The Quantum McKay Correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold resolution clarified. Version to appear in Inventione

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page

Holomorphic anomaly equations and the Igusa cusp form conjecture

Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for ellliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.Comment: 68 page

FiND: Few-shot three-dimensional image-free confocal focusing on point-like emitters

Confocal fluorescence microscopy is widely applied for the study of point-like emitters such as biomolecules, material defects, and quantum light sources. Confocal techniques offer increased optical resolution, dramatic fluorescence background rejection and sub-nanometer localization, useful in super-resolution imaging of fluorescent biomarkers, single-molecule tracking, or the characterization of quantum emitters. However, rapid, noise-robust automated 3D focusing on point-like emitters has been missing for confocal microscopes. Here, we introduce FiND (Focusing in Noisy Domain), an imaging-free, non-trained 3D focusing framework that requires no hardware add-ons or modifications. FiND achieves focusing for signal-to-noise ratios down to 1, with a few-shot operation for signal-to-noise ratios above 5. FiND enables unsupervised, large-scale focusing on a heterogeneous set of quantum emitters. Additionally, we demonstrate the potential of FiND for real-time 3D tracking by following the drift trajectory of a single NV center indefinitely with a positional precision of < 10 nm. Our results show that FiND is a useful focusing framework for the scalable analysis of point-like emitters in biology, material science, and quantum optics.Comment: 17 pages, 7 figure

Strengthening the Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold invariants of [Sym^n(S)].Comment: Revised versio

Bis-spirolabdane Diterpenoids from Leonotis nepetaefolia

Ten new bis-spirolabdane diterpenoids, leonepetaefolins A–E (1, 3, 5, 7, 9) and 15-epi-leonepetaefolins A-E (2, 4, 6, 8, 10), together with eight known labdane diterpenoids (11–18) as well as two known flavonoids apigenin and cirsiliol, were isolated from the leaves of Leonotis nepetaefolia. The structures of the new compounds were determined on the basis of 1D-and 2D-NMR experiments including 1H, 13C, DEPT, 1H-1H COSY, HSQC, HMBC, and NOESY. The absolute configuration of an epimeric mixture of 1 and 2 was determined by X-ray crystallographic analysis. The compounds isolated were evaluated for their binding propensity in several CNS G protein-coupled receptor assays in vitro

Turbulent Mixing in the Interstellar Medium -- an application for Lagrangian Tracer Particles

We use 3-dimensional numerical simulations of self-gravitating compressible turbulent gas in combination with Lagrangian tracer particles to investigate the mixing process of molecular hydrogen (H2) in interstellar clouds. Tracer particles are used to represent shock-compressed dense gas, which is associated with H2. We deposit tracer particles in regions of density contrast in excess of ten times the mean density. Following their trajectories and using probability distribution functions, we find an upper limit for the mixing timescale of H2, which is of order 0.3 Myr. This is significantly smaller than the lifetime of molecular clouds, which demonstrates the importance of the turbulent mixing of H2 as a preliminary stage to star formation.Comment: 10 pages, 5 figures, conference proceedings "Turbulent Mixing and Beyond 2007

Casimir Torques between Anisotropic Boundaries in Nematic Liquid Crystals

Fluctuation-induced interactions between anisotropic objects immersed in a nematic liquid crystal are shown to depend on the relative orientation of these objects. The resulting long-range ``Casimir'' torques are explicitely calculated for a simple geometry where elastic effects are absent. Our study generalizes previous discussions restricted to the case of isotropic walls, and leads to new proposals for experimental tests of Casimir forces and torques in nematics.Comment: 4 pages, 1 figur

Multiple-Surrogate Approach to Helicopter Rotor Blade Vibration Reduction

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77383/1/AIAA-40291-933.pd