76,367 research outputs found
Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties
We define a class of non-compact Fano toric manifolds, called admissible
toric manifolds, for which Floer theory and quantum cohomology are defined. The
class includes Fano toric negative line bundles, and it allows blow-ups along
fixed point sets. We prove closed-string mirror symmetry for this class of
manifolds: the Jacobian ring of the superpotential is the symplectic cohomology
(not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by
localizing at the toric divisors. We give explicit presentations of SH(M) and
QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the
superpotential is Morse (or a milder semisimplicity assumption), we prove that
the wrapped Fukaya category for this class of manifolds satisfies the toric
generation criterion, i.e. is split-generated by the natural Lagrangian torus
fibres of the moment map with suitable holonomies. In particular, the wrapped
category is compactly generated and cohomologically finite. The proof uses a
deformation argument, via a generic generation theorem and an argument about
continuity of eigenspaces. We also prove that for any closed Fano toric
manifold, if the superpotential is Morse (or a milder semisimplicity
assumption) then the Fukaya category satisfies the toric generation criterion.
The key ingredients are non-vanishing results for the open-closed string map,
using tools from the paper by Ritter-Smith (we also prove a conjecture from
that paper that any monotone toric negative line bundle contains a
non-displaceable monotone Lagrangian torus). We also need to extend the class
of Hamiltonians for which the maximum principle holds for symplectic manifolds
conical at infinity, thus extending the class of Hamiltonian circle actions for
which invertible elements can be constructed in SH(M).Comment: 70 pages (51 pages + appendices). Version 2: rewrote the
Introduction, fixed a mistake (Remark 1.15), generation theorem generalized
to all admissible toric manifolds (Section 1.8
Floer theory for negative line bundles via Gromov-Witten invariants
Let M be the total space of a negative line bundle over a closed symplectic
manifold. We prove that the quotient of quantum cohomology by the kernel of a
power of quantum cup product by the first Chern class of the line bundle is
isomorphic to symplectic cohomology. We also prove this for negative vector
bundles and the top Chern class. We explicitly calculate the symplectic and
quantum cohomologies of O(-n) over P^m. For n=1, M is the blow-up of C^{m+1} at
the origin and symplectic cohomology has rank m. The symplectic cohomology
vanishes if and only if the first Chern class of the line bundle is nilpotent
in quantum cohomology. We prove a Kodaira vanishing theorem and a Serre
vanishing theorem for symplectic cohomology. In general, we construct a
representation of \pi_1(Ham(X,\omega)) on the symplectic cohomology of
symplectic manifolds X conical at infinity.Comment: 53 pages; version 3: improved discussion of maximum principle for
negative vector bundles. The final version is published in Advances in
Mathematic
Surgical concepts for reconstruction of the auricle
We compiled and evaluated the world literature on auricular reconstruction, for a total of over 400 publications, more than 200 authors, and over 3,300 reported cases. We found that partial reconstructions were already performed as early as 600 BC; total reconstructions were still considered impracticable in 1830. But since 1891, more than 40 different cartilaginous, osseous, and alloplastic frame materials have been described. Only eight of these were still being applied in the last decade, with autogenous costal cartilage and silicone as the leading substances. Results of the operation can be improved by special surgical manipulations, eg, the "fan-flap" technique. Taking into consideration the complication rate, the number of individual interventions, and the stability of the results, we devised a special point system that makes possible a limited assessment of the different surgical techniques
Conformal Bootstrap With Slightly Broken Higher Spin Symmetry
We consider conformal field theories with slightly broken higher spin
symmetry in arbitrary spacetime dimensions. We analyze the crossing equation in
the double light-cone limit and solve for the anomalous dimensions of higher
spin currents with large spin . The result depends on the
symmetries and the spectrum of the unperturbed conformal field theory. We
reproduce all known results and make further predictions. In particular we make
a prediction for the anomalous dimensions of higher spin currents in the 3d
Ising model.Comment: 41 pages, 2 figures, %\draftmod
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