -operators and certain elements of the non-Archimedean Hecke algebra . 相似文献
17.
We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra
form which we can then ‘quantise’ by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor
space , compactified Minkowski space and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local
form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the
formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we
show that the pull-back of the tautological bundle on pulls back to the basic instanton on and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S
4 as the pull-back of the tautological bundle on our θ-deformed . We likewise quantise the fibration and use it to construct the bundle on θ-deformed that maps over under the transform to the θ-deformed instanton.
The work was mainly completed while S.M. was visiting July-December 2006 at the Isaac Newton Institute, Cambridge, which both
authors thank for support. 相似文献
18.
19.
Manjunath Krishnapur 《Journal of statistical physics》2006,124(6):1399-1423
We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as . For the planar Gaussian analytic function, , we show that this probability is asymptotic to . For the hyperbolic Gaussian analytic functions, , we show that this probability decays like .In the planar case, we also consider the problem posed by Mikhail Sodin2 on moderate and very large deviations in a disk of radius r, as . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479. 相似文献
20.
Simple Systems with Anomalous Dissipation and Energy Cascade 总被引:1,自引:0,他引:1
Jonathan C. Mattingly Toufic Suidan Eric Vanden-Eijnden 《Communications in Mathematical Physics》2007,276(1):189-220
We analyze a class of dynamical systems of the type where f
n
(t) is a forcing term with only for and the coupling coefficients c
n
satisfy a condition ensuring the formal conservation of energy . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined)
and, as a result, may admit unique (statistical) steady states when the forcing term f
n
(t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific
choices of the coupling coefficients c
n
. The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely scales as as n→∞. Here the exponents α depend on the coupling coefficients c
n
and denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions
of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence,
these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in
that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable. 相似文献