4,338 research outputs found
Orbifold Index Cobordism Invariance
We prove cobordism index invariance for pseudo-differential elliptic
operators on closed orbifolds with --theoretical methods.Comment: New proofs for Theorems 4.2 and 4.3, results extended to all actions
by compact Lie group
Massive Orbifold
We study some aspects of 2d supersymmetric sigma models on orbifolds. It
turns out that independently of whether the 2d QFT is conformal the operator
products of twist operators are non-singular, suggesting that massive
(non-conformal) orbifolds also `resolve singularities' just as in the conformal
case. Moreover we recover the OPE of twist operators for conformal theories by
considering the UV limit of the massive orbifold correlation functions.
Alternatively, we can use the OPE of twist fields at the conformal point to
derive conditions for the existence of non-singular solutions to special
non-linear differential equations (such as Painleve III).Comment: 12 page
Classical and quantum ergodicity on orbifolds
We extend to orbifolds classical results on quantum ergodicity due to
Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive,
first-order self-adjoint elliptic pseudodifferential operator P on a compact
orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow
of p implies quantum ergodicity for the operator P. We also prove ergodicity of
the geodesic flow on a compact Riemannian orbifold of negative sectional
curvature.Comment: 14 page
Twisted higher index theory on good orbifolds and fractional quantum numbers
The twisted Connes-Moscovici higher index theorem is generalized to the case
of good orbifolds. The higher index is shown to be a rational number, and in
fact non-integer in specific examples of 2-orbifolds. This results in a
non-commutative geometry model that predicts the occurrence of fractional
quantum numbers in the Hall effect on the hyperbolic plane.Comment: 47 pages, Late
Twisted index theory on good orbifolds, I: noncommutative Bloch theory
This paper, together with Part II, expands the results of math.DG/9803051. In
Part I we study the twisted index theory of elliptic operators on orbifold
covering spaces of compact good orbifolds, which are invariant under a
projective action of the orbifold fundamental group. We apply these results to
obtain qualitative results on real and complex hyperbolic spaces in 2 and 4
dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and
the Ten Martini Problem, on the spectrum of self adjoint elliptic operators
which are invariant under a projective action of a discrete cocompact group.Comment: 34 pages, LaTe
Classical and Quantum Dynamics on Orbifolds
We present two versions of the Egorov theorem for orbifolds. The first one is
a straightforward extension of the classical theorem for smooth manifolds. The
second one considers an orbifold as a singular manifold, the orbit space of a
Lie group action, and deals with the corresponding objects in noncommutative
geometry
Generalized Bergman kernels on symplectic manifolds of bounded geometry
We study the asymptotic behavior of the generalized Bergman kernel of the
renormalized Bochner-Laplacian on high tensor powers of a positive line bundle
on a symplectic manifold of bounded geometry. First, we establish the
off-diagonal exponential estimate for the generalized Bergman kernel. As an
application, we obtain the relation between the generalized Bergman kernel on a
Galois covering of a compact symplectic manifold and the generalized Bergman
kernel on the base. Then we state the full off-diagonal asymptotic expansion of
the generalized Bergman kernel, improving the remainder estimate known in the
compact case to an exponential decay. Finally, we establish the theory of
Berezin-Toeplitz quantization on symplectic orbifolds associated with the
renormalized Bochner-Laplacian.Comment: 33 pages, v.2 is a final update to agree with the published pape
Equivariant quantization of orbifolds
Equivariant quantization is a new theory that highlights the role of
symmetries in the relationship between classical and quantum dynamical systems.
These symmetries are also one of the reasons for the recent interest in
quantization of singular spaces, orbifolds, stratified spaces... In this work,
we prove existence of an equivariant quantization for orbifolds. Our
construction combines an appropriate desingularization of any Riemannian
orbifold by a foliated smooth manifold, with the foliated equivariant
quantization that we built in \cite{PoRaWo}. Further, we suggest definitions of
the common geometric objects on orbifolds, which capture the nature of these
spaces and guarantee, together with the properties of the mentioned foliated
resolution, the needed correspondences between singular objects of the orbifold
and the respective foliated objects of its desingularization.Comment: 13 page
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