116 research outputs found
Topological strings on noncommutative manifolds
We identify a deformation of the N=2 supersymmetric sigma model on a
Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative
deformation of X. We show that for hyperkahler X such deformations allow one to
interpolate continuously between the A-model and the B-model. For generic
values of the noncommutativity and the B-field, properties of the topologically
twisted sigma-models can be described in terms of generalized complex
structures introduced by N. Hitchin. For example, we show that the path
integral for the deformed sigma-model is localized on generalized holomorphic
maps, whereas for the A-model and the B-model it is localized on holomorphic
and constant maps, respectively. The geometry of topological D-branes is also
best described using generalized complex structures. We also derive a
constraint on the Chern character of topological D-branes, which includes
A-branes and B-branes as special cases.Comment: 36 pages, AMS latex. v2: a reference to a related work has been
added. v3: An error in the discussion of the Fourier-Mukai transform for
twisted coherent sheaves has been fixed, resulting in several changes in
Section 2. The rest of the paper is unaffected. v4: an incorrect statement
concerning Lie algebroid cohomology has been fixe
On the logical operators of quantum codes
I show how applying a symplectic Gram-Schmidt orthogonalization to the
normalizer of a quantum code gives a different way of determining the code's
logical operators. This approach may be more natural in the setting where we
produce a quantum code from classical codes because the generator matrices of
the classical codes form the normalizer of the resulting quantum code. This
technique is particularly useful in determining the logical operators of an
entanglement-assisted code produced from two classical binary codes or from one
classical quaternary code. Finally, this approach gives additional formulas for
computing the amount of entanglement that an entanglement-assisted code
requires.Comment: 5 pages, sequel to the findings in arXiv:0804.140
Isotropic A-branes and the stability condition
The existence of a new kind of branes for the open topological A-model is
argued by using the generalized complex geometry of Hitchin and the SYZ picture
of mirror symmetry. Mirror symmetry suggests to consider a bi-vector in the
normal direction of the brane and a new definition of generalized complex
submanifold. Using this definition, it is shown that there exists generalized
complex submanifolds which are isotropic in a symplectic manifold. For certain
target space manifolds this leads to isotropic A-branes, which should be
considered in addition to Lagrangian and coisotropic A-branes. The Fukaya
category should be enlarged with such branes, which might have interesting
consequences for the homological mirror symmetry of Kontsevich. The stability
condition for isotropic A-branes is studied using the worldsheet approach.Comment: 19 page
Symplectic geometry on moduli spaces of J-holomorphic curves
Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface.
We define a 2-form on the space of immersed symplectic surfaces in M, and show
that the form is closed and non-degenerate, up to reparametrizations. Then we
give conditions on a compatible almost complex structure J on (M,\omega) that
ensure that the restriction of the form to the moduli space of simple immersed
J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic
form, and show applications and examples. In particular, we deduce sufficient
conditions for the existence of J-holomorphic Sigma-curves in a given homology
class for a generic J.Comment: 16 page
Multi-mode bosonic Gaussian channels
A complete analysis of multi-mode bosonic Gaussian channels is proposed. We
clarify the structure of unitary dilations of general Gaussian channels
involving any number of bosonic modes and present a normal form. The maximum
number of auxiliary modes that is needed is identified, including all rank
deficient cases, and the specific role of additive classical noise is
highlighted. By using this analysis, we derive a canonical matrix form of the
noisy evolution of n-mode bosonic Gaussian channels and of their weak
complementary counterparts, based on a recent generalization of the normal mode
decomposition for non-symmetric or locality constrained situations. It allows
us to simplify the weak-degradability classification. Moreover, we investigate
the structure of some singular multi-mode channels, like the additive classical
noise channel that can be used to decompose a noisy channel in terms of a less
noisy one in order to find new sets of maps with zero quantum capacity.
Finally, the two-mode case is analyzed in detail. By exploiting the composition
rules of two-mode maps and the fact that anti-degradable channels cannot be
used to transfer quantum information, we identify sets of two-mode bosonic
channels with zero capacity.Comment: 37 pages, 3 figures (minor editing), accepted for publication in New
Journal of Physic
On the geometric quantization of twisted Poisson manifolds
We study the geometric quantization process for twisted Poisson manifolds.
First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for
twisted Poisson manifolds and we use it in order to characterize their
prequantization bundles and to establish their prequantization condition. Next,
we introduce a polarization and we discuss the quantization problem. In each
step, several examples are presented
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