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1.
At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of
as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of
both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where
a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry.
This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed
in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae
for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing
a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples
where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This
is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory
and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create
a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to
the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism
of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed,
in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2]. 相似文献
2.
Indranil Biswas Tomás L. Gómez Norbert Hoffmann Marina Logares 《Communications in Mathematical Physics》2009,290(1):357-369
Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X. 相似文献
3.
《Nuclear Physics B》2001,600(1):81-102
We show that a generalised reduction of D=10 IIB supergravity leads, in a certain limit, to a maximally extended SO(2) gauged supergravity in D=9. We show the scalar potential of this model allows both Minkowski and a new type of domain wall solution to the Bogomol'nyi equations. We relate these vacua to type IIB D-branes. 相似文献
4.
Edward Fan Zuhair Khandker Robert S. Strichartz 《Communications in Mathematical Physics》2009,287(1):351-382
We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the
standard Kigami Laplacian. Our main task is to find a class of potentials analogous to on the line. We describe a class of potentials u with the properties Δu = 1, u attains a minimum value zero, and u → ∞ at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show
how to parameterize the class of potentials by a certain surface in . We obtain estimates for the growth rate of the eigenvalue counting function for −Δ + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction
resembles a Gaussian function.
Research supported by Chinese University Mathematics Alumni.
Research supported by the National Science Foundation through the Research Experiences for Undergraduates program at Cornell.
Research supported in part by the National Science Foundation, grant DMS 0652440. 相似文献
5.
We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four
dimensions, dealing with supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson
transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an
integrable -operator on a supertwistor space, i.e., on regions in . For , we also give a formulation that does not require the choice of a background. 相似文献
6.
D-transport is employed to construct, within the limited setting of a non-graded manifold, a geometrical framework that yields
a generalisation of the ‘supergauge’ transformations of Supergravity. Killing’s equation is shown to be at the origin of the
‘gauged’ supersymmetry transformations. The presence of a field-dependent Lorentz transformation is traced to the fact that, for every given X, the difference of two D-differentiation operators and is a linear transformation that necessarily depends on X.
相似文献
7.
A. G. Ramm 《Journal of statistical physics》2007,127(5):915-934
If A
q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ3 is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave,
and k>0, the wavenumber, is fixed. It is shown that any function , where S
2 is the unit sphere in ℝ3, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density
of the number of small acoustically soft particles D
m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ3. The geometrical shape of a small particle D
m is arbitrary, the boundary S
m of D
m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L
2(S
2× S
2) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L
2(D), i.e., corresponding to an arbitrary refraction coefficient in D.
MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30.
PACS: 03.04.Kf. 相似文献
8.
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
9.
D. Roest 《Fortschritte der Physik》2005,53(2):119-230
We present a pedagogical discussion of the emergence of gauged supergravities from M‐theory. First, a review of maximal supergravity and its global symmetries and supersymmetric solutions is given. Next, different procedures of dimensional reduction are explained: reductions over a torus, a group manifold and a coset manifold and reductions with a twist. Emphasis is placed on the consistency of the truncations, the resulting gaugings and the possibility to generate field equations without an action. Using these techniques, we construct a number of gauged maximal supergravities in diverse dimensions with a string or M‐theory origin. One class consists of the CSO gaugings, which comprise the analytic continuations and group contractions of SO(n) gaugings. We construct the corresponding half‐supersymmetric domain walls and discuss their uplift to D‐ and M‐brane distributions. Furthermore, a number of gauged maximal supergravities are constructed that do not have an action. 相似文献
10.
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
11.
Transport properties of CsHSO4 investigated by impedance spectroscopy and nuclear magnetic resonance
Transport properties of the superprotonic conductor, CsHSO4, have been investigated by impedance spectroscopy and nuclear magnetic resonance (NMR). It has been found that both, conductivity
(σ) and NMR diffusion (D
NMR) are practically isotropic in the high-conductive (superprotonic) phase (above 414 K). The NMR diffusion coefficient, D
NMR
, increases rapidly and discontinuously at the melting point (~490 K). The temperature change of D
NMR in the superprotonic phase is characterized by a smaller activation energy compared to that in the liquid state. The values calculated from the Nernst-Einstein relation practically coincide with D
NMR in the superprotonic phase, i.e., the Haven ratio is close to unity. This indicates that in this phase the proton motion is rather uncorrelated. 相似文献
12.
Dephasing in open quantum chaotic systems has been investigated in the limit of large system sizes to the Fermi wavelength
ratio, L/λF 〉 1. The weak localization correction g
wl to the conductance for a quantum dot coupled to (i) an external closed dot and (ii) a dephasing voltage probe is calculated
in the semiclassical approximation. In addition to the universal algebraic suppression g
wl ∝ (1 + τD/τϕ)−1 with the dwell time τD through the cavity and the dephasing rate τ
ϕ
−1
, we find an exponential suppression of weak localization by a factor of ∝ exp[−
/τϕ], where
is the system-dependent parameter. In the dephasing probe model,
coincides with the Ehrenfest time,
∝ ln[L/λF], for both perfectly and partially transparent dot-lead couplings. In contrast, when dephasing occurs due to the coupling
to an external dot,
∝ ln[L/ξ] depends on the correlation length ξ of the coupling potential instead of λF.
The text was submitted by the authors in English. 相似文献
13.
The principal graph X of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If Δ is the adjacency matrix of X we consider the equation Δ = U + U
−1. When X has square norm ≤ 4 the spectral measure of U can be averaged by using the map u→ u
−1, and we get a probability measure
on the unit circle which does not depend on U. We find explicit formulae for this measure
for the principal graphs of subfactors with index ≤ 4, the (extended) Coxeter-Dynkin graphs of type A, D and E. The moment generating function of
is closely related to Jones’ Θ-series.D.B. was supported by NSF under Grant No. DMS-0301173. 相似文献
14.
Ola Bratteli 《Letters in Mathematical Physics》1982,6(6):499-504
Let δ be a quasi-free derivation of the CAR algebra, and let
be a closed *-derivation which is an extension of δ. We use Price's techniques from [6] to show that if the polynomials in
the linear field operators a(f)→a
*
(f) in D(
) is a core for
, then
is quasi-free. 相似文献
15.
曹贞斌 《中国物理C(英文版)》2018,42(5):053104-053104
We study the maximally supersymmetric AdS backgrounds of matter-coupled N=3 gauged supergravity in four dimensions. We find that to admit supersymmetric AdS vacua, the gauge group can only be of the form G_0×H?SO(3,n) with G_0 =SO(3),SO(3,1) or SL(3,R) and H a compact group of dimension n+3-dim(G_0). We also show that these AdS vacua have no moduli, namely they correspond to critical points in field space. 相似文献
16.
We consider finite-range asymmetric exclusion processes on with non-zero drift. The diffusivity D(t) is expected to be of . We prove that D(t) ≥ Ct
1/3 in the weak (Tauberian) sense that as . The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process,
for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn.
In the nearest neighbor case, we show further that tD(t) is monotone, and hence we can conclude that D(t) ≥ Ct
1/3(log
t)−7/3 in the usual sense.
Supported by the Natural Sciences and Engineering Research Council of Canada.
Partially supported by the Hungarian Scientific Research Fund grants T37685 and K60708. 相似文献
17.
Representations of the Weyl Algebra in Quantum Geometry 总被引:2,自引:0,他引:2
Christian Fleischhack 《Communications in Mathematical Physics》2009,285(1):67-140
The Weyl algebra of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is
studied. It is shown that, in the piecewise analytic category, every regular representation of having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional
assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability
of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms – but neither any domain properties
of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour
of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic
diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of . 相似文献
18.
Matthias Christandl Robert König Graeme Mitchison Renato Renner 《Communications in Mathematical Physics》2007,273(2):473-498
When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor
product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of
this approximation is given by , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general
approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation
of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out U
ν. Then ξμ is close to a convex combination of the coherent states , where and is the highest weight vector in U
μ.
For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second
de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special
symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1].
This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike
their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states
in this set do not form a polytope in general. 相似文献
19.
We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional -modules. As a byproduct we show that there exist exactly two-dimensional vector subspaces with a basis such that deg f = l, deg g = n − l + 1 and f (u)g(u − 1) − f (u − 1)g(u) = (u + 1)
n
.
Supported in part by NSF grant DMS-0601005.
Supported in part by RFFI grant 08-01-00638.
Supported in part by NSF grant DMS-0555327. 相似文献
20.
The Dirac-type equation on
topology is worked out and the complete set of solutions in the particular physical case of the zero-energy modes of the massless field quanta is derived. Unlike the Minkowskian case, the 1/2fermionic vacua on the
manifold is made of nontrivial static modes of defined chirality. 相似文献