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1.
New Frobenius structures on Hurwitz spaces are found. A Hurwitz space is considered as a real manifold; therefore the number of coordinates is twice as large as the number of coordinates on Hurwitz Frobenius manifolds of Dubrovin. Simple branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed Frobenius manifolds. Corresponding solutions to WDVV equations and G-functions are obtained. 相似文献
2.
We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi-Yau manifolds. We introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers [GN1]-[GN6]. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi-Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi-Yau manifolds. Our papers [GN1]-[GN6] and [N3]-[N14] give hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi-Yau manifolds. 相似文献
3.
Roberto Peirone 《Communications in Mathematical Physics》1999,207(1):67-80
I prove that in (sufficiently small) tubular ρ$-neighborhoods of a given C
3 manifold of codimension 1, any two points can be connected by a billiard trajectory, and that in addition there exists such
a trajectory having at most collision points, for a suitable H>0, provided the manifold is of class C
3.
Received: 19 June 1998 / Accepted: 7 April 1999 相似文献
4.
For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl
2 Witten–Reshetikhin–Turaev invariant, Z
K
, at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections
in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to
have an asymptotic expansion in powers of K
−1 which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z
K
at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured
Jones polynomial of a torus knot is obtained, providing a trivial
connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion
of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S
3. The possibility of generalising such results is also discussed.
Received: 26 October 1998 / Accepted: 1 March 1999 相似文献
5.
Oussama Hijazi Sebastián Montiel Xiao Zhang 《Communications in Mathematical Physics》2001,221(2):255-265
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues
of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized
by the existence of real Killing spinors and the minimality of the boundary.
Received: 22 August 2000 / Accepted: 15 March 2001 相似文献
6.
Pedro Freitas 《Communications in Mathematical Physics》2001,217(2):375-382
We consider the problem of minimizing the eigenvalues of the Schr?dinger operator H=−Δ+αF(κ) (α>0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ0.
In the one-dimensional case our results imply in particular that for F(κ)=κ2 the constant potential fails to minimize the principal eigenvalue for α>αc=μ1/(4κ0
2), where μ1 is the first nonzero eigenvalue of −Δ. This complements a result by Exner, Harrell and Loss, showing that the critical value
where the constant potential stops being a minimizer for a class of Schr?dinger operators penalized by curvature is given
by α
c
. Furthermore, we show that the value of μ1/4 remains the infimum for all α >α
c
. Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential.
In higher dimensions we prove a (weak) local version of these results for a general class of potentials F(κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding
eigenvalues of the Laplace–Beltrami operator and is never attained.
Received: 17 July 2000 / Accepted: 11 October 2000 相似文献
7.
N. P. Landsman 《Communications in Mathematical Physics》2001,222(1):97-16
It is well known that a measured groupoid G defines a von Neumann algebra W
*(G), and that a Lie groupoid G canonically defines both a C
*-algebra C
*(G) and a Poisson manifold A
*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C
*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects.
Subsequently, we show that the maps G↦W
*(G), G↦C
*(G), and G↦A
*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence.
Received: 6 December 2000 / Accepted: 19 April 2001 相似文献
8.
George Thompson 《Communications in Mathematical Physics》2000,212(3):649-652
The group SL(2) acts on the space of cohomology groups of any hyper-K?hler manifold X. The χ
y
genus of a hyper-K?hler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ
y
genus.
Received: 3 December 1999 / Accepted: 30 January 2000 相似文献
9.
We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ
n
. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.
The new examples include the instanton algebra and the NC-4-spheres S
4
θ. We construct the noncommutative algebras ?=C
∞ (S
4
θ) of functions on NC-spheres as solutions to the vanishing, ch
j
(e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M
4 (?), e
2=e, e=e
*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the
corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S
3
θ distinct from quantum group deformations SU
q
(2) of SU (2).
We then construct the noncommutative geometry of S
θ
4 as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g
μν on S
4 whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,
where <␣> is the projection on the commutant of 4 × 4 matrices.
We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S
θ
4 so that the previous equation continues to hold without any change.
Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r≥ 2 admits isospectral deformations to noncommutative geometries.
Received: 5 December 2000 / Accepted: 8 March 2001 相似文献
10.
Atsushi Nakayashiki 《Communications in Mathematical Physics》2000,212(1):29-61
The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type ANу(1) is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix elements of the composition of intertwining operators are explicitly determined as rational functions up to an overall scalar function. The integral formula for the trace is presented. 相似文献
11.
We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms | of "Clifford type"-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G · \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K · \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a |: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra. 相似文献
12.
S. A. Merkulov 《Communications in Mathematical Physics》2000,209(1):13-27
It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold X in a Calabi–Yau manifold Y within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of X. Reinterpreting the embedding data X⊂Y within the mathematical framework of the Batalin–Vilkovisky quantization, we find a natural deformation problem which extends
the above moduli space to the full de Rham cohomology group of X.
Received: 29 June 1998 / Accepted: 7 June 1999 相似文献
13.
On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement
the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik–Zamolodchikov
connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case,
the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented;
the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on
non-simply connected groups and to the classification of boundary conditions in such theories.
Received: 11 May 1998 / Accepted: 17 April 1999 相似文献
14.
15.
Rainer Verch 《Communications in Mathematical Physics》2001,223(2):261-288
A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic
spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory
is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates
on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of
a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski
spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes
the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory,
on each spacetime, are trivial.
Received: 1 March 2001 / Accepted: 28 May 2001 相似文献
16.
Various fluid mechanical systems enjoy a hidden, higher-dimensional dynamical Poincaré symmetry, which arises owing to their
descent from a Nambu–Goto action. Also, for the same reason, there are equivalence transformations between different models.
These interconnections, summarized by the diagram below, are discussed in our paper.
Received: 3 February 1999 / Accepted: 9 April 1999 相似文献
17.
Yann Brenier 《Communications in Mathematical Physics》2000,212(1):93-104
A caricature of collisionless plasma involving 2N particles of opposite charge is introduced. The N first particles are called "ions" and don't move. The N other particles are called "electrons". At each time, there is a one-to-one matching between electrons and ions and each pair is linked by a "spring" so that each electron oscillates with fixed frequency )у. The essential point is that the matching between electrons and ions is updated at every discrete time n, n˸,1,2,..., so that the total potential energy of the system stays minimal. This leads to a non trivial interaction which turns out to be a caricature of Coulomb interaction. It is proven that, provided the N ions are equally spaced in a bounded domain D and ), and Nу tend to zero at appropriate rates, the electrons behave as the fluid parcels of an incompressible inviscid liquid moving inside D according to the Euler equations. Our proof relies on a result of P. Lax on the approximation of volume-preserving transformations by permutations. 相似文献
18.
The Local Structure of Zero Mode¶Producing Magnetic Potentials 总被引:1,自引:0,他引:1
Daniel M. Elton 《Communications in Mathematical Physics》2002,229(1):121-139
We consider the class of continuous magnetic potentials on ?3 which decay as o(|x|? 1). Within this class it is shown that the set of potentials whose associated Weyl-Dirac operator produces zero modes with multiplicity m forms a smooth submanifold of co-dimension m 2 when m= 0, 1, 2, and is contained in a smooth submanifold of co-dimension 2m? 1 when m≥ 3. 相似文献
19.
Søren Fournais 《Communications in Mathematical Physics》2001,216(2):375-393
In this paper we study the asymptotic form of the magnetisation and current of large atoms in strong constant magnetic fields.
We prove that the Magnetic Thomas–Fermi theory gives the right magnetisation/current for magnetic field strengths which satisfy
B≤Z
4/3.
Received: 24 April 2000 / Accepted: 21 August 2000 相似文献