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 $\gamma_s$ with large spin $s$. 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